Alternative method : Expand the product
(x² + x/√(1+x²) ) (1 + x/((1+x²)√(1+x²)) )
= x² + x/√(1+x²) + x²•x/(1+x²)³/² + x•x/(1+x²)²
①The first two terms are directly integrable. We get x³/3 and √(1+x²)
②For x²•x/(1+x²)³/² , integrate by parts.
∫ x²•x/(1+x²)³/² dx
= x² • -1/√(1+x²) - ∫ 2x • -1/√(1+x²) dx
= -x²/√(1+x²) + 2√(1+x²)
= (-x²+2x²+2)/√(x²+1)
= (x²+2)/√(x²+1)
③For x•x/(1+x²)², integrate by parts also.
∫ x•x/(1+x²)² dx
= ½∫ x • 2x/(1+x²)² dx
= ½ ( x • -1/(1+x²) - ∫1• -1/(1+x²) dx )
= ½ (∫ 1/(1+x²) dx - x/(1+x²) )
= ½ tan-¹x - x/(2(1+x²))
④ So, ∫ (x² + x/√(1+x²) ) (1 + x/((1+x²)√(1+x²)) ) dx
= [x³/3 + √(1+x²) + (x²+2)/√(1+x²) + ½tan-¹x - x/(2(1+x²)) ]
= [1³/3 + √(1+1²) + (1²+2)/√(1+1²) + ½tan-¹1 - 1/(2(1+1²)) ] - [0³/3 + √(1+0²) + (0²+2)/√(1+0²) + ½tan-¹0 - 0/(2(1+0²)) ]
= ⅓ + √2 + 3/√2 + π/8 - ¼ - 1 - 2
= √2 + 3/2 √2 + π/8 - 2 11/12
= 5√2 /2 + π/8 - 2 11/12
(Approximately 1.01156632)
You can actually integrate all these at once in each step instead of doing them separately. The above is just shown for clarity.