Optimal. Leaf size=701 \[ -\frac {3 h (e h+2 f g) \left (a+b \sin ^{-1}(c x)\right )^2}{32 c^4}+\frac {2 b x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \left (h (d h+2 e g)+f g^2\right )}{9 c}+\frac {b g x \sqrt {1-c^2 x^2} (2 d h+e g) \left (a+b \sin ^{-1}(c x)\right )}{2 c}-\frac {g (2 d h+e g) \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2}+\frac {2 b d g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac {b h x^3 \sqrt {1-c^2 x^2} (e h+2 f g) \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac {2 b f h^2 x^4 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}+\frac {16 b f h^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{75 c^5}+\frac {4 b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \left (h (d h+2 e g)+f g^2\right )}{9 c^3}+\frac {3 b h x \sqrt {1-c^2 x^2} (e h+2 f g) \left (a+b \sin ^{-1}(c x)\right )}{16 c^3}+\frac {8 b f h^2 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{75 c^3}+\frac {1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right )^2 \left (h (d h+2 e g)+f g^2\right )+\frac {1}{2} g x^2 (2 d h+e g) \left (a+b \sin ^{-1}(c x)\right )^2+d g^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} h x^4 (e h+2 f g) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{5} f h^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {16 b^2 f h^2 x}{75 c^4}-\frac {4 b^2 x \left (h (d h+2 e g)+f g^2\right )}{9 c^2}-\frac {3 b^2 h x^2 (e h+2 f g)}{32 c^2}-\frac {8 b^2 f h^2 x^3}{225 c^2}-\frac {2}{27} b^2 x^3 \left (h (d h+2 e g)+f g^2\right )-\frac {1}{4} b^2 g x^2 (2 d h+e g)-2 b^2 d g^2 x-\frac {1}{32} b^2 h x^4 (e h+2 f g)-\frac {2}{125} b^2 f h^2 x^5 \]
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Rubi [A] time = 1.12, antiderivative size = 701, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 8, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4751, 4619, 4677, 8, 4627, 4707, 4641, 30} \[ \frac {2 b x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \left (h (d h+2 e g)+f g^2\right )}{9 c}+\frac {4 b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \left (h (d h+2 e g)+f g^2\right )}{9 c^3}+\frac {b g x \sqrt {1-c^2 x^2} (2 d h+e g) \left (a+b \sin ^{-1}(c x)\right )}{2 c}-\frac {g (2 d h+e g) \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2}+\frac {2 b d g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac {b h x^3 \sqrt {1-c^2 x^2} (e h+2 f g) \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac {3 b h x \sqrt {1-c^2 x^2} (e h+2 f g) \left (a+b \sin ^{-1}(c x)\right )}{16 c^3}-\frac {3 h (e h+2 f g) \left (a+b \sin ^{-1}(c x)\right )^2}{32 c^4}+\frac {2 b f h^2 x^4 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}+\frac {8 b f h^2 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{75 c^3}+\frac {16 b f h^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{75 c^5}+\frac {1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right )^2 \left (h (d h+2 e g)+f g^2\right )+\frac {1}{2} g x^2 (2 d h+e g) \left (a+b \sin ^{-1}(c x)\right )^2+d g^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} h x^4 (e h+2 f g) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{5} f h^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {4 b^2 x \left (h (d h+2 e g)+f g^2\right )}{9 c^2}-\frac {3 b^2 h x^2 (e h+2 f g)}{32 c^2}-\frac {8 b^2 f h^2 x^3}{225 c^2}-\frac {16 b^2 f h^2 x}{75 c^4}-\frac {2}{27} b^2 x^3 \left (h (d h+2 e g)+f g^2\right )-\frac {1}{4} b^2 g x^2 (2 d h+e g)-2 b^2 d g^2 x-\frac {1}{32} b^2 h x^4 (e h+2 f g)-\frac {2}{125} b^2 f h^2 x^5 \]
Antiderivative was successfully verified.
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[Out]
Rule 8
Rule 30
Rule 4619
Rule 4627
Rule 4641
Rule 4677
Rule 4707
Rule 4751
Rubi steps
\begin {align*} \int (g+h x)^2 \left (d+e x+f x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\int \left (d g^2 \left (a+b \sin ^{-1}(c x)\right )^2+g (e g+2 d h) x \left (a+b \sin ^{-1}(c x)\right )^2+\left (f g^2+h (2 e g+d h)\right ) x^2 \left (a+b \sin ^{-1}(c x)\right )^2+h (2 f g+e h) x^3 \left (a+b \sin ^{-1}(c x)\right )^2+f h^2 x^4 \left (a+b \sin ^{-1}(c x)\right )^2\right ) \, dx\\ &=\left (d g^2\right ) \int \left (a+b \sin ^{-1}(c x)\right )^2 \, dx+\left (f h^2\right ) \int x^4 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx+(g (e g+2 d h)) \int x \left (a+b \sin ^{-1}(c x)\right )^2 \, dx+(h (2 f g+e h)) \int x^3 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx+\left (f g^2+h (2 e g+d h)\right ) \int x^2 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx\\ &=d g^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{2} g (e g+2 d h) x^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{3} \left (f g^2+h (2 e g+d h)\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} h (2 f g+e h) x^4 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{5} f h^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2-\left (2 b c d g^2\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx-\frac {1}{5} \left (2 b c f h^2\right ) \int \frac {x^5 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx-(b c g (e g+2 d h)) \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx-\frac {1}{2} (b c h (2 f g+e h)) \int \frac {x^4 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx-\frac {1}{3} \left (2 b c \left (f g^2+h (2 e g+d h)\right )\right ) \int \frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {2 b d g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac {b g (e g+2 d h) x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c}+\frac {2 b \left (f g^2+h (2 e g+d h)\right ) x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c}+\frac {b h (2 f g+e h) x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac {2 b f h^2 x^4 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}+d g^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{2} g (e g+2 d h) x^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{3} \left (f g^2+h (2 e g+d h)\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} h (2 f g+e h) x^4 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{5} f h^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2-\left (2 b^2 d g^2\right ) \int 1 \, dx-\frac {1}{25} \left (2 b^2 f h^2\right ) \int x^4 \, dx-\frac {\left (8 b f h^2\right ) \int \frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{25 c}-\frac {1}{2} \left (b^2 g (e g+2 d h)\right ) \int x \, dx-\frac {(b g (e g+2 d h)) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{2 c}-\frac {1}{8} \left (b^2 h (2 f g+e h)\right ) \int x^3 \, dx-\frac {(3 b h (2 f g+e h)) \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{8 c}-\frac {1}{9} \left (2 b^2 \left (f g^2+h (2 e g+d h)\right )\right ) \int x^2 \, dx-\frac {\left (4 b \left (f g^2+h (2 e g+d h)\right )\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{9 c}\\ &=-2 b^2 d g^2 x-\frac {1}{4} b^2 g (e g+2 d h) x^2-\frac {2}{27} b^2 \left (f g^2+h (2 e g+d h)\right ) x^3-\frac {1}{32} b^2 h (2 f g+e h) x^4-\frac {2}{125} b^2 f h^2 x^5+\frac {2 b d g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac {4 b \left (f g^2+h (2 e g+d h)\right ) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^3}+\frac {b g (e g+2 d h) x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c}+\frac {3 b h (2 f g+e h) x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c^3}+\frac {8 b f h^2 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{75 c^3}+\frac {2 b \left (f g^2+h (2 e g+d h)\right ) x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c}+\frac {b h (2 f g+e h) x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac {2 b f h^2 x^4 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}-\frac {g (e g+2 d h) \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2}+d g^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{2} g (e g+2 d h) x^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{3} \left (f g^2+h (2 e g+d h)\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} h (2 f g+e h) x^4 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{5} f h^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {\left (16 b f h^2\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{75 c^3}-\frac {\left (8 b^2 f h^2\right ) \int x^2 \, dx}{75 c^2}-\frac {(3 b h (2 f g+e h)) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{16 c^3}-\frac {\left (3 b^2 h (2 f g+e h)\right ) \int x \, dx}{16 c^2}-\frac {\left (4 b^2 \left (f g^2+h (2 e g+d h)\right )\right ) \int 1 \, dx}{9 c^2}\\ &=-2 b^2 d g^2 x-\frac {4 b^2 \left (f g^2+h (2 e g+d h)\right ) x}{9 c^2}-\frac {1}{4} b^2 g (e g+2 d h) x^2-\frac {3 b^2 h (2 f g+e h) x^2}{32 c^2}-\frac {8 b^2 f h^2 x^3}{225 c^2}-\frac {2}{27} b^2 \left (f g^2+h (2 e g+d h)\right ) x^3-\frac {1}{32} b^2 h (2 f g+e h) x^4-\frac {2}{125} b^2 f h^2 x^5+\frac {2 b d g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac {16 b f h^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{75 c^5}+\frac {4 b \left (f g^2+h (2 e g+d h)\right ) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^3}+\frac {b g (e g+2 d h) x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c}+\frac {3 b h (2 f g+e h) x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c^3}+\frac {8 b f h^2 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{75 c^3}+\frac {2 b \left (f g^2+h (2 e g+d h)\right ) x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c}+\frac {b h (2 f g+e h) x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac {2 b f h^2 x^4 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}-\frac {g (e g+2 d h) \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2}-\frac {3 h (2 f g+e h) \left (a+b \sin ^{-1}(c x)\right )^2}{32 c^4}+d g^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{2} g (e g+2 d h) x^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{3} \left (f g^2+h (2 e g+d h)\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} h (2 f g+e h) x^4 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{5} f h^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {\left (16 b^2 f h^2\right ) \int 1 \, dx}{75 c^4}\\ &=-2 b^2 d g^2 x-\frac {16 b^2 f h^2 x}{75 c^4}-\frac {4 b^2 \left (f g^2+h (2 e g+d h)\right ) x}{9 c^2}-\frac {1}{4} b^2 g (e g+2 d h) x^2-\frac {3 b^2 h (2 f g+e h) x^2}{32 c^2}-\frac {8 b^2 f h^2 x^3}{225 c^2}-\frac {2}{27} b^2 \left (f g^2+h (2 e g+d h)\right ) x^3-\frac {1}{32} b^2 h (2 f g+e h) x^4-\frac {2}{125} b^2 f h^2 x^5+\frac {2 b d g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac {16 b f h^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{75 c^5}+\frac {4 b \left (f g^2+h (2 e g+d h)\right ) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^3}+\frac {b g (e g+2 d h) x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c}+\frac {3 b h (2 f g+e h) x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c^3}+\frac {8 b f h^2 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{75 c^3}+\frac {2 b \left (f g^2+h (2 e g+d h)\right ) x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c}+\frac {b h (2 f g+e h) x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac {2 b f h^2 x^4 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}-\frac {g (e g+2 d h) \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2}-\frac {3 h (2 f g+e h) \left (a+b \sin ^{-1}(c x)\right )^2}{32 c^4}+d g^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{2} g (e g+2 d h) x^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{3} \left (f g^2+h (2 e g+d h)\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} h (2 f g+e h) x^4 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{5} f h^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2\\ \end {align*}
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Mathematica [A] time = 0.61, size = 534, normalized size = 0.76 \[ -\frac {1}{4} b g (2 d h+e g) \left (-\frac {2 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{b c^2}+b x^2\right )-2 b d g^2 \left (b x-\frac {\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}\right )-\frac {2 b \left (-3 a \sqrt {1-c^2 x^2} \left (c^2 x^2+2\right )+b c x \left (c^2 x^2+6\right )-3 b \sqrt {1-c^2 x^2} \left (c^2 x^2+2\right ) \sin ^{-1}(c x)\right ) \left (h (d h+2 e g)+f g^2\right )}{27 c^3}-\frac {2 b f h^2 \left (-15 a \sqrt {1-c^2 x^2} \left (3 c^4 x^4+4 c^2 x^2+8\right )+b c x \left (9 c^4 x^4+20 c^2 x^2+120\right )-15 b \sqrt {1-c^2 x^2} \left (3 c^4 x^4+4 c^2 x^2+8\right ) \sin ^{-1}(c x)\right )}{1125 c^5}-\frac {1}{32} b h (e h+2 f g) \left (\frac {3 \left (a+b \sin ^{-1}(c x)\right )^2}{b c^4}-\frac {4 x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}-\frac {6 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c^3}+\frac {3 b x^2}{c^2}+b x^4\right )+\frac {1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right )^2 \left (h (d h+2 e g)+f g^2\right )+\frac {1}{2} g x^2 (2 d h+e g) \left (a+b \sin ^{-1}(c x)\right )^2+d g^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} h x^4 (e h+2 f g) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{5} f h^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2 \]
Antiderivative was successfully verified.
[In]
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fricas [A] time = 0.51, size = 1029, normalized size = 1.47 \[ \frac {864 \, {\left (25 \, a^{2} - 2 \, b^{2}\right )} c^{5} f h^{2} x^{5} + 3375 \, {\left (2 \, {\left (8 \, a^{2} - b^{2}\right )} c^{5} f g h + {\left (8 \, a^{2} - b^{2}\right )} c^{5} e h^{2}\right )} x^{4} + 160 \, {\left (25 \, {\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{5} f g^{2} + 50 \, {\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{5} e g h + {\left (25 \, {\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{5} d - 24 \, b^{2} c^{3} f\right )} h^{2}\right )} x^{3} + 3375 \, {\left (8 \, {\left (2 \, a^{2} - b^{2}\right )} c^{5} e g^{2} - 3 \, b^{2} c^{3} e h^{2} + 2 \, {\left (8 \, {\left (2 \, a^{2} - b^{2}\right )} c^{5} d - 3 \, b^{2} c^{3} f\right )} g h\right )} x^{2} + 225 \, {\left (96 \, b^{2} c^{5} f h^{2} x^{5} + 480 \, b^{2} c^{5} d g^{2} x - 120 \, b^{2} c^{3} e g^{2} - 45 \, b^{2} c e h^{2} + 120 \, {\left (2 \, b^{2} c^{5} f g h + b^{2} c^{5} e h^{2}\right )} x^{4} + 160 \, {\left (b^{2} c^{5} f g^{2} + 2 \, b^{2} c^{5} e g h + b^{2} c^{5} d h^{2}\right )} x^{3} - 30 \, {\left (8 \, b^{2} c^{3} d + 3 \, b^{2} c f\right )} g h + 240 \, {\left (b^{2} c^{5} e g^{2} + 2 \, b^{2} c^{5} d g h\right )} x^{2}\right )} \arcsin \left (c x\right )^{2} - 480 \, {\left (200 \, b^{2} c^{3} e g h - 25 \, {\left (9 \, {\left (a^{2} - 2 \, b^{2}\right )} c^{5} d - 4 \, b^{2} c^{3} f\right )} g^{2} + 4 \, {\left (25 \, b^{2} c^{3} d + 12 \, b^{2} c f\right )} h^{2}\right )} x + 450 \, {\left (96 \, a b c^{5} f h^{2} x^{5} + 480 \, a b c^{5} d g^{2} x - 120 \, a b c^{3} e g^{2} - 45 \, a b c e h^{2} + 120 \, {\left (2 \, a b c^{5} f g h + a b c^{5} e h^{2}\right )} x^{4} + 160 \, {\left (a b c^{5} f g^{2} + 2 \, a b c^{5} e g h + a b c^{5} d h^{2}\right )} x^{3} - 30 \, {\left (8 \, a b c^{3} d + 3 \, a b c f\right )} g h + 240 \, {\left (a b c^{5} e g^{2} + 2 \, a b c^{5} d g h\right )} x^{2}\right )} \arcsin \left (c x\right ) + 30 \, {\left (288 \, a b c^{4} f h^{2} x^{4} + 3200 \, a b c^{2} e g h + 450 \, {\left (2 \, a b c^{4} f g h + a b c^{4} e h^{2}\right )} x^{3} + 800 \, {\left (9 \, a b c^{4} d + 2 \, a b c^{2} f\right )} g^{2} + 64 \, {\left (25 \, a b c^{2} d + 12 \, a b f\right )} h^{2} + 32 \, {\left (25 \, a b c^{4} f g^{2} + 50 \, a b c^{4} e g h + {\left (25 \, a b c^{4} d + 12 \, a b c^{2} f\right )} h^{2}\right )} x^{2} + 225 \, {\left (8 \, a b c^{4} e g^{2} + 3 \, a b c^{2} e h^{2} + 2 \, {\left (8 \, a b c^{4} d + 3 \, a b c^{2} f\right )} g h\right )} x + {\left (288 \, b^{2} c^{4} f h^{2} x^{4} + 3200 \, b^{2} c^{2} e g h + 450 \, {\left (2 \, b^{2} c^{4} f g h + b^{2} c^{4} e h^{2}\right )} x^{3} + 800 \, {\left (9 \, b^{2} c^{4} d + 2 \, b^{2} c^{2} f\right )} g^{2} + 64 \, {\left (25 \, b^{2} c^{2} d + 12 \, b^{2} f\right )} h^{2} + 32 \, {\left (25 \, b^{2} c^{4} f g^{2} + 50 \, b^{2} c^{4} e g h + {\left (25 \, b^{2} c^{4} d + 12 \, b^{2} c^{2} f\right )} h^{2}\right )} x^{2} + 225 \, {\left (8 \, b^{2} c^{4} e g^{2} + 3 \, b^{2} c^{2} e h^{2} + 2 \, {\left (8 \, b^{2} c^{4} d + 3 \, b^{2} c^{2} f\right )} g h\right )} x\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{108000 \, c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.49, size = 2200, normalized size = 3.14 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.22, size = 1633, normalized size = 2.33 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{5} \, a^{2} f h^{2} x^{5} + \frac {1}{2} \, a^{2} f g h x^{4} + \frac {1}{4} \, a^{2} e h^{2} x^{4} + \frac {1}{3} \, a^{2} f g^{2} x^{3} + \frac {2}{3} \, a^{2} e g h x^{3} + \frac {1}{3} \, a^{2} d h^{2} x^{3} + b^{2} d g^{2} x \arcsin \left (c x\right )^{2} + \frac {1}{2} \, a^{2} e g^{2} x^{2} + a^{2} d g h x^{2} + \frac {1}{2} \, {\left (2 \, x^{2} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{c^{2}} - \frac {\arcsin \left (c x\right )}{c^{3}}\right )}\right )} a b e g^{2} + \frac {2}{9} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b f g^{2} + {\left (2 \, x^{2} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{c^{2}} - \frac {\arcsin \left (c x\right )}{c^{3}}\right )}\right )} a b d g h + \frac {4}{9} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b e g h + \frac {1}{8} \, {\left (8 \, x^{4} \arcsin \left (c x\right ) + {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} a b f g h + \frac {2}{9} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b d h^{2} + \frac {1}{16} \, {\left (8 \, x^{4} \arcsin \left (c x\right ) + {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} a b e h^{2} + \frac {2}{75} \, {\left (15 \, x^{5} \arcsin \left (c x\right ) + {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} a b f h^{2} - 2 \, b^{2} d g^{2} {\left (x - \frac {\sqrt {-c^{2} x^{2} + 1} \arcsin \left (c x\right )}{c}\right )} + a^{2} d g^{2} x + \frac {2 \, {\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} a b d g^{2}}{c} + \frac {1}{60} \, {\left (12 \, b^{2} f h^{2} x^{5} + 15 \, {\left (2 \, b^{2} f g h + b^{2} e h^{2}\right )} x^{4} + 20 \, {\left (b^{2} f g^{2} + 2 \, b^{2} e g h + b^{2} d h^{2}\right )} x^{3} + 30 \, {\left (b^{2} e g^{2} + 2 \, b^{2} d g h\right )} x^{2}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + \int \frac {{\left (12 \, b^{2} c f h^{2} x^{5} + 15 \, {\left (2 \, b^{2} c f g h + b^{2} c e h^{2}\right )} x^{4} + 20 \, {\left (b^{2} c f g^{2} + 2 \, b^{2} c e g h + b^{2} c d h^{2}\right )} x^{3} + 30 \, {\left (b^{2} c e g^{2} + 2 \, b^{2} c d g h\right )} x^{2}\right )} \sqrt {c x + 1} \sqrt {-c x + 1} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{30 \, {\left (c^{2} x^{2} - 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (g+h\,x\right )}^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\left (f\,x^2+e\,x+d\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 9.25, size = 1935, normalized size = 2.76 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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