Optimal. Leaf size=425 \[ -\frac {3 f h \left (a+b \sin ^{-1}(c x)\right )^2}{32 c^4}+\frac {b x \sqrt {1-c^2 x^2} (d h+e g) \left (a+b \sin ^{-1}(c x)\right )}{2 c}-\frac {(d h+e g) \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2}+\frac {2 b d g \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac {2 b x^2 \sqrt {1-c^2 x^2} (e h+f g) \left (a+b \sin ^{-1}(c x)\right )}{9 c}+\frac {b f h x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac {4 b \sqrt {1-c^2 x^2} (e h+f g) \left (a+b \sin ^{-1}(c x)\right )}{9 c^3}+\frac {3 b f h x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c^3}+\frac {1}{2} x^2 (d h+e g) \left (a+b \sin ^{-1}(c x)\right )^2+d g x \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{3} x^3 (e h+f g) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} f h x^4 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {4 b^2 x (e h+f g)}{9 c^2}-\frac {3 b^2 f h x^2}{32 c^2}-\frac {1}{4} b^2 x^2 (d h+e g)-2 b^2 d g x-\frac {2}{27} b^2 x^3 (e h+f g)-\frac {1}{32} b^2 f h x^4 \]
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Rubi [A] time = 0.70, antiderivative size = 425, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {4751, 4619, 4677, 8, 4627, 4707, 4641, 30} \[ \frac {b x \sqrt {1-c^2 x^2} (d h+e g) \left (a+b \sin ^{-1}(c x)\right )}{2 c}-\frac {(d h+e g) \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2}+\frac {2 b d g \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac {2 b x^2 \sqrt {1-c^2 x^2} (e h+f g) \left (a+b \sin ^{-1}(c x)\right )}{9 c}+\frac {4 b \sqrt {1-c^2 x^2} (e h+f g) \left (a+b \sin ^{-1}(c x)\right )}{9 c^3}+\frac {b f h x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac {3 b f h x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c^3}-\frac {3 f h \left (a+b \sin ^{-1}(c x)\right )^2}{32 c^4}+\frac {1}{2} x^2 (d h+e g) \left (a+b \sin ^{-1}(c x)\right )^2+d g x \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{3} x^3 (e h+f g) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} f h x^4 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {4 b^2 x (e h+f g)}{9 c^2}-\frac {3 b^2 f h x^2}{32 c^2}-\frac {1}{4} b^2 x^2 (d h+e g)-2 b^2 d g x-\frac {2}{27} b^2 x^3 (e h+f g)-\frac {1}{32} b^2 f h x^4 \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 4619
Rule 4627
Rule 4641
Rule 4677
Rule 4707
Rule 4751
Rubi steps
\begin {align*} \int (g+h x) \left (d+e x+f x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\int \left (d g \left (a+b \sin ^{-1}(c x)\right )^2+(e g+d h) x \left (a+b \sin ^{-1}(c x)\right )^2+(f g+e h) x^2 \left (a+b \sin ^{-1}(c x)\right )^2+f h x^3 \left (a+b \sin ^{-1}(c x)\right )^2\right ) \, dx\\ &=(d g) \int \left (a+b \sin ^{-1}(c x)\right )^2 \, dx+(f h) \int x^3 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx+(e g+d h) \int x \left (a+b \sin ^{-1}(c x)\right )^2 \, dx+(f g+e h) \int x^2 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx\\ &=d g x \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{2} (e g+d h) x^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{3} (f g+e h) x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} f h x^4 \left (a+b \sin ^{-1}(c x)\right )^2-(2 b c d g) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx-\frac {1}{2} (b c f h) \int \frac {x^4 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx-(b c (e g+d h)) \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx-\frac {1}{3} (2 b c (f g+e h)) \int \frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {2 b d g \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac {b (e g+d h) x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c}+\frac {2 b (f g+e h) x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c}+\frac {b f h x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+d g x \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{2} (e g+d h) x^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{3} (f g+e h) x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} f h x^4 \left (a+b \sin ^{-1}(c x)\right )^2-\left (2 b^2 d g\right ) \int 1 \, dx-\frac {1}{8} \left (b^2 f h\right ) \int x^3 \, dx-\frac {(3 b f h) \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{8 c}-\frac {1}{2} \left (b^2 (e g+d h)\right ) \int x \, dx-\frac {(b (e g+d h)) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{2 c}-\frac {1}{9} \left (2 b^2 (f g+e h)\right ) \int x^2 \, dx-\frac {(4 b (f g+e h)) \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 x-\frac {1}{4} b^2 (e g+d h) x^2-\frac {2}{27} b^2 (f g+e h) x^3-\frac {1}{32} b^2 f h x^4+\frac {2 b d g \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac {4 b (f g+e h) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^3}+\frac {3 b f h x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c^3}+\frac {b (e g+d h) x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c}+\frac {2 b (f g+e h) x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c}+\frac {b f h x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}-\frac {(e g+d h) \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2}+d g x \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{2} (e g+d h) x^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{3} (f g+e h) x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} f h x^4 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {(3 b f h) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{16 c^3}-\frac {\left (3 b^2 f h\right ) \int x \, dx}{16 c^2}-\frac {\left (4 b^2 (f g+e h)\right ) \int 1 \, dx}{9 c^2}\\ &=-2 b^2 d g x-\frac {4 b^2 (f g+e h) x}{9 c^2}-\frac {3 b^2 f h x^2}{32 c^2}-\frac {1}{4} b^2 (e g+d h) x^2-\frac {2}{27} b^2 (f g+e h) x^3-\frac {1}{32} b^2 f h x^4+\frac {2 b d g \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac {4 b (f g+e h) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^3}+\frac {3 b f h x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c^3}+\frac {b (e g+d h) x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c}+\frac {2 b (f g+e h) x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c}+\frac {b f h x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}-\frac {3 f h \left (a+b \sin ^{-1}(c x)\right )^2}{32 c^4}-\frac {(e g+d h) \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2}+d g x \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{2} (e g+d h) x^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{3} (f g+e h) x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} f h x^4 \left (a+b \sin ^{-1}(c x)\right )^2\\ \end {align*}
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Mathematica [A] time = 0.39, size = 364, normalized size = 0.86 \[ -\frac {1}{4} b (d h+e g) \left (-\frac {2 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{b c^2}+b x^2\right )-2 b d g \left (b x-\frac {\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}\right )-\frac {2 b (e h+f g) \left (-3 a \sqrt {1-c^2 x^2} \left (c^2 x^2+2\right )+b c x \left (c^2 x^2+6\right )-3 b \sqrt {1-c^2 x^2} \left (c^2 x^2+2\right ) \sin ^{-1}(c x)\right )}{27 c^3}-\frac {1}{32} b f h \left (\frac {3 \left (a+b \sin ^{-1}(c x)\right )^2}{b c^4}-\frac {4 x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}-\frac {6 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c^3}+\frac {3 b x^2}{c^2}+b x^4\right )+\frac {1}{2} x^2 (d h+e g) \left (a+b \sin ^{-1}(c x)\right )^2+d g x \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{3} x^3 (e h+f g) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} f h x^4 \left (a+b \sin ^{-1}(c x)\right )^2 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 579, normalized size = 1.36 \[ \frac {27 \, {\left (8 \, a^{2} - b^{2}\right )} c^{4} f h x^{4} + 32 \, {\left ({\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{4} f g + {\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{4} e h\right )} x^{3} + 27 \, {\left (8 \, {\left (2 \, a^{2} - b^{2}\right )} c^{4} e g + {\left (8 \, {\left (2 \, a^{2} - b^{2}\right )} c^{4} d - 3 \, b^{2} c^{2} f\right )} h\right )} x^{2} + 9 \, {\left (24 \, b^{2} c^{4} f h x^{4} + 96 \, b^{2} c^{4} d g x - 24 \, b^{2} c^{2} e g + 32 \, {\left (b^{2} c^{4} f g + b^{2} c^{4} e h\right )} x^{3} + 48 \, {\left (b^{2} c^{4} e g + b^{2} c^{4} d h\right )} x^{2} - 3 \, {\left (8 \, b^{2} c^{2} d + 3 \, b^{2} f\right )} h\right )} \arcsin \left (c x\right )^{2} - 96 \, {\left (4 \, b^{2} c^{2} e h - {\left (9 \, {\left (a^{2} - 2 \, b^{2}\right )} c^{4} d - 4 \, b^{2} c^{2} f\right )} g\right )} x + 18 \, {\left (24 \, a b c^{4} f h x^{4} + 96 \, a b c^{4} d g x - 24 \, a b c^{2} e g + 32 \, {\left (a b c^{4} f g + a b c^{4} e h\right )} x^{3} + 48 \, {\left (a b c^{4} e g + a b c^{4} d h\right )} x^{2} - 3 \, {\left (8 \, a b c^{2} d + 3 \, a b f\right )} h\right )} \arcsin \left (c x\right ) + 6 \, {\left (18 \, a b c^{3} f h x^{3} + 64 \, a b c e h + 32 \, {\left (a b c^{3} f g + a b c^{3} e h\right )} x^{2} + 32 \, {\left (9 \, a b c^{3} d + 2 \, a b c f\right )} g + 9 \, {\left (8 \, a b c^{3} e g + {\left (8 \, a b c^{3} d + 3 \, a b c f\right )} h\right )} x + {\left (18 \, b^{2} c^{3} f h x^{3} + 64 \, b^{2} c e h + 32 \, {\left (b^{2} c^{3} f g + b^{2} c^{3} e h\right )} x^{2} + 32 \, {\left (9 \, b^{2} c^{3} d + 2 \, b^{2} c f\right )} g + 9 \, {\left (8 \, b^{2} c^{3} e g + {\left (8 \, b^{2} c^{3} d + 3 \, b^{2} c f\right )} h\right )} x\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{864 \, c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.45, size = 1165, normalized size = 2.74 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.16, size = 870, normalized size = 2.05 \[ \frac {\frac {a^{2} \left (\frac {h f \,c^{4} x^{4}}{4}+\frac {\left (e c h +c f g \right ) c^{3} x^{3}}{3}+\frac {\left (d \,c^{2} h +e \,c^{2} g \right ) c^{2} x^{2}}{2}+d \,c^{4} g x \right )}{c^{3}}+\frac {b^{2} \left (\frac {h f \left (8 \arcsin \left (c x \right )^{2} c^{4} x^{4}+4 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{3} x^{3}-16 \arcsin \left (c x \right )^{2} c^{2} x^{2}-c^{4} x^{4}-10 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x +5 \arcsin \left (c x \right )^{2}+5 c^{2} x^{2}-4\right )}{32}+\frac {d \,c^{2} h \left (2 \arcsin \left (c x \right )^{2} c^{2} x^{2}+2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x -\arcsin \left (c x \right )^{2}-c^{2} x^{2}\right )}{4}+\frac {e \,c^{2} g \left (2 \arcsin \left (c x \right )^{2} c^{2} x^{2}+2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x -\arcsin \left (c x \right )^{2}-c^{2} x^{2}\right )}{4}+\frac {e c h \left (9 \arcsin \left (c x \right )^{2} c^{3} x^{3}+6 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-27 c x \arcsin \left (c x \right )^{2}-2 c^{3} x^{3}-42 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}+42 c x \right )}{27}+\frac {c f g \left (9 \arcsin \left (c x \right )^{2} c^{3} x^{3}+6 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-27 c x \arcsin \left (c x \right )^{2}-2 c^{3} x^{3}-42 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}+42 c x \right )}{27}+d \,c^{3} g \left (c x \arcsin \left (c x \right )^{2}-2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+\frac {h f \left (2 \arcsin \left (c x \right )^{2} c^{2} x^{2}+2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x -\arcsin \left (c x \right )^{2}-c^{2} x^{2}\right )}{4}+e c h \left (c x \arcsin \left (c x \right )^{2}-2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+c f g \left (c x \arcsin \left (c x \right )^{2}-2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )\right )}{c^{3}}+\frac {2 a b \left (\frac {\arcsin \left (c x \right ) h f \,c^{4} x^{4}}{4}+\frac {\arcsin \left (c x \right ) c^{4} x^{3} e h}{3}+\frac {\arcsin \left (c x \right ) c^{4} x^{3} f g}{3}+\frac {\arcsin \left (c x \right ) c^{4} x^{2} d h}{2}+\frac {\arcsin \left (c x \right ) c^{4} x^{2} e g}{2}+\arcsin \left (c x \right ) d \,c^{4} g x -\frac {h f \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )}{4}-\frac {\left (4 e c h +4 c f g \right ) \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{12}-\frac {\left (6 d \,c^{2} h +6 e \,c^{2} g \right ) \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{12}+d \,c^{3} g \sqrt {-c^{2} x^{2}+1}\right )}{c^{3}}}{c} \]
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 \[ \frac {1}{4} \, a^{2} f h x^{4} + \frac {1}{3} \, a^{2} f g x^{3} + \frac {1}{3} \, a^{2} e h x^{3} + b^{2} d g x \arcsin \left (c x\right )^{2} + \frac {1}{2} \, a^{2} e g x^{2} + \frac {1}{2} \, a^{2} d h x^{2} + \frac {1}{2} \, {\left (2 \, x^{2} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{c^{2}} - \frac {\arcsin \left (c x\right )}{c^{3}}\right )}\right )} a b e g + \frac {2}{9} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b f g + \frac {1}{2} \, {\left (2 \, x^{2} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{c^{2}} - \frac {\arcsin \left (c x\right )}{c^{3}}\right )}\right )} a b d h + \frac {2}{9} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b e h + \frac {1}{16} \, {\left (8 \, x^{4} \arcsin \left (c x\right ) + {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} a b f h - 2 \, b^{2} d g {\left (x - \frac {\sqrt {-c^{2} x^{2} + 1} \arcsin \left (c x\right )}{c}\right )} + a^{2} d g x + \frac {2 \, {\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} a b d g}{c} + \frac {1}{12} \, {\left (3 \, b^{2} f h x^{4} + 4 \, {\left (b^{2} f g + b^{2} e h\right )} x^{3} + 6 \, {\left (b^{2} e g + b^{2} d h\right )} x^{2}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + \int \frac {{\left (3 \, b^{2} c f h x^{4} + 4 \, {\left (b^{2} c f g + b^{2} c e h\right )} x^{3} + 6 \, {\left (b^{2} c e g + b^{2} c d h\right )} x^{2}\right )} \sqrt {c x + 1} \sqrt {-c x + 1} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{6 \, {\left (c^{2} x^{2} - 1\right )}}\,{d x} \]
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 \[ \int \left (g+h\,x\right )\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\left (f\,x^2+e\,x+d\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.43, size = 1059, normalized size = 2.49 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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