Optimal. Leaf size=425 \[ -2 b^2 d g x-\frac {4 b^2 (f g+e h) x}{9 c^2}-\frac {3 b^2 f h x^2}{32 c^2}-\frac {1}{4} b^2 (e g+d h) x^2-\frac {2}{27} b^2 (f g+e h) x^3-\frac {1}{32} b^2 f h x^4+\frac {2 b d g \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{c}+\frac {4 b (f g+e h) \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{9 c^3}+\frac {3 b f h x \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{16 c^3}+\frac {b (e g+d h) x \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{2 c}+\frac {2 b (f g+e h) x^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{9 c}+\frac {b f h x^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{8 c}-\frac {3 f h (a+b \text {ArcSin}(c x))^2}{32 c^4}-\frac {(e g+d h) (a+b \text {ArcSin}(c x))^2}{4 c^2}+d g x (a+b \text {ArcSin}(c x))^2+\frac {1}{2} (e g+d h) x^2 (a+b \text {ArcSin}(c x))^2+\frac {1}{3} (f g+e h) x^3 (a+b \text {ArcSin}(c x))^2+\frac {1}{4} f h x^4 (a+b \text {ArcSin}(c x))^2 \]
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Rubi [A]
time = 0.48, 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 = {4835, 4715,
4767, 8, 4723, 4795, 4737, 30} \begin {gather*} -\frac {3 f h (a+b \text {ArcSin}(c x))^2}{32 c^4}+\frac {b x \sqrt {1-c^2 x^2} (d h+e g) (a+b \text {ArcSin}(c x))}{2 c}-\frac {(d h+e g) (a+b \text {ArcSin}(c x))^2}{4 c^2}+\frac {2 b d g \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{c}+\frac {2 b x^2 \sqrt {1-c^2 x^2} (e h+f g) (a+b \text {ArcSin}(c x))}{9 c}+\frac {b f h x^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{8 c}+\frac {4 b \sqrt {1-c^2 x^2} (e h+f g) (a+b \text {ArcSin}(c x))}{9 c^3}+\frac {3 b f h x \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{16 c^3}+\frac {1}{2} x^2 (d h+e g) (a+b \text {ArcSin}(c x))^2+d g x (a+b \text {ArcSin}(c x))^2+\frac {1}{3} x^3 (e h+f g) (a+b \text {ArcSin}(c x))^2+\frac {1}{4} f h x^4 (a+b \text {ArcSin}(c x))^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 \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 4715
Rule 4723
Rule 4737
Rule 4767
Rule 4795
Rule 4835
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 = 0.25, size = 364, normalized size = 0.86 \begin {gather*} d g x (a+b \text {ArcSin}(c x))^2+\frac {1}{2} (e g+d h) x^2 (a+b \text {ArcSin}(c x))^2+\frac {1}{3} (f g+e h) x^3 (a+b \text {ArcSin}(c x))^2+\frac {1}{4} f h x^4 (a+b \text {ArcSin}(c x))^2-\frac {2 b (f g+e h) \left (-3 a \sqrt {1-c^2 x^2} \left (2+c^2 x^2\right )+b c x \left (6+c^2 x^2\right )-3 b \sqrt {1-c^2 x^2} \left (2+c^2 x^2\right ) \text {ArcSin}(c x)\right )}{27 c^3}-2 b d g \left (b x-\frac {\sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{c}\right )-\frac {1}{32} b f h \left (\frac {3 b x^2}{c^2}+b x^4-\frac {6 x \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{c^3}-\frac {4 x^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{c}+\frac {3 (a+b \text {ArcSin}(c x))^2}{b c^4}\right )-\frac {1}{4} b (e g+d h) \left (b x^2-\frac {2 x \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{c}+\frac {(a+b \text {ArcSin}(c x))^2}{b c^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(869\) vs.
\(2(383)=766\).
time = 0.25, size = 870, normalized size = 2.05 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.07, size = 600, normalized size = 1.41 \begin {gather*} \frac {27 \, {\left (8 \, a^{2} - b^{2}\right )} c^{4} f h x^{4} + 32 \, {\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{4} f g x^{3} + 27 \, {\left (8 \, {\left (2 \, a^{2} - b^{2}\right )} c^{4} d - 3 \, b^{2} c^{2} f\right )} h x^{2} + 96 \, {\left (9 \, {\left (a^{2} - 2 \, b^{2}\right )} c^{4} d - 4 \, b^{2} c^{2} f\right )} g x + 9 \, {\left (24 \, b^{2} c^{4} f h x^{4} + 32 \, b^{2} c^{4} f g x^{3} + 48 \, b^{2} c^{4} d h x^{2} + 96 \, b^{2} c^{4} d g x - 3 \, {\left (8 \, b^{2} c^{2} d + 3 \, b^{2} f\right )} h + 8 \, {\left (4 \, b^{2} c^{4} h x^{3} + 6 \, b^{2} c^{4} g x^{2} - 3 \, b^{2} c^{2} g\right )} e\right )} \arcsin \left (c x\right )^{2} + 18 \, {\left (24 \, a b c^{4} f h x^{4} + 32 \, a b c^{4} f g x^{3} + 48 \, a b c^{4} d h x^{2} + 96 \, a b c^{4} d g x - 3 \, {\left (8 \, a b c^{2} d + 3 \, a b f\right )} h + 8 \, {\left (4 \, a b c^{4} h x^{3} + 6 \, a b c^{4} g x^{2} - 3 \, a b c^{2} g\right )} e\right )} \arcsin \left (c x\right ) + 8 \, {\left (4 \, {\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{4} h x^{3} + 27 \, {\left (2 \, a^{2} - b^{2}\right )} c^{4} g x^{2} - 48 \, b^{2} c^{2} h x\right )} e + 6 \, {\left (18 \, a b c^{3} f h x^{3} + 32 \, a b c^{3} f g x^{2} + 9 \, {\left (8 \, a b c^{3} d + 3 \, a b c f\right )} h x + 32 \, {\left (9 \, a b c^{3} d + 2 \, a b c f\right )} g + {\left (18 \, b^{2} c^{3} f h x^{3} + 32 \, b^{2} c^{3} f g x^{2} + 9 \, {\left (8 \, b^{2} c^{3} d + 3 \, b^{2} c f\right )} h x + 32 \, {\left (9 \, b^{2} c^{3} d + 2 \, b^{2} c f\right )} g + 8 \, {\left (4 \, b^{2} c^{3} h x^{2} + 9 \, b^{2} c^{3} g x + 8 \, b^{2} c h\right )} e\right )} \arcsin \left (c x\right ) + 8 \, {\left (4 \, a b c^{3} h x^{2} + 9 \, a b c^{3} g x + 8 \, a b c h\right )} e\right )} \sqrt {-c^{2} x^{2} + 1}}{864 \, c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1059 vs.
\(2 (416) = 832\).
time = 0.61, size = 1059, normalized size = 2.49 \begin {gather*} \begin {cases} a^{2} d g x + \frac {a^{2} d h x^{2}}{2} + \frac {a^{2} e g x^{2}}{2} + \frac {a^{2} e h x^{3}}{3} + \frac {a^{2} f g x^{3}}{3} + \frac {a^{2} f h x^{4}}{4} + 2 a b d g x \operatorname {asin}{\left (c x \right )} + a b d h x^{2} \operatorname {asin}{\left (c x \right )} + a b e g x^{2} \operatorname {asin}{\left (c x \right )} + \frac {2 a b e h x^{3} \operatorname {asin}{\left (c x \right )}}{3} + \frac {2 a b f g x^{3} \operatorname {asin}{\left (c x \right )}}{3} + \frac {a b f h x^{4} \operatorname {asin}{\left (c x \right )}}{2} + \frac {2 a b d g \sqrt {- c^{2} x^{2} + 1}}{c} + \frac {a b d h x \sqrt {- c^{2} x^{2} + 1}}{2 c} + \frac {a b e g x \sqrt {- c^{2} x^{2} + 1}}{2 c} + \frac {2 a b e h x^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c} + \frac {2 a b f g x^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c} + \frac {a b f h x^{3} \sqrt {- c^{2} x^{2} + 1}}{8 c} - \frac {a b d h \operatorname {asin}{\left (c x \right )}}{2 c^{2}} - \frac {a b e g \operatorname {asin}{\left (c x \right )}}{2 c^{2}} + \frac {4 a b e h \sqrt {- c^{2} x^{2} + 1}}{9 c^{3}} + \frac {4 a b f g \sqrt {- c^{2} x^{2} + 1}}{9 c^{3}} + \frac {3 a b f h x \sqrt {- c^{2} x^{2} + 1}}{16 c^{3}} - \frac {3 a b f h \operatorname {asin}{\left (c x \right )}}{16 c^{4}} + b^{2} d g x \operatorname {asin}^{2}{\left (c x \right )} - 2 b^{2} d g x + \frac {b^{2} d h x^{2} \operatorname {asin}^{2}{\left (c x \right )}}{2} - \frac {b^{2} d h x^{2}}{4} + \frac {b^{2} e g x^{2} \operatorname {asin}^{2}{\left (c x \right )}}{2} - \frac {b^{2} e g x^{2}}{4} + \frac {b^{2} e h x^{3} \operatorname {asin}^{2}{\left (c x \right )}}{3} - \frac {2 b^{2} e h x^{3}}{27} + \frac {b^{2} f g x^{3} \operatorname {asin}^{2}{\left (c x \right )}}{3} - \frac {2 b^{2} f g x^{3}}{27} + \frac {b^{2} f h x^{4} \operatorname {asin}^{2}{\left (c x \right )}}{4} - \frac {b^{2} f h x^{4}}{32} + \frac {2 b^{2} d g \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{c} + \frac {b^{2} d h x \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{2 c} + \frac {b^{2} e g x \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{2 c} + \frac {2 b^{2} e h x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{9 c} + \frac {2 b^{2} f g x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{9 c} + \frac {b^{2} f h x^{3} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{8 c} - \frac {b^{2} d h \operatorname {asin}^{2}{\left (c x \right )}}{4 c^{2}} - \frac {b^{2} e g \operatorname {asin}^{2}{\left (c x \right )}}{4 c^{2}} - \frac {4 b^{2} e h x}{9 c^{2}} - \frac {4 b^{2} f g x}{9 c^{2}} - \frac {3 b^{2} f h x^{2}}{32 c^{2}} + \frac {4 b^{2} e h \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{9 c^{3}} + \frac {4 b^{2} f g \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{9 c^{3}} + \frac {3 b^{2} f h x \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{16 c^{3}} - \frac {3 b^{2} f h \operatorname {asin}^{2}{\left (c x \right )}}{32 c^{4}} & \text {for}\: c \neq 0 \\a^{2} \left (d g x + \frac {d h x^{2}}{2} + \frac {e g x^{2}}{2} + \frac {e h x^{3}}{3} + \frac {f g x^{3}}{3} + \frac {f h x^{4}}{4}\right ) & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1145 vs.
\(2 (383) = 766\).
time = 0.45, size = 1145, normalized size = 2.69 \begin {gather*} \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} + 2 \, a b d g x \arcsin \left (c x\right ) + \frac {{\left (c^{2} x^{2} - 1\right )} b^{2} f g x \arcsin \left (c x\right )^{2}}{3 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} b^{2} e h x \arcsin \left (c x\right )^{2}}{3 \, c^{2}} + \frac {\sqrt {-c^{2} x^{2} + 1} b^{2} e g x \arcsin \left (c x\right )}{2 \, c} + \frac {\sqrt {-c^{2} x^{2} + 1} b^{2} d h x \arcsin \left (c x\right )}{2 \, c} + a^{2} d g x - 2 \, b^{2} d g x + \frac {2 \, {\left (c^{2} x^{2} - 1\right )} a b f g x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )} a b e h x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} b^{2} e g \arcsin \left (c x\right )^{2}}{2 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} b^{2} d h \arcsin \left (c x\right )^{2}}{2 \, c^{2}} + \frac {b^{2} f g x \arcsin \left (c x\right )^{2}}{3 \, c^{2}} + \frac {b^{2} e h x \arcsin \left (c x\right )^{2}}{3 \, c^{2}} + \frac {\sqrt {-c^{2} x^{2} + 1} a b e g x}{2 \, c} + \frac {\sqrt {-c^{2} x^{2} + 1} a b d h x}{2 \, c} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d g \arcsin \left (c x\right )}{c} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{2} f h x \arcsin \left (c x\right )}{8 \, c^{3}} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )} b^{2} f g x}{27 \, c^{2}} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )} b^{2} e h x}{27 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} a b e g \arcsin \left (c x\right )}{c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} a b d h \arcsin \left (c x\right )}{c^{2}} + \frac {2 \, a b f g x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {2 \, a b e h x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {b^{2} e g \arcsin \left (c x\right )^{2}}{4 \, c^{2}} + \frac {b^{2} d h \arcsin \left (c x\right )^{2}}{4 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b^{2} f h \arcsin \left (c x\right )^{2}}{4 \, c^{4}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a b d g}{c} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b f h x}{8 \, c^{3}} - \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{2} f g \arcsin \left (c x\right )}{9 \, c^{3}} - \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{2} e h \arcsin \left (c x\right )}{9 \, c^{3}} + \frac {5 \, \sqrt {-c^{2} x^{2} + 1} b^{2} f h x \arcsin \left (c x\right )}{16 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )} a^{2} e g}{2 \, c^{2}} - \frac {{\left (c^{2} x^{2} - 1\right )} b^{2} e g}{4 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} a^{2} d h}{2 \, c^{2}} - \frac {{\left (c^{2} x^{2} - 1\right )} b^{2} d h}{4 \, c^{2}} - \frac {14 \, b^{2} f g x}{27 \, c^{2}} - \frac {14 \, b^{2} e h x}{27 \, c^{2}} + \frac {a b e g \arcsin \left (c x\right )}{2 \, c^{2}} + \frac {a b d h \arcsin \left (c x\right )}{2 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} a b f h \arcsin \left (c x\right )}{2 \, c^{4}} + \frac {{\left (c^{2} x^{2} - 1\right )} b^{2} f h \arcsin \left (c x\right )^{2}}{2 \, c^{4}} - \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b f g}{9 \, c^{3}} - \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b e h}{9 \, c^{3}} + \frac {5 \, \sqrt {-c^{2} x^{2} + 1} a b f h x}{16 \, c^{3}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{2} f g \arcsin \left (c x\right )}{3 \, c^{3}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{2} e h \arcsin \left (c x\right )}{3 \, c^{3}} - \frac {b^{2} e g}{8 \, c^{2}} - \frac {b^{2} d h}{8 \, c^{2}} - \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b^{2} f h}{32 \, c^{4}} + \frac {{\left (c^{2} x^{2} - 1\right )} a b f h \arcsin \left (c x\right )}{c^{4}} + \frac {5 \, b^{2} f h \arcsin \left (c x\right )^{2}}{32 \, c^{4}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a b f g}{3 \, c^{3}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a b e h}{3 \, c^{3}} - \frac {5 \, {\left (c^{2} x^{2} - 1\right )} b^{2} f h}{32 \, c^{4}} + \frac {5 \, a b f h \arcsin \left (c x\right )}{16 \, c^{4}} - \frac {17 \, b^{2} f h}{256 \, c^{4}} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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