Optimal. Leaf size=528 \[ \frac {(f+g x)^3 \left (c^2 f x+g\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {g^4 \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \left (a+b \sin ^{-1}(c x)\right )}{c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {f g \left (1-c^2 x^2\right ) \left (2 c^2 f^2-5 g^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {(f+g x) \left (2 c^2 f x \left (c^2 f^2-2 g^2\right )+g \left (c^2 f^2-3 g^2\right )\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} (c f-2 g) (c f+g)^3 \log (1-c x)}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} (c f-g)^3 (c f+2 g) \log (c x+1)}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {b g^4 \sqrt {1-c^2 x^2} \sin ^{-1}(c x)^2}{2 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {b (f+g x)^2 \left (c^2 f^2+2 c^2 f g x+g^2\right )}{6 c^3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {b f g^3 x \sqrt {1-c^2 x^2}}{3 c^3 d^2 \sqrt {d-c^2 d x^2}} \]
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Rubi [A] time = 0.75, antiderivative size = 754, normalized size of antiderivative = 1.43, number of steps used = 13, number of rules used = 10, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.323, Rules used = {4777, 739, 819, 641, 216, 4761, 774, 633, 31, 4641} \[ \frac {f g \left (1-c^2 x^2\right ) \left (2 c^2 f^2-5 g^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {(f+g x) \left (2 c^2 f x \left (c^2 f^2-2 g^2\right )+g \left (c^2 f^2-3 g^2\right )\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {(f+g x)^3 \left (c^2 f x+g\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {g^4 \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \left (a+b \sin ^{-1}(c x)\right )}{c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {b f g x \sqrt {1-c^2 x^2} \left (2 c^2 f^2-5 g^2\right )}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 b f g x \sqrt {1-c^2 x^2} \left (c^2 f^2-2 g^2\right )}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b (f+g x)^2 \left (c^2 f^2+2 c^2 f g x+g^2\right )}{6 c^3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {2 b f g^3 x \sqrt {1-c^2 x^2}}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b g \sqrt {1-c^2 x^2} (c f+g)^3 \log (1-c x)}{6 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {b g \sqrt {1-c^2 x^2} (c f-g)^3 \log (c x+1)}{6 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} (2 c f-3 g) (c f+g)^3 \log (1-c x)}{6 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} (c f-g)^3 (2 c f+3 g) \log (c x+1)}{6 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {b g^4 \sqrt {1-c^2 x^2} \sin ^{-1}(c x)^2}{2 c^5 d^2 \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 216
Rule 633
Rule 641
Rule 739
Rule 774
Rule 819
Rule 4641
Rule 4761
Rule 4777
Rubi steps
\begin {align*} \int \frac {(f+g x)^4 \left (a+b \sin ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^{5/2}} \, dx &=\frac {\sqrt {1-c^2 x^2} \int \frac {(f+g x)^4 \left (a+b \sin ^{-1}(c x)\right )}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {(f+g x) \left (g \left (c^2 f^2-3 g^2\right )+2 c^2 f \left (c^2 f^2-2 g^2\right ) x\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (g+c^2 f x\right ) (f+g x)^3 \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {f g \left (2 c^2 f^2-5 g^2\right ) \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {g^4 \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \left (a+b \sin ^{-1}(c x)\right )}{c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b c \sqrt {1-c^2 x^2}\right ) \int \left (\frac {f g \left (2 c^2 f^2-5 g^2\right )}{3 c^4}+\frac {\left (g+c^2 f x\right ) (f+g x)^3}{3 c^2 \left (1-c^2 x^2\right )^2}+\frac {(f+g x) \left (g \left (c^2 f^2-3 g^2\right )+2 c^2 f \left (c^2 f^2-2 g^2\right ) x\right )}{3 c^4 \left (1-c^2 x^2\right )}+\frac {g^4 \sin ^{-1}(c x)}{c^5 \sqrt {1-c^2 x^2}}\right ) \, dx}{d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b f g \left (2 c^2 f^2-5 g^2\right ) x \sqrt {1-c^2 x^2}}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(f+g x) \left (g \left (c^2 f^2-3 g^2\right )+2 c^2 f \left (c^2 f^2-2 g^2\right ) x\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (g+c^2 f x\right ) (f+g x)^3 \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {f g \left (2 c^2 f^2-5 g^2\right ) \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {g^4 \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \left (a+b \sin ^{-1}(c x)\right )}{c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \int \frac {(f+g x) \left (g \left (c^2 f^2-3 g^2\right )+2 c^2 f \left (c^2 f^2-2 g^2\right ) x\right )}{1-c^2 x^2} \, dx}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \int \frac {\left (g+c^2 f x\right ) (f+g x)^3}{\left (1-c^2 x^2\right )^2} \, dx}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b g^4 \sqrt {1-c^2 x^2}\right ) \int \frac {\sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{c^4 d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b (f+g x)^2 \left (c^2 f^2+g^2+2 c^2 f g x\right )}{6 c^3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {b f g \left (2 c^2 f^2-5 g^2\right ) x \sqrt {1-c^2 x^2}}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 b f g \left (c^2 f^2-2 g^2\right ) x \sqrt {1-c^2 x^2}}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b g^4 \sqrt {1-c^2 x^2} \sin ^{-1}(c x)^2}{2 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {(f+g x) \left (g \left (c^2 f^2-3 g^2\right )+2 c^2 f \left (c^2 f^2-2 g^2\right ) x\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (g+c^2 f x\right ) (f+g x)^3 \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {f g \left (2 c^2 f^2-5 g^2\right ) \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {g^4 \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \left (a+b \sin ^{-1}(c x)\right )}{c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b \sqrt {1-c^2 x^2}\right ) \int \frac {-c^2 f g \left (c^2 f^2-3 g^2\right )-2 c^2 f g \left (c^2 f^2-2 g^2\right )-c^2 \left (g^2 \left (c^2 f^2-3 g^2\right )+2 c^2 f^2 \left (c^2 f^2-2 g^2\right )\right ) x}{1-c^2 x^2} \, dx}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b \sqrt {1-c^2 x^2}\right ) \int \frac {(f+g x) \left (2 g \left (c^2 f^2+g^2\right )+4 c^2 f g^2 x\right )}{1-c^2 x^2} \, dx}{6 c^3 d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b (f+g x)^2 \left (c^2 f^2+g^2+2 c^2 f g x\right )}{6 c^3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {2 b f g^3 x \sqrt {1-c^2 x^2}}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b f g \left (2 c^2 f^2-5 g^2\right ) x \sqrt {1-c^2 x^2}}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 b f g \left (c^2 f^2-2 g^2\right ) x \sqrt {1-c^2 x^2}}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b g^4 \sqrt {1-c^2 x^2} \sin ^{-1}(c x)^2}{2 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {(f+g x) \left (g \left (c^2 f^2-3 g^2\right )+2 c^2 f \left (c^2 f^2-2 g^2\right ) x\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (g+c^2 f x\right ) (f+g x)^3 \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {f g \left (2 c^2 f^2-5 g^2\right ) \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {g^4 \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \left (a+b \sin ^{-1}(c x)\right )}{c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \int \frac {-4 c^2 f g^3-2 c^2 f g \left (c^2 f^2+g^2\right )-c^2 \left (4 c^2 f^2 g^2+2 g^2 \left (c^2 f^2+g^2\right )\right ) x}{1-c^2 x^2} \, dx}{6 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b (2 c f-3 g) (c f+g)^3 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{c-c^2 x} \, dx}{6 c^3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b (c f-g)^3 (2 c f+3 g) \sqrt {1-c^2 x^2}\right ) \int \frac {1}{-c-c^2 x} \, dx}{6 c^3 d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b (f+g x)^2 \left (c^2 f^2+g^2+2 c^2 f g x\right )}{6 c^3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {2 b f g^3 x \sqrt {1-c^2 x^2}}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b f g \left (2 c^2 f^2-5 g^2\right ) x \sqrt {1-c^2 x^2}}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 b f g \left (c^2 f^2-2 g^2\right ) x \sqrt {1-c^2 x^2}}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b g^4 \sqrt {1-c^2 x^2} \sin ^{-1}(c x)^2}{2 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {(f+g x) \left (g \left (c^2 f^2-3 g^2\right )+2 c^2 f \left (c^2 f^2-2 g^2\right ) x\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (g+c^2 f x\right ) (f+g x)^3 \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {f g \left (2 c^2 f^2-5 g^2\right ) \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {g^4 \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \left (a+b \sin ^{-1}(c x)\right )}{c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {b (2 c f-3 g) (c f+g)^3 \sqrt {1-c^2 x^2} \log (1-c x)}{6 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {b (c f-g)^3 (2 c f+3 g) \sqrt {1-c^2 x^2} \log (1+c x)}{6 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b (c f-g)^3 g \sqrt {1-c^2 x^2}\right ) \int \frac {1}{-c-c^2 x} \, dx}{6 c^3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b g (c f+g)^3 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{c-c^2 x} \, dx}{6 c^3 d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b (f+g x)^2 \left (c^2 f^2+g^2+2 c^2 f g x\right )}{6 c^3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {2 b f g^3 x \sqrt {1-c^2 x^2}}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b f g \left (2 c^2 f^2-5 g^2\right ) x \sqrt {1-c^2 x^2}}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 b f g \left (c^2 f^2-2 g^2\right ) x \sqrt {1-c^2 x^2}}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b g^4 \sqrt {1-c^2 x^2} \sin ^{-1}(c x)^2}{2 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {(f+g x) \left (g \left (c^2 f^2-3 g^2\right )+2 c^2 f \left (c^2 f^2-2 g^2\right ) x\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (g+c^2 f x\right ) (f+g x)^3 \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {f g \left (2 c^2 f^2-5 g^2\right ) \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {g^4 \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \left (a+b \sin ^{-1}(c x)\right )}{c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {b (2 c f-3 g) (c f+g)^3 \sqrt {1-c^2 x^2} \log (1-c x)}{6 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {b g (c f+g)^3 \sqrt {1-c^2 x^2} \log (1-c x)}{6 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {b (c f-g)^3 g \sqrt {1-c^2 x^2} \log (1+c x)}{6 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {b (c f-g)^3 (2 c f+3 g) \sqrt {1-c^2 x^2} \log (1+c x)}{6 c^5 d^2 \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 3.21, size = 868, normalized size = 1.64 \[ \frac {b \left (4 c x \sin ^{-1}(c x)+\frac {\frac {2 c x \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}}-1}{\sqrt {1-c^2 x^2}}+4 \sqrt {1-c^2 x^2} \log \left (\sqrt {1-c^2 x^2}\right )\right ) f^4}{6 c d^2 \sqrt {d \left (1-c^2 x^2\right )}}+\frac {b g \left (8 \sin ^{-1}(c x)+\cos \left (3 \sin ^{-1}(c x)\right ) \left (\log \left (\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right )-\log \left (\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )+\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right )\right )+3 \sqrt {1-c^2 x^2} \left (\log \left (\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right )-\log \left (\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )+\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right )\right )-2 \sin \left (2 \sin ^{-1}(c x)\right )\right ) f^3}{6 c^2 d \left (d \left (1-c^2 x^2\right )\right )^{3/2}}+\frac {b g^2 \left (-2 c x \sin ^{-1}(c x)+\frac {\frac {2 c x \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}}-1}{\sqrt {1-c^2 x^2}}-2 \sqrt {1-c^2 x^2} \log \left (\sqrt {1-c^2 x^2}\right )\right ) f^2}{c^3 d^2 \sqrt {d \left (1-c^2 x^2\right )}}-\frac {b g^3 \left (12 \cos \left (2 \sin ^{-1}(c x)\right ) \sin ^{-1}(c x)+4 \sin ^{-1}(c x)+5 \cos \left (3 \sin ^{-1}(c x)\right ) \log \left (\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right )+15 \sqrt {1-c^2 x^2} \left (\log \left (\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right )-\log \left (\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )+\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right )\right )-5 \cos \left (3 \sin ^{-1}(c x)\right ) \log \left (\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )+\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right )+2 \sin \left (2 \sin ^{-1}(c x)\right )\right ) f}{6 c^4 d \left (d \left (1-c^2 x^2\right )\right )^{3/2}}+\sqrt {-d \left (c^2 x^2-1\right )} \left (\frac {a c^4 x f^4+4 a c^2 g f^3+6 a c^2 g^2 x f^2+4 a g^3 f+a g^4 x}{3 c^4 d^3 \left (c^2 x^2-1\right )^2}-\frac {2 a \left (c^4 x f^4-3 c^2 g^2 x f^2-6 g^3 f-2 g^4 x\right )}{3 c^4 d^3 \left (c^2 x^2-1\right )}\right )-\frac {a g^4 \tan ^{-1}\left (\frac {c x \sqrt {-d \left (c^2 x^2-1\right )}}{\sqrt {d} \left (c^2 x^2-1\right )}\right )}{c^5 d^{5/2}}+\frac {b g^4 \left (\sqrt {1-c^2 x^2} \left (3 \sin ^{-1}(c x)^2-8 \log \left (\sqrt {1-c^2 x^2}\right )\right )-\frac {\frac {2 \sin ^{-1}(c x) \sin \left (3 \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}+1}{\sqrt {1-c^2 x^2}}\right )}{6 c^5 d^2 \sqrt {d \left (1-c^2 x^2\right )}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.14, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (a g^{4} x^{4} + 4 \, a f g^{3} x^{3} + 6 \, a f^{2} g^{2} x^{2} + 4 \, a f^{3} g x + a f^{4} + {\left (b g^{4} x^{4} + 4 \, b f g^{3} x^{3} + 6 \, b f^{2} g^{2} x^{2} + 4 \, b f^{3} g x + b f^{4}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{c^{6} d^{3} x^{6} - 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} - d^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (g x + f\right )}^{4} {\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.69, size = 6743, normalized size = 12.77 \[ \text {output too large to display} \]
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 \[ \frac {1}{6} \, b c f^{4} {\left (\frac {1}{c^{4} d^{\frac {5}{2}} x^{2} - c^{2} d^{\frac {5}{2}}} + \frac {2 \, \log \left (c x + 1\right )}{c^{2} d^{\frac {5}{2}}} + \frac {2 \, \log \left (c x - 1\right )}{c^{2} d^{\frac {5}{2}}}\right )} + \frac {1}{3} \, b f^{4} {\left (\frac {2 \, x}{\sqrt {-c^{2} d x^{2} + d} d^{2}} + \frac {x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d}\right )} \arcsin \left (c x\right ) + \frac {1}{3} \, {\left (x {\left (\frac {3 \, x^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2} d} - \frac {2}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{4} d}\right )} - \frac {x}{\sqrt {-c^{2} d x^{2} + d} c^{4} d^{2}} + \frac {3 \, \arcsin \left (c x\right )}{c^{5} d^{\frac {5}{2}}}\right )} a g^{4} + \frac {1}{3} \, a f^{4} {\left (\frac {2 \, x}{\sqrt {-c^{2} d x^{2} + d} d^{2}} + \frac {x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d}\right )} + \frac {4}{3} \, a f g^{3} {\left (\frac {3 \, x^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2} d} - \frac {2}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{4} d}\right )} - 2 \, a f^{2} g^{2} {\left (\frac {x}{\sqrt {-c^{2} d x^{2} + d} c^{2} d^{2}} - \frac {x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2} d}\right )} - \sqrt {d} \int \frac {{\left (b g^{4} x^{4} + 4 \, b f g^{3} x^{3} + 6 \, b f^{2} g^{2} x^{2} + 4 \, b f^{3} g x\right )} \sqrt {c x + 1} \sqrt {-c x + 1} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{c^{6} d^{3} x^{6} - 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} - d^{3}}\,{d x} + \frac {4 \, a f^{3} g}{3 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2} d} \]
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 \[ \int \frac {{\left (f+g\,x\right )}^4\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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