Optimal. Leaf size=410 \[ \frac {(f+g x)^2 \left (c^2 f x+g\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {2 (c f+g) (c f-g) \left (c^2 f x+g\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {b g \sqrt {1-c^2 x^2} (c f-g)^2 \log (c x+1)}{12 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} (c f+g) (c f-g)^2 \log (c x+1)}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} (c f+g)^2 (c f-g) \log (1-c x)}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {b g \sqrt {1-c^2 x^2} (c f+g)^2 \log (1-c x)}{12 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {b (f+g x) \left (c^2 f^2+2 c^2 f g x+g^2\right )}{6 c^3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}} \]
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Rubi [A] time = 0.43, antiderivative size = 410, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {4777, 723, 637, 4761, 819, 633, 31} \[ \frac {2 (c f+g) (c f-g) \left (c^2 f x+g\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {(f+g x)^2 \left (c^2 f x+g\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {b (f+g x) \left (c^2 f^2+2 c^2 f g x+g^2\right )}{6 c^3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {b g \sqrt {1-c^2 x^2} (c f-g)^2 \log (c x+1)}{12 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} (c f+g) (c f-g)^2 \log (c x+1)}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} (c f+g)^2 (c f-g) \log (1-c x)}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {b g \sqrt {1-c^2 x^2} (c f+g)^2 \log (1-c x)}{12 c^4 d^2 \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 633
Rule 637
Rule 723
Rule 819
Rule 4761
Rule 4777
Rubi steps
\begin {align*} \int \frac {(f+g x)^3 \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)^3 \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 {2 (c f-g) (c f+g) \left (g+c^2 f 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)^2 \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 {\left (b c \sqrt {1-c^2 x^2}\right ) \int \left (\frac {\left (g+c^2 f x\right ) (f+g x)^2}{3 c^2 \left (1-c^2 x^2\right )^2}+\frac {2 (c f-g) (c f+g) \left (g+c^2 f x\right )}{3 c^4 \left (1-c^2 x^2\right )}\right ) \, dx}{d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {2 (c f-g) (c f+g) \left (g+c^2 f 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)^2 \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 {\left (b \sqrt {1-c^2 x^2}\right ) \int \frac {\left (g+c^2 f x\right ) (f+g x)^2}{\left (1-c^2 x^2\right )^2} \, dx}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 b (c f-g) (c f+g) \sqrt {1-c^2 x^2}\right ) \int \frac {g+c^2 f x}{1-c^2 x^2} \, dx}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b (f+g x) \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 (c f-g) (c f+g) \left (g+c^2 f 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)^2 \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 {\left (b \sqrt {1-c^2 x^2}\right ) \int \frac {g \left (c^2 f^2+g^2\right )+2 c^2 f g^2 x}{1-c^2 x^2} \, dx}{6 c^3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b (c f-g)^2 (c f+g) \sqrt {1-c^2 x^2}\right ) \int \frac {1}{-c-c^2 x} \, dx}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b (c f-g) (c f+g)^2 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{c-c^2 x} \, dx}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b (f+g x) \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 (c f-g) (c f+g) \left (g+c^2 f 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)^2 \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 {b (c f-g) (c f+g)^2 \sqrt {1-c^2 x^2} \log (1-c x)}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {b (c f-g)^2 (c f+g) \sqrt {1-c^2 x^2} \log (1+c x)}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b (c f-g)^2 g \sqrt {1-c^2 x^2}\right ) \int \frac {1}{-c-c^2 x} \, dx}{12 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b g (c f+g)^2 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{c-c^2 x} \, dx}{12 c^2 d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b (f+g x) \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 (c f-g) (c f+g) \left (g+c^2 f 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)^2 \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 {b (c f-g) (c f+g)^2 \sqrt {1-c^2 x^2} \log (1-c x)}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {b g (c f+g)^2 \sqrt {1-c^2 x^2} \log (1-c x)}{12 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {b (c f-g)^2 g \sqrt {1-c^2 x^2} \log (1+c x)}{12 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {b (c f-g)^2 (c f+g) \sqrt {1-c^2 x^2} \log (1+c x)}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [C] time = 1.30, size = 366, normalized size = 0.89 \[ \frac {\sqrt {d-c^2 d x^2} \left (-\sqrt {-c^2} \left (4 a c^6 f^3 x^3-6 a c^4 f^3 x-6 a c^4 f g^2 x^3-6 a c^2 f^2 g-6 a c^2 g^3 x^2+4 a g^3-b c f \left (1-c^2 x^2\right )^{3/2} \left (2 c^2 f^2-3 g^2\right ) \log \left (c^2 x^2-1\right )+3 b c f g^2 \sqrt {1-c^2 x^2}+b c g^3 x \sqrt {1-c^2 x^2}+b c^3 f^3 \sqrt {1-c^2 x^2}+3 b c^3 f^2 g x \sqrt {1-c^2 x^2}+2 b \sin ^{-1}(c x) \left (2 c^6 f^3 x^3-3 c^4 f x \left (f^2+g^2 x^2\right )-3 c^2 g \left (f^2+g^2 x^2\right )+2 g^3\right )\right )+i b c g \left (1-c^2 x^2\right )^{3/2} \left (3 c^2 f^2-5 g^2\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt {-c^2} x\right )\right |1\right )\right )}{6 c^4 \sqrt {-c^2} d^3 \left (c^2 x^2-1\right )^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 16.39, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (a g^{3} x^{3} + 3 \, a f g^{2} x^{2} + 3 \, a f^{2} g x + a f^{3} + {\left (b g^{3} x^{3} + 3 \, b f g^{2} x^{2} + 3 \, b f^{2} g x + b f^{3}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{c^{6} d^{3} x^{6} - 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} - d^{3}}, x\right ) \]
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 \[ \int \frac {{\left (g x + f\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.18, size = 5098, normalized size = 12.43 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {{\left (f+g\,x\right )}^3\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]
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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