Optimal. Leaf size=315 \[ \frac {\left (c^2 f x \left (c^2 f^2+3 g^2\right )+g \left (3 c^2 f^2+g^2\right )\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {3 f g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c^3 d \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} (c f-g)^3 \log (c x+1)}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} (c f+g)^3 \log (1-c x)}{2 c^4 d \sqrt {d-c^2 d x^2}}-\frac {b g^3 x \sqrt {1-c^2 x^2}}{c^3 d \sqrt {d-c^2 d x^2}} \]
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Rubi [A] time = 0.56, antiderivative size = 315, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.323, Rules used = {4777, 4775, 637, 4761, 12, 633, 31, 4641, 4677, 8} \[ \frac {\left (c^2 f x \left (c^2 f^2+3 g^2\right )+g \left (3 c^2 f^2+g^2\right )\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {3 f g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c^3 d \sqrt {d-c^2 d x^2}}+\frac {g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} (c f-g)^3 \log (c x+1)}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} (c f+g)^3 \log (1-c x)}{2 c^4 d \sqrt {d-c^2 d x^2}}-\frac {b g^3 x \sqrt {1-c^2 x^2}}{c^3 d \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 8
Rule 12
Rule 31
Rule 633
Rule 637
Rule 4641
Rule 4677
Rule 4761
Rule 4775
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 )^{3/2}} \, 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 )^{3/2}} \, dx}{d \sqrt {d-c^2 d x^2}}\\ &=\frac {\sqrt {1-c^2 x^2} \int \left (\frac {\left (c^2 f^3+3 f g^2+g \left (3 c^2 f^2+g^2\right ) x\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {3 f g^2 \left (a+b \sin ^{-1}(c x)\right )}{c^2 \sqrt {1-c^2 x^2}}-\frac {g^3 x \left (a+b \sin ^{-1}(c x)\right )}{c^2 \sqrt {1-c^2 x^2}}\right ) \, dx}{d \sqrt {d-c^2 d x^2}}\\ &=\frac {\sqrt {1-c^2 x^2} \int \frac {\left (c^2 f^3+3 f g^2+g \left (3 c^2 f^2+g^2\right ) x\right ) \left (a+b \sin ^{-1}(c x)\right )}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {\left (3 f g^2 \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {\left (g^3 \sqrt {1-c^2 x^2}\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{c^2 d \sqrt {d-c^2 d x^2}}\\ &=\frac {\left (g \left (3 c^2 f^2+g^2\right )+c^2 f \left (c^2 f^2+3 g^2\right ) x\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {3 f g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c^3 d \sqrt {d-c^2 d x^2}}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \int \frac {g \left (3 c^2 f^2+g^2\right )+c^2 f \left (c^2 f^2+3 g^2\right ) x}{c^2 \left (1-c^2 x^2\right )} \, dx}{c d \sqrt {d-c^2 d x^2}}-\frac {\left (b g^3 \sqrt {1-c^2 x^2}\right ) \int 1 \, dx}{c^3 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {b g^3 x \sqrt {1-c^2 x^2}}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {\left (g \left (3 c^2 f^2+g^2\right )+c^2 f \left (c^2 f^2+3 g^2\right ) x\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {3 f g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c^3 d \sqrt {d-c^2 d x^2}}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \int \frac {g \left (3 c^2 f^2+g^2\right )+c^2 f \left (c^2 f^2+3 g^2\right ) x}{1-c^2 x^2} \, dx}{c^3 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {b g^3 x \sqrt {1-c^2 x^2}}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {\left (g \left (3 c^2 f^2+g^2\right )+c^2 f \left (c^2 f^2+3 g^2\right ) x\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {3 f g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c^3 d \sqrt {d-c^2 d x^2}}-\frac {\left (b (c f-g)^3 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{-c-c^2 x} \, dx}{2 c^2 d \sqrt {d-c^2 d x^2}}-\frac {\left (b (c f+g)^3 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{c-c^2 x} \, dx}{2 c^2 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {b g^3 x \sqrt {1-c^2 x^2}}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {\left (g \left (3 c^2 f^2+g^2\right )+c^2 f \left (c^2 f^2+3 g^2\right ) x\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {3 f g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c^3 d \sqrt {d-c^2 d x^2}}+\frac {b (c f+g)^3 \sqrt {1-c^2 x^2} \log (1-c x)}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {b (c f-g)^3 \sqrt {1-c^2 x^2} \log (1+c x)}{2 c^4 d \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 1.10, size = 194, normalized size = 0.62 \[ \frac {\sqrt {1-c^2 x^2} \left (2 g^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {3 c f g^2 \left (a+b \sin ^{-1}(c x)\right )^2}{b}+(c f-g)^3 \left (2 b \log \left (\sin \left (\frac {1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )\right )-\cot \left (\frac {1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right ) \left (a+b \sin ^{-1}(c x)\right )\right )+(c f+g)^3 \left (\tan \left (\frac {1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right ) \left (a+b \sin ^{-1}(c x)\right )+2 b \log \left (\cos \left (\frac {1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )\right )\right )-2 b c g^3 x\right )}{2 c^4 d \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a g^{3} x^{3} + 3 \, a f g^{2} x^{2} + 3 \, a f^{2} g x + a f^{3} + {\left (b g^{3} x^{3} + 3 \, b f g^{2} x^{2} + 3 \, b f^{2} g x + b f^{3}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [C] time = 1.20, size = 1158, normalized size = 3.68 \[ -\frac {a \,g^{3} x^{2}}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {2 a \,g^{3}}{d \,c^{4} \sqrt {-c^{2} d \,x^{2}+d}}+\frac {3 a f \,g^{2} x}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {3 a f \,g^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{c^{2} d \sqrt {c^{2} d}}+\frac {3 a \,f^{2} g}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {a \,f^{3} x}{d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) x \,f^{3}}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) f^{2} g}{c^{2} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right ) f^{3}}{c \,d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right ) g^{3}}{c^{4} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right ) f^{3}}{c \,d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right ) g^{3}}{c^{4} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{3} \arcsin \left (c x \right )}{c^{4} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right ) f^{2} g}{c^{2} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right ) f \,g^{2}}{c^{3} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right ) f^{2} g}{c^{2} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right ) f \,g^{2}}{c^{3} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{3} \sqrt {-c^{2} x^{2}+1}\, x}{c^{3} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {i b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f^{3} \arcsin \left (c x \right )}{c \,d^{2} \left (c^{2} x^{2}-1\right )}-\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) x f \,g^{2}}{c^{2} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{3} \arcsin \left (c x \right ) x^{2}}{c^{2} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2} f \,g^{2}}{2 c^{3} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {3 i b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f \arcsin \left (c x \right ) g^{2}}{c^{3} d^{2} \left (c^{2} x^{2}-1\right )} \]
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 \[ \int \frac {{\left (f+g\,x\right )}^3\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]
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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