Optimal. Leaf size=213 \[ \frac {\left (x \left (c^2 f^2+g^2\right )+2 f g\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c^3 d \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} (c f-g)^2 \log (c x+1)}{2 c^3 d \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} (c f+g)^2 \log (1-c x)}{2 c^3 d \sqrt {d-c^2 d x^2}} \]
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Rubi [A] time = 0.43, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {4777, 4775, 637, 4761, 633, 31, 4641} \[ \frac {\left (x \left (c^2 f^2+g^2\right )+2 f g\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c^3 d \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} (c f-g)^2 \log (c x+1)}{2 c^3 d \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} (c f+g)^2 \log (1-c x)}{2 c^3 d \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 633
Rule 637
Rule 4641
Rule 4761
Rule 4775
Rule 4777
Rubi steps
\begin {align*} \int \frac {(f+g x)^2 \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)^2 \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^2+g^2+2 c^2 f g x\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {g^2 \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^2+g^2+2 c^2 f g 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 (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 (2 f g+\left (c^2 f^2+g^2\right ) x\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {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 {2 f g+\left (c^2 f^2+g^2\right ) x}{1-c^2 x^2} \, dx}{c d \sqrt {d-c^2 d x^2}}\\ &=\frac {\left (2 f g+\left (c^2 f^2+g^2\right ) x\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {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)^2 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{-c-c^2 x} \, dx}{2 c d \sqrt {d-c^2 d x^2}}-\frac {\left (b (c f+g)^2 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{c-c^2 x} \, dx}{2 c d \sqrt {d-c^2 d x^2}}\\ &=\frac {\left (2 f g+\left (c^2 f^2+g^2\right ) x\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {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)^2 \sqrt {1-c^2 x^2} \log (1-c x)}{2 c^3 d \sqrt {d-c^2 d x^2}}+\frac {b (c f-g)^2 \sqrt {1-c^2 x^2} \log (1+c x)}{2 c^3 d \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 0.77, size = 156, normalized size = 0.73 \[ \frac {\sqrt {1-c^2 x^2} \left ((g-c f)^2 \left (2 b \log \left (\sin \left (\frac {1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )\right )-\cot \left (\frac {1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right ) \left (a+b \sin ^{-1}(c x)\right )\right )+(c f+g)^2 \left (\tan \left (\frac {1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right ) \left (a+b \sin ^{-1}(c x)\right )+2 b \log \left (\cos \left (\frac {1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )\right )\right )-\frac {g^2 \left (a+b \sin ^{-1}(c x)\right )^2}{b}\right )}{2 c^3 d \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-c^{2} d x^{2} + d} {\left (a g^{2} x^{2} + 2 \, a f g x + a f^{2} + {\left (b g^{2} x^{2} + 2 \, b f g x + b f^{2}\right )} \arcsin \left (c x\right )\right )}}{c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}}, x\right ) \]
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 = 0.85, size = 867, normalized size = 4.07 \[ \frac {a \,g^{2} x}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {a \,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 {2 a f g}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {a \,f^{2} x}{d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, g^{2} \arcsin \left (c x \right )^{2}}{2 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 )}\, \arcsin \left (c x \right ) g^{2}}{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 )}\, \arcsin \left (c x \right ) f^{2}}{c \,d^{2} \left (c^{2} x^{2}-1\right )}-\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \left (-c^{2} x^{2}+1\right ) f g}{c^{2} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) x^{2} f g}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) x \,f^{2}}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) x \,g^{2}}{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^{2}}{c \,d^{2} \left (c^{2} x^{2}-1\right )}-\frac {2 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}{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 ) g^{2}}{c^{3} 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^{2}}{c \,d^{2} \left (c^{2} x^{2}-1\right )}+\frac {2 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}{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 ) 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] time = 0.00, size = 0, normalized size = 0.00 \[ a g^{2} {\left (\frac {x}{\sqrt {-c^{2} d x^{2} + d} c^{2} d} - \frac {\arcsin \left (c x\right )}{c^{3} d^{\frac {3}{2}}}\right )} + \frac {b f^{2} x \arcsin \left (c x\right )}{\sqrt {-c^{2} d x^{2} + d} d} + \frac {a f^{2} x}{\sqrt {-c^{2} d x^{2} + d} d} + \sqrt {d} \int \frac {{\left (b g^{2} x^{2} + 2 \, b f g x\right )} \sqrt {c x + 1} \sqrt {-c x + 1} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}}\,{d x} - \frac {b f^{2} \log \left (x^{2} - \frac {1}{c^{2}}\right )}{2 \, c d^{\frac {3}{2}}} + \frac {2 \, a f g}{\sqrt {-c^{2} d x^{2} + d} c^{2} d} \]
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 \[ \int \frac {{\left (f+g\,x\right )}^2\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (f + g x\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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