Optimal. Leaf size=96 \[ \frac {x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{(c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac {b \left (1-c^2 x^2\right )^{3/2} \log \left (1-c^2 x^2\right )}{2 c (c d x+d)^{3/2} (f-c f x)^{3/2}} \]
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Rubi [A] time = 0.17, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {4673, 4651, 260} \[ \frac {x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{(c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac {b \left (1-c^2 x^2\right )^{3/2} \log \left (1-c^2 x^2\right )}{2 c (c d x+d)^{3/2} (f-c f x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 260
Rule 4651
Rule 4673
Rubi steps
\begin {align*} \int \frac {a+b \sin ^{-1}(c x)}{(d+c d x)^{3/2} (f-c f x)^{3/2}} \, dx &=\frac {\left (1-c^2 x^2\right )^{3/2} \int \frac {a+b \sin ^{-1}(c x)}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{(d+c d x)^{3/2} (f-c f x)^{3/2}}\\ &=\frac {x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{(d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac {\left (b c \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac {x}{1-c^2 x^2} \, dx}{(d+c d x)^{3/2} (f-c f x)^{3/2}}\\ &=\frac {x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{(d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {b \left (1-c^2 x^2\right )^{3/2} \log \left (1-c^2 x^2\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.51, size = 105, normalized size = 1.09 \[ \frac {\sqrt {c d x+d} \left (2 a c x+b \sqrt {1-c^2 x^2} \log (-f (c x+1))+b \sqrt {1-c^2 x^2} \log (f-c f x)+2 b c x \sin ^{-1}(c x)\right )}{2 c d^2 f (c x+1) \sqrt {f-c f x}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c d x + d} \sqrt {-c f x + f} {\left (b \arcsin \left (c x\right ) + a\right )}}{c^{4} d^{2} f^{2} x^{4} - 2 \, c^{2} d^{2} f^{2} x^{2} + d^{2} f^{2}}, x\right ) \]
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 \[ \int \frac {b \arcsin \left (c x\right ) + a}{{\left (c d x + d\right )}^{\frac {3}{2}} {\left (-c f x + f\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.33, size = 0, normalized size = 0.00 \[ \int \frac {a +b \arcsin \left (c x \right )}{\left (c d x +d \right )^{\frac {3}{2}} \left (-c f x +f \right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 86, normalized size = 0.90 \[ \frac {b x \arcsin \left (c x\right )}{\sqrt {-c^{2} d f x^{2} + d f} d f} + \frac {a x}{\sqrt {-c^{2} d f x^{2} + d f} d f} - \frac {b \sqrt {\frac {1}{d f}} \log \left (x^{2} - \frac {1}{c^{2}}\right )}{2 \, c d f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{{\left (d+c\,d\,x\right )}^{3/2}\,{\left (f-c\,f\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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