Optimal. Leaf size=96 \[ \frac {x \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))}{(d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {b \left (1-c^2 x^2\right )^{3/2} \log \left (1-c^2 x^2\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}} \]
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Rubi [A]
time = 0.12, 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 = {4763, 4745,
266} \begin {gather*} \frac {x \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))}{(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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 266
Rule 4745
Rule 4763
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 = 0.24, size = 105, normalized size = 1.09 \begin {gather*} \frac {\sqrt {d+c d x} \left (2 a c x+2 b c x \text {ArcSin}(c x)+b \sqrt {1-c^2 x^2} \log (-f (1+c x))+b \sqrt {1-c^2 x^2} \log (f-c f x)\right )}{2 c d^2 f (1+c x) \sqrt {f-c f x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.13, 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 = 0.49, size = 86, normalized size = 0.90 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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