Optimal. Leaf size=324 \[ -\frac {4 b f^4 \left (1-c^2 x^2\right )^{5/2}}{3 c (1+c x) (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {b f^4 \left (1-c^2 x^2\right )^{5/2} \text {ArcSin}(c x)^2}{2 c (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {2 f^4 (1-c x)^3 \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {2 f^4 (1-c x) \left (1-c^2 x^2\right )^2 (a+b \text {ArcSin}(c x))}{c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {f^4 \left (1-c^2 x^2\right )^{5/2} \text {ArcSin}(c x) (a+b \text {ArcSin}(c x))}{c (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {8 b f^4 \left (1-c^2 x^2\right )^{5/2} \log (1+c x)}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}} \]
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Rubi [A]
time = 0.25, antiderivative size = 324, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 9, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4763, 683, 667,
222, 4845, 641, 45, 31, 4737} \begin {gather*} \frac {2 f^4 (1-c x) \left (1-c^2 x^2\right )^2 (a+b \text {ArcSin}(c x))}{c (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac {2 f^4 (1-c x)^3 \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))}{3 c (c d x+d)^{5/2} (f-c f x)^{5/2}}+\frac {f^4 \left (1-c^2 x^2\right )^{5/2} \text {ArcSin}(c x) (a+b \text {ArcSin}(c x))}{c (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac {b f^4 \left (1-c^2 x^2\right )^{5/2} \text {ArcSin}(c x)^2}{2 c (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac {4 b f^4 \left (1-c^2 x^2\right )^{5/2}}{3 c (c x+1) (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac {8 b f^4 \left (1-c^2 x^2\right )^{5/2} \log (c x+1)}{3 c (c d x+d)^{5/2} (f-c f x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 45
Rule 222
Rule 641
Rule 667
Rule 683
Rule 4737
Rule 4763
Rule 4845
Rubi steps
\begin {align*} \int \frac {(f-c f x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{(d+c d x)^{5/2}} \, dx &=\frac {\left (1-c^2 x^2\right )^{5/2} \int \frac {(f-c f x)^4 \left (a+b \sin ^{-1}(c x)\right )}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{(d+c d x)^{5/2} (f-c f x)^{5/2}}\\ &=-\frac {2 f^4 (1-c x)^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {2 f^4 (1-c x) \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {f^4 \left (1-c^2 x^2\right )^{5/2} \sin ^{-1}(c x) \left (a+b \sin ^{-1}(c x)\right )}{c (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {\left (b c \left (1-c^2 x^2\right )^{5/2}\right ) \int \left (-\frac {2 f^4 (1-c x)^3}{3 c \left (1-c^2 x^2\right )^2}+\frac {2 f^4 (1-c x)}{c \left (1-c^2 x^2\right )}+\frac {f^4 \sin ^{-1}(c x)}{c \sqrt {1-c^2 x^2}}\right ) \, dx}{(d+c d x)^{5/2} (f-c f x)^{5/2}}\\ &=-\frac {2 f^4 (1-c x)^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {2 f^4 (1-c x) \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {f^4 \left (1-c^2 x^2\right )^{5/2} \sin ^{-1}(c x) \left (a+b \sin ^{-1}(c x)\right )}{c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {\left (2 b f^4 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {(1-c x)^3}{\left (1-c^2 x^2\right )^2} \, dx}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {\left (b f^4 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {\sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{(d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {\left (2 b f^4 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {1-c x}{1-c^2 x^2} \, dx}{(d+c d x)^{5/2} (f-c f x)^{5/2}}\\ &=-\frac {b f^4 \left (1-c^2 x^2\right )^{5/2} \sin ^{-1}(c x)^2}{2 c (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {2 f^4 (1-c x)^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {2 f^4 (1-c x) \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {f^4 \left (1-c^2 x^2\right )^{5/2} \sin ^{-1}(c x) \left (a+b \sin ^{-1}(c x)\right )}{c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {\left (2 b f^4 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {1-c x}{(1+c x)^2} \, dx}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {\left (2 b f^4 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {1}{1+c x} \, dx}{(d+c d x)^{5/2} (f-c f x)^{5/2}}\\ &=-\frac {b f^4 \left (1-c^2 x^2\right )^{5/2} \sin ^{-1}(c x)^2}{2 c (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {2 f^4 (1-c x)^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {2 f^4 (1-c x) \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {f^4 \left (1-c^2 x^2\right )^{5/2} \sin ^{-1}(c x) \left (a+b \sin ^{-1}(c x)\right )}{c (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {2 b f^4 \left (1-c^2 x^2\right )^{5/2} \log (1+c x)}{c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {\left (2 b f^4 \left (1-c^2 x^2\right )^{5/2}\right ) \int \left (\frac {1}{-1-c x}+\frac {2}{(1+c x)^2}\right ) \, dx}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}}\\ &=-\frac {4 b f^4 \left (1-c^2 x^2\right )^{5/2}}{3 c (1+c x) (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {b f^4 \left (1-c^2 x^2\right )^{5/2} \sin ^{-1}(c x)^2}{2 c (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {2 f^4 (1-c x)^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {2 f^4 (1-c x) \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {f^4 \left (1-c^2 x^2\right )^{5/2} \sin ^{-1}(c x) \left (a+b \sin ^{-1}(c x)\right )}{c (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {8 b f^4 \left (1-c^2 x^2\right )^{5/2} \log (1+c x)}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 3.21, size = 599, normalized size = 1.85 \begin {gather*} \frac {f \left (\frac {16 a (1+2 c x) \sqrt {d+c d x} \sqrt {f-c f x}}{(1+c x)^2}-12 a \sqrt {d} \sqrt {f} \text {ArcTan}\left (\frac {c x \sqrt {d+c d x} \sqrt {f-c f x}}{\sqrt {d} \sqrt {f} \left (-1+c^2 x^2\right )}\right )-\frac {b \sqrt {d+c d x} \sqrt {f-c f x} \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )-\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right ) \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right ) \left (-8+6 \text {ArcSin}(c x)+9 \text {ArcSin}(c x)^2-84 \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )\right )+\cos \left (\frac {3}{2} \text {ArcSin}(c x)\right ) \left ((14-3 \text {ArcSin}(c x)) \text {ArcSin}(c x)+28 \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )\right )+2 \left (-4+2 \left (2+7 \sqrt {1-c^2 x^2}\right ) \text {ArcSin}(c x)+3 \left (2+\sqrt {1-c^2 x^2}\right ) \text {ArcSin}(c x)^2-28 \left (2+\sqrt {1-c^2 x^2}\right ) \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )\right ) \sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )}{(-1+c x) \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )^4}-\frac {2 b \sqrt {d+c d x} \sqrt {f-c f x} \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )-\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right ) \left (\cos \left (\frac {3}{2} \text {ArcSin}(c x)\right ) \left (\text {ArcSin}(c x)+2 \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )\right )-\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right ) \left (4+3 \text {ArcSin}(c x)+6 \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )\right )+2 \left (-2+\left (2+\sqrt {1-c^2 x^2}\right ) \text {ArcSin}(c x)-2 \left (2+\sqrt {1-c^2 x^2}\right ) \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )\right ) \sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )}{(-1+c x) \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )^4}\right )}{12 c d^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.28, size = 0, normalized size = 0.00 \[\int \frac {\left (-c f x +f \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )}{\left (c d x +d \right )^{\frac {5}{2}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \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 = -0.00 \begin {gather*} \int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (f-c\,f\,x\right )}^{3/2}}{{\left (d+c\,d\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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