Optimal. Leaf size=463 \[ -\frac {3 b d^4 x \left (1-c^2 x^2\right )^{3/2}}{2 (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {b c d^4 x^2 \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac {5 b d^4 (1+c x)^2 \left (1-c^2 x^2\right )^{3/2}}{4 c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {15 b d^4 \left (1-c^2 x^2\right )^{3/2} \text {ArcSin}(c x)^2}{4 c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {2 d^4 (1+c x)^3 \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))}{c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {15 d^4 \left (1-c^2 x^2\right )^2 (a+b \text {ArcSin}(c x))}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {5 d^4 (1+c x) \left (1-c^2 x^2\right )^2 (a+b \text {ArcSin}(c x))}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac {15 d^4 \left (1-c^2 x^2\right )^{3/2} \text {ArcSin}(c x) (a+b \text {ArcSin}(c x))}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {8 b d^4 \left (1-c^2 x^2\right )^{3/2} \log (1-c x)}{c (d+c d x)^{3/2} (f-c f x)^{3/2}} \]
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Rubi [A]
time = 0.26, antiderivative size = 463, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 9, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4763, 683, 685,
655, 222, 4845, 641, 45, 4737} \begin {gather*} \frac {5 d^4 (c x+1) \left (1-c^2 x^2\right )^2 (a+b \text {ArcSin}(c x))}{2 c (c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac {15 d^4 \left (1-c^2 x^2\right )^2 (a+b \text {ArcSin}(c x))}{2 c (c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac {2 d^4 (c x+1)^3 \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))}{c (c d x+d)^{3/2} (f-c f x)^{3/2}}-\frac {15 d^4 \left (1-c^2 x^2\right )^{3/2} \text {ArcSin}(c x) (a+b \text {ArcSin}(c x))}{2 c (c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac {15 b d^4 \left (1-c^2 x^2\right )^{3/2} \text {ArcSin}(c x)^2}{4 c (c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac {b c d^4 x^2 \left (1-c^2 x^2\right )^{3/2}}{(c d x+d)^{3/2} (f-c f x)^{3/2}}-\frac {5 b d^4 (c x+1)^2 \left (1-c^2 x^2\right )^{3/2}}{4 c (c d x+d)^{3/2} (f-c f x)^{3/2}}-\frac {3 b d^4 x \left (1-c^2 x^2\right )^{3/2}}{2 (c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac {8 b d^4 \left (1-c^2 x^2\right )^{3/2} \log (1-c x)}{c (c d x+d)^{3/2} (f-c f x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 222
Rule 641
Rule 655
Rule 683
Rule 685
Rule 4737
Rule 4763
Rule 4845
Rubi steps
\begin {align*} \int \frac {(d+c d x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{(f-c f x)^{3/2}} \, dx &=\frac {\left (1-c^2 x^2\right )^{3/2} \int \frac {(d+c d x)^4 \left (a+b \sin ^{-1}(c x)\right )}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{(d+c d x)^{3/2} (f-c f x)^{3/2}}\\ &=\frac {2 d^4 (1+c x)^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {15 d^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {5 d^4 (1+c x) \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac {15 d^4 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x) \left (a+b \sin ^{-1}(c x)\right )}{2 c (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 \left (\frac {15 d^4}{2 c}+\frac {5 d^4 (1+c x)}{2 c}+\frac {2 d^4 (1+c x)^3}{c \left (1-c^2 x^2\right )}-\frac {15 d^4 \sin ^{-1}(c x)}{2 c \sqrt {1-c^2 x^2}}\right ) \, dx}{(d+c d x)^{3/2} (f-c f x)^{3/2}}\\ &=-\frac {15 b d^4 x \left (1-c^2 x^2\right )^{3/2}}{2 (d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac {5 b d^4 (1+c x)^2 \left (1-c^2 x^2\right )^{3/2}}{4 c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {2 d^4 (1+c x)^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {15 d^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {5 d^4 (1+c x) \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac {15 d^4 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x) \left (a+b \sin ^{-1}(c x)\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac {\left (2 b d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac {(1+c x)^3}{1-c^2 x^2} \, dx}{(d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {\left (15 b d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac {\sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{2 (d+c d x)^{3/2} (f-c f x)^{3/2}}\\ &=-\frac {15 b d^4 x \left (1-c^2 x^2\right )^{3/2}}{2 (d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac {5 b d^4 (1+c x)^2 \left (1-c^2 x^2\right )^{3/2}}{4 c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {15 b d^4 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)^2}{4 c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {2 d^4 (1+c x)^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {15 d^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {5 d^4 (1+c x) \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac {15 d^4 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x) \left (a+b \sin ^{-1}(c x)\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac {\left (2 b d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac {(1+c x)^2}{1-c x} \, dx}{(d+c d x)^{3/2} (f-c f x)^{3/2}}\\ &=-\frac {15 b d^4 x \left (1-c^2 x^2\right )^{3/2}}{2 (d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac {5 b d^4 (1+c x)^2 \left (1-c^2 x^2\right )^{3/2}}{4 c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {15 b d^4 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)^2}{4 c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {2 d^4 (1+c x)^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {15 d^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {5 d^4 (1+c x) \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac {15 d^4 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x) \left (a+b \sin ^{-1}(c x)\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac {\left (2 b d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \int \left (-3-c x+\frac {4}{1-c x}\right ) \, dx}{(d+c d x)^{3/2} (f-c f x)^{3/2}}\\ &=-\frac {3 b d^4 x \left (1-c^2 x^2\right )^{3/2}}{2 (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {b c d^4 x^2 \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac {5 b d^4 (1+c x)^2 \left (1-c^2 x^2\right )^{3/2}}{4 c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {15 b d^4 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)^2}{4 c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {2 d^4 (1+c x)^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {15 d^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {5 d^4 (1+c x) \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac {15 d^4 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x) \left (a+b \sin ^{-1}(c x)\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {8 b d^4 \left (1-c^2 x^2\right )^{3/2} \log (1-c x)}{c (d+c d x)^{3/2} (f-c f x)^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 2.52, size = 768, normalized size = 1.66 \begin {gather*} \frac {d^2 \left (\frac {8 a \sqrt {d+c d x} \sqrt {f-c f x} \left (-24+7 c x+c^2 x^2\right )}{-1+c x}+120 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 {8 b (1+c x) \sqrt {d+c d x} \sqrt {f-c f x} \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right ) \left ((-4+\text {ArcSin}(c x)) \text {ArcSin}(c x)-8 \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )-\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )\right )-\left (\text {ArcSin}(c x) (4+\text {ArcSin}(c x))-8 \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 )}{\sqrt {1-c^2 x^2} \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 )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )^2}-\frac {32 b (1+c x) \sqrt {d+c d x} \sqrt {f-c f x} \left (\text {ArcSin}(c x)^2 \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )-\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )+\left (c x-4 \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )-\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )-\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )-\text {ArcSin}(c x) \left (\left (2+\sqrt {1-c^2 x^2}\right ) \cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )-\left (-2+\sqrt {1-c^2 x^2}\right ) \sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )\right )}{\sqrt {1-c^2 x^2} \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 )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )^2}+\frac {b (1+c x) \sqrt {d+c d x} \sqrt {f-c f x} \left (-20 \text {ArcSin}(c x)^2 \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )-\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )+2 \left (-16 c x+\cos (2 \text {ArcSin}(c x))+32 \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )-\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )-\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )+2 \text {ArcSin}(c x) \left (24 \cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+7 \cos \left (\frac {3}{2} \text {ArcSin}(c x)\right )+\cos \left (\frac {5}{2} \text {ArcSin}(c x)\right )+24 \sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )-7 \sin \left (\frac {3}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {5}{2} \text {ArcSin}(c x)\right )\right )\right )}{\sqrt {1-c^2 x^2} \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 )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )^2}\right )}{16 c f^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.30, size = 0, normalized size = 0.00 \[\int \frac {\left (c d x +d \right )^{\frac {5}{2}} \left (a +b \arcsin \left (c x \right )\right )}{\left (-c f x +f \right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 (d+c\,d\,x\right )}^{5/2}}{{\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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