Optimal. Leaf size=518 \[ -\frac {3 i b f^4 x \left (1+c^2 x^2\right )^{3/2}}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {b c f^4 x^2 \left (1+c^2 x^2\right )^{3/2}}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {5 b f^4 (1-i c x)^2 \left (1+c^2 x^2\right )^{3/2}}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {15 b f^4 \left (1+c^2 x^2\right )^{3/2} \sinh ^{-1}(c x)^2}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {2 i f^4 (1-i c x)^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {15 i f^4 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {5 i f^4 (1-i c x) \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {15 f^4 \left (1+c^2 x^2\right )^{3/2} \sinh ^{-1}(c x) \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {8 b f^4 \left (1+c^2 x^2\right )^{3/2} \log (i-c x)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}} \]
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Rubi [A]
time = 0.28, antiderivative size = 518, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 9, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {5796, 683, 685,
655, 221, 5837, 641, 45, 5783} \begin {gather*} \frac {5 i f^4 (1-i c x) \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {15 i f^4 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {2 i f^4 (1-i c x)^3 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {15 f^4 \left (c^2 x^2+1\right )^{3/2} \sinh ^{-1}(c x) \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {b c f^4 x^2 \left (c^2 x^2+1\right )^{3/2}}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {5 b f^4 (1-i c x)^2 \left (c^2 x^2+1\right )^{3/2}}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {3 i b f^4 x \left (c^2 x^2+1\right )^{3/2}}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {8 b f^4 \left (c^2 x^2+1\right )^{3/2} \log (-c x+i)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {15 b f^4 \left (c^2 x^2+1\right )^{3/2} \sinh ^{-1}(c x)^2}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 221
Rule 641
Rule 655
Rule 683
Rule 685
Rule 5783
Rule 5796
Rule 5837
Rubi steps
\begin {align*} \int \frac {(f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{(d+i c d x)^{3/2}} \, dx &=\frac {\left (1+c^2 x^2\right )^{3/2} \int \frac {(f-i c f x)^4 \left (a+b \sinh ^{-1}(c x)\right )}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}\\ &=\frac {2 i f^4 (1-i c x)^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {15 i f^4 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {5 i f^4 (1-i c x) \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {15 f^4 \left (1+c^2 x^2\right )^{3/2} \sinh ^{-1}(c x) \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {\left (b c \left (1+c^2 x^2\right )^{3/2}\right ) \int \left (\frac {15 i f^4}{2 c}+\frac {5 i f^4 (1-i c x)}{2 c}+\frac {2 i f^4 (1-i c x)^3}{c \left (1+c^2 x^2\right )}-\frac {15 f^4 \sinh ^{-1}(c x)}{2 c \sqrt {1+c^2 x^2}}\right ) \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}\\ &=-\frac {15 i b f^4 x \left (1+c^2 x^2\right )^{3/2}}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {5 b f^4 (1-i c x)^2 \left (1+c^2 x^2\right )^{3/2}}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {2 i f^4 (1-i c x)^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {15 i f^4 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {5 i f^4 (1-i c x) \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {15 f^4 \left (1+c^2 x^2\right )^{3/2} \sinh ^{-1}(c x) \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {\left (2 i b f^4 \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac {(1-i c x)^3}{1+c^2 x^2} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {\left (15 b f^4 \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac {\sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}\\ &=-\frac {15 i b f^4 x \left (1+c^2 x^2\right )^{3/2}}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {5 b f^4 (1-i c x)^2 \left (1+c^2 x^2\right )^{3/2}}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {15 b f^4 \left (1+c^2 x^2\right )^{3/2} \sinh ^{-1}(c x)^2}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {2 i f^4 (1-i c x)^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {15 i f^4 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {5 i f^4 (1-i c x) \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {15 f^4 \left (1+c^2 x^2\right )^{3/2} \sinh ^{-1}(c x) \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {\left (2 i b f^4 \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac {(1-i c x)^2}{1+i c x} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}\\ &=-\frac {15 i b f^4 x \left (1+c^2 x^2\right )^{3/2}}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {5 b f^4 (1-i c x)^2 \left (1+c^2 x^2\right )^{3/2}}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {15 b f^4 \left (1+c^2 x^2\right )^{3/2} \sinh ^{-1}(c x)^2}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {2 i f^4 (1-i c x)^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {15 i f^4 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {5 i f^4 (1-i c x) \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {15 f^4 \left (1+c^2 x^2\right )^{3/2} \sinh ^{-1}(c x) \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {\left (2 i b f^4 \left (1+c^2 x^2\right )^{3/2}\right ) \int \left (-3+i c x+\frac {4}{1+i c x}\right ) \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}\\ &=-\frac {3 i b f^4 x \left (1+c^2 x^2\right )^{3/2}}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {b c f^4 x^2 \left (1+c^2 x^2\right )^{3/2}}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {5 b f^4 (1-i c x)^2 \left (1+c^2 x^2\right )^{3/2}}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {15 b f^4 \left (1+c^2 x^2\right )^{3/2} \sinh ^{-1}(c x)^2}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {2 i f^4 (1-i c x)^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {15 i f^4 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {5 i f^4 (1-i c x) \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {15 f^4 \left (1+c^2 x^2\right )^{3/2} \sinh ^{-1}(c x) \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {8 b f^4 \left (1+c^2 x^2\right )^{3/2} \log (i-c x)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 2.52, size = 779, normalized size = 1.50 \begin {gather*} \frac {\frac {4 a f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (24+7 i c x+c^2 x^2\right )}{d^2 (-i+c x)}-\frac {60 a f^{5/2} \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )}{d^{3/2}}-\frac {4 b f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (\sinh ^{-1}(c x) \left (-4 i \cosh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )-4 \sinh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )+\sinh ^{-1}(c x)^2 \left (\cosh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )+i \sinh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )+2 \left (4 i \text {ArcTan}\left (\tanh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )+\log \left (1+c^2 x^2\right )\right ) \left (\cosh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )+i \sinh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )\right )}{d^2 \sqrt {1+c^2 x^2} \left (\cosh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )+i \sinh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )}+\frac {16 b f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (-\sinh ^{-1}(c x)^2 \left (\cosh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )+i \sinh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )+\left (c x-4 \text {ArcTan}\left (\coth \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )-i \log \left (1+c^2 x^2\right )\right ) \left (-i \cosh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )+\sinh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )+\sinh ^{-1}(c x) \left (i \left (2+\sqrt {1+c^2 x^2}\right ) \cosh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )-\left (-2+\sqrt {1+c^2 x^2}\right ) \sinh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )\right )}{d^2 \sqrt {1+c^2 x^2} \left (\cosh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )+i \sinh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )}+\frac {b f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (-10 \sinh ^{-1}(c x)^2 \left (\cosh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )+i \sinh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )-\left (\cosh \left (2 \sinh ^{-1}(c x)\right )+8 \left (2 i c x+4 i \text {ArcTan}\left (\tanh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )+\log \left (1+c^2 x^2\right )\right )\right ) \left (\cosh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )+i \sinh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )+2 \sinh ^{-1}(c x) \left (\sinh \left (\frac {1}{2} \sinh ^{-1}(c x)\right ) \left (8-8 \sqrt {1+c^2 x^2}+i \sinh \left (2 \sinh ^{-1}(c x)\right )\right )+\cosh \left (\frac {1}{2} \sinh ^{-1}(c x)\right ) \left (8 i \left (1+\sqrt {1+c^2 x^2}\right )+\sinh \left (2 \sinh ^{-1}(c x)\right )\right )\right )\right )}{d^2 \sqrt {1+c^2 x^2} \left (\cosh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )+i \sinh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )}}{8 c} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (-i c f x +f \right )^{\frac {5}{2}} \left (a +b \arcsinh \left (c x \right )\right )}{\left (i c d x +d \right )^{\frac {3}{2}}}\, dx\]
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 {asinh}\left (c\,x\right )\right )\,{\left (f-c\,f\,x\,1{}\mathrm {i}\right )}^{5/2}}{{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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