Optimal. Leaf size=517 \[ -\frac{5 i d^4 (1+i c x) \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{15 i d^4 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{2 i d^4 (1+i c x)^3 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{15 d^4 \left (c^2 x^2+1\right )^{3/2} \sinh ^{-1}(c x) \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{b c d^4 x^2 \left (c^2 x^2+1\right )^{3/2}}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{5 b d^4 (1+i c x)^2 \left (c^2 x^2+1\right )^{3/2}}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{3 i b d^4 x \left (c^2 x^2+1\right )^{3/2}}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{8 b d^4 \left (c^2 x^2+1\right )^{3/2} \log (c x+i)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{15 b d^4 \left (c^2 x^2+1\right )^{3/2} \sinh ^{-1}(c x)^2}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}} \]
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Rubi [A] time = 0.422032, antiderivative size = 517, normalized size of antiderivative = 1., 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 = {5712, 669, 671, 641, 215, 5819, 627, 43, 5675} \[ -\frac{5 i d^4 (1+i c x) \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{15 i d^4 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{2 i d^4 (1+i c x)^3 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{15 d^4 \left (c^2 x^2+1\right )^{3/2} \sinh ^{-1}(c x) \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{b c d^4 x^2 \left (c^2 x^2+1\right )^{3/2}}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{5 b d^4 (1+i c x)^2 \left (c^2 x^2+1\right )^{3/2}}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{3 i b d^4 x \left (c^2 x^2+1\right )^{3/2}}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{8 b d^4 \left (c^2 x^2+1\right )^{3/2} \log (c x+i)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{15 b d^4 \left (c^2 x^2+1\right )^{3/2} \sinh ^{-1}(c x)^2}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 5712
Rule 669
Rule 671
Rule 641
Rule 215
Rule 5819
Rule 627
Rule 43
Rule 5675
Rubi steps
\begin{align*} \int \frac{(d+i c d x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{(f-i c f x)^{3/2}} \, dx &=\frac{\left (1+c^2 x^2\right )^{3/2} \int \frac{(d+i c d 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 d^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 d^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 d^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 d^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 d^4}{2 c}-\frac{5 i d^4 (1+i c x)}{2 c}-\frac{2 i d^4 (1+i c x)^3}{c \left (1+c^2 x^2\right )}-\frac{15 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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*}
Mathematica [A] time = 3.88668, size = 781, normalized size = 1.51 \[ \frac{\frac{4 a d^2 \left (c^2 x^2-7 i c x+24\right ) \sqrt{d+i c d x} \sqrt{f-i c f x}}{f^2 (c x+i)}-\frac{60 a d^{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 )}{f^{3/2}}+\frac{4 b d^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \left (2 \left (\sinh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )+i \cosh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right ) \left (4 \tan ^{-1}\left (\tanh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )+i \log \left (c^2 x^2+1\right )\right )+\sinh ^{-1}(c x)^2 \left (-\left (\cosh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )-i \sinh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )\right )+4 \sinh ^{-1}(c x) \left (\sinh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )-i \cosh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )\right )}{f^2 \sqrt{c^2 x^2+1} \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 d^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \left (2 \sinh ^{-1}(c x) \left (\sinh \left (\frac{1}{2} \sinh ^{-1}(c x)\right ) \left (-8 \sqrt{c^2 x^2+1}-i \sinh \left (2 \sinh ^{-1}(c x)\right )+8\right )+\left (\sinh \left (2 \sinh ^{-1}(c x)\right )-8 i \left (\sqrt{c^2 x^2+1}+1\right )\right ) \cosh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )+\left (\sinh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )+i \cosh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right ) \left (8 i \log \left (c^2 x^2+1\right )+16 c x+i \cosh \left (2 \sinh ^{-1}(c x)\right )+32 \tan ^{-1}\left (\tanh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )\right )-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 )\right )}{f^2 \sqrt{c^2 x^2+1} \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 d^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \left (\sinh ^{-1}(c x) \left (-\left (\sqrt{c^2 x^2+1}-2\right ) \sinh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )-i \left (\sqrt{c^2 x^2+1}+2\right ) \cosh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )+\left (\sinh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )+i \cosh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right ) \left (i \log \left (c^2 x^2+1\right )+c x-4 \tan ^{-1}\left (\coth \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )\right )+\sinh ^{-1}(c x)^2 \left (-\left (\cosh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )-i \sinh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )\right )\right )}{f^2 \sqrt{c^2 x^2+1} \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} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.306, size = 0, normalized size = 0. \begin{align*} \int{(a+b{\it Arcsinh} \left ( cx \right ) ) \left ( d+icdx \right ) ^{{\frac{5}{2}}} \left ( f-icfx \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b c^{2} d^{2} x^{2} - 2 i \, b c d^{2} x - b d^{2}\right )} \sqrt{i \, c d x + d} \sqrt{-i \, c f x + f} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) +{\left (a c^{2} d^{2} x^{2} - 2 i \, a c d^{2} x - a d^{2}\right )} \sqrt{i \, c d x + d} \sqrt{-i \, c f x + f}}{c^{2} f^{2} x^{2} + 2 i \, c f^{2} x - f^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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