Optimal. Leaf size=282 \[ \frac{2 f x \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{f (c x+i) \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{i b f \left (c^2 x^2+1\right )^{5/2}}{6 c (-c x+i) (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{b f \left (c^2 x^2+1\right )^{5/2} \log \left (c^2 x^2+1\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{i b f \left (c^2 x^2+1\right )^{5/2} \tan ^{-1}(c x)}{6 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \]
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Rubi [A] time = 0.306403, antiderivative size = 282, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.229, Rules used = {5712, 639, 191, 5819, 627, 44, 203, 260} \[ \frac{2 f x \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{f (c x+i) \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{i b f \left (c^2 x^2+1\right )^{5/2}}{6 c (-c x+i) (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{b f \left (c^2 x^2+1\right )^{5/2} \log \left (c^2 x^2+1\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{i b f \left (c^2 x^2+1\right )^{5/2} \tan ^{-1}(c x)}{6 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 5712
Rule 639
Rule 191
Rule 5819
Rule 627
Rule 44
Rule 203
Rule 260
Rubi steps
\begin{align*} \int \frac{a+b \sinh ^{-1}(c x)}{(d+i c d x)^{5/2} (f-i c f x)^{3/2}} \, dx &=\frac{\left (1+c^2 x^2\right )^{5/2} \int \frac{(f-i c f x) \left (a+b \sinh ^{-1}(c x)\right )}{\left (1+c^2 x^2\right )^{5/2}} \, dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}\\ &=\frac{f (i+c x) \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{2 f x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{\left (b c \left (1+c^2 x^2\right )^{5/2}\right ) \int \left (\frac{f (i+c x)}{3 c \left (1+c^2 x^2\right )^2}+\frac{2 f x}{3 \left (1+c^2 x^2\right )}\right ) \, dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}\\ &=\frac{f (i+c x) \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{2 f x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{\left (b f \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac{i+c x}{\left (1+c^2 x^2\right )^2} \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{\left (2 b c f \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac{x}{1+c^2 x^2} \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}\\ &=\frac{f (i+c x) \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{2 f x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{b f \left (1+c^2 x^2\right )^{5/2} \log \left (1+c^2 x^2\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{\left (b f \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac{1}{(-i+c x)^2 (i+c x)} \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}\\ &=\frac{f (i+c x) \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{2 f x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{b f \left (1+c^2 x^2\right )^{5/2} \log \left (1+c^2 x^2\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{\left (b f \left (1+c^2 x^2\right )^{5/2}\right ) \int \left (-\frac{i}{2 (-i+c x)^2}+\frac{i}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}\\ &=\frac{i b f \left (1+c^2 x^2\right )^{5/2}}{6 c (i-c x) (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{f (i+c x) \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{2 f x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{b f \left (1+c^2 x^2\right )^{5/2} \log \left (1+c^2 x^2\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{\left (i b f \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac{1}{1+c^2 x^2} \, dx}{6 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}\\ &=\frac{i b f \left (1+c^2 x^2\right )^{5/2}}{6 c (i-c x) (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{f (i+c x) \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{2 f x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{i b f \left (1+c^2 x^2\right )^{5/2} \tan ^{-1}(c x)}{6 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{b f \left (1+c^2 x^2\right )^{5/2} \log \left (1+c^2 x^2\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.607269, size = 201, normalized size = 0.71 \[ \frac{\sqrt{f-i c f x} \left (8 i a c^2 x^2+8 a c x+4 i a-5 i b c x \sqrt{c^2 x^2+1} \log (d+i c d x)+3 b (-1-i c x) \sqrt{c^2 x^2+1} \log (d (-1+i c x))-5 b \sqrt{c^2 x^2+1} \log (d+i c d x)+2 b \sqrt{c^2 x^2+1}+4 b \left (2 i c^2 x^2+2 c x+i\right ) \sinh ^{-1}(c x)\right )}{12 d^2 f^2 \left (c^3 x^2+c\right ) \sqrt{d+i c d x}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.25, 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*} \text{result too large to display} \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: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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