Optimal. Leaf size=459 \[ \frac{3 d x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{8 \left (c^2 x^2+1\right )}+\frac{3 d (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b c \left (c^2 x^2+1\right )^{3/2}}+\frac{i d \left (c^2 x^2+1\right ) (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{5 c}+\frac{1}{4} d x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{i b c^4 d x^5 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{25 \left (c^2 x^2+1\right )^{3/2}}-\frac{b c^3 d x^4 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{16 \left (c^2 x^2+1\right )^{3/2}}-\frac{2 i b c^2 d x^3 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{15 \left (c^2 x^2+1\right )^{3/2}}-\frac{5 b c d x^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{16 \left (c^2 x^2+1\right )^{3/2}}-\frac{i b d x (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{5 \left (c^2 x^2+1\right )^{3/2}} \]
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Rubi [A] time = 0.441307, antiderivative size = 459, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.257, Rules used = {5712, 5821, 5684, 5682, 5675, 30, 14, 5717, 194} \[ \frac{3 d x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{8 \left (c^2 x^2+1\right )}+\frac{3 d (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b c \left (c^2 x^2+1\right )^{3/2}}+\frac{i d \left (c^2 x^2+1\right ) (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{5 c}+\frac{1}{4} d x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{i b c^4 d x^5 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{25 \left (c^2 x^2+1\right )^{3/2}}-\frac{b c^3 d x^4 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{16 \left (c^2 x^2+1\right )^{3/2}}-\frac{2 i b c^2 d x^3 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{15 \left (c^2 x^2+1\right )^{3/2}}-\frac{5 b c d x^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{16 \left (c^2 x^2+1\right )^{3/2}}-\frac{i b d x (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{5 \left (c^2 x^2+1\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 5712
Rule 5821
Rule 5684
Rule 5682
Rule 5675
Rule 30
Rule 14
Rule 5717
Rule 194
Rubi steps
\begin{align*} \int (d+i c d x)^{5/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\frac{\left ((d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int (d+i c d x) \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{\left (1+c^2 x^2\right )^{3/2}}\\ &=\frac{\left ((d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \left (d \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+i c d x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )\right ) \, dx}{\left (1+c^2 x^2\right )^{3/2}}\\ &=\frac{\left (d (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{\left (1+c^2 x^2\right )^{3/2}}+\frac{\left (i c d (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{\left (1+c^2 x^2\right )^{3/2}}\\ &=\frac{1}{4} d x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{i d (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{5 c}+\frac{\left (3 d (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{4 \left (1+c^2 x^2\right )^{3/2}}-\frac{\left (i b d (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \left (1+c^2 x^2\right )^2 \, dx}{5 \left (1+c^2 x^2\right )^{3/2}}-\frac{\left (b c d (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int x \left (1+c^2 x^2\right ) \, dx}{4 \left (1+c^2 x^2\right )^{3/2}}\\ &=\frac{1}{4} d x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{3 d x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{8 \left (1+c^2 x^2\right )}+\frac{i d (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{5 c}+\frac{\left (3 d (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \frac{a+b \sinh ^{-1}(c x)}{\sqrt{1+c^2 x^2}} \, dx}{8 \left (1+c^2 x^2\right )^{3/2}}-\frac{\left (i b d (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \left (1+2 c^2 x^2+c^4 x^4\right ) \, dx}{5 \left (1+c^2 x^2\right )^{3/2}}-\frac{\left (b c d (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \left (x+c^2 x^3\right ) \, dx}{4 \left (1+c^2 x^2\right )^{3/2}}-\frac{\left (3 b c d (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int x \, dx}{8 \left (1+c^2 x^2\right )^{3/2}}\\ &=-\frac{i b d x (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{5 \left (1+c^2 x^2\right )^{3/2}}-\frac{5 b c d x^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{16 \left (1+c^2 x^2\right )^{3/2}}-\frac{2 i b c^2 d x^3 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{15 \left (1+c^2 x^2\right )^{3/2}}-\frac{b c^3 d x^4 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{16 \left (1+c^2 x^2\right )^{3/2}}-\frac{i b c^4 d x^5 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{25 \left (1+c^2 x^2\right )^{3/2}}+\frac{1}{4} d x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{3 d x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{8 \left (1+c^2 x^2\right )}+\frac{i d (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{5 c}+\frac{3 d (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b c \left (1+c^2 x^2\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 1.6868, size = 683, normalized size = 1.49 \[ \frac{3600 a d^{5/2} f^{3/2} \sqrt{c^2 x^2+1} \log \left (c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right )+1920 i a c^4 d^2 f x^4 \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+2400 a c^3 d^2 f x^3 \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+3840 i a c^2 d^2 f x^2 \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+6000 a c d^2 f x \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+1920 i a d^2 f \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+60 b d^2 f \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x) \left (5 \left (4 i \sqrt{c^2 x^2+1}+8 \sinh \left (2 \sinh ^{-1}(c x)\right )+\sinh \left (4 \sinh ^{-1}(c x)\right )\right )+10 i \cosh \left (3 \sinh ^{-1}(c x)\right )+2 i \cosh \left (5 \sinh ^{-1}(c x)\right )\right )-1200 i b c d^2 f x \sqrt{d+i c d x} \sqrt{f-i c f x}+1800 b d^2 f \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^2-200 i b d^2 f \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh \left (3 \sinh ^{-1}(c x)\right )-24 i b d^2 f \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh \left (5 \sinh ^{-1}(c x)\right )-1200 b d^2 f \sqrt{d+i c d x} \sqrt{f-i c f x} \cosh \left (2 \sinh ^{-1}(c x)\right )-75 b d^2 f \sqrt{d+i c d x} \sqrt{f-i c f x} \cosh \left (4 \sinh ^{-1}(c x)\right )}{9600 c \sqrt{c^2 x^2+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.247, size = 0, normalized size = 0. \begin{align*} \int \left ( d+icdx \right ) ^{{\frac{5}{2}}} \left ( f-icfx \right ) ^{{\frac{3}{2}}} \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) \, 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 ({\left (i \, b c^{3} d^{2} f x^{3} + b c^{2} d^{2} f x^{2} + i \, b c d^{2} f x + b d^{2} f\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 (i \, a c^{3} d^{2} f x^{3} + a c^{2} d^{2} f x^{2} + i \, a c d^{2} f x + a d^{2} f\right )} \sqrt{i \, c d x + d} \sqrt{-i \, c f x + f}, 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: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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