Optimal. Leaf size=88 \[ \frac{2 b \cosh (c+d x) (b \sinh (c+d x))^{3/2}}{5 d}+\frac{6 i b^2 E\left (\left .\frac{1}{2} \left (i c+i d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{b \sinh (c+d x)}}{5 d \sqrt{i \sinh (c+d x)}} \]
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Rubi [A] time = 0.036614, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2635, 2640, 2639} \[ \frac{2 b \cosh (c+d x) (b \sinh (c+d x))^{3/2}}{5 d}+\frac{6 i b^2 E\left (\left .\frac{1}{2} \left (i c+i d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{b \sinh (c+d x)}}{5 d \sqrt{i \sinh (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int (b \sinh (c+d x))^{5/2} \, dx &=\frac{2 b \cosh (c+d x) (b \sinh (c+d x))^{3/2}}{5 d}-\frac{1}{5} \left (3 b^2\right ) \int \sqrt{b \sinh (c+d x)} \, dx\\ &=\frac{2 b \cosh (c+d x) (b \sinh (c+d x))^{3/2}}{5 d}-\frac{\left (3 b^2 \sqrt{b \sinh (c+d x)}\right ) \int \sqrt{i \sinh (c+d x)} \, dx}{5 \sqrt{i \sinh (c+d x)}}\\ &=\frac{6 i b^2 E\left (\left .\frac{1}{2} \left (i c-\frac{\pi }{2}+i d x\right )\right |2\right ) \sqrt{b \sinh (c+d x)}}{5 d \sqrt{i \sinh (c+d x)}}+\frac{2 b \cosh (c+d x) (b \sinh (c+d x))^{3/2}}{5 d}\\ \end{align*}
Mathematica [A] time = 0.119512, size = 68, normalized size = 0.77 \[ \frac{b^2 \sqrt{b \sinh (c+d x)} \left (\sinh (2 (c+d x))-\frac{6 i E\left (\left .\frac{1}{4} (-2 i c-2 i d x+\pi )\right |2\right )}{\sqrt{i \sinh (c+d x)}}\right )}{5 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 170, normalized size = 1.9 \begin{align*} -{\frac{{b}^{3}}{5\,d\cosh \left ( dx+c \right ) } \left ( 6\,\sqrt{1-i\sinh \left ( dx+c \right ) }\sqrt{2}\sqrt{1+i\sinh \left ( dx+c \right ) }\sqrt{i\sinh \left ( dx+c \right ) }{\it EllipticE} \left ( \sqrt{1-i\sinh \left ( dx+c \right ) },1/2\,\sqrt{2} \right ) -3\,\sqrt{1-i\sinh \left ( dx+c \right ) }\sqrt{2}\sqrt{1+i\sinh \left ( dx+c \right ) }\sqrt{i\sinh \left ( dx+c \right ) }{\it EllipticF} \left ( \sqrt{1-i\sinh \left ( dx+c \right ) },1/2\,\sqrt{2} \right ) -2\, \left ( \cosh \left ( dx+c \right ) \right ) ^{4}+2\, \left ( \cosh \left ( dx+c \right ) \right ) ^{2} \right ){\frac{1}{\sqrt{b\sinh \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \sinh \left (d x + c\right )\right )^{\frac{5}{2}}\,{d x} \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 (\sqrt{b \sinh \left (d x + c\right )} b^{2} \sinh \left (d x + c\right )^{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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \sinh \left (d x + c\right )\right )^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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