Optimal. Leaf size=80 \[ \frac{2 \sinh ^{\frac{3}{2}}(a+b x) \cosh (a+b x)}{5 b}+\frac{6 i \sqrt{\sinh (a+b x)} E\left (\left .\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right )\right |2\right )}{5 b \sqrt{i \sinh (a+b x)}} \]
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Rubi [A] time = 0.0320144, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {2635, 2640, 2639} \[ \frac{2 \sinh ^{\frac{3}{2}}(a+b x) \cosh (a+b x)}{5 b}+\frac{6 i \sqrt{\sinh (a+b x)} E\left (\left .\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right )\right |2\right )}{5 b \sqrt{i \sinh (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int \sinh ^{\frac{5}{2}}(a+b x) \, dx &=\frac{2 \cosh (a+b x) \sinh ^{\frac{3}{2}}(a+b x)}{5 b}-\frac{3}{5} \int \sqrt{\sinh (a+b x)} \, dx\\ &=\frac{2 \cosh (a+b x) \sinh ^{\frac{3}{2}}(a+b x)}{5 b}-\frac{\left (3 \sqrt{\sinh (a+b x)}\right ) \int \sqrt{i \sinh (a+b x)} \, dx}{5 \sqrt{i \sinh (a+b x)}}\\ &=\frac{6 i E\left (\left .\frac{1}{2} \left (i a-\frac{\pi }{2}+i b x\right )\right |2\right ) \sqrt{\sinh (a+b x)}}{5 b \sqrt{i \sinh (a+b x)}}+\frac{2 \cosh (a+b x) \sinh ^{\frac{3}{2}}(a+b x)}{5 b}\\ \end{align*}
Mathematica [A] time = 0.0733094, size = 68, normalized size = 0.85 \[ \frac{\sinh (a+b x) \sinh (2 (a+b x))-6 \sqrt{i \sinh (a+b x)} E\left (\left .\frac{1}{4} (-2 i a-2 i b x+\pi )\right |2\right )}{5 b \sqrt{\sinh (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 164, normalized size = 2.1 \begin{align*}{\frac{1}{b\cosh \left ( bx+a \right ) } \left ( -{\frac{6\,\sqrt{2}}{5}\sqrt{1-i\sinh \left ( bx+a \right ) }\sqrt{1+i\sinh \left ( bx+a \right ) }\sqrt{i\sinh \left ( bx+a \right ) }{\it EllipticE} \left ( \sqrt{1-i\sinh \left ( bx+a \right ) },{\frac{\sqrt{2}}{2}} \right ) }+{\frac{3\,\sqrt{2}}{5}\sqrt{1-i\sinh \left ( bx+a \right ) }\sqrt{1+i\sinh \left ( bx+a \right ) }\sqrt{i\sinh \left ( bx+a \right ) }{\it EllipticF} \left ( \sqrt{1-i\sinh \left ( bx+a \right ) },{\frac{\sqrt{2}}{2}} \right ) }+{\frac{2\, \left ( \cosh \left ( bx+a \right ) \right ) ^{4}}{5}}-{\frac{2\, \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{5}} \right ){\frac{1}{\sqrt{\sinh \left ( bx+a \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 \sinh \left (b x + a\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 (\sinh \left (b x + a\right )^{\frac{5}{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 \sinh \left (b x + a\right )^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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