Integrand size = 18, antiderivative size = 87 \[ \int x \cosh ^{\frac {5}{2}}(a+b x) \sinh (a+b x) \, dx=\frac {2 x \cosh ^{\frac {7}{2}}(a+b x)}{7 b}+\frac {20 i \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right )}{147 b^2}-\frac {20 \sqrt {\cosh (a+b x)} \sinh (a+b x)}{147 b^2}-\frac {4 \cosh ^{\frac {5}{2}}(a+b x) \sinh (a+b x)}{49 b^2} \]
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Time = 0.05 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5481, 2715, 2720} \[ \int x \cosh ^{\frac {5}{2}}(a+b x) \sinh (a+b x) \, dx=\frac {20 i \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right )}{147 b^2}-\frac {4 \sinh (a+b x) \cosh ^{\frac {5}{2}}(a+b x)}{49 b^2}-\frac {20 \sinh (a+b x) \sqrt {\cosh (a+b x)}}{147 b^2}+\frac {2 x \cosh ^{\frac {7}{2}}(a+b x)}{7 b} \]
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Rule 2715
Rule 2720
Rule 5481
Rubi steps \begin{align*} \text {integral}& = \frac {2 x \cosh ^{\frac {7}{2}}(a+b x)}{7 b}-\frac {2 \int \cosh ^{\frac {7}{2}}(a+b x) \, dx}{7 b} \\ & = \frac {2 x \cosh ^{\frac {7}{2}}(a+b x)}{7 b}-\frac {4 \cosh ^{\frac {5}{2}}(a+b x) \sinh (a+b x)}{49 b^2}-\frac {10 \int \cosh ^{\frac {3}{2}}(a+b x) \, dx}{49 b} \\ & = \frac {2 x \cosh ^{\frac {7}{2}}(a+b x)}{7 b}-\frac {20 \sqrt {\cosh (a+b x)} \sinh (a+b x)}{147 b^2}-\frac {4 \cosh ^{\frac {5}{2}}(a+b x) \sinh (a+b x)}{49 b^2}-\frac {10 \int \frac {1}{\sqrt {\cosh (a+b x)}} \, dx}{147 b} \\ & = \frac {2 x \cosh ^{\frac {7}{2}}(a+b x)}{7 b}+\frac {20 i \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right )}{147 b^2}-\frac {20 \sqrt {\cosh (a+b x)} \sinh (a+b x)}{147 b^2}-\frac {4 \cosh ^{\frac {5}{2}}(a+b x) \sinh (a+b x)}{49 b^2} \\ \end{align*}
Time = 0.39 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.89 \[ \int x \cosh ^{\frac {5}{2}}(a+b x) \sinh (a+b x) \, dx=\frac {40 i \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right )+\sqrt {\cosh (a+b x)} (63 b x \cosh (a+b x)+21 b x \cosh (3 (a+b x))-46 \sinh (a+b x)-6 \sinh (3 (a+b x)))}{294 b^2} \]
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\[\int x \cosh \left (b x +a \right )^{\frac {5}{2}} \sinh \left (b x +a \right )d x\]
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Exception generated. \[ \int x \cosh ^{\frac {5}{2}}(a+b x) \sinh (a+b x) \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int x \cosh ^{\frac {5}{2}}(a+b x) \sinh (a+b x) \, dx=\text {Timed out} \]
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\[ \int x \cosh ^{\frac {5}{2}}(a+b x) \sinh (a+b x) \, dx=\int { x \cosh \left (b x + a\right )^{\frac {5}{2}} \sinh \left (b x + a\right ) \,d x } \]
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\[ \int x \cosh ^{\frac {5}{2}}(a+b x) \sinh (a+b x) \, dx=\int { x \cosh \left (b x + a\right )^{\frac {5}{2}} \sinh \left (b x + a\right ) \,d x } \]
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Timed out. \[ \int x \cosh ^{\frac {5}{2}}(a+b x) \sinh (a+b x) \, dx=\int x\,{\mathrm {cosh}\left (a+b\,x\right )}^{5/2}\,\mathrm {sinh}\left (a+b\,x\right ) \,d x \]
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