Integrand size = 16, antiderivative size = 149 \[ \int \frac {\sinh (a+b x)}{(c+d x)^{5/2}} \, dx=-\frac {4 b \cosh (a+b x)}{3 d^2 \sqrt {c+d x}}-\frac {2 b^{3/2} e^{-a+\frac {b c}{d}} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{3 d^{5/2}}+\frac {2 b^{3/2} e^{a-\frac {b c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{3 d^{5/2}}-\frac {2 \sinh (a+b x)}{3 d (c+d x)^{3/2}} \]
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Time = 0.18 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {3378, 3389, 2211, 2235, 2236} \[ \int \frac {\sinh (a+b x)}{(c+d x)^{5/2}} \, dx=-\frac {2 \sqrt {\pi } b^{3/2} e^{\frac {b c}{d}-a} \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{3 d^{5/2}}+\frac {2 \sqrt {\pi } b^{3/2} e^{a-\frac {b c}{d}} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{3 d^{5/2}}-\frac {4 b \cosh (a+b x)}{3 d^2 \sqrt {c+d x}}-\frac {2 \sinh (a+b x)}{3 d (c+d x)^{3/2}} \]
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Rule 2211
Rule 2235
Rule 2236
Rule 3378
Rule 3389
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sinh (a+b x)}{3 d (c+d x)^{3/2}}+\frac {(2 b) \int \frac {\cosh (a+b x)}{(c+d x)^{3/2}} \, dx}{3 d} \\ & = -\frac {4 b \cosh (a+b x)}{3 d^2 \sqrt {c+d x}}-\frac {2 \sinh (a+b x)}{3 d (c+d x)^{3/2}}+\frac {\left (4 b^2\right ) \int \frac {\sinh (a+b x)}{\sqrt {c+d x}} \, dx}{3 d^2} \\ & = -\frac {4 b \cosh (a+b x)}{3 d^2 \sqrt {c+d x}}-\frac {2 \sinh (a+b x)}{3 d (c+d x)^{3/2}}+\frac {\left (2 b^2\right ) \int \frac {e^{-i (i a+i b x)}}{\sqrt {c+d x}} \, dx}{3 d^2}-\frac {\left (2 b^2\right ) \int \frac {e^{i (i a+i b x)}}{\sqrt {c+d x}} \, dx}{3 d^2} \\ & = -\frac {4 b \cosh (a+b x)}{3 d^2 \sqrt {c+d x}}-\frac {2 \sinh (a+b x)}{3 d (c+d x)^{3/2}}-\frac {\left (4 b^2\right ) \text {Subst}\left (\int e^{i \left (i a-\frac {i b c}{d}\right )-\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{3 d^3}+\frac {\left (4 b^2\right ) \text {Subst}\left (\int e^{-i \left (i a-\frac {i b c}{d}\right )+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{3 d^3} \\ & = -\frac {4 b \cosh (a+b x)}{3 d^2 \sqrt {c+d x}}-\frac {2 b^{3/2} e^{-a+\frac {b c}{d}} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{3 d^{5/2}}+\frac {2 b^{3/2} e^{a-\frac {b c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{3 d^{5/2}}-\frac {2 \sinh (a+b x)}{3 d (c+d x)^{3/2}} \\ \end{align*}
Time = 0.47 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.08 \[ \int \frac {\sinh (a+b x)}{(c+d x)^{5/2}} \, dx=\frac {2 b \left (\frac {e^a \left (-e^{b x}+e^{-\frac {b c}{d}} \sqrt {-\frac {b (c+d x)}{d}} \Gamma \left (\frac {1}{2},-\frac {b (c+d x)}{d}\right )\right )}{d \sqrt {c+d x}}+\frac {e^{-a-b x} \left (-1+e^{b \left (\frac {c}{d}+x\right )} \sqrt {\frac {b (c+d x)}{d}} \Gamma \left (\frac {1}{2},\frac {b (c+d x)}{d}\right )\right )}{d \sqrt {c+d x}}\right )}{3 d}-\frac {2 \sinh (a+b x)}{3 d (c+d x)^{3/2}} \]
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\[\int \frac {\sinh \left (b x +a \right )}{\left (d x +c \right )^{\frac {5}{2}}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 532 vs. \(2 (111) = 222\).
Time = 0.27 (sec) , antiderivative size = 532, normalized size of antiderivative = 3.57 \[ \int \frac {\sinh (a+b x)}{(c+d x)^{5/2}} \, dx=-\frac {2 \, \sqrt {\pi } {\left ({\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \cosh \left (b x + a\right ) \cosh \left (-\frac {b c - a d}{d}\right ) - {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (-\frac {b c - a d}{d}\right ) + {\left ({\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \cosh \left (-\frac {b c - a d}{d}\right ) - {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \sinh \left (-\frac {b c - a d}{d}\right )\right )} \sinh \left (b x + a\right )\right )} \sqrt {\frac {b}{d}} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) + 2 \, \sqrt {\pi } {\left ({\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \cosh \left (b x + a\right ) \cosh \left (-\frac {b c - a d}{d}\right ) + {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (-\frac {b c - a d}{d}\right ) + {\left ({\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \cosh \left (-\frac {b c - a d}{d}\right ) + {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \sinh \left (-\frac {b c - a d}{d}\right )\right )} \sinh \left (b x + a\right )\right )} \sqrt {-\frac {b}{d}} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) + {\left (2 \, b d x + {\left (2 \, b d x + 2 \, b c + d\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (2 \, b d x + 2 \, b c + d\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (2 \, b d x + 2 \, b c + d\right )} \sinh \left (b x + a\right )^{2} + 2 \, b c - d\right )} \sqrt {d x + c}}{3 \, {\left ({\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )} \cosh \left (b x + a\right ) + {\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )} \sinh \left (b x + a\right )\right )}} \]
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\[ \int \frac {\sinh (a+b x)}{(c+d x)^{5/2}} \, dx=\int \frac {\sinh {\left (a + b x \right )}}{\left (c + d x\right )^{\frac {5}{2}}}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.77 \[ \int \frac {\sinh (a+b x)}{(c+d x)^{5/2}} \, dx=-\frac {\frac {{\left (\frac {\sqrt {\frac {{\left (d x + c\right )} b}{d}} e^{\left (-a + \frac {b c}{d}\right )} \Gamma \left (-\frac {1}{2}, \frac {{\left (d x + c\right )} b}{d}\right )}{\sqrt {d x + c}} + \frac {\sqrt {-\frac {{\left (d x + c\right )} b}{d}} e^{\left (a - \frac {b c}{d}\right )} \Gamma \left (-\frac {1}{2}, -\frac {{\left (d x + c\right )} b}{d}\right )}{\sqrt {d x + c}}\right )} b}{d} + \frac {2 \, \sinh \left (b x + a\right )}{{\left (d x + c\right )}^{\frac {3}{2}}}}{3 \, d} \]
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\[ \int \frac {\sinh (a+b x)}{(c+d x)^{5/2}} \, dx=\int { \frac {\sinh \left (b x + a\right )}{{\left (d x + c\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sinh (a+b x)}{(c+d x)^{5/2}} \, dx=\int \frac {\mathrm {sinh}\left (a+b\,x\right )}{{\left (c+d\,x\right )}^{5/2}} \,d x \]
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