Integrand size = 18, antiderivative size = 277 \[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^{5/2}} \, dx=\frac {b^{3/2} e^{-a+\frac {b c}{d}} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 d^{5/2}}-\frac {b^{3/2} e^{-3 a+\frac {3 b c}{d}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 d^{5/2}}-\frac {b^{3/2} e^{a-\frac {b c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 d^{5/2}}+\frac {b^{3/2} e^{3 a-\frac {3 b c}{d}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 d^{5/2}}-\frac {4 b \cosh (a+b x) \sinh ^2(a+b x)}{d^2 \sqrt {c+d x}}-\frac {2 \sinh ^3(a+b x)}{3 d (c+d x)^{3/2}} \]
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Time = 0.46 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3395, 3389, 2211, 2235, 2236, 3393} \[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^{5/2}} \, dx=\frac {\sqrt {\pi } b^{3/2} e^{\frac {b c}{d}-a} \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 d^{5/2}}-\frac {\sqrt {3 \pi } b^{3/2} e^{\frac {3 b c}{d}-3 a} \text {erf}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 d^{5/2}}-\frac {\sqrt {\pi } b^{3/2} e^{a-\frac {b c}{d}} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 d^{5/2}}+\frac {\sqrt {3 \pi } b^{3/2} e^{3 a-\frac {3 b c}{d}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 d^{5/2}}-\frac {4 b \sinh ^2(a+b x) \cosh (a+b x)}{d^2 \sqrt {c+d x}}-\frac {2 \sinh ^3(a+b x)}{3 d (c+d x)^{3/2}} \]
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Rule 2211
Rule 2235
Rule 2236
Rule 3389
Rule 3393
Rule 3395
Rubi steps \begin{align*} \text {integral}& = -\frac {4 b \cosh (a+b x) \sinh ^2(a+b x)}{d^2 \sqrt {c+d x}}-\frac {2 \sinh ^3(a+b x)}{3 d (c+d x)^{3/2}}+\frac {\left (8 b^2\right ) \int \frac {\sinh (a+b x)}{\sqrt {c+d x}} \, dx}{d^2}+\frac {\left (12 b^2\right ) \int \frac {\sinh ^3(a+b x)}{\sqrt {c+d x}} \, dx}{d^2} \\ & = -\frac {4 b \cosh (a+b x) \sinh ^2(a+b x)}{d^2 \sqrt {c+d x}}-\frac {2 \sinh ^3(a+b x)}{3 d (c+d x)^{3/2}}+\frac {\left (12 i b^2\right ) \int \left (\frac {3 i \sinh (a+b x)}{4 \sqrt {c+d x}}-\frac {i \sinh (3 a+3 b x)}{4 \sqrt {c+d x}}\right ) \, dx}{d^2}+\frac {\left (4 b^2\right ) \int \frac {e^{-i (i a+i b x)}}{\sqrt {c+d x}} \, dx}{d^2}-\frac {\left (4 b^2\right ) \int \frac {e^{i (i a+i b x)}}{\sqrt {c+d x}} \, dx}{d^2} \\ & = -\frac {4 b \cosh (a+b x) \sinh ^2(a+b x)}{d^2 \sqrt {c+d x}}-\frac {2 \sinh ^3(a+b x)}{3 d (c+d x)^{3/2}}-\frac {\left (8 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 )}{d^3}+\frac {\left (8 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 )}{d^3}+\frac {\left (3 b^2\right ) \int \frac {\sinh (3 a+3 b x)}{\sqrt {c+d x}} \, dx}{d^2}-\frac {\left (9 b^2\right ) \int \frac {\sinh (a+b x)}{\sqrt {c+d x}} \, dx}{d^2} \\ & = -\frac {4 b^{3/2} e^{-a+\frac {b c}{d}} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{5/2}}+\frac {4 b^{3/2} e^{a-\frac {b c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{5/2}}-\frac {4 b \cosh (a+b x) \sinh ^2(a+b x)}{d^2 \sqrt {c+d x}}-\frac {2 \sinh ^3(a+b x)}{3 d (c+d x)^{3/2}}+\frac {\left (3 b^2\right ) \int \frac {e^{-i (3 i a+3 i b x)}}{\sqrt {c+d x}} \, dx}{2 d^2}-\frac {\left (3 b^2\right ) \int \frac {e^{i (3 i a+3 i b x)}}{\sqrt {c+d x}} \, dx}{2 d^2}-\frac {\left (9 b^2\right ) \int \frac {e^{-i (i a+i b x)}}{\sqrt {c+d x}} \, dx}{2 d^2}+\frac {\left (9 b^2\right ) \int \frac {e^{i (i a+i b x)}}{\sqrt {c+d x}} \, dx}{2 d^2} \\ & = -\frac {4 b^{3/2} e^{-a+\frac {b c}{d}} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{5/2}}+\frac {4 b^{3/2} e^{a-\frac {b c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{5/2}}-\frac {4 b \cosh (a+b x) \sinh ^2(a+b x)}{d^2 \sqrt {c+d x}}-\frac {2 \sinh ^3(a+b x)}{3 d (c+d x)^{3/2}}-\frac {\left (3 b^2\right ) \text {Subst}\left (\int e^{i \left (3 i a-\frac {3 i b c}{d}\right )-\frac {3 b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{d^3}+\frac {\left (3 b^2\right ) \text {Subst}\left (\int e^{-i \left (3 i a-\frac {3 i b c}{d}\right )+\frac {3 b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{d^3}+\frac {\left (9 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 )}{d^3}-\frac {\left (9 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 )}{d^3} \\ & = \frac {b^{3/2} e^{-a+\frac {b c}{d}} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 d^{5/2}}-\frac {b^{3/2} e^{-3 a+\frac {3 b c}{d}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 d^{5/2}}-\frac {b^{3/2} e^{a-\frac {b c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 d^{5/2}}+\frac {b^{3/2} e^{3 a-\frac {3 b c}{d}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 d^{5/2}}-\frac {4 b \cosh (a+b x) \sinh ^2(a+b x)}{d^2 \sqrt {c+d x}}-\frac {2 \sinh ^3(a+b x)}{3 d (c+d x)^{3/2}} \\ \end{align*}
Time = 2.02 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.91 \[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^{5/2}} \, dx=\frac {e^{-3 \left (a+\frac {b c}{d}\right )} \left (-3 \sqrt {3} d e^{6 a} \left (-\frac {b (c+d x)}{d}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {3 b (c+d x)}{d}\right )+3 d e^{4 a+\frac {2 b c}{d}} \left (-\frac {b (c+d x)}{d}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {b (c+d x)}{d}\right )-3 d e^{2 a+\frac {4 b c}{d}} \left (\frac {b (c+d x)}{d}\right )^{3/2} \Gamma \left (\frac {1}{2},\frac {b (c+d x)}{d}\right )+3 \sqrt {3} d e^{\frac {6 b c}{d}} \left (\frac {b (c+d x)}{d}\right )^{3/2} \Gamma \left (\frac {1}{2},\frac {3 b (c+d x)}{d}\right )-4 e^{3 \left (a+\frac {b c}{d}\right )} \sinh ^2(a+b x) (6 b (c+d x) \cosh (a+b x)+d \sinh (a+b x))\right )}{6 d^2 (c+d x)^{3/2}} \]
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\[\int \frac {\sinh \left (b x +a \right )^{3}}{\left (d x +c \right )^{\frac {5}{2}}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 2059 vs. \(2 (209) = 418\).
Time = 0.30 (sec) , antiderivative size = 2059, normalized size of antiderivative = 7.43 \[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^{5/2}} \, dx=\text {Too large to display} \]
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\[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^{5/2}} \, dx=\int \frac {\sinh ^{3}{\left (a + b x \right )}}{\left (c + d x\right )^{\frac {5}{2}}}\, dx \]
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Time = 0.32 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.71 \[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^{5/2}} \, dx=\frac {3 \, {\left (\frac {\sqrt {3} \left (\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {3}{2}} e^{\left (\frac {3 \, {\left (b c - a d\right )}}{d}\right )} \Gamma \left (-\frac {3}{2}, \frac {3 \, {\left (d x + c\right )} b}{d}\right )}{{\left (d x + c\right )}^{\frac {3}{2}}} - \frac {\sqrt {3} \left (-\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {3}{2}} e^{\left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )} \Gamma \left (-\frac {3}{2}, -\frac {3 \, {\left (d x + c\right )} b}{d}\right )}{{\left (d x + c\right )}^{\frac {3}{2}}} - \frac {\left (\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {3}{2}} e^{\left (-a + \frac {b c}{d}\right )} \Gamma \left (-\frac {3}{2}, \frac {{\left (d x + c\right )} b}{d}\right )}{{\left (d x + c\right )}^{\frac {3}{2}}} + \frac {\left (-\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {3}{2}} e^{\left (a - \frac {b c}{d}\right )} \Gamma \left (-\frac {3}{2}, -\frac {{\left (d x + c\right )} b}{d}\right )}{{\left (d x + c\right )}^{\frac {3}{2}}}\right )}}{8 \, d} \]
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\[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^{5/2}} \, dx=\int { \frac {\sinh \left (b x + a\right )^{3}}{{\left (d x + c\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^{5/2}} \, dx=\int \frac {{\mathrm {sinh}\left (a+b\,x\right )}^3}{{\left (c+d\,x\right )}^{5/2}} \,d x \]
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