Optimal. Leaf size=325 \[ \frac{9 \sqrt{\pi } d^{3/2} e^{\frac{b c}{d}-a} \text{Erf}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{32 b^{5/2}}-\frac{\sqrt{\frac{\pi }{3}} d^{3/2} e^{\frac{3 b c}{d}-3 a} \text{Erf}\left (\frac{\sqrt{3} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{96 b^{5/2}}-\frac{9 \sqrt{\pi } d^{3/2} e^{a-\frac{b c}{d}} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{32 b^{5/2}}+\frac{\sqrt{\frac{\pi }{3}} d^{3/2} e^{3 a-\frac{3 b c}{d}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{96 b^{5/2}}-\frac{d \sqrt{c+d x} \sinh ^3(a+b x)}{6 b^2}+\frac{d \sqrt{c+d x} \sinh (a+b x)}{b^2}-\frac{2 (c+d x)^{3/2} \cosh (a+b x)}{3 b}+\frac{(c+d x)^{3/2} \sinh ^2(a+b x) \cosh (a+b x)}{3 b} \]
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Rubi [A] time = 0.799132, antiderivative size = 325, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {3311, 3296, 3308, 2180, 2204, 2205, 3312} \[ \frac{9 \sqrt{\pi } d^{3/2} e^{\frac{b c}{d}-a} \text{Erf}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{32 b^{5/2}}-\frac{\sqrt{\frac{\pi }{3}} d^{3/2} e^{\frac{3 b c}{d}-3 a} \text{Erf}\left (\frac{\sqrt{3} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{96 b^{5/2}}-\frac{9 \sqrt{\pi } d^{3/2} e^{a-\frac{b c}{d}} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{32 b^{5/2}}+\frac{\sqrt{\frac{\pi }{3}} d^{3/2} e^{3 a-\frac{3 b c}{d}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{96 b^{5/2}}-\frac{d \sqrt{c+d x} \sinh ^3(a+b x)}{6 b^2}+\frac{d \sqrt{c+d x} \sinh (a+b x)}{b^2}-\frac{2 (c+d x)^{3/2} \cosh (a+b x)}{3 b}+\frac{(c+d x)^{3/2} \sinh ^2(a+b x) \cosh (a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 3311
Rule 3296
Rule 3308
Rule 2180
Rule 2204
Rule 2205
Rule 3312
Rubi steps
\begin{align*} \int (c+d x)^{3/2} \sinh ^3(a+b x) \, dx &=\frac{(c+d x)^{3/2} \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac{d \sqrt{c+d x} \sinh ^3(a+b x)}{6 b^2}-\frac{2}{3} \int (c+d x)^{3/2} \sinh (a+b x) \, dx+\frac{d^2 \int \frac{\sinh ^3(a+b x)}{\sqrt{c+d x}} \, dx}{12 b^2}\\ &=-\frac{2 (c+d x)^{3/2} \cosh (a+b x)}{3 b}+\frac{(c+d x)^{3/2} \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac{d \sqrt{c+d x} \sinh ^3(a+b x)}{6 b^2}+\frac{d \int \sqrt{c+d x} \cosh (a+b x) \, dx}{b}+\frac{\left (i d^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}{12 b^2}\\ &=-\frac{2 (c+d x)^{3/2} \cosh (a+b x)}{3 b}+\frac{d \sqrt{c+d x} \sinh (a+b x)}{b^2}+\frac{(c+d x)^{3/2} \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac{d \sqrt{c+d x} \sinh ^3(a+b x)}{6 b^2}+\frac{d^2 \int \frac{\sinh (3 a+3 b x)}{\sqrt{c+d x}} \, dx}{48 b^2}-\frac{d^2 \int \frac{\sinh (a+b x)}{\sqrt{c+d x}} \, dx}{16 b^2}-\frac{d^2 \int \frac{\sinh (a+b x)}{\sqrt{c+d x}} \, dx}{2 b^2}\\ &=-\frac{2 (c+d x)^{3/2} \cosh (a+b x)}{3 b}+\frac{d \sqrt{c+d x} \sinh (a+b x)}{b^2}+\frac{(c+d x)^{3/2} \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac{d \sqrt{c+d x} \sinh ^3(a+b x)}{6 b^2}+\frac{d^2 \int \frac{e^{-i (3 i a+3 i b x)}}{\sqrt{c+d x}} \, dx}{96 b^2}-\frac{d^2 \int \frac{e^{i (3 i a+3 i b x)}}{\sqrt{c+d x}} \, dx}{96 b^2}-\frac{d^2 \int \frac{e^{-i (i a+i b x)}}{\sqrt{c+d x}} \, dx}{32 b^2}+\frac{d^2 \int \frac{e^{i (i a+i b x)}}{\sqrt{c+d x}} \, dx}{32 b^2}-\frac{d^2 \int \frac{e^{-i (i a+i b x)}}{\sqrt{c+d x}} \, dx}{4 b^2}+\frac{d^2 \int \frac{e^{i (i a+i b x)}}{\sqrt{c+d x}} \, dx}{4 b^2}\\ &=-\frac{2 (c+d x)^{3/2} \cosh (a+b x)}{3 b}+\frac{d \sqrt{c+d x} \sinh (a+b x)}{b^2}+\frac{(c+d x)^{3/2} \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac{d \sqrt{c+d x} \sinh ^3(a+b x)}{6 b^2}-\frac{d \operatorname{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 )}{48 b^2}+\frac{d \operatorname{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 )}{48 b^2}+\frac{d \operatorname{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 )}{16 b^2}-\frac{d \operatorname{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 )}{16 b^2}+\frac{d \operatorname{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 )}{2 b^2}-\frac{d \operatorname{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 )}{2 b^2}\\ &=-\frac{2 (c+d x)^{3/2} \cosh (a+b x)}{3 b}+\frac{9 d^{3/2} e^{-a+\frac{b c}{d}} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{32 b^{5/2}}-\frac{d^{3/2} e^{-3 a+\frac{3 b c}{d}} \sqrt{\frac{\pi }{3}} \text{erf}\left (\frac{\sqrt{3} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{96 b^{5/2}}-\frac{9 d^{3/2} e^{a-\frac{b c}{d}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{32 b^{5/2}}+\frac{d^{3/2} e^{3 a-\frac{3 b c}{d}} \sqrt{\frac{\pi }{3}} \text{erfi}\left (\frac{\sqrt{3} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{96 b^{5/2}}+\frac{d \sqrt{c+d x} \sinh (a+b x)}{b^2}+\frac{(c+d x)^{3/2} \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac{d \sqrt{c+d x} \sinh ^3(a+b x)}{6 b^2}\\ \end{align*}
Mathematica [A] time = 3.89138, size = 243, normalized size = 0.75 \[ \frac{d^2 \left (\sqrt{3} \sqrt{-\frac{b (c+d x)}{d}} \text{Gamma}\left (\frac{5}{2},-\frac{3 b (c+d x)}{d}\right ) \left (\sinh \left (3 a-\frac{3 b c}{d}\right )+\cosh \left (3 a-\frac{3 b c}{d}\right )\right )+\left (\sinh \left (a-\frac{b c}{d}\right )-\cosh \left (a-\frac{b c}{d}\right )\right ) \left (81 \sqrt{-\frac{b (c+d x)}{d}} \text{Gamma}\left (\frac{5}{2},-\frac{b (c+d x)}{d}\right ) \left (\sinh \left (2 a-\frac{2 b c}{d}\right )+\cosh \left (2 a-\frac{2 b c}{d}\right )\right )+\sqrt{\frac{b (c+d x)}{d}} \left (\sqrt{3} \text{Gamma}\left (\frac{5}{2},\frac{3 b (c+d x)}{d}\right ) \left (\sinh \left (2 a-\frac{2 b c}{d}\right )-\cosh \left (2 a-\frac{2 b c}{d}\right )\right )+81 \text{Gamma}\left (\frac{5}{2},\frac{b (c+d x)}{d}\right )\right )\right )\right )}{216 b^3 \sqrt{c+d x}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.078, size = 0, normalized size = 0. \begin{align*} \int \left ( dx+c \right ) ^{{\frac{3}{2}}} \left ( \sinh \left ( bx+a \right ) \right ) ^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.79347, size = 581, normalized size = 1.79 \begin{align*} \frac{\frac{\sqrt{3} \sqrt{\pi } d^{2} \operatorname{erf}\left (\sqrt{3} \sqrt{d x + c} \sqrt{-\frac{b}{d}}\right ) e^{\left (3 \, a - \frac{3 \, b c}{d}\right )}}{b^{2} \sqrt{-\frac{b}{d}}} - \frac{\sqrt{3} \sqrt{\pi } d^{2} \operatorname{erf}\left (\sqrt{3} \sqrt{d x + c} \sqrt{\frac{b}{d}}\right ) e^{\left (-3 \, a + \frac{3 \, b c}{d}\right )}}{b^{2} \sqrt{\frac{b}{d}}} - \frac{81 \, \sqrt{\pi } d^{2} \operatorname{erf}\left (\sqrt{d x + c} \sqrt{-\frac{b}{d}}\right ) e^{\left (a - \frac{b c}{d}\right )}}{b^{2} \sqrt{-\frac{b}{d}}} + \frac{81 \, \sqrt{\pi } d^{2} \operatorname{erf}\left (\sqrt{d x + c} \sqrt{\frac{b}{d}}\right ) e^{\left (-a + \frac{b c}{d}\right )}}{b^{2} \sqrt{\frac{b}{d}}} - \frac{54 \,{\left (2 \,{\left (d x + c\right )}^{\frac{3}{2}} b d e^{\left (\frac{b c}{d}\right )} + 3 \, \sqrt{d x + c} d^{2} e^{\left (\frac{b c}{d}\right )}\right )} e^{\left (-a - \frac{{\left (d x + c\right )} b}{d}\right )}}{b^{2}} + \frac{6 \,{\left (2 \,{\left (d x + c\right )}^{\frac{3}{2}} b d e^{\left (\frac{3 \, b c}{d}\right )} + \sqrt{d x + c} d^{2} e^{\left (\frac{3 \, b c}{d}\right )}\right )} e^{\left (-3 \, a - \frac{3 \,{\left (d x + c\right )} b}{d}\right )}}{b^{2}} + \frac{6 \,{\left (2 \,{\left (d x + c\right )}^{\frac{3}{2}} b d e^{\left (3 \, a\right )} - \sqrt{d x + c} d^{2} e^{\left (3 \, a\right )}\right )} e^{\left (\frac{3 \,{\left (d x + c\right )} b}{d} - \frac{3 \, b c}{d}\right )}}{b^{2}} - \frac{54 \,{\left (2 \,{\left (d x + c\right )}^{\frac{3}{2}} b d e^{a} - 3 \, \sqrt{d x + c} d^{2} e^{a}\right )} e^{\left (\frac{{\left (d x + c\right )} b}{d} - \frac{b c}{d}\right )}}{b^{2}}}{288 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 3.30136, size = 3622, normalized size = 11.14 \begin{align*} \text{result too large to display} \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 (d x + c\right )}^{\frac{3}{2}} \sinh \left (b x + a\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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