Optimal. Leaf size=239 \[ \frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} e^{\frac{2 b c}{d}-2 a} \text{Erf}\left (\frac{\sqrt{2} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{256 b^{7/2}}-\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} e^{2 a-\frac{2 b c}{d}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{256 b^{7/2}}+\frac{15 d^2 \sqrt{c+d x} \sinh (2 a+2 b x)}{64 b^3}-\frac{5 d (c+d x)^{3/2} \sinh ^2(a+b x)}{8 b^2}+\frac{(c+d x)^{5/2} \sinh (a+b x) \cosh (a+b x)}{2 b}-\frac{5 d (c+d x)^{3/2}}{16 b^2}-\frac{(c+d x)^{7/2}}{7 d} \]
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Rubi [A] time = 0.448625, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {3311, 32, 3312, 3296, 3308, 2180, 2204, 2205} \[ \frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} e^{\frac{2 b c}{d}-2 a} \text{Erf}\left (\frac{\sqrt{2} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{256 b^{7/2}}-\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} e^{2 a-\frac{2 b c}{d}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{256 b^{7/2}}+\frac{15 d^2 \sqrt{c+d x} \sinh (2 a+2 b x)}{64 b^3}-\frac{5 d (c+d x)^{3/2} \sinh ^2(a+b x)}{8 b^2}+\frac{(c+d x)^{5/2} \sinh (a+b x) \cosh (a+b x)}{2 b}-\frac{5 d (c+d x)^{3/2}}{16 b^2}-\frac{(c+d x)^{7/2}}{7 d} \]
Antiderivative was successfully verified.
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Rule 3311
Rule 32
Rule 3312
Rule 3296
Rule 3308
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int (c+d x)^{5/2} \sinh ^2(a+b x) \, dx &=\frac{(c+d x)^{5/2} \cosh (a+b x) \sinh (a+b x)}{2 b}-\frac{5 d (c+d x)^{3/2} \sinh ^2(a+b x)}{8 b^2}-\frac{1}{2} \int (c+d x)^{5/2} \, dx+\frac{\left (15 d^2\right ) \int \sqrt{c+d x} \sinh ^2(a+b x) \, dx}{16 b^2}\\ &=-\frac{(c+d x)^{7/2}}{7 d}+\frac{(c+d x)^{5/2} \cosh (a+b x) \sinh (a+b x)}{2 b}-\frac{5 d (c+d x)^{3/2} \sinh ^2(a+b x)}{8 b^2}-\frac{\left (15 d^2\right ) \int \left (\frac{1}{2} \sqrt{c+d x}-\frac{1}{2} \sqrt{c+d x} \cosh (2 a+2 b x)\right ) \, dx}{16 b^2}\\ &=-\frac{5 d (c+d x)^{3/2}}{16 b^2}-\frac{(c+d x)^{7/2}}{7 d}+\frac{(c+d x)^{5/2} \cosh (a+b x) \sinh (a+b x)}{2 b}-\frac{5 d (c+d x)^{3/2} \sinh ^2(a+b x)}{8 b^2}+\frac{\left (15 d^2\right ) \int \sqrt{c+d x} \cosh (2 a+2 b x) \, dx}{32 b^2}\\ &=-\frac{5 d (c+d x)^{3/2}}{16 b^2}-\frac{(c+d x)^{7/2}}{7 d}+\frac{(c+d x)^{5/2} \cosh (a+b x) \sinh (a+b x)}{2 b}-\frac{5 d (c+d x)^{3/2} \sinh ^2(a+b x)}{8 b^2}+\frac{15 d^2 \sqrt{c+d x} \sinh (2 a+2 b x)}{64 b^3}-\frac{\left (15 d^3\right ) \int \frac{\sinh (2 a+2 b x)}{\sqrt{c+d x}} \, dx}{128 b^3}\\ &=-\frac{5 d (c+d x)^{3/2}}{16 b^2}-\frac{(c+d x)^{7/2}}{7 d}+\frac{(c+d x)^{5/2} \cosh (a+b x) \sinh (a+b x)}{2 b}-\frac{5 d (c+d x)^{3/2} \sinh ^2(a+b x)}{8 b^2}+\frac{15 d^2 \sqrt{c+d x} \sinh (2 a+2 b x)}{64 b^3}-\frac{\left (15 d^3\right ) \int \frac{e^{-i (2 i a+2 i b x)}}{\sqrt{c+d x}} \, dx}{256 b^3}+\frac{\left (15 d^3\right ) \int \frac{e^{i (2 i a+2 i b x)}}{\sqrt{c+d x}} \, dx}{256 b^3}\\ &=-\frac{5 d (c+d x)^{3/2}}{16 b^2}-\frac{(c+d x)^{7/2}}{7 d}+\frac{(c+d x)^{5/2} \cosh (a+b x) \sinh (a+b x)}{2 b}-\frac{5 d (c+d x)^{3/2} \sinh ^2(a+b x)}{8 b^2}+\frac{15 d^2 \sqrt{c+d x} \sinh (2 a+2 b x)}{64 b^3}+\frac{\left (15 d^2\right ) \operatorname{Subst}\left (\int e^{i \left (2 i a-\frac{2 i b c}{d}\right )-\frac{2 b x^2}{d}} \, dx,x,\sqrt{c+d x}\right )}{128 b^3}-\frac{\left (15 d^2\right ) \operatorname{Subst}\left (\int e^{-i \left (2 i a-\frac{2 i b c}{d}\right )+\frac{2 b x^2}{d}} \, dx,x,\sqrt{c+d x}\right )}{128 b^3}\\ &=-\frac{5 d (c+d x)^{3/2}}{16 b^2}-\frac{(c+d x)^{7/2}}{7 d}+\frac{15 d^{5/2} e^{-2 a+\frac{2 b c}{d}} \sqrt{\frac{\pi }{2}} \text{erf}\left (\frac{\sqrt{2} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{256 b^{7/2}}-\frac{15 d^{5/2} e^{2 a-\frac{2 b c}{d}} \sqrt{\frac{\pi }{2}} \text{erfi}\left (\frac{\sqrt{2} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{256 b^{7/2}}+\frac{(c+d x)^{5/2} \cosh (a+b x) \sinh (a+b x)}{2 b}-\frac{5 d (c+d x)^{3/2} \sinh ^2(a+b x)}{8 b^2}+\frac{15 d^2 \sqrt{c+d x} \sinh (2 a+2 b x)}{64 b^3}\\ \end{align*}
Mathematica [A] time = 6.59064, size = 190, normalized size = 0.79 \[ \frac{\sqrt{c+d x} \left (-7 \sqrt{2} d^4 \sqrt{-\frac{b^2 (c+d x)^2}{d^2}} \text{Gamma}\left (\frac{7}{2},-\frac{2 b (c+d x)}{d}\right ) \left (\sinh \left (2 a-\frac{2 b c}{d}\right )+\cosh \left (2 a-\frac{2 b c}{d}\right )\right )-b (c+d x) \left (7 \sqrt{2} d^3 \text{Gamma}\left (\frac{7}{2},\frac{2 b (c+d x)}{d}\right ) \left (\cosh \left (2 a-\frac{2 b c}{d}\right )-\sinh \left (2 a-\frac{2 b c}{d}\right )\right )+64 b^3 (c+d x)^3 \sqrt{\frac{b (c+d x)}{d}}\right )\right )}{448 b^3 d^2 \left (\frac{b (c+d x)}{d}\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.062, size = 0, normalized size = 0. \begin{align*} \int \left ( dx+c \right ) ^{{\frac{5}{2}}} \left ( \sinh \left ( bx+a \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.74935, size = 379, normalized size = 1.59 \begin{align*} -\frac{512 \,{\left (d x + c\right )}^{\frac{7}{2}} + \frac{105 \, \sqrt{2} \sqrt{\pi } d^{3} \operatorname{erf}\left (\sqrt{2} \sqrt{d x + c} \sqrt{-\frac{b}{d}}\right ) e^{\left (2 \, a - \frac{2 \, b c}{d}\right )}}{b^{3} \sqrt{-\frac{b}{d}}} - \frac{105 \, \sqrt{2} \sqrt{\pi } d^{3} \operatorname{erf}\left (\sqrt{2} \sqrt{d x + c} \sqrt{\frac{b}{d}}\right ) e^{\left (-2 \, a + \frac{2 \, b c}{d}\right )}}{b^{3} \sqrt{\frac{b}{d}}} + \frac{28 \,{\left (16 \,{\left (d x + c\right )}^{\frac{5}{2}} b^{2} d e^{\left (\frac{2 \, b c}{d}\right )} + 20 \,{\left (d x + c\right )}^{\frac{3}{2}} b d^{2} e^{\left (\frac{2 \, b c}{d}\right )} + 15 \, \sqrt{d x + c} d^{3} e^{\left (\frac{2 \, b c}{d}\right )}\right )} e^{\left (-2 \, a - \frac{2 \,{\left (d x + c\right )} b}{d}\right )}}{b^{3}} - \frac{28 \,{\left (16 \,{\left (d x + c\right )}^{\frac{5}{2}} b^{2} d e^{\left (2 \, a\right )} - 20 \,{\left (d x + c\right )}^{\frac{3}{2}} b d^{2} e^{\left (2 \, a\right )} + 15 \, \sqrt{d x + c} d^{3} e^{\left (2 \, a\right )}\right )} e^{\left (\frac{2 \,{\left (d x + c\right )} b}{d} - \frac{2 \, b c}{d}\right )}}{b^{3}}}{3584 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 3.19787, size = 2319, normalized size = 9.7 \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{5}{2}} \sinh \left (b x + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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