Optimal. Leaf size=211 \[ \frac{3 \sqrt{\frac{\pi }{2}} d^{3/2} e^{\frac{2 b c}{d}-2 a} \text{Erf}\left (\frac{\sqrt{2} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{64 b^{5/2}}+\frac{3 \sqrt{\frac{\pi }{2}} d^{3/2} e^{2 a-\frac{2 b c}{d}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{64 b^{5/2}}-\frac{3 d \sqrt{c+d x} \cosh ^2(a+b x)}{8 b^2}+\frac{(c+d x)^{3/2} \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac{3 d \sqrt{c+d x}}{16 b^2}+\frac{(c+d x)^{5/2}}{5 d} \]
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Rubi [A] time = 0.303136, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {3311, 32, 3312, 3307, 2180, 2204, 2205} \[ \frac{3 \sqrt{\frac{\pi }{2}} d^{3/2} e^{\frac{2 b c}{d}-2 a} \text{Erf}\left (\frac{\sqrt{2} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{64 b^{5/2}}+\frac{3 \sqrt{\frac{\pi }{2}} d^{3/2} e^{2 a-\frac{2 b c}{d}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{64 b^{5/2}}-\frac{3 d \sqrt{c+d x} \cosh ^2(a+b x)}{8 b^2}+\frac{(c+d x)^{3/2} \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac{3 d \sqrt{c+d x}}{16 b^2}+\frac{(c+d x)^{5/2}}{5 d} \]
Antiderivative was successfully verified.
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Rule 3311
Rule 32
Rule 3312
Rule 3307
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int (c+d x)^{3/2} \cosh ^2(a+b x) \, dx &=-\frac{3 d \sqrt{c+d x} \cosh ^2(a+b x)}{8 b^2}+\frac{(c+d x)^{3/2} \cosh (a+b x) \sinh (a+b x)}{2 b}+\frac{1}{2} \int (c+d x)^{3/2} \, dx+\frac{\left (3 d^2\right ) \int \frac{\cosh ^2(a+b x)}{\sqrt{c+d x}} \, dx}{16 b^2}\\ &=\frac{(c+d x)^{5/2}}{5 d}-\frac{3 d \sqrt{c+d x} \cosh ^2(a+b x)}{8 b^2}+\frac{(c+d x)^{3/2} \cosh (a+b x) \sinh (a+b x)}{2 b}+\frac{\left (3 d^2\right ) \int \left (\frac{1}{2 \sqrt{c+d x}}+\frac{\cosh (2 a+2 b x)}{2 \sqrt{c+d x}}\right ) \, dx}{16 b^2}\\ &=\frac{3 d \sqrt{c+d x}}{16 b^2}+\frac{(c+d x)^{5/2}}{5 d}-\frac{3 d \sqrt{c+d x} \cosh ^2(a+b x)}{8 b^2}+\frac{(c+d x)^{3/2} \cosh (a+b x) \sinh (a+b x)}{2 b}+\frac{\left (3 d^2\right ) \int \frac{\cosh (2 a+2 b x)}{\sqrt{c+d x}} \, dx}{32 b^2}\\ &=\frac{3 d \sqrt{c+d x}}{16 b^2}+\frac{(c+d x)^{5/2}}{5 d}-\frac{3 d \sqrt{c+d x} \cosh ^2(a+b x)}{8 b^2}+\frac{(c+d x)^{3/2} \cosh (a+b x) \sinh (a+b x)}{2 b}+\frac{\left (3 d^2\right ) \int \frac{e^{-i (2 i a+2 i b x)}}{\sqrt{c+d x}} \, dx}{64 b^2}+\frac{\left (3 d^2\right ) \int \frac{e^{i (2 i a+2 i b x)}}{\sqrt{c+d x}} \, dx}{64 b^2}\\ &=\frac{3 d \sqrt{c+d x}}{16 b^2}+\frac{(c+d x)^{5/2}}{5 d}-\frac{3 d \sqrt{c+d x} \cosh ^2(a+b x)}{8 b^2}+\frac{(c+d x)^{3/2} \cosh (a+b x) \sinh (a+b x)}{2 b}+\frac{(3 d) \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 )}{32 b^2}+\frac{(3 d) \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 )}{32 b^2}\\ &=\frac{3 d \sqrt{c+d x}}{16 b^2}+\frac{(c+d x)^{5/2}}{5 d}-\frac{3 d \sqrt{c+d x} \cosh ^2(a+b x)}{8 b^2}+\frac{3 d^{3/2} 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 )}{64 b^{5/2}}+\frac{3 d^{3/2} 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 )}{64 b^{5/2}}+\frac{(c+d x)^{3/2} \cosh (a+b x) \sinh (a+b x)}{2 b}\\ \end{align*}
Mathematica [A] time = 0.58322, size = 163, normalized size = 0.77 \[ \frac{5 \sqrt{2} d^3 \sqrt{\frac{b (c+d x)}{d}} \text{Gamma}\left (\frac{5}{2},\frac{2 b (c+d x)}{d}\right ) \left (\sinh \left (2 a-\frac{2 b c}{d}\right )-\cosh \left (2 a-\frac{2 b c}{d}\right )\right )+5 \sqrt{2} d^3 \sqrt{-\frac{b (c+d x)}{d}} \text{Gamma}\left (\frac{5}{2},-\frac{2 b (c+d x)}{d}\right ) \left (\sinh \left (2 a-\frac{2 b c}{d}\right )+\cosh \left (2 a-\frac{2 b c}{d}\right )\right )+32 b^3 (c+d x)^3}{160 b^3 d \sqrt{c+d x}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.08, size = 0, normalized size = 0. \begin{align*} \int \left ( dx+c \right ) ^{{\frac{3}{2}}} \left ( \cosh \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 = 1.57011, size = 323, normalized size = 1.53 \begin{align*} \frac{128 \,{\left (d x + c\right )}^{\frac{5}{2}} + \frac{15 \, \sqrt{2} \sqrt{\pi } d^{2} \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^{2} \sqrt{-\frac{b}{d}}} + \frac{15 \, \sqrt{2} \sqrt{\pi } d^{2} \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^{2} \sqrt{\frac{b}{d}}} - \frac{20 \,{\left (4 \,{\left (d x + c\right )}^{\frac{3}{2}} b d e^{\left (\frac{2 \, b c}{d}\right )} + 3 \, \sqrt{d x + c} d^{2} e^{\left (\frac{2 \, b c}{d}\right )}\right )} e^{\left (-2 \, a - \frac{2 \,{\left (d x + c\right )} b}{d}\right )}}{b^{2}} + \frac{20 \,{\left (4 \,{\left (d x + c\right )}^{\frac{3}{2}} b d e^{\left (2 \, a\right )} - 3 \, \sqrt{d x + c} d^{2} e^{\left (2 \, a\right )}\right )} e^{\left (\frac{2 \,{\left (d x + c\right )} b}{d} - \frac{2 \, b c}{d}\right )}}{b^{2}}}{640 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.21844, size = 1789, normalized size = 8.48 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c + d x\right )^{\frac{3}{2}} \cosh ^{2}{\left (a + b x \right )}\, dx \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}} \cosh \left (b x + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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