Optimal. Leaf size=149 \[ \frac{2 \sqrt{\pi } b^{3/2} e^{\frac{b c}{d}-a} \text{Erf}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{3 d^{5/2}}+\frac{2 \sqrt{\pi } b^{3/2} e^{a-\frac{b c}{d}} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{3 d^{5/2}}-\frac{4 b \sinh (a+b x)}{3 d^2 \sqrt{c+d x}}-\frac{2 \cosh (a+b x)}{3 d (c+d x)^{3/2}} \]
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Rubi [A] time = 0.248108, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {3297, 3307, 2180, 2204, 2205} \[ \frac{2 \sqrt{\pi } b^{3/2} e^{\frac{b c}{d}-a} \text{Erf}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{3 d^{5/2}}+\frac{2 \sqrt{\pi } b^{3/2} e^{a-\frac{b c}{d}} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{3 d^{5/2}}-\frac{4 b \sinh (a+b x)}{3 d^2 \sqrt{c+d x}}-\frac{2 \cosh (a+b x)}{3 d (c+d x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3307
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int \frac{\cosh (a+b x)}{(c+d x)^{5/2}} \, dx &=-\frac{2 \cosh (a+b x)}{3 d (c+d x)^{3/2}}+\frac{(2 b) \int \frac{\sinh (a+b x)}{(c+d x)^{3/2}} \, dx}{3 d}\\ &=-\frac{2 \cosh (a+b x)}{3 d (c+d x)^{3/2}}-\frac{4 b \sinh (a+b x)}{3 d^2 \sqrt{c+d x}}+\frac{\left (4 b^2\right ) \int \frac{\cosh (a+b x)}{\sqrt{c+d x}} \, dx}{3 d^2}\\ &=-\frac{2 \cosh (a+b x)}{3 d (c+d x)^{3/2}}-\frac{4 b \sinh (a+b x)}{3 d^2 \sqrt{c+d x}}+\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{2 \cosh (a+b x)}{3 d (c+d x)^{3/2}}-\frac{4 b \sinh (a+b x)}{3 d^2 \sqrt{c+d x}}+\frac{\left (4 b^2\right ) \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 )}{3 d^3}+\frac{\left (4 b^2\right ) \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 )}{3 d^3}\\ &=-\frac{2 \cosh (a+b x)}{3 d (c+d x)^{3/2}}+\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{4 b \sinh (a+b x)}{3 d^2 \sqrt{c+d x}}\\ \end{align*}
Mathematica [A] time = 0.708778, size = 150, normalized size = 1.01 \[ \frac{e^{-a} \left (-2 d e^{2 a-\frac{b c}{d}} \left (-\frac{b (c+d x)}{d}\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-\frac{b (c+d x)}{d}\right )-e^{-b x} \left (2 d e^{b \left (\frac{c}{d}+x\right )} \left (\frac{b (c+d x)}{d}\right )^{3/2} \text{Gamma}\left (\frac{1}{2},b \left (\frac{c}{d}+x\right )\right )+2 b \left (e^{2 (a+b x)}-1\right ) (c+d x)+d \left (e^{2 (a+b x)}+1\right )\right )\right )}{3 d^2 (c+d x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.043, size = 0, normalized size = 0. \begin{align*} \int{\cosh \left ( bx+a \right ) \left ( dx+c \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.20593, size = 155, normalized size = 1.04 \begin{align*} \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 \, \cosh \left (b x + a\right )}{{\left (d x + c\right )}^{\frac{3}{2}}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.15731, size = 1224, normalized size = 8.21 \begin{align*} \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 )}} \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 \frac{\cosh{\left (a + b x \right )}}{\left (c + d x\right )^{\frac{5}{2}}}\, 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 \frac{\cosh \left (b x + a\right )}{{\left (d x + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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