Optimal. Leaf size=116 \[ \frac{2 \sqrt{\pi } e^{-\frac{a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{b^{3/2} e n^{3/2}}-\frac{2 (d+e x)}{b e n \sqrt{a+b \log \left (c (d+e x)^n\right )}} \]
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Rubi [A] time = 0.101096, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {2389, 2297, 2300, 2180, 2204} \[ \frac{2 \sqrt{\pi } e^{-\frac{a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{b^{3/2} e n^{3/2}}-\frac{2 (d+e x)}{b e n \sqrt{a+b \log \left (c (d+e x)^n\right )}} \]
Antiderivative was successfully verified.
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Rule 2389
Rule 2297
Rule 2300
Rule 2180
Rule 2204
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (a+b \log \left (c x^n\right )\right )^{3/2}} \, dx,x,d+e x\right )}{e}\\ &=-\frac{2 (d+e x)}{b e n \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{b e n}\\ &=-\frac{2 (d+e x)}{b e n \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{\left (2 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{x}{n}}}{\sqrt{a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b e n^2}\\ &=-\frac{2 (d+e x)}{b e n \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{\left (4 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int e^{-\frac{a}{b n}+\frac{x^2}{b n}} \, dx,x,\sqrt{a+b \log \left (c (d+e x)^n\right )}\right )}{b^2 e n^2}\\ &=\frac{2 e^{-\frac{a}{b n}} \sqrt{\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text{erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{b^{3/2} e n^{3/2}}-\frac{2 (d+e x)}{b e n \sqrt{a+b \log \left (c (d+e x)^n\right )}}\\ \end{align*}
Mathematica [A] time = 0.116222, size = 139, normalized size = 1.2 \[ -\frac{2 e^{-\frac{a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \left (e^{\frac{a}{b n}} \left (c (d+e x)^n\right )^{\frac{1}{n}}-\sqrt{-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}} \text{Gamma}\left (\frac{1}{2},-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )\right )}{b e n \sqrt{a+b \log \left (c (d+e x)^n\right )}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.526, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{-{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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{1}{\left (a + b \log{\left (c \left (d + e x\right )^{n} \right )}\right )^{\frac{3}{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{1}{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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