Optimal. Leaf size=143 \[ \frac{3 \sqrt{\pi } b^{3/2} n^{3/2} e^{-\frac{a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{4 e}+\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{e}-\frac{3 b n (d+e x) \sqrt{a+b \log \left (c (d+e x)^n\right )}}{2 e} \]
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Rubi [A] time = 0.106902, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {2389, 2296, 2300, 2180, 2204} \[ \frac{3 \sqrt{\pi } b^{3/2} n^{3/2} e^{-\frac{a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{4 e}+\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{e}-\frac{3 b n (d+e x) \sqrt{a+b \log \left (c (d+e x)^n\right )}}{2 e} \]
Antiderivative was successfully verified.
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Rule 2389
Rule 2296
Rule 2300
Rule 2180
Rule 2204
Rubi steps
\begin{align*} \int \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^{3/2} \, dx,x,d+e x\right )}{e}\\ &=\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{e}-\frac{(3 b n) \operatorname{Subst}\left (\int \sqrt{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{2 e}\\ &=-\frac{3 b n (d+e x) \sqrt{a+b \log \left (c (d+e x)^n\right )}}{2 e}+\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{e}+\frac{\left (3 b^2 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{4 e}\\ &=-\frac{3 b n (d+e x) \sqrt{a+b \log \left (c (d+e x)^n\right )}}{2 e}+\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{e}+\frac{\left (3 b^2 n (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 )}{4 e}\\ &=-\frac{3 b n (d+e x) \sqrt{a+b \log \left (c (d+e x)^n\right )}}{2 e}+\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{e}+\frac{\left (3 b n (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 )}{2 e}\\ &=\frac{3 b^{3/2} e^{-\frac{a}{b n}} n^{3/2} \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 )}{4 e}-\frac{3 b n (d+e x) \sqrt{a+b \log \left (c (d+e x)^n\right )}}{2 e}+\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{e}\\ \end{align*}
Mathematica [A] time = 0.0669751, size = 127, normalized size = 0.89 \[ \frac{(d+e x) \left (3 \sqrt{\pi } b^{3/2} n^{3/2} e^{-\frac{a}{b n}} \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )+2 \sqrt{a+b \log \left (c (d+e x)^n\right )} \left (2 a+2 b \log \left (c (d+e x)^n\right )-3 b n\right )\right )}{4 e} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.507, 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{\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 \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{\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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