Optimal. Leaf size=220 \[ \frac{2 \sqrt{\pi } e^{-\frac{a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{b^{3/2} e^2 n^{3/2}}+\frac{2 \sqrt{2 \pi } g e^{-\frac{2 a}{b n}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{b^{3/2} e^2 n^{3/2}}-\frac{2 (d+e x) (f+g x)}{b e n \sqrt{a+b \log \left (c (d+e x)^n\right )}} \]
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Rubi [A] time = 0.402761, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2400, 2401, 2389, 2300, 2180, 2204, 2390, 2310} \[ \frac{2 \sqrt{\pi } e^{-\frac{a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{b^{3/2} e^2 n^{3/2}}+\frac{2 \sqrt{2 \pi } g e^{-\frac{2 a}{b n}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{b^{3/2} e^2 n^{3/2}}-\frac{2 (d+e x) (f+g x)}{b e n \sqrt{a+b \log \left (c (d+e x)^n\right )}} \]
Antiderivative was successfully verified.
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Rule 2400
Rule 2401
Rule 2389
Rule 2300
Rule 2180
Rule 2204
Rule 2390
Rule 2310
Rubi steps
\begin{align*} \int \frac{f+g x}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx &=-\frac{2 (d+e x) (f+g x)}{b e n \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{4 \int \frac{f+g x}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{b n}-\frac{(2 (e f-d g)) \int \frac{1}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{b e n}\\ &=-\frac{2 (d+e x) (f+g x)}{b e n \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{4 \int \left (\frac{e f-d g}{e \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{g (d+e x)}{e \sqrt{a+b \log \left (c (d+e x)^n\right )}}\right ) \, dx}{b n}-\frac{(2 (e f-d g)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{b e^2 n}\\ &=-\frac{2 (d+e x) (f+g x)}{b e n \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{(4 g) \int \frac{d+e x}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{b e n}+\frac{(4 (e f-d g)) \int \frac{1}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{b e n}-\frac{\left (2 (e f-d g) (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^2 n^2}\\ &=-\frac{2 (d+e x) (f+g x)}{b e n \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{(4 g) \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{b e^2 n}+\frac{(4 (e f-d g)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{b e^2 n}-\frac{\left (4 (e f-d g) (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^2 n^2}\\ &=-\frac{2 e^{-\frac{a}{b n}} (e f-d g) \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^2 n^{3/2}}-\frac{2 (d+e x) (f+g x)}{b e n \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{\left (4 g (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{2 x}{n}}}{\sqrt{a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b e^2 n^2}+\frac{\left (4 (e f-d g) (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^2 n^2}\\ &=-\frac{2 e^{-\frac{a}{b n}} (e f-d g) \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^2 n^{3/2}}-\frac{2 (d+e x) (f+g x)}{b e n \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{\left (8 g (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int e^{-\frac{2 a}{b n}+\frac{2 x^2}{b n}} \, dx,x,\sqrt{a+b \log \left (c (d+e x)^n\right )}\right )}{b^2 e^2 n^2}+\frac{\left (8 (e f-d g) (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^2 n^2}\\ &=\frac{2 e^{-\frac{a}{b n}} (e f-d g) \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^2 n^{3/2}}+\frac{2 e^{-\frac{2 a}{b n}} g \sqrt{2 \pi } (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text{erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{b^{3/2} e^2 n^{3/2}}-\frac{2 (d+e x) (f+g x)}{b e n \sqrt{a+b \log \left (c (d+e x)^n\right )}}\\ \end{align*}
Mathematica [A] time = 0.807175, size = 338, normalized size = 1.54 \[ \frac{2 e^{-\frac{2 a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-2/n} \left (\sqrt{b} \sqrt{n} e^{\frac{a}{b n}} \left (c (d+e x)^n\right )^{\frac{1}{n}} \left ((d g+e f) \sqrt{-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}} \text{Gamma}\left (\frac{1}{2},-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )-e e^{\frac{a}{b n}} (f+g x) \left (c (d+e x)^n\right )^{\frac{1}{n}}\right )-2 \sqrt{\pi } d g e^{\frac{a}{b n}} \left (c (d+e x)^n\right )^{\frac{1}{n}} \sqrt{a+b \log \left (c (d+e x)^n\right )} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )+\sqrt{2 \pi } g (d+e x) \sqrt{a+b \log \left (c (d+e x)^n\right )} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )\right )}{b^{3/2} e^2 n^{3/2} \sqrt{a+b \log \left (c (d+e x)^n\right )}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.466, size = 0, normalized size = 0. \begin{align*} \int{(gx+f) \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{g x + f}{{\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{f + g x}{\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{g x + f}{{\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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