Optimal. Leaf size=91 \[ \frac{(f x+g)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f}-\frac{b n (d f-e g)^2 \log (d+e x)}{2 e^2 f}+\frac{b n x (d f-e g)}{2 e}-\frac{b n (f x+g)^2}{4 f} \]
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Rubi [A] time = 0.0639724, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {2412, 2395, 43} \[ \frac{(f x+g)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f}-\frac{b n (d f-e g)^2 \log (d+e x)}{2 e^2 f}+\frac{b n x (d f-e g)}{2 e}-\frac{b n (f x+g)^2}{4 f} \]
Antiderivative was successfully verified.
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Rule 2412
Rule 2395
Rule 43
Rubi steps
\begin{align*} \int \left (f+\frac{g}{x}\right ) x \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx &=\int (g+f x) \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx\\ &=\frac{(g+f x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f}-\frac{(b e n) \int \frac{(g+f x)^2}{d+e x} \, dx}{2 f}\\ &=\frac{(g+f x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f}-\frac{(b e n) \int \left (\frac{f (-d f+e g)}{e^2}+\frac{(-d f+e g)^2}{e^2 (d+e x)}+\frac{f (g+f x)}{e}\right ) \, dx}{2 f}\\ &=\frac{b (d f-e g) n x}{2 e}-\frac{b n (g+f x)^2}{4 f}-\frac{b (d f-e g)^2 n \log (d+e x)}{2 e^2 f}+\frac{(g+f x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f}\\ \end{align*}
Mathematica [A] time = 0.0555758, size = 101, normalized size = 1.11 \[ \frac{1}{2} a f x^2+a g x+\frac{1}{2} b f x^2 \log \left (c (d+e x)^n\right )+\frac{b g (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac{b d^2 f n \log (d+e x)}{2 e^2}+\frac{b d f n x}{2 e}-\frac{1}{4} b f n x^2-b g n x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.086, size = 101, normalized size = 1.1 \begin{align*} agx+{\frac{af{x}^{2}}{2}}+bg\ln \left ( c \left ( ex+d \right ) ^{n} \right ) x-bgnx+{\frac{bdgn\ln \left ( ex+d \right ) }{e}}+{\frac{bf{x}^{2}\ln \left ( c{{\rm e}^{n\ln \left ( ex+d \right ) }} \right ) }{2}}-{\frac{nbf{x}^{2}}{4}}-{\frac{{d}^{2}bfn\ln \left ( ex+d \right ) }{2\,{e}^{2}}}+{\frac{bdfnx}{2\,e}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05571, size = 138, normalized size = 1.52 \begin{align*} -b e g n{\left (\frac{x}{e} - \frac{d \log \left (e x + d\right )}{e^{2}}\right )} - \frac{1}{4} \, b e f n{\left (\frac{2 \, d^{2} \log \left (e x + d\right )}{e^{3}} + \frac{e x^{2} - 2 \, d x}{e^{2}}\right )} + \frac{1}{2} \, b f x^{2} \log \left ({\left (e x + d\right )}^{n} c\right ) + \frac{1}{2} \, a f x^{2} + b g x \log \left ({\left (e x + d\right )}^{n} c\right ) + a g x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.95898, size = 267, normalized size = 2.93 \begin{align*} -\frac{{\left (b e^{2} f n - 2 \, a e^{2} f\right )} x^{2} - 2 \,{\left (2 \, a e^{2} g +{\left (b d e f - 2 \, b e^{2} g\right )} n\right )} x - 2 \,{\left (b e^{2} f n x^{2} + 2 \, b e^{2} g n x -{\left (b d^{2} f - 2 \, b d e g\right )} n\right )} \log \left (e x + d\right ) - 2 \,{\left (b e^{2} f x^{2} + 2 \, b e^{2} g x\right )} \log \left (c\right )}{4 \, e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.91779, size = 148, normalized size = 1.63 \begin{align*} \begin{cases} \frac{a f x^{2}}{2} + a g x - \frac{b d^{2} f n \log{\left (d + e x \right )}}{2 e^{2}} + \frac{b d f n x}{2 e} + \frac{b d g n \log{\left (d + e x \right )}}{e} + \frac{b f n x^{2} \log{\left (d + e x \right )}}{2} - \frac{b f n x^{2}}{4} + \frac{b f x^{2} \log{\left (c \right )}}{2} + b g n x \log{\left (d + e x \right )} - b g n x + b g x \log{\left (c \right )} & \text{for}\: e \neq 0 \\\left (a + b \log{\left (c d^{n} \right )}\right ) \left (\frac{f x^{2}}{2} + g x\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.2317, size = 251, normalized size = 2.76 \begin{align*} \frac{1}{2} \,{\left (x e + d\right )}^{2} b f n e^{\left (-2\right )} \log \left (x e + d\right ) -{\left (x e + d\right )} b d f n e^{\left (-2\right )} \log \left (x e + d\right ) - \frac{1}{4} \,{\left (x e + d\right )}^{2} b f n e^{\left (-2\right )} +{\left (x e + d\right )} b d f n e^{\left (-2\right )} +{\left (x e + d\right )} b g n e^{\left (-1\right )} \log \left (x e + d\right ) + \frac{1}{2} \,{\left (x e + d\right )}^{2} b f e^{\left (-2\right )} \log \left (c\right ) -{\left (x e + d\right )} b d f e^{\left (-2\right )} \log \left (c\right ) -{\left (x e + d\right )} b g n e^{\left (-1\right )} + \frac{1}{2} \,{\left (x e + d\right )}^{2} a f e^{\left (-2\right )} -{\left (x e + d\right )} a d f e^{\left (-2\right )} +{\left (x e + d\right )} b g e^{\left (-1\right )} \log \left (c\right ) +{\left (x e + d\right )} a g e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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