Optimal. Leaf size=65 \[ \frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-2 a b n x-\frac{2 b^2 n (d+e x) \log \left (c (d+e x)^n\right )}{e}+2 b^2 n^2 x \]
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Rubi [A] time = 0.0347364, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {2389, 2296, 2295} \[ \frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-2 a b n x-\frac{2 b^2 n (d+e x) \log \left (c (d+e x)^n\right )}{e}+2 b^2 n^2 x \]
Antiderivative was successfully verified.
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Rule 2389
Rule 2296
Rule 2295
Rubi steps
\begin{align*} \int \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e}\\ &=\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{(2 b n) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e}\\ &=-2 a b n x+\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{\left (2 b^2 n\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e}\\ &=-2 a b n x+2 b^2 n^2 x-\frac{2 b^2 n (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}\\ \end{align*}
Mathematica [A] time = 0.0085129, size = 59, normalized size = 0.91 \[ \frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-2 b n \left (a x+\frac{b (d+e x) \log \left (c (d+e x)^n\right )}{e}-b n x\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.078, size = 130, normalized size = 2. \begin{align*} x{a}^{2}+{b}^{2}x \left ( \ln \left ( c{{\rm e}^{n\ln \left ( ex+d \right ) }} \right ) \right ) ^{2}+{\frac{{b}^{2}d \left ( \ln \left ( c{{\rm e}^{n\ln \left ( ex+d \right ) }} \right ) \right ) ^{2}}{e}}+2\,{b}^{2}{n}^{2}x-2\,{b}^{2}nx\ln \left ( c{{\rm e}^{n\ln \left ( ex+d \right ) }} \right ) -2\,{\frac{\ln \left ( ex+d \right ){b}^{2}d{n}^{2}}{e}}+2\,ab\ln \left ( c \left ( ex+d \right ) ^{n} \right ) x-2\,abnx+2\,{\frac{\ln \left ( ex+d \right ) abdn}{e}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.29862, size = 177, normalized size = 2.72 \begin{align*} -2 \, a b e n{\left (\frac{x}{e} - \frac{d \log \left (e x + d\right )}{e^{2}}\right )} + b^{2} x \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 2 \, a b x \log \left ({\left (e x + d\right )}^{n} c\right ) -{\left (2 \, e n{\left (\frac{x}{e} - \frac{d \log \left (e x + d\right )}{e^{2}}\right )} \log \left ({\left (e x + d\right )}^{n} c\right ) + \frac{{\left (d \log \left (e x + d\right )^{2} - 2 \, e x + 2 \, d \log \left (e x + d\right )\right )} n^{2}}{e}\right )} b^{2} + a^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.97066, size = 311, normalized size = 4.78 \begin{align*} \frac{b^{2} e x \log \left (c\right )^{2} +{\left (b^{2} e n^{2} x + b^{2} d n^{2}\right )} \log \left (e x + d\right )^{2} - 2 \,{\left (b^{2} e n - a b e\right )} x \log \left (c\right ) +{\left (2 \, b^{2} e n^{2} - 2 \, a b e n + a^{2} e\right )} x - 2 \,{\left (b^{2} d n^{2} - a b d n +{\left (b^{2} e n^{2} - a b e n\right )} x -{\left (b^{2} e n x + b^{2} d n\right )} \log \left (c\right )\right )} \log \left (e x + d\right )}{e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.50867, size = 211, normalized size = 3.25 \begin{align*} \begin{cases} a^{2} x + \frac{2 a b d n \log{\left (d + e x \right )}}{e} + 2 a b n x \log{\left (d + e x \right )} - 2 a b n x + 2 a b x \log{\left (c \right )} + \frac{b^{2} d n^{2} \log{\left (d + e x \right )}^{2}}{e} - \frac{2 b^{2} d n^{2} \log{\left (d + e x \right )}}{e} + \frac{2 b^{2} d n \log{\left (c \right )} \log{\left (d + e x \right )}}{e} + b^{2} n^{2} x \log{\left (d + e x \right )}^{2} - 2 b^{2} n^{2} x \log{\left (d + e x \right )} + 2 b^{2} n^{2} x + 2 b^{2} n x \log{\left (c \right )} \log{\left (d + e x \right )} - 2 b^{2} n x \log{\left (c \right )} + b^{2} x \log{\left (c \right )}^{2} & \text{for}\: e \neq 0 \\x \left (a + b \log{\left (c d^{n} \right )}\right )^{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.30928, size = 240, normalized size = 3.69 \begin{align*}{\left (x e + d\right )} b^{2} n^{2} e^{\left (-1\right )} \log \left (x e + d\right )^{2} - 2 \,{\left (x e + d\right )} b^{2} n^{2} e^{\left (-1\right )} \log \left (x e + d\right ) + 2 \,{\left (x e + d\right )} b^{2} n e^{\left (-1\right )} \log \left (x e + d\right ) \log \left (c\right ) + 2 \,{\left (x e + d\right )} b^{2} n^{2} e^{\left (-1\right )} + 2 \,{\left (x e + d\right )} a b n e^{\left (-1\right )} \log \left (x e + d\right ) - 2 \,{\left (x e + d\right )} b^{2} n e^{\left (-1\right )} \log \left (c\right ) +{\left (x e + d\right )} b^{2} e^{\left (-1\right )} \log \left (c\right )^{2} - 2 \,{\left (x e + d\right )} a b n e^{\left (-1\right )} + 2 \,{\left (x e + d\right )} a b e^{\left (-1\right )} \log \left (c\right ) +{\left (x e + d\right )} a^{2} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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