Optimal. Leaf size=80 \[ d^2 \log (x) \left (a+b \log \left (c x^n\right )\right )+2 d e x \left (a+b \log \left (c x^n\right )\right )+\frac{1}{2} e^2 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} b d^2 n \log ^2(x)-\frac{1}{4} b n (4 d+e x)^2 \]
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Rubi [A] time = 0.0714581, antiderivative size = 63, normalized size of antiderivative = 0.79, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {43, 2334, 2301} \[ \frac{1}{2} \left (2 d^2 \log (x)+4 d e x+e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} b d^2 n \log ^2(x)-\frac{1}{4} b n (4 d+e x)^2 \]
Antiderivative was successfully verified.
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Rule 43
Rule 2334
Rule 2301
Rubi steps
\begin{align*} \int \frac{(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{1}{2} \left (4 d e x+e^2 x^2+2 d^2 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (\frac{1}{2} e (4 d+e x)+\frac{d^2 \log (x)}{x}\right ) \, dx\\ &=-\frac{1}{4} b n (4 d+e x)^2+\frac{1}{2} \left (4 d e x+e^2 x^2+2 d^2 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-\left (b d^2 n\right ) \int \frac{\log (x)}{x} \, dx\\ &=-\frac{1}{4} b n (4 d+e x)^2-\frac{1}{2} b d^2 n \log ^2(x)+\frac{1}{2} \left (4 d e x+e^2 x^2+2 d^2 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}
Mathematica [A] time = 0.0457333, size = 83, normalized size = 1.04 \[ \frac{d^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 b n}+\frac{1}{2} e^2 x^2 \left (a+b \log \left (c x^n\right )\right )+2 a d e x+2 b d e x \log \left (c x^n\right )-2 b d e n x-\frac{1}{4} b e^2 n x^2 \]
Antiderivative was successfully verified.
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Maple [C] time = 0.239, size = 410, normalized size = 5.1 \begin{align*} \left ({\frac{b{e}^{2}{x}^{2}}{2}}+2\,bdex+b{d}^{2}\ln \left ( x \right ) \right ) \ln \left ({x}^{n} \right ) -{\frac{b{d}^{2}n \left ( \ln \left ( x \right ) \right ) ^{2}}{2}}-{\frac{i}{4}}\pi \,b{e}^{2}{x}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}-{\frac{i}{2}}\ln \left ( x \right ) \pi \,b{d}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}-i\pi \,bdex \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}-{\frac{i}{2}}\ln \left ( x \right ) \pi \,b{d}^{2}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) +{\frac{i}{2}}\ln \left ( x \right ) \pi \,b{d}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +i\pi \,bdex{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-i\pi \,bdex{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) +i\pi \,bdex \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +{\frac{\ln \left ( c \right ) b{e}^{2}{x}^{2}}{2}}-{\frac{b{e}^{2}n{x}^{2}}{4}}+2\,\ln \left ( c \right ) bdex+{\frac{a{e}^{2}{x}^{2}}{2}}-2\,bdenx+2\,adex+{\frac{i}{2}}\ln \left ( x \right ) \pi \,b{d}^{2}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-{\frac{i}{4}}\pi \,b{e}^{2}{x}^{2}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) +{\frac{i}{4}}\pi \,b{e}^{2}{x}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +{\frac{i}{4}}\pi \,b{e}^{2}{x}^{2}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+\ln \left ( x \right ) \ln \left ( c \right ) b{d}^{2}+\ln \left ( x \right ) a{d}^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07129, size = 113, normalized size = 1.41 \begin{align*} -\frac{1}{4} \, b e^{2} n x^{2} + \frac{1}{2} \, b e^{2} x^{2} \log \left (c x^{n}\right ) - 2 \, b d e n x + \frac{1}{2} \, a e^{2} x^{2} + 2 \, b d e x \log \left (c x^{n}\right ) + 2 \, a d e x + \frac{b d^{2} \log \left (c x^{n}\right )^{2}}{2 \, n} + a d^{2} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.994326, size = 244, normalized size = 3.05 \begin{align*} \frac{1}{2} \, b d^{2} n \log \left (x\right )^{2} - \frac{1}{4} \,{\left (b e^{2} n - 2 \, a e^{2}\right )} x^{2} - 2 \,{\left (b d e n - a d e\right )} x + \frac{1}{2} \,{\left (b e^{2} x^{2} + 4 \, b d e x\right )} \log \left (c\right ) + \frac{1}{2} \,{\left (b e^{2} n x^{2} + 4 \, b d e n x + 2 \, b d^{2} \log \left (c\right ) + 2 \, a d^{2}\right )} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.38067, size = 128, normalized size = 1.6 \begin{align*} a d^{2} \log{\left (x \right )} + 2 a d e x + \frac{a e^{2} x^{2}}{2} + \frac{b d^{2} n \log{\left (x \right )}^{2}}{2} + b d^{2} \log{\left (c \right )} \log{\left (x \right )} + 2 b d e n x \log{\left (x \right )} - 2 b d e n x + 2 b d e x \log{\left (c \right )} + \frac{b e^{2} n x^{2} \log{\left (x \right )}}{2} - \frac{b e^{2} n x^{2}}{4} + \frac{b e^{2} x^{2} \log{\left (c \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.32801, size = 135, normalized size = 1.69 \begin{align*} \frac{1}{2} \, b n x^{2} e^{2} \log \left (x\right ) + 2 \, b d n x e \log \left (x\right ) + \frac{1}{2} \, b d^{2} n \log \left (x\right )^{2} - \frac{1}{4} \, b n x^{2} e^{2} - 2 \, b d n x e + \frac{1}{2} \, b x^{2} e^{2} \log \left (c\right ) + 2 \, b d x e \log \left (c\right ) + b d^{2} \log \left (c\right ) \log \left (x\right ) + \frac{1}{2} \, a x^{2} e^{2} + 2 \, a d x e + a d^{2} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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