Optimal. Leaf size=91 \[ -\frac{d^2 \left (a+b \log \left (c x^n\right )\right )}{2 x^2}+2 d e \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac{1}{2} e^2 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{b d^2 n}{4 x^2}-b d e n \log ^2(x)-\frac{1}{4} b e^2 n x^2 \]
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Rubi [A] time = 0.0986328, antiderivative size = 71, normalized size of antiderivative = 0.78, number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {266, 43, 2334, 12, 14, 2301} \[ -\frac{1}{2} \left (\frac{d^2}{x^2}-4 d e \log (x)-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{b d^2 n}{4 x^2}-b d e n \log ^2(x)-\frac{1}{4} b e^2 n x^2 \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rule 2334
Rule 12
Rule 14
Rule 2301
Rubi steps
\begin{align*} \int \frac{\left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx &=-\frac{1}{2} \left (\frac{d^2}{x^2}-e^2 x^2-4 d e \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{-d^2+e^2 x^4+4 d e x^2 \log (x)}{2 x^3} \, dx\\ &=-\frac{1}{2} \left (\frac{d^2}{x^2}-e^2 x^2-4 d e \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} (b n) \int \frac{-d^2+e^2 x^4+4 d e x^2 \log (x)}{x^3} \, dx\\ &=-\frac{1}{2} \left (\frac{d^2}{x^2}-e^2 x^2-4 d e \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} (b n) \int \left (\frac{-d^2+e^2 x^4}{x^3}+\frac{4 d e \log (x)}{x}\right ) \, dx\\ &=-\frac{1}{2} \left (\frac{d^2}{x^2}-e^2 x^2-4 d e \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} (b n) \int \frac{-d^2+e^2 x^4}{x^3} \, dx-(2 b d e n) \int \frac{\log (x)}{x} \, dx\\ &=-b d e n \log ^2(x)-\frac{1}{2} \left (\frac{d^2}{x^2}-e^2 x^2-4 d e \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} (b n) \int \left (-\frac{d^2}{x^3}+e^2 x\right ) \, dx\\ &=-\frac{b d^2 n}{4 x^2}-\frac{1}{4} b e^2 n x^2-b d e n \log ^2(x)-\frac{1}{2} \left (\frac{d^2}{x^2}-e^2 x^2-4 d e \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}
Mathematica [A] time = 0.0584254, size = 83, normalized size = 0.91 \[ \frac{1}{4} \left (-\frac{2 d^2 \left (a+b \log \left (c x^n\right )\right )}{x^2}+\frac{4 d e \left (a+b \log \left (c x^n\right )\right )^2}{b n}+2 e^2 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{b d^2 n}{x^2}-b e^2 n x^2\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.237, size = 433, normalized size = 4.8 \begin{align*} -{\frac{b \left ( -{e}^{2}{x}^{4}-4\,de\ln \left ( x \right ){x}^{2}+{d}^{2} \right ) \ln \left ({x}^{n} \right ) }{2\,{x}^{2}}}-{\frac{-4\,i\ln \left ( x \right ) \pi \,bde{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{x}^{2}+i\pi \,b{d}^{2}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-i\pi \,b{e}^{2}{x}^{4} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +i\pi \,b{e}^{2}{x}^{4} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}-i\pi \,b{d}^{2}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) +i\pi \,b{d}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +4\,i\ln \left ( x \right ) \pi \,bde \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}{x}^{2}+i\pi \,b{e}^{2}{x}^{4}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -2\,\ln \left ( c \right ) b{e}^{2}{x}^{4}+4\,i\ln \left ( x \right ) \pi \,bde{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ){x}^{2}-i\pi \,b{d}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}-4\,i\ln \left ( x \right ) \pi \,bde \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ){x}^{2}-i\pi \,b{e}^{2}{x}^{4}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+4\,bden \left ( \ln \left ( x \right ) \right ) ^{2}{x}^{2}+b{e}^{2}n{x}^{4}-8\,\ln \left ( x \right ) \ln \left ( c \right ) bde{x}^{2}-2\,a{e}^{2}{x}^{4}-8\,\ln \left ( x \right ) ade{x}^{2}+2\,\ln \left ( c \right ) b{d}^{2}+b{d}^{2}n+2\,a{d}^{2}}{4\,{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0803, size = 123, normalized size = 1.35 \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 ) + \frac{1}{2} \, a e^{2} x^{2} + \frac{b d e \log \left (c x^{n}\right )^{2}}{n} + 2 \, a d e \log \left (x\right ) - \frac{b d^{2} n}{4 \, x^{2}} - \frac{b d^{2} \log \left (c x^{n}\right )}{2 \, x^{2}} - \frac{a d^{2}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.234, size = 244, normalized size = 2.68 \begin{align*} \frac{4 \, b d e n x^{2} \log \left (x\right )^{2} -{\left (b e^{2} n - 2 \, a e^{2}\right )} x^{4} - b d^{2} n - 2 \, a d^{2} + 2 \,{\left (b e^{2} x^{4} - b d^{2}\right )} \log \left (c\right ) + 2 \,{\left (b e^{2} n x^{4} + 4 \, b d e x^{2} \log \left (c\right ) + 4 \, a d e x^{2} - b d^{2} n\right )} \log \left (x\right )}{4 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.54678, size = 136, normalized size = 1.49 \begin{align*} - \frac{a d^{2}}{2 x^{2}} + 2 a d e \log{\left (x \right )} + \frac{a e^{2} x^{2}}{2} - \frac{b d^{2} n \log{\left (x \right )}}{2 x^{2}} - \frac{b d^{2} n}{4 x^{2}} - \frac{b d^{2} \log{\left (c \right )}}{2 x^{2}} + b d e n \log{\left (x \right )}^{2} + 2 b d e \log{\left (c \right )} \log{\left (x \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 = 1.29697, size = 151, normalized size = 1.66 \begin{align*} \frac{2 \, b n x^{4} e^{2} \log \left (x\right ) + 4 \, b d n x^{2} e \log \left (x\right )^{2} - b n x^{4} e^{2} + 2 \, b x^{4} e^{2} \log \left (c\right ) + 8 \, b d x^{2} e \log \left (c\right ) \log \left (x\right ) + 2 \, a x^{4} e^{2} + 8 \, a d x^{2} e \log \left (x\right ) - 2 \, b d^{2} n \log \left (x\right ) - b d^{2} n - 2 \, b d^{2} \log \left (c\right ) - 2 \, a d^{2}}{4 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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