Optimal. Leaf size=84 \[ \frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )-\frac{1}{8} e r x^2 \left (2 a+2 b \log \left (c x^n\right )-b n\right )-\frac{1}{4} b n x^2 \left (d+e \log \left (f x^r\right )\right )+\frac{1}{8} b e n r x^2 \]
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Rubi [A] time = 0.0518652, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {2304, 2366, 12} \[ \frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )-\frac{1}{8} e r x^2 \left (2 a+2 b \log \left (c x^n\right )-b n\right )-\frac{1}{4} b n x^2 \left (d+e \log \left (f x^r\right )\right )+\frac{1}{8} b e n r x^2 \]
Antiderivative was successfully verified.
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Rule 2304
Rule 2366
Rule 12
Rubi steps
\begin{align*} \int x \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right ) \, dx &=-\frac{1}{4} b n x^2 \left (d+e \log \left (f x^r\right )\right )+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )-(e r) \int \frac{1}{4} x \left (2 a \left (1-\frac{b n}{2 a}\right )+2 b \log \left (c x^n\right )\right ) \, dx\\ &=-\frac{1}{4} b n x^2 \left (d+e \log \left (f x^r\right )\right )+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )-\frac{1}{4} (e r) \int x \left (2 a \left (1-\frac{b n}{2 a}\right )+2 b \log \left (c x^n\right )\right ) \, dx\\ &=\frac{1}{8} b e n r x^2-\frac{1}{8} e r x^2 \left (2 a-b n+2 b \log \left (c x^n\right )\right )-\frac{1}{4} b n x^2 \left (d+e \log \left (f x^r\right )\right )+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )\\ \end{align*}
Mathematica [A] time = 0.0622799, size = 68, normalized size = 0.81 \[ \frac{1}{4} x^2 \left (e (2 a-b n) \log \left (f x^r\right )+2 a d-a e r+b \log \left (c x^n\right ) \left (2 d+2 e \log \left (f x^r\right )-e r\right )-b d n+b e n r\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.18, size = 1640, normalized size = 19.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.18, size = 138, normalized size = 1.64 \begin{align*} -\frac{1}{4} \, b d n x^{2} - \frac{1}{4} \, a e r x^{2} + \frac{1}{2} \, b d x^{2} \log \left (c x^{n}\right ) + \frac{1}{2} \, a e x^{2} \log \left (f x^{r}\right ) + \frac{1}{4} \,{\left ({\left (r - \log \left (f\right )\right )} x^{2} - x^{2} \log \left (x^{r}\right )\right )} b e n + \frac{1}{2} \, a d x^{2} - \frac{1}{4} \,{\left (r x^{2} - 2 \, x^{2} \log \left (f x^{r}\right )\right )} b e \log \left (c x^{n}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.801929, size = 324, normalized size = 3.86 \begin{align*} \frac{1}{2} \, b e n r x^{2} \log \left (x\right )^{2} - \frac{1}{4} \,{\left (b e r - 2 \, b d\right )} x^{2} \log \left (c\right ) - \frac{1}{4} \,{\left (b d n - 2 \, a d -{\left (b e n - a e\right )} r\right )} x^{2} + \frac{1}{4} \,{\left (2 \, b e x^{2} \log \left (c\right ) -{\left (b e n - 2 \, a e\right )} x^{2}\right )} \log \left (f\right ) + \frac{1}{2} \,{\left (b e r x^{2} \log \left (c\right ) + b e n x^{2} \log \left (f\right ) +{\left (b d n -{\left (b e n - a e\right )} r\right )} x^{2}\right )} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 12.2933, size = 199, normalized size = 2.37 \begin{align*} \frac{a d x^{2}}{2} + \frac{a e r x^{2} \log{\left (x \right )}}{2} - \frac{a e r x^{2}}{4} + \frac{a e x^{2} \log{\left (f \right )}}{2} + \frac{b d n x^{2} \log{\left (x \right )}}{2} - \frac{b d n x^{2}}{4} + \frac{b d x^{2} \log{\left (c \right )}}{2} + \frac{b e n r x^{2} \log{\left (x \right )}^{2}}{2} - \frac{b e n r x^{2} \log{\left (x \right )}}{2} + \frac{b e n r x^{2}}{4} + \frac{b e n x^{2} \log{\left (f \right )} \log{\left (x \right )}}{2} - \frac{b e n x^{2} \log{\left (f \right )}}{4} + \frac{b e r x^{2} \log{\left (c \right )} \log{\left (x \right )}}{2} - \frac{b e r x^{2} \log{\left (c \right )}}{4} + \frac{b e x^{2} \log{\left (c \right )} \log{\left (f \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.40135, size = 217, normalized size = 2.58 \begin{align*} \frac{1}{2} \, b n r x^{2} e \log \left (x\right )^{2} - \frac{1}{2} \, b n r x^{2} e \log \left (x\right ) + \frac{1}{2} \, b r x^{2} e \log \left (c\right ) \log \left (x\right ) + \frac{1}{2} \, b n x^{2} e \log \left (f\right ) \log \left (x\right ) + \frac{1}{4} \, b n r x^{2} e - \frac{1}{4} \, b r x^{2} e \log \left (c\right ) - \frac{1}{4} \, b n x^{2} e \log \left (f\right ) + \frac{1}{2} \, b x^{2} e \log \left (c\right ) \log \left (f\right ) + \frac{1}{2} \, b d n x^{2} \log \left (x\right ) + \frac{1}{2} \, a r x^{2} e \log \left (x\right ) - \frac{1}{4} \, b d n x^{2} - \frac{1}{4} \, a r x^{2} e + \frac{1}{2} \, b d x^{2} \log \left (c\right ) + \frac{1}{2} \, a x^{2} e \log \left (f\right ) + \frac{1}{2} \, a d x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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