Optimal. Leaf size=84 \[ \frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )-\frac{1}{27} e r x^3 \left (3 a+3 b \log \left (c x^n\right )-b n\right )-\frac{1}{9} b n x^3 \left (d+e \log \left (f x^r\right )\right )+\frac{1}{27} b e n r x^3 \]
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Rubi [A] time = 0.0748924, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2304, 2366, 12} \[ \frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )-\frac{1}{27} e r x^3 \left (3 a+3 b \log \left (c x^n\right )-b n\right )-\frac{1}{9} b n x^3 \left (d+e \log \left (f x^r\right )\right )+\frac{1}{27} b e n r x^3 \]
Antiderivative was successfully verified.
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Rule 2304
Rule 2366
Rule 12
Rubi steps
\begin{align*} \int x^2 \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right ) \, dx &=-\frac{1}{9} b n x^3 \left (d+e \log \left (f x^r\right )\right )+\frac{1}{3} x^3 \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}{9} x^2 \left (3 a \left (1-\frac{b n}{3 a}\right )+3 b \log \left (c x^n\right )\right ) \, dx\\ &=-\frac{1}{9} b n x^3 \left (d+e \log \left (f x^r\right )\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )-\frac{1}{9} (e r) \int x^2 \left (3 a \left (1-\frac{b n}{3 a}\right )+3 b \log \left (c x^n\right )\right ) \, dx\\ &=\frac{1}{27} b e n r x^3-\frac{1}{27} e r x^3 \left (3 a-b n+3 b \log \left (c x^n\right )\right )-\frac{1}{9} b n x^3 \left (d+e \log \left (f x^r\right )\right )+\frac{1}{3} x^3 \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 = 0.0717404, size = 71, normalized size = 0.85 \[ \frac{1}{27} x^3 \left ((9 a e-3 b e n) \log \left (f x^r\right )+9 a d-3 a e r+3 b \log \left (c x^n\right ) \left (3 d+3 e \log \left (f x^r\right )-e r\right )-3 b d n+2 b e n r\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.188, 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 = 1.19134, size = 140, normalized size = 1.67 \begin{align*} -\frac{1}{9} \, b d n x^{3} - \frac{1}{9} \, a e r x^{3} + \frac{1}{3} \, b d x^{3} \log \left (c x^{n}\right ) + \frac{1}{3} \, a e x^{3} \log \left (f x^{r}\right ) + \frac{1}{3} \, a d x^{3} + \frac{1}{27} \,{\left ({\left (2 \, r - 3 \, \log \left (f\right )\right )} x^{3} - 3 \, x^{3} \log \left (x^{r}\right )\right )} b e n - \frac{1}{9} \,{\left (r x^{3} - 3 \, x^{3} \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 = 0.872477, size = 347, normalized size = 4.13 \begin{align*} \frac{1}{3} \, b e n r x^{3} \log \left (x\right )^{2} - \frac{1}{9} \,{\left (b e r - 3 \, b d\right )} x^{3} \log \left (c\right ) - \frac{1}{27} \,{\left (3 \, b d n - 9 \, a d -{\left (2 \, b e n - 3 \, a e\right )} r\right )} x^{3} + \frac{1}{9} \,{\left (3 \, b e x^{3} \log \left (c\right ) -{\left (b e n - 3 \, a e\right )} x^{3}\right )} \log \left (f\right ) + \frac{1}{9} \,{\left (3 \, b e r x^{3} \log \left (c\right ) + 3 \, b e n x^{3} \log \left (f\right ) +{\left (3 \, b d n -{\left (2 \, b e n - 3 \, a e\right )} r\right )} x^{3}\right )} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 34.7266, size = 202, normalized size = 2.4 \begin{align*} \frac{a d x^{3}}{3} + \frac{a e r x^{3} \log{\left (x \right )}}{3} - \frac{a e r x^{3}}{9} + \frac{a e x^{3} \log{\left (f \right )}}{3} + \frac{b d n x^{3} \log{\left (x \right )}}{3} - \frac{b d n x^{3}}{9} + \frac{b d x^{3} \log{\left (c \right )}}{3} + \frac{b e n r x^{3} \log{\left (x \right )}^{2}}{3} - \frac{2 b e n r x^{3} \log{\left (x \right )}}{9} + \frac{2 b e n r x^{3}}{27} + \frac{b e n x^{3} \log{\left (f \right )} \log{\left (x \right )}}{3} - \frac{b e n x^{3} \log{\left (f \right )}}{9} + \frac{b e r x^{3} \log{\left (c \right )} \log{\left (x \right )}}{3} - \frac{b e r x^{3} \log{\left (c \right )}}{9} + \frac{b e x^{3} \log{\left (c \right )} \log{\left (f \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.2946, size = 217, normalized size = 2.58 \begin{align*} \frac{1}{3} \, b n r x^{3} e \log \left (x\right )^{2} - \frac{2}{9} \, b n r x^{3} e \log \left (x\right ) + \frac{1}{3} \, b r x^{3} e \log \left (c\right ) \log \left (x\right ) + \frac{1}{3} \, b n x^{3} e \log \left (f\right ) \log \left (x\right ) + \frac{2}{27} \, b n r x^{3} e - \frac{1}{9} \, b r x^{3} e \log \left (c\right ) - \frac{1}{9} \, b n x^{3} e \log \left (f\right ) + \frac{1}{3} \, b x^{3} e \log \left (c\right ) \log \left (f\right ) + \frac{1}{3} \, b d n x^{3} \log \left (x\right ) + \frac{1}{3} \, a r x^{3} e \log \left (x\right ) - \frac{1}{9} \, b d n x^{3} - \frac{1}{9} \, a r x^{3} e + \frac{1}{3} \, b d x^{3} \log \left (c\right ) + \frac{1}{3} \, a x^{3} e \log \left (f\right ) + \frac{1}{3} \, a d x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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