Optimal. Leaf size=57 \[ -\frac {e r \left (a+b \log \left (c x^n\right )\right )^3}{6 b^2 n^2}+\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{2 b n} \]
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Rubi [A]
time = 0.05, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {2338, 2413, 12,
2339, 30} \begin {gather*} \frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{2 b n}-\frac {e r \left (a+b \log \left (c x^n\right )\right )^3}{6 b^2 n^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 2338
Rule 2339
Rule 2413
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x} \, dx &=\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{2 b n}-(e r) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 b n x} \, dx\\ &=\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{2 b n}-\frac {(e r) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx}{2 b n}\\ &=\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{2 b n}-\frac {(e r) \text {Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )}{2 b^2 n^2}\\ &=-\frac {e r \left (a+b \log \left (c x^n\right )\right )^3}{6 b^2 n^2}+\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{2 b n}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 72, normalized size = 1.26 \begin {gather*} \frac {1}{6} \log (x) \left (2 b e n r \log ^2(x)+6 \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )-3 \log (x) \left (b d n+a e r+b e r \log \left (c x^n\right )+b e n \log \left (f x^r\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.24, size = 1597, normalized size = 28.02
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1597\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 76, normalized size = 1.33 \begin {gather*} \frac {b e \log \left (c x^{n}\right ) \log \left (f x^{r}\right )^{2}}{2 \, r} - \frac {b n e \log \left (f x^{r}\right )^{3}}{6 \, r^{2}} + \frac {b d \log \left (c x^{n}\right )^{2}}{2 \, n} + \frac {a e \log \left (f x^{r}\right )^{2}}{2 \, r} + a d \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 68, normalized size = 1.19 \begin {gather*} \frac {1}{3} \, b n r e \log \left (x\right )^{3} + \frac {1}{2} \, {\left (b r e \log \left (c\right ) + b n e \log \left (f\right ) + b d n + a r e\right )} \log \left (x\right )^{2} + {\left (b d \log \left (c\right ) + a d + {\left (b e \log \left (c\right ) + a e\right )} \log \left (f\right )\right )} \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \log {\left (c x^{n} \right )}\right ) \left (d + e \log {\left (f x^{r} \right )}\right )}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.20, size = 85, normalized size = 1.49 \begin {gather*} \frac {1}{3} \, b n r e \log \left (x\right )^{3} + \frac {1}{2} \, b r e \log \left (c\right ) \log \left (x\right )^{2} + \frac {1}{2} \, b n e \log \left (f\right ) \log \left (x\right )^{2} + b e \log \left (c\right ) \log \left (f\right ) \log \left (x\right ) + \frac {1}{2} \, b d n \log \left (x\right )^{2} + \frac {1}{2} \, a r e \log \left (x\right )^{2} + b d \log \left (c\right ) \log \left (x\right ) + a e \log \left (f\right ) \log \left (x\right ) + a d \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.87, size = 73, normalized size = 1.28 \begin {gather*} a\,d\,\ln \left (x\right )+\frac {b\,d\,{\ln \left (c\,x^n\right )}^2}{2\,n}+\frac {a\,e\,{\ln \left (f\,x^r\right )}^2}{2\,r}-\frac {b\,e\,r\,{\ln \left (c\,x^n\right )}^3}{6\,n^2}+\frac {b\,e\,{\ln \left (c\,x^n\right )}^2\,\ln \left (f\,x^r\right )}{2\,n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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