Optimal. Leaf size=57 \[ -\frac {e r \left (a+b \log \left (c x^n\right )\right )^4}{12 b^2 n^2}+\frac {\left (a+b \log \left (c x^n\right )\right )^3 \left (d+e \log \left (f x^r\right )\right )}{3 b n} \]
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Rubi [A]
time = 0.07, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2339, 30, 2413,
12} \begin {gather*} \frac {\left (a+b \log \left (c x^n\right )\right )^3 \left (d+e \log \left (f x^r\right )\right )}{3 b n}-\frac {e r \left (a+b \log \left (c x^n\right )\right )^4}{12 b^2 n^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 2339
Rule 2413
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{x} \, dx &=\frac {\left (a+b \log \left (c x^n\right )\right )^3 \left (d+e \log \left (f x^r\right )\right )}{3 b n}-(e r) \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{3 b n x} \, dx\\ &=\frac {\left (a+b \log \left (c x^n\right )\right )^3 \left (d+e \log \left (f x^r\right )\right )}{3 b n}-\frac {(e r) \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x} \, dx}{3 b n}\\ &=\frac {\left (a+b \log \left (c x^n\right )\right )^3 \left (d+e \log \left (f x^r\right )\right )}{3 b n}-\frac {(e r) \text {Subst}\left (\int x^3 \, dx,x,a+b \log \left (c x^n\right )\right )}{3 b^2 n^2}\\ &=-\frac {e r \left (a+b \log \left (c x^n\right )\right )^4}{12 b^2 n^2}+\frac {\left (a+b \log \left (c x^n\right )\right )^3 \left (d+e \log \left (f x^r\right )\right )}{3 b n}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(129\) vs. \(2(57)=114\).
time = 0.07, size = 129, normalized size = 2.26 \begin {gather*} \frac {1}{12} \log (x) \left (-3 b^2 e n^2 r \log ^3(x)+12 \left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )+4 b n \log ^2(x) \left (b d n+2 a e r+2 b e r \log \left (c x^n\right )+b e n \log \left (f x^r\right )\right )-6 \log (x) \left (a+b \log \left (c x^n\right )\right ) \left (2 b d n+a e r+b e r \log \left (c x^n\right )+2 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 = 0.70, size = 9164, normalized size = 160.77
method | result | size |
risch | \(\text {Expression too large to display}\) | \(9164\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 168 vs.
\(2 (55) = 110\).
time = 0.29, size = 168, normalized size = 2.95 \begin {gather*} \frac {b^{2} e \log \left (c x^{n}\right )^{2} \log \left (f x^{r}\right )^{2}}{2 \, r} + \frac {b^{2} d \log \left (c x^{n}\right )^{3}}{3 \, n} + \frac {a b e \log \left (c x^{n}\right ) \log \left (f x^{r}\right )^{2}}{r} - \frac {a b n e \log \left (f x^{r}\right )^{3}}{3 \, r^{2}} - \frac {1}{12} \, {\left (\frac {4 \, n \log \left (c x^{n}\right ) \log \left (f x^{r}\right )^{3}}{r^{2}} - \frac {n^{2} \log \left (f x^{r}\right )^{4}}{r^{3}}\right )} b^{2} e + \frac {a b d \log \left (c x^{n}\right )^{2}}{n} + \frac {a^{2} e \log \left (f x^{r}\right )^{2}}{2 \, r} + a^{2} d \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 182 vs.
\(2 (55) = 110\).
time = 0.35, size = 182, normalized size = 3.19 \begin {gather*} \frac {1}{4} \, b^{2} n^{2} r e \log \left (x\right )^{4} + \frac {1}{3} \, {\left (2 \, b^{2} n r e \log \left (c\right ) + b^{2} n^{2} e \log \left (f\right ) + b^{2} d n^{2} + 2 \, a b n r e\right )} \log \left (x\right )^{3} + \frac {1}{2} \, {\left (b^{2} r e \log \left (c\right )^{2} + 2 \, a b d n + a^{2} r e + 2 \, {\left (b^{2} d n + a b r e\right )} \log \left (c\right ) + 2 \, {\left (b^{2} n e \log \left (c\right ) + a b n e\right )} \log \left (f\right )\right )} \log \left (x\right )^{2} + {\left (b^{2} d \log \left (c\right )^{2} + 2 \, a b d \log \left (c\right ) + a^{2} d + {\left (b^{2} e \log \left (c\right )^{2} + 2 \, a b e \log \left (c\right ) + a^{2} 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 )^{2} \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 [B] Leaf count of result is larger than twice the leaf count of optimal. 223 vs.
\(2 (55) = 110\).
time = 4.62, size = 223, normalized size = 3.91 \begin {gather*} \frac {1}{4} \, b^{2} n^{2} r e \log \left (x\right )^{4} + \frac {2}{3} \, b^{2} n r e \log \left (c\right ) \log \left (x\right )^{3} + \frac {1}{3} \, b^{2} n^{2} e \log \left (f\right ) \log \left (x\right )^{3} + \frac {1}{2} \, b^{2} r e \log \left (c\right )^{2} \log \left (x\right )^{2} + b^{2} n e \log \left (c\right ) \log \left (f\right ) \log \left (x\right )^{2} + \frac {1}{3} \, b^{2} d n^{2} \log \left (x\right )^{3} + \frac {2}{3} \, a b n r e \log \left (x\right )^{3} + b^{2} e \log \left (c\right )^{2} \log \left (f\right ) \log \left (x\right ) + b^{2} d n \log \left (c\right ) \log \left (x\right )^{2} + a b r e \log \left (c\right ) \log \left (x\right )^{2} + a b n e \log \left (f\right ) \log \left (x\right )^{2} + b^{2} d \log \left (c\right )^{2} \log \left (x\right ) + 2 \, a b e \log \left (c\right ) \log \left (f\right ) \log \left (x\right ) + a b d n \log \left (x\right )^{2} + \frac {1}{2} \, a^{2} r e \log \left (x\right )^{2} + 2 \, a b d \log \left (c\right ) \log \left (x\right ) + a^{2} e \log \left (f\right ) \log \left (x\right ) + a^{2} d \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.99, size = 124, normalized size = 2.18 \begin {gather*} \ln \left (f\,x^r\right )\,\left (\frac {b^2\,e\,{\ln \left (c\,x^n\right )}^3}{3\,n}+\frac {a\,b\,e\,{\ln \left (c\,x^n\right )}^2}{n}\right )+\frac {{\ln \left (c\,x^n\right )}^3\,\left (b^2\,d\,n-a\,b\,e\,r\right )}{3\,n^2}+a^2\,d\,\ln \left (x\right )+\frac {a^2\,e\,{\ln \left (f\,x^r\right )}^2}{2\,r}+\frac {a\,b\,d\,{\ln \left (c\,x^n\right )}^2}{n}-\frac {b^2\,e\,r\,{\ln \left (c\,x^n\right )}^4}{12\,n^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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