Optimal. Leaf size=83 \[ -\frac {b e n r}{27 x^3}-\frac {e r \left (3 a+b n+3 b \log \left (c x^n\right )\right )}{27 x^3}-\frac {b n \left (d+e \log \left (f x^r\right )\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{3 x^3} \]
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Rubi [A]
time = 0.05, antiderivative size = 83, normalized size of antiderivative = 1.00, 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 = {2341, 2413, 12}
\begin {gather*} -\frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{3 x^3}-\frac {e r \left (3 a+3 b \log \left (c x^n\right )+b n\right )}{27 x^3}-\frac {b n \left (d+e \log \left (f x^r\right )\right )}{9 x^3}-\frac {b e n r}{27 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2341
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^4} \, dx &=-\frac {b n \left (d+e \log \left (f x^r\right )\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{3 x^3}-(e r) \int \frac {-3 a \left (1+\frac {b n}{3 a}\right )-3 b \log \left (c x^n\right )}{9 x^4} \, dx\\ &=-\frac {b n \left (d+e \log \left (f x^r\right )\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{3 x^3}-\frac {1}{9} (e r) \int \frac {-3 a \left (1+\frac {b n}{3 a}\right )-3 b \log \left (c x^n\right )}{x^4} \, dx\\ &=-\frac {b e n r}{27 x^3}-\frac {e r \left (3 a+b n+3 b \log \left (c x^n\right )\right )}{27 x^3}-\frac {b n \left (d+e \log \left (f x^r\right )\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{3 x^3}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 69, normalized size = 0.83 \begin {gather*} -\frac {9 a d+3 b d n+3 a e r+2 b e n r+3 e (3 a+b n) \log \left (f x^r\right )+3 b \log \left (c x^n\right ) \left (3 d+e r+3 e \log \left (f x^r\right )\right )}{27 x^3} \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 = 0.16, size = 1451, normalized size = 17.48
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1451\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 103, normalized size = 1.24 \begin {gather*} -\frac {1}{9} \, b {\left (\frac {r}{x^{3}} + \frac {3 \, \log \left (f x^{r}\right )}{x^{3}}\right )} e \log \left (c x^{n}\right ) - \frac {b n {\left (2 \, r + 3 \, \log \left (f\right ) + 3 \, \log \left (x^{r}\right )\right )} e}{27 \, x^{3}} - \frac {b d n}{9 \, x^{3}} - \frac {a r e}{9 \, x^{3}} - \frac {b d \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac {a e \log \left (f x^{r}\right )}{3 \, x^{3}} - \frac {a d}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 112, normalized size = 1.35 \begin {gather*} -\frac {9 \, b n r e \log \left (x\right )^{2} + 3 \, b d n + {\left (2 \, b n + 3 \, a\right )} r e + 9 \, a d + 3 \, {\left (b r e + 3 \, b d\right )} \log \left (c\right ) + 3 \, {\left (3 \, b e \log \left (c\right ) + {\left (b n + 3 \, a\right )} e\right )} \log \left (f\right ) + 3 \, {\left (3 \, b r e \log \left (c\right ) + 3 \, b n e \log \left (f\right ) + 3 \, b d n + {\left (2 \, b n + 3 \, a\right )} r e\right )} \log \left (x\right )}{27 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.66, size = 129, normalized size = 1.55 \begin {gather*} - \frac {a d}{3 x^{3}} - \frac {a e r}{9 x^{3}} - \frac {a e \log {\left (f x^{r} \right )}}{3 x^{3}} - \frac {b d n}{9 x^{3}} - \frac {b d \log {\left (c x^{n} \right )}}{3 x^{3}} - \frac {2 b e n r}{27 x^{3}} - \frac {b e n \log {\left (f x^{r} \right )}}{9 x^{3}} - \frac {b e r \log {\left (c x^{n} \right )}}{9 x^{3}} - \frac {b e \log {\left (c x^{n} \right )} \log {\left (f x^{r} \right )}}{3 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 6.28, size = 121, normalized size = 1.46 \begin {gather*} -\frac {9 \, b n r e \log \left (x\right )^{2} + 6 \, b n r e \log \left (x\right ) + 9 \, b r e \log \left (c\right ) \log \left (x\right ) + 9 \, b n e \log \left (f\right ) \log \left (x\right ) + 2 \, b n r e + 3 \, b r e \log \left (c\right ) + 3 \, b n e \log \left (f\right ) + 9 \, b e \log \left (c\right ) \log \left (f\right ) + 9 \, b d n \log \left (x\right ) + 9 \, a r e \log \left (x\right ) + 3 \, b d n + 3 \, a r e + 9 \, b d \log \left (c\right ) + 9 \, a e \log \left (f\right ) + 9 \, a d}{27 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.93, size = 83, normalized size = 1.00 \begin {gather*} -\ln \left (f\,x^r\right )\,\left (\frac {a\,e}{3\,x^3}+\frac {b\,e\,n}{9\,x^3}+\frac {b\,e\,\ln \left (c\,x^n\right )}{3\,x^3}\right )-\frac {\frac {a\,d}{3}+\frac {b\,d\,n}{9}+\frac {a\,e\,r}{9}+\frac {2\,b\,e\,n\,r}{27}}{x^3}-\frac {b\,\ln \left (c\,x^n\right )\,\left (3\,d+e\,r\right )}{9\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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