Optimal. Leaf size=31 \[ \text {Int}\left ((g x)^q \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ),x\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int (g x)^q \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ) \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int (g x)^q \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ) \, dx &=\int (g x)^q \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ) \, dx\\ \end {align*}
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Mathematica [A] Leaf count is larger than twice the leaf count of optimal. \(304\) vs. \(2(31)=62\).
time = 0.21, size = 304, normalized size = 9.81 \begin {gather*} \frac {x (g x)^q \left (-a k m+2 b k m n-a k m q-b k m n \, _3F_2\left (1,\frac {1}{m}+\frac {q}{m},\frac {1}{m}+\frac {q}{m};1+\frac {1}{m}+\frac {q}{m},1+\frac {1}{m}+\frac {q}{m};-\frac {f x^m}{e}\right )-b k m \log \left (c x^n\right )-b k m q \log \left (c x^n\right )+k m \, _2F_1\left (1,\frac {1+q}{m};\frac {1+m+q}{m};-\frac {f x^m}{e}\right ) \left (a-b n+a q+b (1+q) \log \left (c x^n\right )\right )+a \log \left (d \left (e+f x^m\right )^k\right )-b n \log \left (d \left (e+f x^m\right )^k\right )+2 a q \log \left (d \left (e+f x^m\right )^k\right )-b n q \log \left (d \left (e+f x^m\right )^k\right )+a q^2 \log \left (d \left (e+f x^m\right )^k\right )+b \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )+2 b q \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )+b q^2 \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )\right )}{(1+q)^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 0, normalized size = 0.00 \[\int \left (g x \right )^{q} \left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (d \left (e +f \,x^{m}\right )^{k}\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \ln \left (d\,{\left (e+f\,x^m\right )}^k\right )\,{\left (g\,x\right )}^q\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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