Optimal. Leaf size=28 \[ \text {Int}\left (\frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2},x\right ) \]
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Rubi [A]
time = 0.04, 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 \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx &=\int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx\\ \end {align*}
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Mathematica [A] Leaf count is larger than twice the leaf count of optimal. \(108\) vs. \(2(28)=56\).
time = 0.07, size = 108, normalized size = 3.86 \begin {gather*} \frac {x (f x)^m \left (-b n \, _3F_2\left (2,\frac {1}{2}+\frac {m}{2},\frac {1}{2}+\frac {m}{2};\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};-\frac {e x^2}{d}\right )+(1+m) \, _2F_1\left (2,\frac {1+m}{2};\frac {3+m}{2};-\frac {e x^2}{d}\right ) \left (a+b \log \left (c x^n\right )\right )\right )}{d^2 (1+m)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (f x \right )^{m} \left (a +b \ln \left (c \,x^{n}\right )\right )}{\left (e \,x^{2}+d \right )^{2}}\, dx\]
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 [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (f x\right )^{m} \left (a + b \log {\left (c x^{n} \right )}\right )}{\left (d + e x^{2}\right )^{2}}\, dx \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 = -0.04 \begin {gather*} \int \frac {{\left (f\,x\right )}^m\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (e\,x^2+d\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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