Optimal. Leaf size=162 \[ -\frac {3 b n}{4 d^3 x^2}+\frac {a+b \log \left (c x^n\right )}{4 d x^2 \left (d+e x^2\right )^2}+\frac {6 a-b n+6 b \log \left (c x^n\right )}{8 d^2 x^2 \left (d+e x^2\right )}-\frac {12 a-5 b n+12 b \log \left (c x^n\right )}{8 d^3 x^2}+\frac {e \log \left (1+\frac {d}{e x^2}\right ) \left (12 a-5 b n+12 b \log \left (c x^n\right )\right )}{8 d^4}-\frac {3 b e n \text {Li}_2\left (-\frac {d}{e x^2}\right )}{4 d^4} \]
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Rubi [A]
time = 0.24, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2385, 2380,
2341, 2379, 2438} \begin {gather*} -\frac {3 b e n \text {PolyLog}\left (2,-\frac {d}{e x^2}\right )}{4 d^4}+\frac {e \log \left (\frac {d}{e x^2}+1\right ) \left (12 a+12 b \log \left (c x^n\right )-5 b n\right )}{8 d^4}-\frac {12 a+12 b \log \left (c x^n\right )-5 b n}{8 d^3 x^2}+\frac {6 a+6 b \log \left (c x^n\right )-b n}{8 d^2 x^2 \left (d+e x^2\right )}+\frac {a+b \log \left (c x^n\right )}{4 d x^2 \left (d+e x^2\right )^2}-\frac {3 b n}{4 d^3 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2341
Rule 2379
Rule 2380
Rule 2385
Rule 2438
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e x^2\right )^3} \, dx &=\frac {a+b \log \left (c x^n\right )}{4 d x^2 \left (d+e x^2\right )^2}-\frac {\int \frac {-6 a+b n-6 b \log \left (c x^n\right )}{x^3 \left (d+e x^2\right )^2} \, dx}{4 d}\\ &=\frac {a+b \log \left (c x^n\right )}{4 d x^2 \left (d+e x^2\right )^2}+\frac {6 a-b n+6 b \log \left (c x^n\right )}{8 d^2 x^2 \left (d+e x^2\right )}+\frac {\int \frac {-6 b n-4 (-6 a+b n)+24 b \log \left (c x^n\right )}{x^3 \left (d+e x^2\right )} \, dx}{8 d^2}\\ &=\frac {a+b \log \left (c x^n\right )}{4 d x^2 \left (d+e x^2\right )^2}+\frac {6 a-b n+6 b \log \left (c x^n\right )}{8 d^2 x^2 \left (d+e x^2\right )}+\frac {\int \left (\frac {-6 b n-4 (-6 a+b n)+24 b \log \left (c x^n\right )}{d x^3}-\frac {e \left (-6 b n-4 (-6 a+b n)+24 b \log \left (c x^n\right )\right )}{d^2 x}+\frac {e^2 x \left (-6 b n-4 (-6 a+b n)+24 b \log \left (c x^n\right )\right )}{d^2 \left (d+e x^2\right )}\right ) \, dx}{8 d^2}\\ &=\frac {a+b \log \left (c x^n\right )}{4 d x^2 \left (d+e x^2\right )^2}+\frac {6 a-b n+6 b \log \left (c x^n\right )}{8 d^2 x^2 \left (d+e x^2\right )}+\frac {\int \frac {-6 b n-4 (-6 a+b n)+24 b \log \left (c x^n\right )}{x^3} \, dx}{8 d^3}-\frac {e \int \frac {-6 b n-4 (-6 a+b n)+24 b \log \left (c x^n\right )}{x} \, dx}{8 d^4}+\frac {e^2 \int \frac {x \left (-6 b n-4 (-6 a+b n)+24 b \log \left (c x^n\right )\right )}{d+e x^2} \, dx}{8 d^4}\\ &=-\frac {3 b n}{4 d^3 x^2}+\frac {a+b \log \left (c x^n\right )}{4 d x^2 \left (d+e x^2\right )^2}+\frac {6 a-b n+6 b \log \left (c x^n\right )}{8 d^2 x^2 \left (d+e x^2\right )}-\frac {12 a-5 b n+12 b \log \left (c x^n\right )}{8 d^3 x^2}-\frac {e \left (12 a-5 b n+12 b \log \left (c x^n\right )\right )^2}{96 b d^4 n}+\frac {e \left (12 a-5 b n+12 b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x^2}{d}\right )}{8 d^4}-\frac {(3 b e n) \int \frac {\log \left (1+\frac {e x^2}{d}\right )}{x} \, dx}{2 d^4}\\ &=-\frac {3 b n}{4 d^3 x^2}+\frac {a+b \log \left (c x^n\right )}{4 d x^2 \left (d+e x^2\right )^2}+\frac {6 a-b n+6 b \log \left (c x^n\right )}{8 d^2 x^2 \left (d+e x^2\right )}-\frac {12 a-5 b n+12 b \log \left (c x^n\right )}{8 d^3 x^2}-\frac {e \left (12 a-5 b n+12 b \log \left (c x^n\right )\right )^2}{96 b d^4 n}+\frac {e \left (12 a-5 b n+12 b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x^2}{d}\right )}{8 d^4}+\frac {3 b e n \text {Li}_2\left (-\frac {e x^2}{d}\right )}{4 d^4}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.83, size = 507, normalized size = 3.13 \begin {gather*} \frac {-\frac {8 d \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{x^2}-\frac {4 d^2 e \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2}-\frac {16 d e \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{d+e x^2}-48 e \log (x) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )+24 e \left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \log \left (d+e x^2\right )+b n \left (\frac {9 e^{3/2} x \log (x)}{-i \sqrt {d}+\sqrt {e} x}-24 e \log ^2(x)-\frac {4 d (1+2 \log (x))}{x^2}+e \left (\frac {d}{d+i \sqrt {d} \sqrt {e} x}+\log (x)-\frac {d \log (x)}{\left (\sqrt {d}+i \sqrt {e} x\right )^2}-\log \left (i \sqrt {d}-\sqrt {e} x\right )\right )-9 e \log \left (i \sqrt {d}-\sqrt {e} x\right )+e \left (\frac {d}{d-i \sqrt {d} \sqrt {e} x}+\log (x)-\frac {d \log (x)}{\left (\sqrt {d}-i \sqrt {e} x\right )^2}-\log \left (i \sqrt {d}+\sqrt {e} x\right )\right )+\frac {-9 i e^{3/2} x \log (x)+9 i e \left (i \sqrt {d}+\sqrt {e} x\right ) \log \left (i \sqrt {d}+\sqrt {e} x\right )}{\sqrt {d}-i \sqrt {e} x}+24 e \left (\log (x) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )+\text {Li}_2\left (-\frac {i \sqrt {e} x}{\sqrt {d}}\right )\right )+24 e \left (\log (x) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )+\text {Li}_2\left (\frac {i \sqrt {e} x}{\sqrt {d}}\right )\right )\right )}{16 d^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.13, size = 1030, normalized size = 6.36
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1030\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
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 [F]
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 [F]
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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a+b\,\ln \left (c\,x^n\right )}{x^3\,{\left (e\,x^2+d\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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