Optimal. Leaf size=156 \[ \frac {b e^2 g n^2 \log (x)}{d^2}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{2 x^2}-\frac {e n (d+e x) \left (b f+a g+2 b g \log \left (c (d+e x)^n\right )\right )}{2 d^2 x}-\frac {e^2 n \left (b f+a g+2 b g \log \left (c (d+e x)^n\right )\right ) \log \left (1-\frac {d}{d+e x}\right )}{2 d^2}+\frac {b e^2 g n^2 \text {Li}_2\left (\frac {d}{d+e x}\right )}{d^2} \]
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Rubi [A]
time = 0.23, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {2483, 2458,
2389, 2379, 2438, 2351, 31} \begin {gather*} \frac {b e^2 g n^2 \text {PolyLog}\left (2,\frac {d}{d+e x}\right )}{d^2}-\frac {e^2 n \log \left (1-\frac {d}{d+e x}\right ) \left (a g+2 b g \log \left (c (d+e x)^n\right )+b f\right )}{2 d^2}-\frac {e n (d+e x) \left (a g+2 b g \log \left (c (d+e x)^n\right )+b f\right )}{2 d^2 x}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (g \log \left (c (d+e x)^n\right )+f\right )}{2 x^2}+\frac {b e^2 g n^2 \log (x)}{d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 2351
Rule 2379
Rule 2389
Rule 2438
Rule 2458
Rule 2483
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{x^3} \, dx &=-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{2 x^2}+\frac {1}{2} (b e n) \int \frac {f+g \log \left (c (d+e x)^n\right )}{x^2 (d+e x)} \, dx+\frac {1}{2} (e g n) \int \frac {a+b \log \left (c (d+e x)^n\right )}{x^2 (d+e x)} \, dx\\ &=-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{2 x^2}+\frac {1}{2} (b n) \text {Subst}\left (\int \frac {f+g \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+e x\right )+\frac {1}{2} (g n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+e x\right )\\ &=-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{2 x^2}+\frac {(b n) \text {Subst}\left (\int \frac {f+g \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+e x\right )}{2 d}-\frac {(b e n) \text {Subst}\left (\int \frac {f+g \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )} \, dx,x,d+e x\right )}{2 d}+\frac {(g n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+e x\right )}{2 d}-\frac {(e g n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )} \, dx,x,d+e x\right )}{2 d}\\ &=-\frac {e g n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 d^2 x}-\frac {b e n (d+e x) \left (f+g \log \left (c (d+e x)^n\right )\right )}{2 d^2 x}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{2 x^2}-\frac {(b e n) \text {Subst}\left (\int \frac {f+g \log \left (c x^n\right )}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e x\right )}{2 d^2}+\frac {\left (b e^2 n\right ) \text {Subst}\left (\int \frac {f+g \log \left (c x^n\right )}{x} \, dx,x,d+e x\right )}{2 d^2}-\frac {(e g n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e x\right )}{2 d^2}+\frac {\left (e^2 g n\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x} \, dx,x,d+e x\right )}{2 d^2}+2 \frac {\left (b e g n^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e x\right )}{2 d^2}\\ &=\frac {b e^2 g n^2 \log (x)}{d^2}-\frac {e g n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 d^2 x}-\frac {e^2 g n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 d^2}+\frac {e^2 g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 b d^2}-\frac {b e n (d+e x) \left (f+g \log \left (c (d+e x)^n\right )\right )}{2 d^2 x}-\frac {b e^2 n \log \left (-\frac {e x}{d}\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{2 d^2}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{2 x^2}+\frac {b e^2 \left (f+g \log \left (c (d+e x)^n\right )\right )^2}{4 d^2 g}+2 \frac {\left (b e^2 g n^2\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{d}\right )}{x} \, dx,x,d+e x\right )}{2 d^2}\\ &=\frac {b e^2 g n^2 \log (x)}{d^2}-\frac {e g n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 d^2 x}-\frac {e^2 g n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 d^2}+\frac {e^2 g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 b d^2}-\frac {b e n (d+e x) \left (f+g \log \left (c (d+e x)^n\right )\right )}{2 d^2 x}-\frac {b e^2 n \log \left (-\frac {e x}{d}\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{2 d^2}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{2 x^2}+\frac {b e^2 \left (f+g \log \left (c (d+e x)^n\right )\right )^2}{4 d^2 g}-\frac {b e^2 g n^2 \text {Li}_2\left (1+\frac {e x}{d}\right )}{d^2}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 254, normalized size = 1.63 \begin {gather*} -\frac {a f}{2 x^2}+\frac {1}{2} b e f n \left (-\frac {1}{d x}-\frac {e \log (x)}{d^2}+\frac {e \log (d+e x)}{d^2}\right )+\frac {1}{2} a e g n \left (-\frac {1}{d x}-\frac {e \log (x)}{d^2}+\frac {e \log (d+e x)}{d^2}\right )-\frac {b f \log \left (c (d+e x)^n\right )}{2 x^2}-\frac {a g \log \left (c (d+e x)^n\right )}{2 x^2}-\frac {b g \log ^2\left (c (d+e x)^n\right )}{2 x^2}+b e g n \left (\frac {e n \left (\frac {\log (x)}{d}-\frac {\log (d+e x)}{d}\right )}{d}-\frac {\log \left (c (d+e x)^n\right )}{d x}-\frac {e \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )}{d^2}+\frac {e \log ^2\left (c (d+e x)^n\right )}{2 d^2 n}-\frac {e n \text {Li}_2\left (\frac {d+e x}{d}\right )}{d^2}\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
4.
time = 0.26, size = 1201, normalized size = 7.70
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1201\) |
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right ) \left (f + g \log {\left (c \left (d + e x\right )^{n} \right )}\right )}{x^{3}}\, dx \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 {\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )\,\left (f+g\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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