Optimal. Leaf size=96 \[ \frac {e n \log \left (1-\frac {d}{d+e x}\right ) \left (a g+2 b g \log \left (c (d+e x)^n\right )+b f\right )}{d}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (g \log \left (c (d+e x)^n\right )+f\right )}{x}-\frac {2 b e g n^2 \text {Li}_2\left (\frac {d}{d+e x}\right )}{d} \]
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Rubi [A] time = 0.35, antiderivative size = 169, normalized size of antiderivative = 1.76, number of steps used = 11, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2439, 2411, 2344, 2301, 2317, 2391} \[ \frac {2 b e g n^2 \text {PolyLog}\left (2,\frac {e x}{d}+1\right )}{d}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (g \log \left (c (d+e x)^n\right )+f\right )}{x}+\frac {e g n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d}-\frac {e g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 b d}+\frac {b e n \log \left (-\frac {e x}{d}\right ) \left (g \log \left (c (d+e x)^n\right )+f\right )}{d}-\frac {b e \left (g \log \left (c (d+e x)^n\right )+f\right )^2}{2 d g} \]
Antiderivative was successfully verified.
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Rule 2301
Rule 2317
Rule 2344
Rule 2391
Rule 2411
Rule 2439
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^2} \, 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 )}{x}+(b e n) \int \frac {f+g \log \left (c (d+e x)^n\right )}{x (d+e x)} \, dx+(e g n) \int \frac {a+b \log \left (c (d+e x)^n\right )}{x (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 )}{x}+(b n) \operatorname {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 )+(g n) \operatorname {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 )\\ &=-\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}+\frac {(b n) \operatorname {Subst}\left (\int \frac {f+g \log \left (c x^n\right )}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e x\right )}{d}-\frac {(b e n) \operatorname {Subst}\left (\int \frac {f+g \log \left (c x^n\right )}{x} \, dx,x,d+e x\right )}{d}+\frac {(g n) \operatorname {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e x\right )}{d}-\frac {(e g n) \operatorname {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x} \, dx,x,d+e x\right )}{d}\\ &=\frac {e g n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d}-\frac {e g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 b d}+\frac {b e n \log \left (-\frac {e x}{d}\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{d}-\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}-\frac {b e \left (f+g \log \left (c (d+e x)^n\right )\right )^2}{2 d g}-2 \frac {\left (b e g n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{d}\right )}{x} \, dx,x,d+e x\right )}{d}\\ &=\frac {e g n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d}-\frac {e g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 b d}+\frac {b e n \log \left (-\frac {e x}{d}\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{d}-\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}-\frac {b e \left (f+g \log \left (c (d+e x)^n\right )\right )^2}{2 d g}+\frac {2 b e g n^2 \text {Li}_2\left (1+\frac {e x}{d}\right )}{d}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 180, normalized size = 1.88 \[ -\frac {a g \log \left (c (d+e x)^n\right )}{x}+\frac {a e g n \log (x)}{d}-\frac {a e g n \log (d+e x)}{d}-\frac {a f}{x}-\frac {b f \log \left (c (d+e x)^n\right )}{x}-\frac {b e g \log ^2\left (c (d+e x)^n\right )}{d}-\frac {b g \log ^2\left (c (d+e x)^n\right )}{x}+\frac {2 b e g n \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )}{d}+\frac {b e f n \log (x)}{d}-\frac {b e f n \log (d+e x)}{d}+\frac {2 b e g n^2 \text {Li}_2\left (\frac {d+e x}{d}\right )}{d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b g \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + a f + {\left (b f + a g\right )} \log \left ({\left (e x + d\right )}^{n} c\right )}{x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} {\left (g \log \left ({\left (e x + d\right )}^{n} c\right ) + f\right )}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.34, size = 931, normalized size = 9.70 \[ \text {result too large to display} \]
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 \[ -b e f n {\left (\frac {\log \left (e x + d\right )}{d} - \frac {\log \relax (x)}{d}\right )} - a e g n {\left (\frac {\log \left (e x + d\right )}{d} - \frac {\log \relax (x)}{d}\right )} - b g {\left (\frac {\log \left ({\left (e x + d\right )}^{n}\right )^{2}}{x} - \int \frac {e x \log \relax (c)^{2} + d \log \relax (c)^{2} + 2 \, {\left ({\left (e n + e \log \relax (c)\right )} x + d \log \relax (c)\right )} \log \left ({\left (e x + d\right )}^{n}\right )}{e x^{3} + d x^{2}}\,{d x}\right )} - \frac {b f \log \left ({\left (e x + d\right )}^{n} c\right )}{x} - \frac {a g \log \left ({\left (e x + d\right )}^{n} c\right )}{x} - \frac {a f}{x} \]
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 \[ \int \frac {\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )\,\left (f+g\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}{x^2} \,d x \]
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 \[ \int \frac {\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right ) \left (f + g \log {\left (c \left (d + e x\right )^{n} \right )}\right )}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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