Optimal. Leaf size=169 \[ -\frac {\log \left (\frac {d x^{-r}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3 r}-\frac {e x^r \left (a+b \log \left (c x^n\right )\right )}{d^3 r \left (d+e x^r\right )}+\frac {a+b \log \left (c x^n\right )}{2 d r \left (d+e x^r\right )^2}+\frac {b n \text {Li}_2\left (-\frac {d x^{-r}}{e}\right )}{d^3 r^2}+\frac {3 b n \log \left (d+e x^r\right )}{2 d^3 r^2}-\frac {b n \log (x)}{2 d^3 r}-\frac {b n}{2 d^2 r^2 \left (d+e x^r\right )} \]
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Rubi [A] time = 0.41, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {2349, 2345, 2391, 2335, 260, 2338, 266, 44} \[ \frac {b n \text {PolyLog}\left (2,-\frac {d x^{-r}}{e}\right )}{d^3 r^2}-\frac {e x^r \left (a+b \log \left (c x^n\right )\right )}{d^3 r \left (d+e x^r\right )}-\frac {\log \left (\frac {d x^{-r}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3 r}+\frac {a+b \log \left (c x^n\right )}{2 d r \left (d+e x^r\right )^2}-\frac {b n}{2 d^2 r^2 \left (d+e x^r\right )}+\frac {3 b n \log \left (d+e x^r\right )}{2 d^3 r^2}-\frac {b n \log (x)}{2 d^3 r} \]
Antiderivative was successfully verified.
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Rule 44
Rule 260
Rule 266
Rule 2335
Rule 2338
Rule 2345
Rule 2349
Rule 2391
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )^3} \, dx &=\frac {\int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )^2} \, dx}{d}-\frac {e \int \frac {x^{-1+r} \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^3} \, dx}{d}\\ &=\frac {a+b \log \left (c x^n\right )}{2 d r \left (d+e x^r\right )^2}+\frac {\int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )} \, dx}{d^2}-\frac {e \int \frac {x^{-1+r} \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx}{d^2}-\frac {(b n) \int \frac {1}{x \left (d+e x^r\right )^2} \, dx}{2 d r}\\ &=\frac {a+b \log \left (c x^n\right )}{2 d r \left (d+e x^r\right )^2}-\frac {e x^r \left (a+b \log \left (c x^n\right )\right )}{d^3 r \left (d+e x^r\right )}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x^{-r}}{e}\right )}{d^3 r}-\frac {(b n) \operatorname {Subst}\left (\int \frac {1}{x (d+e x)^2} \, dx,x,x^r\right )}{2 d r^2}+\frac {(b n) \int \frac {\log \left (1+\frac {d x^{-r}}{e}\right )}{x} \, dx}{d^3 r}+\frac {(b e n) \int \frac {x^{-1+r}}{d+e x^r} \, dx}{d^3 r}\\ &=\frac {a+b \log \left (c x^n\right )}{2 d r \left (d+e x^r\right )^2}-\frac {e x^r \left (a+b \log \left (c x^n\right )\right )}{d^3 r \left (d+e x^r\right )}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x^{-r}}{e}\right )}{d^3 r}+\frac {b n \log \left (d+e x^r\right )}{d^3 r^2}+\frac {b n \text {Li}_2\left (-\frac {d x^{-r}}{e}\right )}{d^3 r^2}-\frac {(b n) \operatorname {Subst}\left (\int \left (\frac {1}{d^2 x}-\frac {e}{d (d+e x)^2}-\frac {e}{d^2 (d+e x)}\right ) \, dx,x,x^r\right )}{2 d r^2}\\ &=-\frac {b n}{2 d^2 r^2 \left (d+e x^r\right )}-\frac {b n \log (x)}{2 d^3 r}+\frac {a+b \log \left (c x^n\right )}{2 d r \left (d+e x^r\right )^2}-\frac {e x^r \left (a+b \log \left (c x^n\right )\right )}{d^3 r \left (d+e x^r\right )}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x^{-r}}{e}\right )}{d^3 r}+\frac {3 b n \log \left (d+e x^r\right )}{2 d^3 r^2}+\frac {b n \text {Li}_2\left (-\frac {d x^{-r}}{e}\right )}{d^3 r^2}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 170, normalized size = 1.01 \[ \frac {\frac {d^2 r \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2}+\frac {d \left (2 a r+2 b r \log \left (c x^n\right )-b n\right )}{d+e x^r}-2 a r \log \left (d-d x^r\right )+2 b r \left (n \log (x)-\log \left (c x^n\right )\right ) \log \left (d-d x^r\right )+2 b n \left (\text {Li}_2\left (\frac {e x^r}{d}+1\right )+\left (\log \left (-\frac {e x^r}{d}\right )-r \log (x)\right ) \log \left (d+e x^r\right )+\frac {1}{2} r^2 \log ^2(x)\right )+3 b n \log \left (d-d x^r\right )}{2 d^3 r^2} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.75, size = 401, normalized size = 2.37 \[ \frac {b d^{2} n r^{2} \log \relax (x)^{2} + 3 \, b d^{2} r \log \relax (c) - b d^{2} n + 3 \, a d^{2} r + {\left (b e^{2} n r^{2} \log \relax (x)^{2} + {\left (2 \, b e^{2} r^{2} \log \relax (c) - 3 \, b e^{2} n r + 2 \, a e^{2} r^{2}\right )} \log \relax (x)\right )} x^{2 \, r} + {\left (2 \, b d e n r^{2} \log \relax (x)^{2} + 2 \, b d e r \log \relax (c) - b d e n + 2 \, a d e r + 4 \, {\left (b d e r^{2} \log \relax (c) - b d e n r + a d e r^{2}\right )} \log \relax (x)\right )} x^{r} - 2 \, {\left (b e^{2} n x^{2 \, r} + 2 \, b d e n x^{r} + b d^{2} n\right )} {\rm Li}_2\left (-\frac {e x^{r} + d}{d} + 1\right ) - {\left (2 \, b d^{2} r \log \relax (c) - 3 \, b d^{2} n + 2 \, a d^{2} r + {\left (2 \, b e^{2} r \log \relax (c) - 3 \, b e^{2} n + 2 \, a e^{2} r\right )} x^{2 \, r} + 2 \, {\left (2 \, b d e r \log \relax (c) - 3 \, b d e n + 2 \, a d e r\right )} x^{r}\right )} \log \left (e x^{r} + d\right ) + 2 \, {\left (b d^{2} r^{2} \log \relax (c) + a d^{2} r^{2}\right )} \log \relax (x) - 2 \, {\left (b e^{2} n r x^{2 \, r} \log \relax (x) + 2 \, b d e n r x^{r} \log \relax (x) + b d^{2} n r \log \relax (x)\right )} \log \left (\frac {e x^{r} + d}{d}\right )}{2 \, {\left (d^{3} e^{2} r^{2} x^{2 \, r} + 2 \, d^{4} e r^{2} x^{r} + d^{5} r^{2}\right )}} \]
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 \[ \int \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{r} + d\right )}^{3} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.30, size = 1012, normalized size = 5.99 \[ \text {result too large to display} \]
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 \[ \frac {1}{2} \, a {\left (\frac {2 \, e x^{r} + 3 \, d}{d^{2} e^{2} r x^{2 \, r} + 2 \, d^{3} e r x^{r} + d^{4} r} + \frac {2 \, \log \relax (x)}{d^{3}} - \frac {2 \, \log \left (\frac {e x^{r} + d}{e}\right )}{d^{3} r}\right )} + b \int \frac {\log \relax (c) + \log \left (x^{n}\right )}{e^{3} x x^{3 \, r} + 3 \, d e^{2} x x^{2 \, r} + 3 \, d^{2} e x x^{r} + d^{3} x}\,{d x} \]
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 \[ \int \frac {a+b\,\ln \left (c\,x^n\right )}{x\,{\left (d+e\,x^r\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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