Optimal. Leaf size=62 \[ \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 d (d+e x)^2}-\frac {b n}{2 e^2 (d+e x)}-\frac {b n \log (d+e x)}{2 d e^2} \]
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Rubi [A] time = 0.05, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2335, 43} \[ \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 d (d+e x)^2}-\frac {b n}{2 e^2 (d+e x)}-\frac {b n \log (d+e x)}{2 d e^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2335
Rubi steps
\begin {align*} \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^3} \, dx &=\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 d (d+e x)^2}-\frac {(b n) \int \frac {x}{(d+e x)^2} \, dx}{2 d}\\ &=\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 d (d+e x)^2}-\frac {(b n) \int \left (-\frac {d}{e (d+e x)^2}+\frac {1}{e (d+e x)}\right ) \, dx}{2 d}\\ &=-\frac {b n}{2 e^2 (d+e x)}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 d (d+e x)^2}-\frac {b n \log (d+e x)}{2 d e^2}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 75, normalized size = 1.21 \[ \frac {b n \log (x)-\frac {a d (d+2 e x)+b d (d+2 e x) \log \left (c x^n\right )+b d n (d+e x)+b n (d+e x)^2 \log (d+e x)}{(d+e x)^2}}{2 d e^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.58, size = 115, normalized size = 1.85 \[ \frac {b e^{2} n x^{2} \log \relax (x) - b d^{2} n - a d^{2} - {\left (b d e n + 2 \, a d e\right )} x - {\left (b e^{2} n x^{2} + 2 \, b d e n x + b d^{2} n\right )} \log \left (e x + d\right ) - {\left (2 \, b d e x + b d^{2}\right )} \log \relax (c)}{2 \, {\left (d e^{4} x^{2} + 2 \, d^{2} e^{3} x + d^{3} e^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.39, size = 122, normalized size = 1.97 \[ -\frac {b n x^{2} e^{2} \log \left (x e + d\right ) + 2 \, b d n x e \log \left (x e + d\right ) - b n x^{2} e^{2} \log \relax (x) + b d n x e + b d^{2} n \log \left (x e + d\right ) + 2 \, b d x e \log \relax (c) + b d^{2} n + 2 \, a d x e + b d^{2} \log \relax (c) + a d^{2}}{2 \, {\left (d x^{2} e^{4} + 2 \, d^{2} x e^{3} + d^{3} e^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.22, size = 349, normalized size = 5.63 \[ -\frac {\left (2 e x +d \right ) b \ln \left (x^{n}\right )}{2 \left (e x +d \right )^{2} e^{2}}-\frac {-2 i \pi b d e x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+2 i \pi b d e x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+2 i \pi b d e x \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-2 i \pi b d e x \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-2 b \,e^{2} n \,x^{2} \ln \left (-x \right )+2 b \,e^{2} n \,x^{2} \ln \left (e x +d \right )-i \pi b \,d^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i \pi b \,d^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b \,d^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b \,d^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-4 b d e n x \ln \left (-x \right )+4 b d e n x \ln \left (e x +d \right )-2 b \,d^{2} n \ln \left (-x \right )+2 b \,d^{2} n \ln \left (e x +d \right )+2 b d e n x +4 b d e x \ln \relax (c )+4 a d e x +2 b \,d^{2} n +2 b \,d^{2} \ln \relax (c )+2 a \,d^{2}}{4 \left (e x +d \right )^{2} d \,e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.61, size = 114, normalized size = 1.84 \[ -\frac {1}{2} \, b n {\left (\frac {1}{e^{3} x + d e^{2}} + \frac {\log \left (e x + d\right )}{d e^{2}} - \frac {\log \relax (x)}{d e^{2}}\right )} - \frac {{\left (2 \, e x + d\right )} b \log \left (c x^{n}\right )}{2 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}} - \frac {{\left (2 \, e x + d\right )} a}{2 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.03, size = 108, normalized size = 1.74 \[ -\frac {a\,d+x\,\left (2\,a\,e+b\,e\,n\right )+b\,d\,n}{2\,d^2\,e^2+4\,d\,e^3\,x+2\,e^4\,x^2}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {b\,d}{2\,e^2}+\frac {b\,x}{e}\right )}{d^2+2\,d\,e\,x+e^2\,x^2}-\frac {b\,n\,\mathrm {atanh}\left (\frac {2\,e\,x}{d}+1\right )}{d\,e^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.04, size = 456, normalized size = 7.35 \[ \begin {cases} \tilde {\infty } \left (- \frac {a}{x} - \frac {b n \log {\relax (x )}}{x} - \frac {b n}{x} - \frac {b \log {\relax (c )}}{x}\right ) & \text {for}\: d = 0 \wedge e = 0 \\\frac {- \frac {a}{x} - \frac {b n \log {\relax (x )}}{x} - \frac {b n}{x} - \frac {b \log {\relax (c )}}{x}}{e^{3}} & \text {for}\: d = 0 \\\frac {\frac {a x^{2}}{2} + \frac {b n x^{2} \log {\relax (x )}}{2} - \frac {b n x^{2}}{4} + \frac {b x^{2} \log {\relax (c )}}{2}}{d^{3}} & \text {for}\: e = 0 \\- \frac {a d^{2}}{2 d^{3} e^{2} + 4 d^{2} e^{3} x + 2 d e^{4} x^{2}} - \frac {2 a d e x}{2 d^{3} e^{2} + 4 d^{2} e^{3} x + 2 d e^{4} x^{2}} - \frac {b d^{2} n \log {\left (\frac {d}{e} + x \right )}}{2 d^{3} e^{2} + 4 d^{2} e^{3} x + 2 d e^{4} x^{2}} - \frac {b d^{2} n}{2 d^{3} e^{2} + 4 d^{2} e^{3} x + 2 d e^{4} x^{2}} - \frac {2 b d e n x \log {\left (\frac {d}{e} + x \right )}}{2 d^{3} e^{2} + 4 d^{2} e^{3} x + 2 d e^{4} x^{2}} - \frac {b d e n x}{2 d^{3} e^{2} + 4 d^{2} e^{3} x + 2 d e^{4} x^{2}} + \frac {b e^{2} n x^{2} \log {\relax (x )}}{2 d^{3} e^{2} + 4 d^{2} e^{3} x + 2 d e^{4} x^{2}} - \frac {b e^{2} n x^{2} \log {\left (\frac {d}{e} + x \right )}}{2 d^{3} e^{2} + 4 d^{2} e^{3} x + 2 d e^{4} x^{2}} + \frac {b e^{2} x^{2} \log {\relax (c )}}{2 d^{3} e^{2} + 4 d^{2} e^{3} x + 2 d e^{4} x^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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