Optimal. Leaf size=84 \[ -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {2 d e \left (a+b \log \left (c x^n\right )\right )}{x}+e^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {b n (d+4 e x)^2}{4 x^2}-\frac {1}{2} b e^2 n \log ^2(x) \]
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Rubi [A] time = 0.08, antiderivative size = 67, normalized size of antiderivative = 0.80, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {43, 2334, 37, 2301} \[ -\frac {1}{2} \left (\frac {d^2}{x^2}+\frac {4 d e}{x}-2 e^2 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b n (d+4 e x)^2}{4 x^2}-\frac {1}{2} b e^2 n \log ^2(x) \]
Antiderivative was successfully verified.
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Rule 37
Rule 43
Rule 2301
Rule 2334
Rubi steps
\begin {align*} \int \frac {(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx &=-\frac {1}{2} \left (\frac {d^2}{x^2}+\frac {4 d e}{x}-2 e^2 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (-\frac {d (d+4 e x)}{2 x^3}+\frac {e^2 \log (x)}{x}\right ) \, dx\\ &=-\frac {1}{2} \left (\frac {d^2}{x^2}+\frac {4 d e}{x}-2 e^2 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {1}{2} (b d n) \int \frac {d+4 e x}{x^3} \, dx-\left (b e^2 n\right ) \int \frac {\log (x)}{x} \, dx\\ &=-\frac {b n (d+4 e x)^2}{4 x^2}-\frac {1}{2} b e^2 n \log ^2(x)-\frac {1}{2} \left (\frac {d^2}{x^2}+\frac {4 d e}{x}-2 e^2 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}
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Mathematica [A] time = 0.06, size = 84, normalized size = 1.00 \[ -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {2 d e \left (a+b \log \left (c x^n\right )\right )}{x}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 b n}-\frac {b d^2 n}{4 x^2}-\frac {2 b d e n}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 101, normalized size = 1.20 \[ \frac {2 \, b e^{2} n x^{2} \log \relax (x)^{2} - b d^{2} n - 2 \, a d^{2} - 8 \, {\left (b d e n + a d e\right )} x - 2 \, {\left (4 \, b d e x + b d^{2}\right )} \log \relax (c) + 2 \, {\left (2 \, b e^{2} x^{2} \log \relax (c) - 4 \, b d e n x + 2 \, a e^{2} x^{2} - b d^{2} n\right )} \log \relax (x)}{4 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 105, normalized size = 1.25 \[ \frac {2 \, b n x^{2} e^{2} \log \relax (x)^{2} - 8 \, b d n x e \log \relax (x) + 4 \, b x^{2} e^{2} \log \relax (c) \log \relax (x) - 8 \, b d n x e - 8 \, b d x e \log \relax (c) - 2 \, b d^{2} n \log \relax (x) + 4 \, a x^{2} e^{2} \log \relax (x) - b d^{2} n - 8 \, a d x e - 2 \, b d^{2} \log \relax (c) - 2 \, a d^{2}}{4 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.24, size = 418, normalized size = 4.98 \[ -\frac {\left (-2 e^{2} x^{2} \ln \relax (x )+4 d e x +d^{2}\right ) b \ln \left (x^{n}\right )}{2 x^{2}}-\frac {2 i \pi b \,e^{2} x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) \ln \relax (x )-2 i \pi b \,e^{2} x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (x )-2 i \pi b \,e^{2} x^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (x )+2 i \pi b \,e^{2} x^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \relax (x )+2 b \,e^{2} n \,x^{2} \ln \relax (x )^{2}-4 i \pi b d e x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+4 i \pi b d e x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+4 i \pi b d e x \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-4 i \pi b d e x \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-4 b \,e^{2} x^{2} \ln \relax (c ) \ln \relax (x )-4 a \,e^{2} x^{2} \ln \relax (x )-i \pi b \,d^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i \pi b \,d^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b \,d^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b \,d^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+8 b d e n x +8 b d e x \ln \relax (c )+8 a d e x +b \,d^{2} n +2 b \,d^{2} \ln \relax (c )+2 a \,d^{2}}{4 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 90, normalized size = 1.07 \[ \frac {b e^{2} \log \left (c x^{n}\right )^{2}}{2 \, n} + a e^{2} \log \relax (x) - \frac {2 \, b d e n}{x} - \frac {2 \, b d e \log \left (c x^{n}\right )}{x} - \frac {b d^{2} n}{4 \, x^{2}} - \frac {2 \, a d e}{x} - \frac {b d^{2} \log \left (c x^{n}\right )}{2 \, x^{2}} - \frac {a d^{2}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.69, size = 99, normalized size = 1.18 \[ \ln \relax (x)\,\left (a\,e^2+\frac {3\,b\,e^2\,n}{2}\right )-\frac {a\,d^2+x\,\left (4\,a\,d\,e+4\,b\,d\,e\,n\right )+\frac {b\,d^2\,n}{2}}{2\,x^2}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {b\,d^2}{2}+2\,b\,d\,e\,x+\frac {3\,b\,e^2\,x^2}{2}\right )}{x^2}+\frac {b\,e^2\,{\ln \left (c\,x^n\right )}^2}{2\,n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.47, size = 99, normalized size = 1.18 \[ - \frac {a d^{2}}{2 x^{2}} - \frac {2 a d e}{x} + a e^{2} \log {\relax (x )} + b d^{2} \left (- \frac {n}{4 x^{2}} - \frac {\log {\left (c x^{n} \right )}}{2 x^{2}}\right ) + 2 b d e \left (- \frac {n}{x} - \frac {\log {\left (c x^{n} \right )}}{x}\right ) - b e^{2} \left (\begin {cases} - \log {\relax (c )} \log {\relax (x )} & \text {for}\: n = 0 \\- \frac {\log {\left (c x^{n} \right )}^{2}}{2 n} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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