Optimal. Leaf size=84 \[ -\frac {b n (d+4 e x)^2}{4 x^2}-\frac {1}{2} b e^2 n \log ^2(x)-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {2 d e \left (a+b \log \left (c x^n\right )\right )}{x}+e^2 \log (x) \left (a+b \log \left (c x^n\right )\right ) \]
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Rubi [A]
time = 0.06, antiderivative size = 84, normalized size of antiderivative = 1.00, 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 = {45, 2372, 37,
2338} \begin {gather*} -\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) \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 2338
Rule 2372
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 = 0.04, size = 84, normalized size = 1.00 \begin {gather*} -\frac {b d^2 n}{4 x^2}-\frac {2 b d e n}{x}-\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} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.16, size = 418, normalized size = 4.98
method | result | size |
risch | \(-\frac {b \left (-2 e^{2} \ln \left (x \right ) x^{2}+4 d e x +d^{2}\right ) \ln \left (x^{n}\right )}{2 x^{2}}-\frac {-4 i \pi b d e x \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-4 i \pi b d e x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )-i \pi b \,d^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i \pi b \,d^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b \,d^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+i \pi b \,d^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+2 i \ln \left (x \right ) \pi b \,e^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) x^{2}-2 i \ln \left (x \right ) \pi b \,e^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{2}+4 i \pi b d e x \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-2 i \ln \left (x \right ) \pi b \,e^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{2}+4 i \pi b d e x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+2 i \ln \left (x \right ) \pi b \,e^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3} x^{2}+2 e^{2} b n \ln \left (x \right )^{2} x^{2}-4 \ln \left (x \right ) \ln \left (c \right ) b \,e^{2} x^{2}-4 \ln \left (x \right ) a \,e^{2} x^{2}+8 \ln \left (c \right ) b d e x +8 b d e n x +2 d^{2} b \ln \left (c \right )+8 a d e x +b \,d^{2} n +2 a \,d^{2}}{4 x^{2}}\) | \(418\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 91, normalized size = 1.08 \begin {gather*} -\frac {2 \, b d n e}{x} - \frac {2 \, b d e \log \left (c x^{n}\right )}{x} + \frac {b e^{2} \log \left (c x^{n}\right )^{2}}{2 \, n} + a e^{2} \log \left (x\right ) - \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 99, normalized size = 1.18 \begin {gather*} \frac {2 \, b n x^{2} e^{2} \log \left (x\right )^{2} - b d^{2} n - 2 \, a d^{2} - 8 \, {\left (b d n + a d\right )} x e - 2 \, {\left (4 \, b d x e + b d^{2}\right )} \log \left (c\right ) - 2 \, {\left (4 \, b d n x e - 2 \, b x^{2} e^{2} \log \left (c\right ) + b d^{2} n - 2 \, a x^{2} e^{2}\right )} \log \left (x\right )}{4 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 3.14, size = 99, normalized size = 1.18 \begin {gather*} - \frac {a d^{2}}{2 x^{2}} - \frac {2 a d e}{x} + a e^{2} \log {\left (x \right )} + 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 {\left (c \right )} \log {\left (x \right )} & \text {for}\: n = 0 \\- \frac {\log {\left (c x^{n} \right )}^{2}}{2 n} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.72, size = 105, normalized size = 1.25 \begin {gather*} \frac {2 \, b n x^{2} e^{2} \log \left (x\right )^{2} - 8 \, b d n x e \log \left (x\right ) + 4 \, b x^{2} e^{2} \log \left (c\right ) \log \left (x\right ) - 8 \, b d n x e - 8 \, b d x e \log \left (c\right ) - 2 \, b d^{2} n \log \left (x\right ) + 4 \, a x^{2} e^{2} \log \left (x\right ) - b d^{2} n - 8 \, a d x e - 2 \, b d^{2} \log \left (c\right ) - 2 \, a d^{2}}{4 \, x^{2}} \end {gather*}
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 \begin {gather*} \ln \left (x\right )\,\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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