Optimal. Leaf size=44 \[ -\frac {b d n}{x}-b e n x-\frac {d \left (a+b \log \left (c x^n\right )\right )}{x}+e x \left (a+b \log \left (c x^n\right )\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {14, 2372}
\begin {gather*} -\frac {d \left (a+b \log \left (c x^n\right )\right )}{x}+e x \left (a+b \log \left (c x^n\right )\right )-\frac {b d n}{x}-b e n x \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2372
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx &=-\left (\frac {d}{x}-e x\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (e-\frac {d}{x^2}\right ) \, dx\\ &=-\frac {b d n}{x}-b e n x-\left (\frac {d}{x}-e x\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 49, normalized size = 1.11 \begin {gather*} -\frac {a d}{x}-\frac {b d n}{x}+a e x-b e n x-\frac {b d \log \left (c x^n\right )}{x}+b e x \log \left (c x^n\right ) \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 = 0.05, size = 249, normalized size = 5.66
method | result | size |
risch | \(-\frac {b \left (-e \,x^{2}+d \right ) \ln \left (x^{n}\right )}{x}-\frac {i \pi b e \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )-i \pi b e \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b e \,x^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b e \,x^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b d \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b d \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-2 \ln \left (c \right ) b e \,x^{2}+2 b e n \,x^{2}-2 a e \,x^{2}+2 d b \ln \left (c \right )+2 b d n +2 a d}{2 x}\) | \(249\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 52, normalized size = 1.18 \begin {gather*} -b n x e + b x e \log \left (c x^{n}\right ) + a x e - \frac {b d n}{x} - \frac {b d \log \left (c x^{n}\right )}{x} - \frac {a d}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 60, normalized size = 1.36 \begin {gather*} -\frac {{\left (b n - a\right )} x^{2} e + b d n + a d - {\left (b x^{2} e - b d\right )} \log \left (c\right ) - {\left (b n x^{2} e - b d n\right )} \log \left (x\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.23, size = 46, normalized size = 1.05 \begin {gather*} - \frac {a d}{x} + a e x - \frac {b d n}{x} - \frac {b d \log {\left (c x^{n} \right )}}{x} - b e n x + b e x \log {\left (c x^{n} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.74, size = 62, normalized size = 1.41 \begin {gather*} \frac {b n x^{2} e \log \left (x\right ) - b n x^{2} e + b x^{2} e \log \left (c\right ) + a x^{2} e - b d n \log \left (x\right ) - b d n - b d \log \left (c\right ) - a d}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.34, size = 51, normalized size = 1.16 \begin {gather*} e\,x\,\left (a-b\,n\right )-\ln \left (c\,x^n\right )\,\left (\frac {b\,e\,x^2+b\,d}{x}-2\,b\,e\,x\right )-\frac {a\,d+b\,d\,n}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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