Optimal. Leaf size=57 \[ -\frac {b d n}{25 x^5}-\frac {b e n}{9 x^3}-\frac {d \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{3 x^3} \]
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Rubi [A]
time = 0.03, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {14, 2372, 12}
\begin {gather*} -\frac {d \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {b d n}{25 x^5}-\frac {b e n}{9 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
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^6} \, dx &=-\frac {1}{15} \left (\frac {3 d}{x^5}+\frac {5 e}{x^3}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {-3 d-5 e x^2}{15 x^6} \, dx\\ &=-\frac {1}{15} \left (\frac {3 d}{x^5}+\frac {5 e}{x^3}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{15} (b n) \int \frac {-3 d-5 e x^2}{x^6} \, dx\\ &=-\frac {1}{15} \left (\frac {3 d}{x^5}+\frac {5 e}{x^3}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{15} (b n) \int \left (-\frac {3 d}{x^6}-\frac {5 e}{x^4}\right ) \, dx\\ &=-\frac {b d n}{25 x^5}-\frac {b e n}{9 x^3}-\frac {1}{15} \left (\frac {3 d}{x^5}+\frac {5 e}{x^3}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 69, normalized size = 1.21 \begin {gather*} -\frac {a d}{5 x^5}-\frac {b d n}{25 x^5}-\frac {a e}{3 x^3}-\frac {b e n}{9 x^3}-\frac {b d \log \left (c x^n\right )}{5 x^5}-\frac {b e \log \left (c x^n\right )}{3 x^3} \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 = 251, normalized size = 4.40
method | result | size |
risch | \(-\frac {b \left (5 e \,x^{2}+3 d \right ) \ln \left (x^{n}\right )}{15 x^{5}}-\frac {-75 i \pi b e \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+75 i \pi b e \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+75 i \pi b e \,x^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-75 i \pi b e \,x^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+150 \ln \left (c \right ) b e \,x^{2}+50 b e n \,x^{2}+150 a e \,x^{2}-45 i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+45 i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+45 i \pi b d \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-45 i \pi b d \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+90 d b \ln \left (c \right )+18 b d n +90 a d}{450 x^{5}}\) | \(251\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 60, normalized size = 1.05 \begin {gather*} -\frac {b n e}{9 \, x^{3}} - \frac {b e \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac {a e}{3 \, x^{3}} - \frac {b d n}{25 \, x^{5}} - \frac {b d \log \left (c x^{n}\right )}{5 \, x^{5}} - \frac {a d}{5 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 65, normalized size = 1.14 \begin {gather*} -\frac {25 \, {\left (b n + 3 \, a\right )} x^{2} e + 9 \, b d n + 45 \, a d + 15 \, {\left (5 \, b x^{2} e + 3 \, b d\right )} \log \left (c\right ) + 15 \, {\left (5 \, b n x^{2} e + 3 \, b d n\right )} \log \left (x\right )}{225 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.75, size = 68, normalized size = 1.19 \begin {gather*} - \frac {a d}{5 x^{5}} - \frac {a e}{3 x^{3}} - \frac {b d n}{25 x^{5}} - \frac {b d \log {\left (c x^{n} \right )}}{5 x^{5}} - \frac {b e n}{9 x^{3}} - \frac {b e \log {\left (c x^{n} \right )}}{3 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.63, size = 66, normalized size = 1.16 \begin {gather*} -\frac {75 \, b n x^{2} e \log \left (x\right ) + 25 \, b n x^{2} e + 75 \, b x^{2} e \log \left (c\right ) + 75 \, a x^{2} e + 45 \, b d n \log \left (x\right ) + 9 \, b d n + 45 \, b d \log \left (c\right ) + 45 \, a d}{225 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.61, size = 53, normalized size = 0.93 \begin {gather*} -\frac {\left (5\,a\,e+\frac {5\,b\,e\,n}{3}\right )\,x^2+3\,a\,d+\frac {3\,b\,d\,n}{5}}{15\,x^5}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {b\,e\,x^2}{3}+\frac {b\,d}{5}\right )}{x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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