Optimal. Leaf size=95 \[ -\frac {b d^2 n}{25 x^5}-\frac {b d e n}{8 x^4}-\frac {b e^2 n}{9 x^3}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {d e \left (a+b \log \left (c x^n\right )\right )}{2 x^4}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{3 x^3} \]
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Rubi [A]
time = 0.05, antiderivative size = 95, 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, 12,
14} \begin {gather*} -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {d e \left (a+b \log \left (c x^n\right )\right )}{2 x^4}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {b d^2 n}{25 x^5}-\frac {b d e n}{8 x^4}-\frac {b e^2 n}{9 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 45
Rule 2372
Rubi steps
\begin {align*} \int \frac {(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx &=-\frac {1}{30} \left (\frac {6 d^2}{x^5}+\frac {15 d e}{x^4}+\frac {10 e^2}{x^3}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {-6 d^2-15 d e x-10 e^2 x^2}{30 x^6} \, dx\\ &=-\frac {1}{30} \left (\frac {6 d^2}{x^5}+\frac {15 d e}{x^4}+\frac {10 e^2}{x^3}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{30} (b n) \int \frac {-6 d^2-15 d e x-10 e^2 x^2}{x^6} \, dx\\ &=-\frac {1}{30} \left (\frac {6 d^2}{x^5}+\frac {15 d e}{x^4}+\frac {10 e^2}{x^3}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{30} (b n) \int \left (-\frac {6 d^2}{x^6}-\frac {15 d e}{x^5}-\frac {10 e^2}{x^4}\right ) \, dx\\ &=-\frac {b d^2 n}{25 x^5}-\frac {b d e n}{8 x^4}-\frac {b e^2 n}{9 x^3}-\frac {1}{30} \left (\frac {6 d^2}{x^5}+\frac {15 d e}{x^4}+\frac {10 e^2}{x^3}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 80, normalized size = 0.84 \begin {gather*} -\frac {60 a \left (6 d^2+15 d e x+10 e^2 x^2\right )+b n \left (72 d^2+225 d e x+200 e^2 x^2\right )+60 b \left (6 d^2+15 d e x+10 e^2 x^2\right ) \log \left (c x^n\right )}{1800 x^5} \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 = 0.12, size = 403, normalized size = 4.24
method | result | size |
risch | \(-\frac {b \left (10 e^{2} x^{2}+15 d e x +6 d^{2}\right ) \ln \left (x^{n}\right )}{30 x^{5}}-\frac {-300 i \pi b \,e^{2} x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )-180 i \pi b \,d^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )-450 i \pi b d e x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+180 i \pi b \,d^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+600 \ln \left (c \right ) b \,e^{2} x^{2}+200 b \,e^{2} n \,x^{2}+600 a \,e^{2} x^{2}-450 i \pi b d e x \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-300 i \pi b \,e^{2} x^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-180 i \pi b \,d^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+300 i \pi b \,e^{2} x^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+900 \ln \left (c \right ) b d e x +225 b d e n x +900 a d e x +450 i \pi b d e x \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+450 i \pi b d e x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+300 i \pi b \,e^{2} x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+180 i \pi b \,d^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+360 d^{2} b \ln \left (c \right )+72 b \,d^{2} n +360 a \,d^{2}}{1800 x^{5}}\) | \(403\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 100, normalized size = 1.05 \begin {gather*} -\frac {b n e^{2}}{9 \, x^{3}} - \frac {b d n e}{8 \, x^{4}} - \frac {b e^{2} \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac {b d e \log \left (c x^{n}\right )}{2 \, x^{4}} - \frac {b d^{2} n}{25 \, x^{5}} - \frac {a e^{2}}{3 \, x^{3}} - \frac {a d e}{2 \, x^{4}} - \frac {b d^{2} \log \left (c x^{n}\right )}{5 \, x^{5}} - \frac {a d^{2}}{5 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 102, normalized size = 1.07 \begin {gather*} -\frac {72 \, b d^{2} n + 200 \, {\left (b n + 3 \, a\right )} x^{2} e^{2} + 360 \, a d^{2} + 225 \, {\left (b d n + 4 \, a d\right )} x e + 60 \, {\left (10 \, b x^{2} e^{2} + 15 \, b d x e + 6 \, b d^{2}\right )} \log \left (c\right ) + 60 \, {\left (10 \, b n x^{2} e^{2} + 15 \, b d n x e + 6 \, b d^{2} n\right )} \log \left (x\right )}{1800 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.73, size = 117, normalized size = 1.23 \begin {gather*} - \frac {a d^{2}}{5 x^{5}} - \frac {a d e}{2 x^{4}} - \frac {a e^{2}}{3 x^{3}} - \frac {b d^{2} n}{25 x^{5}} - \frac {b d^{2} \log {\left (c x^{n} \right )}}{5 x^{5}} - \frac {b d e n}{8 x^{4}} - \frac {b d e \log {\left (c x^{n} \right )}}{2 x^{4}} - \frac {b e^{2} n}{9 x^{3}} - \frac {b e^{2} \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 = 1.51, size = 108, normalized size = 1.14 \begin {gather*} -\frac {600 \, b n x^{2} e^{2} \log \left (x\right ) + 900 \, b d n x e \log \left (x\right ) + 200 \, b n x^{2} e^{2} + 225 \, b d n x e + 600 \, b x^{2} e^{2} \log \left (c\right ) + 900 \, b d x e \log \left (c\right ) + 360 \, b d^{2} n \log \left (x\right ) + 72 \, b d^{2} n + 600 \, a x^{2} e^{2} + 900 \, a d x e + 360 \, b d^{2} \log \left (c\right ) + 360 \, a d^{2}}{1800 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.63, size = 85, normalized size = 0.89 \begin {gather*} -\frac {x^2\,\left (10\,a\,e^2+\frac {10\,b\,e^2\,n}{3}\right )+6\,a\,d^2+x\,\left (15\,a\,d\,e+\frac {15\,b\,d\,e\,n}{4}\right )+\frac {6\,b\,d^2\,n}{5}}{30\,x^5}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {b\,d^2}{5}+\frac {b\,d\,e\,x}{2}+\frac {b\,e^2\,x^2}{3}\right )}{x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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