Optimal. Leaf size=127 \[ -\frac {b d^3 n}{49 x^7}-\frac {3 b d^2 e n}{25 x^5}-\frac {b d e^2 n}{3 x^3}-\frac {b e^3 n}{x}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {d e^2 \left (a+b \log \left (c x^n\right )\right )}{x^3}-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )}{x} \]
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Rubi [A]
time = 0.07, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {276, 2372, 12,
14} \begin {gather*} -\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {d e^2 \left (a+b \log \left (c x^n\right )\right )}{x^3}-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {b d^3 n}{49 x^7}-\frac {3 b d^2 e n}{25 x^5}-\frac {b d e^2 n}{3 x^3}-\frac {b e^3 n}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 276
Rule 2372
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx &=-\frac {1}{35} \left (\frac {5 d^3}{x^7}+\frac {21 d^2 e}{x^5}+\frac {35 d e^2}{x^3}+\frac {35 e^3}{x}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {-5 d^3-21 d^2 e x^2-35 d e^2 x^4-35 e^3 x^6}{35 x^8} \, dx\\ &=-\frac {1}{35} \left (\frac {5 d^3}{x^7}+\frac {21 d^2 e}{x^5}+\frac {35 d e^2}{x^3}+\frac {35 e^3}{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{35} (b n) \int \frac {-5 d^3-21 d^2 e x^2-35 d e^2 x^4-35 e^3 x^6}{x^8} \, dx\\ &=-\frac {1}{35} \left (\frac {5 d^3}{x^7}+\frac {21 d^2 e}{x^5}+\frac {35 d e^2}{x^3}+\frac {35 e^3}{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{35} (b n) \int \left (-\frac {5 d^3}{x^8}-\frac {21 d^2 e}{x^6}-\frac {35 d e^2}{x^4}-\frac {35 e^3}{x^2}\right ) \, dx\\ &=-\frac {b d^3 n}{49 x^7}-\frac {3 b d^2 e n}{25 x^5}-\frac {b d e^2 n}{3 x^3}-\frac {b e^3 n}{x}-\frac {1}{35} \left (\frac {5 d^3}{x^7}+\frac {21 d^2 e}{x^5}+\frac {35 d e^2}{x^3}+\frac {35 e^3}{x}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 127, normalized size = 1.00 \begin {gather*} -\frac {b d^3 n}{49 x^7}-\frac {3 b d^2 e n}{25 x^5}-\frac {b d e^2 n}{3 x^3}-\frac {b e^3 n}{x}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {d e^2 \left (a+b \log \left (c x^n\right )\right )}{x^3}-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )}{x} \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 = 0.14, size = 587, normalized size = 4.62
method | result | size |
risch | \(-\frac {b \left (35 e^{3} x^{6}+35 d \,e^{2} x^{4}+21 d^{2} e \,x^{2}+5 d^{3}\right ) \ln \left (x^{n}\right )}{35 x^{7}}-\frac {7350 \ln \left (c \right ) b \,e^{3} x^{6}+7350 x^{6} a \,e^{3}+3675 i \pi b \,e^{3} x^{6} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+2205 i \pi b \,d^{2} x^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} e +3675 i \pi b d \,e^{2} x^{4} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+3675 i \pi b d \,e^{2} x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+7350 x^{4} a d \,e^{2}+4410 a \,d^{2} x^{2} e +1050 a \,d^{3}-3675 i \pi b d \,e^{2} x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )-3675 i \pi b \,e^{3} x^{6} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-3675 i \pi b \,e^{3} x^{6} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+150 b \,d^{3} n +1050 d^{3} b \ln \left (c \right )+4410 \ln \left (c \right ) b \,d^{2} x^{2} e +7350 \ln \left (c \right ) b d \,e^{2} x^{4}-2205 i \pi b \,d^{2} e \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+7350 b \,e^{3} n \,x^{6}-2205 i \pi b \,d^{2} e \,x^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+3675 i \pi b \,e^{3} x^{6} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-3675 i \pi b d \,e^{2} x^{4} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+2450 b d \,e^{2} n \,x^{4}+882 b \,d^{2} e n \,x^{2}+525 i \pi b \,d^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+525 i \pi b \,d^{3} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+2205 i \pi b \,d^{2} e \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-525 i \pi b \,d^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )-525 i \pi b \,d^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{7350 x^{7}}\) | \(587\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 140, normalized size = 1.10 \begin {gather*} -\frac {b n e^{3}}{x} - \frac {b e^{3} \log \left (c x^{n}\right )}{x} - \frac {a e^{3}}{x} - \frac {b d n e^{2}}{3 \, x^{3}} - \frac {b d e^{2} \log \left (c x^{n}\right )}{x^{3}} - \frac {a d e^{2}}{x^{3}} - \frac {3 \, b d^{2} n e}{25 \, x^{5}} - \frac {3 \, b d^{2} e \log \left (c x^{n}\right )}{5 \, x^{5}} - \frac {3 \, a d^{2} e}{5 \, x^{5}} - \frac {b d^{3} n}{49 \, x^{7}} - \frac {b d^{3} \log \left (c x^{n}\right )}{7 \, x^{7}} - \frac {a d^{3}}{7 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 149, normalized size = 1.17 \begin {gather*} -\frac {3675 \, {\left (b n + a\right )} x^{6} e^{3} + 1225 \, {\left (b d n + 3 \, a d\right )} x^{4} e^{2} + 75 \, b d^{3} n + 525 \, a d^{3} + 441 \, {\left (b d^{2} n + 5 \, a d^{2}\right )} x^{2} e + 105 \, {\left (35 \, b x^{6} e^{3} + 35 \, b d x^{4} e^{2} + 21 \, b d^{2} x^{2} e + 5 \, b d^{3}\right )} \log \left (c\right ) + 105 \, {\left (35 \, b n x^{6} e^{3} + 35 \, b d n x^{4} e^{2} + 21 \, b d^{2} n x^{2} e + 5 \, b d^{3} n\right )} \log \left (x\right )}{3675 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.57, size = 158, normalized size = 1.24 \begin {gather*} - \frac {a d^{3}}{7 x^{7}} - \frac {3 a d^{2} e}{5 x^{5}} - \frac {a d e^{2}}{x^{3}} - \frac {a e^{3}}{x} - \frac {b d^{3} n}{49 x^{7}} - \frac {b d^{3} \log {\left (c x^{n} \right )}}{7 x^{7}} - \frac {3 b d^{2} e n}{25 x^{5}} - \frac {3 b d^{2} e \log {\left (c x^{n} \right )}}{5 x^{5}} - \frac {b d e^{2} n}{3 x^{3}} - \frac {b d e^{2} \log {\left (c x^{n} \right )}}{x^{3}} - \frac {b e^{3} n}{x} - \frac {b e^{3} \log {\left (c x^{n} \right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 7.91, size = 166, normalized size = 1.31 \begin {gather*} -\frac {3675 \, b n x^{6} e^{3} \log \left (x\right ) + 3675 \, b n x^{6} e^{3} + 3675 \, b x^{6} e^{3} \log \left (c\right ) + 3675 \, b d n x^{4} e^{2} \log \left (x\right ) + 3675 \, a x^{6} e^{3} + 1225 \, b d n x^{4} e^{2} + 3675 \, b d x^{4} e^{2} \log \left (c\right ) + 2205 \, b d^{2} n x^{2} e \log \left (x\right ) + 3675 \, a d x^{4} e^{2} + 441 \, b d^{2} n x^{2} e + 2205 \, b d^{2} x^{2} e \log \left (c\right ) + 2205 \, a d^{2} x^{2} e + 525 \, b d^{3} n \log \left (x\right ) + 75 \, b d^{3} n + 525 \, b d^{3} \log \left (c\right ) + 525 \, a d^{3}}{3675 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.80, size = 123, normalized size = 0.97 \begin {gather*} -\frac {x^6\,\left (35\,a\,e^3+35\,b\,e^3\,n\right )+5\,a\,d^3+x^2\,\left (21\,a\,d^2\,e+\frac {21\,b\,d^2\,e\,n}{5}\right )+x^4\,\left (35\,a\,d\,e^2+\frac {35\,b\,d\,e^2\,n}{3}\right )+\frac {5\,b\,d^3\,n}{7}}{35\,x^7}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {b\,d^3}{7}+\frac {3\,b\,d^2\,e\,x^2}{5}+b\,d\,e^2\,x^4+b\,e^3\,x^6\right )}{x^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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