Optimal. Leaf size=131 \[ -\frac {b d^3 n}{16 x^4}-\frac {3 b d^2 e n}{4 x^2}-\frac {1}{4} b e^3 n x^2-\frac {3}{2} b d e^2 n \log ^2(x)-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{4 x^4}-\frac {3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{2 x^2}+\frac {1}{2} e^3 x^2 \left (a+b \log \left (c x^n\right )\right )+3 d e^2 \log (x) \left (a+b \log \left (c x^n\right )\right ) \]
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Rubi [A]
time = 0.09, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {272, 45, 2372,
12, 14, 2338} \begin {gather*} -\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{4 x^4}-\frac {3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{2 x^2}+3 d e^2 \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac {1}{2} e^3 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {b d^3 n}{16 x^4}-\frac {3 b d^2 e n}{4 x^2}-\frac {3}{2} b d e^2 n \log ^2(x)-\frac {1}{4} b e^3 n x^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 45
Rule 272
Rule 2338
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^5} \, dx &=-\frac {1}{4} \left (\frac {d^3}{x^4}+\frac {6 d^2 e}{x^2}-2 e^3 x^2-12 d e^2 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {-d^3-6 d^2 e x^2+2 e^3 x^6+12 d e^2 x^4 \log (x)}{4 x^5} \, dx\\ &=-\frac {1}{4} \left (\frac {d^3}{x^4}+\frac {6 d^2 e}{x^2}-2 e^3 x^2-12 d e^2 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} (b n) \int \frac {-d^3-6 d^2 e x^2+2 e^3 x^6+12 d e^2 x^4 \log (x)}{x^5} \, dx\\ &=-\frac {1}{4} \left (\frac {d^3}{x^4}+\frac {6 d^2 e}{x^2}-2 e^3 x^2-12 d e^2 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} (b n) \int \left (\frac {-d^3-6 d^2 e x^2+2 e^3 x^6}{x^5}+\frac {12 d e^2 \log (x)}{x}\right ) \, dx\\ &=-\frac {1}{4} \left (\frac {d^3}{x^4}+\frac {6 d^2 e}{x^2}-2 e^3 x^2-12 d e^2 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} (b n) \int \frac {-d^3-6 d^2 e x^2+2 e^3 x^6}{x^5} \, dx-\left (3 b d e^2 n\right ) \int \frac {\log (x)}{x} \, dx\\ &=-\frac {3}{2} b d e^2 n \log ^2(x)-\frac {1}{4} \left (\frac {d^3}{x^4}+\frac {6 d^2 e}{x^2}-2 e^3 x^2-12 d e^2 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} (b n) \int \left (-\frac {d^3}{x^5}-\frac {6 d^2 e}{x^3}+2 e^3 x\right ) \, dx\\ &=-\frac {b d^3 n}{16 x^4}-\frac {3 b d^2 e n}{4 x^2}-\frac {1}{4} b e^3 n x^2-\frac {3}{2} b d e^2 n \log ^2(x)-\frac {1}{4} \left (\frac {d^3}{x^4}+\frac {6 d^2 e}{x^2}-2 e^3 x^2-12 d e^2 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 115, normalized size = 0.88 \begin {gather*} \frac {1}{16} \left (-\frac {b d^3 n}{x^4}-\frac {12 b d^2 e n}{x^2}-4 b e^3 n x^2-\frac {4 d^3 \left (a+b \log \left (c x^n\right )\right )}{x^4}-\frac {24 d^2 e \left (a+b \log \left (c x^n\right )\right )}{x^2}+8 e^3 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {24 d e^2 \left (a+b \log \left (c x^n\right )\right )^2}{b 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 = 0.17, size = 602, normalized size = 4.60
method | result | size |
risch | \(-\frac {b \left (-2 e^{3} x^{6}-12 d \,e^{2} \ln \left (x \right ) x^{4}+6 d^{2} e \,x^{2}+d^{3}\right ) \ln \left (x^{n}\right )}{4 x^{4}}-\frac {-8 \ln \left (c \right ) b \,e^{3} x^{6}-8 x^{6} a \,e^{3}-4 i \pi b \,e^{3} x^{6} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-24 i \ln \left (x \right ) \pi b d \,e^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{4}+24 a \,d^{2} x^{2} e +24 i \ln \left (x \right ) \pi b d \,e^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3} x^{4}+4 a \,d^{3}+4 i \pi b \,e^{3} x^{6} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+24 i \ln \left (x \right ) \pi b d \,e^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) x^{4}+4 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 )+b \,d^{3} n +4 d^{3} b \ln \left (c \right )+24 \ln \left (c \right ) b \,d^{2} x^{2} e -12 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 )-24 i \ln \left (x \right ) \pi b d \,e^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{4}+4 b \,e^{3} n \,x^{6}-12 i \pi b \,d^{2} e \,x^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-4 i \pi b \,e^{3} x^{6} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+12 b \,d^{2} e n \,x^{2}-48 \ln \left (x \right ) a d \,e^{2} x^{4}-48 \ln \left (x \right ) \ln \left (c \right ) b d \,e^{2} x^{4}+24 e^{2} d b n \ln \left (x \right )^{2} x^{4}+2 i \pi b \,d^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+2 i \pi b \,d^{3} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+12 i \pi b \,d^{2} x^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} e +12 i \pi b \,d^{2} e \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-2 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 )-2 i \pi b \,d^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{16 x^{4}}\) | \(602\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 131, normalized size = 1.00 \begin {gather*} -\frac {1}{4} \, b n x^{2} e^{3} + \frac {1}{2} \, b x^{2} e^{3} \log \left (c x^{n}\right ) + \frac {1}{2} \, a x^{2} e^{3} + \frac {3 \, b d e^{2} \log \left (c x^{n}\right )^{2}}{2 \, n} + 3 \, a d e^{2} \log \left (x\right ) - \frac {3 \, b d^{2} n e}{4 \, x^{2}} - \frac {3 \, b d^{2} e \log \left (c x^{n}\right )}{2 \, x^{2}} - \frac {3 \, a d^{2} e}{2 \, x^{2}} - \frac {b d^{3} n}{16 \, x^{4}} - \frac {b d^{3} \log \left (c x^{n}\right )}{4 \, x^{4}} - \frac {a d^{3}}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 150, normalized size = 1.15 \begin {gather*} \frac {24 \, b d n x^{4} e^{2} \log \left (x\right )^{2} - 4 \, {\left (b n - 2 \, a\right )} x^{6} e^{3} - b d^{3} n - 4 \, a d^{3} - 12 \, {\left (b d^{2} n + 2 \, a d^{2}\right )} x^{2} e + 4 \, {\left (2 \, b x^{6} e^{3} - 6 \, b d^{2} x^{2} e - b d^{3}\right )} \log \left (c\right ) + 4 \, {\left (2 \, b n x^{6} e^{3} + 12 \, b d x^{4} e^{2} \log \left (c\right ) + 12 \, a d x^{4} e^{2} - 6 \, b d^{2} n x^{2} e - b d^{3} n\right )} \log \left (x\right )}{16 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.59, size = 209, normalized size = 1.60 \begin {gather*} \begin {cases} - \frac {a d^{3}}{4 x^{4}} - \frac {3 a d^{2} e}{2 x^{2}} + \frac {3 a d e^{2} \log {\left (c x^{n} \right )}}{n} + \frac {a e^{3} x^{2}}{2} - \frac {b d^{3} n}{16 x^{4}} - \frac {b d^{3} \log {\left (c x^{n} \right )}}{4 x^{4}} - \frac {3 b d^{2} e n}{4 x^{2}} - \frac {3 b d^{2} e \log {\left (c x^{n} \right )}}{2 x^{2}} + \frac {3 b d e^{2} \log {\left (c x^{n} \right )}^{2}}{2 n} - \frac {b e^{3} n x^{2}}{4} + \frac {b e^{3} x^{2} \log {\left (c x^{n} \right )}}{2} & \text {for}\: n \neq 0 \\\left (a + b \log {\left (c \right )}\right ) \left (- \frac {d^{3}}{4 x^{4}} - \frac {3 d^{2} e}{2 x^{2}} + 3 d e^{2} \log {\left (x \right )} + \frac {e^{3} x^{2}}{2}\right ) & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.92, size = 162, normalized size = 1.24 \begin {gather*} \frac {8 \, b n x^{6} e^{3} \log \left (x\right ) + 24 \, b d n x^{4} e^{2} \log \left (x\right )^{2} - 4 \, b n x^{6} e^{3} + 8 \, b x^{6} e^{3} \log \left (c\right ) + 48 \, b d x^{4} e^{2} \log \left (c\right ) \log \left (x\right ) + 8 \, a x^{6} e^{3} + 48 \, a d x^{4} e^{2} \log \left (x\right ) - 24 \, b d^{2} n x^{2} e \log \left (x\right ) - 12 \, b d^{2} n x^{2} e - 24 \, b d^{2} x^{2} e \log \left (c\right ) - 24 \, a d^{2} x^{2} e - 4 \, b d^{3} n \log \left (x\right ) - b d^{3} n - 4 \, b d^{3} \log \left (c\right ) - 4 \, a d^{3}}{16 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.66, size = 149, normalized size = 1.14 \begin {gather*} \ln \left (x\right )\,\left (3\,a\,d\,e^2+\frac {9\,b\,d\,e^2\,n}{4}\right )-\ln \left (c\,x^n\right )\,\left (\frac {\frac {b\,d^3}{4}+\frac {3\,b\,d^2\,e\,x^2}{2}+\frac {9\,b\,d\,e^2\,x^4}{4}+b\,e^3\,x^6}{x^4}-\frac {3\,b\,e^3\,x^2}{2}\right )-\frac {a\,d^3+x^2\,\left (6\,a\,d^2\,e+3\,b\,d^2\,e\,n\right )+\frac {b\,d^3\,n}{4}}{4\,x^4}+\frac {e^3\,x^2\,\left (2\,a-b\,n\right )}{4}+\frac {3\,b\,d\,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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