Optimal. Leaf size=74 \[ \frac {1}{105} \left (35 d^2 x^3+42 d e x^5+15 e^2 x^7\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{9} b d^2 n x^3-\frac {2}{25} b d e n x^5-\frac {1}{49} b e^2 n x^7 \]
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Rubi [A] time = 0.07, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {270, 2334} \[ \frac {1}{105} \left (35 d^2 x^3+42 d e x^5+15 e^2 x^7\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{9} b d^2 n x^3-\frac {2}{25} b d e n x^5-\frac {1}{49} b e^2 n x^7 \]
Antiderivative was successfully verified.
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Rule 270
Rule 2334
Rubi steps
\begin {align*} \int x^2 \left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {1}{105} \left (35 d^2 x^3+42 d e x^5+15 e^2 x^7\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (\frac {d^2 x^2}{3}+\frac {2}{5} d e x^4+\frac {e^2 x^6}{7}\right ) \, dx\\ &=-\frac {1}{9} b d^2 n x^3-\frac {2}{25} b d e n x^5-\frac {1}{49} b e^2 n x^7+\frac {1}{105} \left (35 d^2 x^3+42 d e x^5+15 e^2 x^7\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 95, normalized size = 1.28 \[ \frac {1}{3} d^2 x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {2}{5} d e x^5 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{7} e^2 x^7 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{9} b d^2 n x^3-\frac {2}{25} b d e n x^5-\frac {1}{49} b e^2 n x^7 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 118, normalized size = 1.59 \[ -\frac {1}{49} \, {\left (b e^{2} n - 7 \, a e^{2}\right )} x^{7} - \frac {2}{25} \, {\left (b d e n - 5 \, a d e\right )} x^{5} - \frac {1}{9} \, {\left (b d^{2} n - 3 \, a d^{2}\right )} x^{3} + \frac {1}{105} \, {\left (15 \, b e^{2} x^{7} + 42 \, b d e x^{5} + 35 \, b d^{2} x^{3}\right )} \log \relax (c) + \frac {1}{105} \, {\left (15 \, b e^{2} n x^{7} + 42 \, b d e n x^{5} + 35 \, b d^{2} n x^{3}\right )} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 123, normalized size = 1.66 \[ \frac {1}{7} \, b n x^{7} e^{2} \log \relax (x) - \frac {1}{49} \, b n x^{7} e^{2} + \frac {1}{7} \, b x^{7} e^{2} \log \relax (c) + \frac {2}{5} \, b d n x^{5} e \log \relax (x) + \frac {1}{7} \, a x^{7} e^{2} - \frac {2}{25} \, b d n x^{5} e + \frac {2}{5} \, b d x^{5} e \log \relax (c) + \frac {2}{5} \, a d x^{5} e + \frac {1}{3} \, b d^{2} n x^{3} \log \relax (x) - \frac {1}{9} \, b d^{2} n x^{3} + \frac {1}{3} \, b d^{2} x^{3} \log \relax (c) + \frac {1}{3} \, a d^{2} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.22, size = 434, normalized size = 5.86 \[ -\frac {i \pi b \,e^{2} x^{7} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{14}+\frac {i \pi b \,e^{2} x^{7} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{14}+\frac {i \pi b \,e^{2} x^{7} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{14}-\frac {i \pi b \,e^{2} x^{7} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{14}-\frac {i \pi b d e \,x^{5} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{5}+\frac {i \pi b d e \,x^{5} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{5}+\frac {i \pi b d e \,x^{5} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{5}-\frac {i \pi b d e \,x^{5} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{5}-\frac {b \,e^{2} n \,x^{7}}{49}+\frac {b \,e^{2} x^{7} \ln \relax (c )}{7}+\frac {a \,e^{2} x^{7}}{7}-\frac {i \pi b \,d^{2} x^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{6}+\frac {i \pi b \,d^{2} x^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{6}+\frac {i \pi b \,d^{2} x^{3} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{6}-\frac {i \pi b \,d^{2} x^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{6}-\frac {2 b d e n \,x^{5}}{25}+\frac {2 b d e \,x^{5} \ln \relax (c )}{5}+\frac {2 a d e \,x^{5}}{5}-\frac {b \,d^{2} n \,x^{3}}{9}+\frac {b \,d^{2} x^{3} \ln \relax (c )}{3}+\frac {a \,d^{2} x^{3}}{3}+\frac {\left (15 e^{2} x^{4}+42 d e \,x^{2}+35 d^{2}\right ) b \,x^{3} \ln \left (x^{n}\right )}{105} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 100, normalized size = 1.35 \[ -\frac {1}{49} \, b e^{2} n x^{7} + \frac {1}{7} \, b e^{2} x^{7} \log \left (c x^{n}\right ) + \frac {1}{7} \, a e^{2} x^{7} - \frac {2}{25} \, b d e n x^{5} + \frac {2}{5} \, b d e x^{5} \log \left (c x^{n}\right ) + \frac {2}{5} \, a d e x^{5} - \frac {1}{9} \, b d^{2} n x^{3} + \frac {1}{3} \, b d^{2} x^{3} \log \left (c x^{n}\right ) + \frac {1}{3} \, a d^{2} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.59, size = 82, normalized size = 1.11 \[ \ln \left (c\,x^n\right )\,\left (\frac {b\,d^2\,x^3}{3}+\frac {2\,b\,d\,e\,x^5}{5}+\frac {b\,e^2\,x^7}{7}\right )+\frac {d^2\,x^3\,\left (3\,a-b\,n\right )}{9}+\frac {e^2\,x^7\,\left (7\,a-b\,n\right )}{49}+\frac {2\,d\,e\,x^5\,\left (5\,a-b\,n\right )}{25} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 6.18, size = 158, normalized size = 2.14 \[ \frac {a d^{2} x^{3}}{3} + \frac {2 a d e x^{5}}{5} + \frac {a e^{2} x^{7}}{7} + \frac {b d^{2} n x^{3} \log {\relax (x )}}{3} - \frac {b d^{2} n x^{3}}{9} + \frac {b d^{2} x^{3} \log {\relax (c )}}{3} + \frac {2 b d e n x^{5} \log {\relax (x )}}{5} - \frac {2 b d e n x^{5}}{25} + \frac {2 b d e x^{5} \log {\relax (c )}}{5} + \frac {b e^{2} n x^{7} \log {\relax (x )}}{7} - \frac {b e^{2} n x^{7}}{49} + \frac {b e^{2} x^{7} \log {\relax (c )}}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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