Optimal. Leaf size=178 \[ \frac {1}{2} d^2 x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} b d^2 n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {2}{3} d e x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac {4}{9} b d e n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{4} e^2 x^4 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{8} b e^2 n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{4} b^2 d^2 n^2 x^2+\frac {4}{27} b^2 d e n^2 x^3+\frac {1}{32} b^2 e^2 n^2 x^4 \]
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Rubi [A] time = 0.18, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2353, 2305, 2304} \[ \frac {1}{2} d^2 x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} b d^2 n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {2}{3} d e x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac {4}{9} b d e n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{4} e^2 x^4 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{8} b e^2 n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{4} b^2 d^2 n^2 x^2+\frac {4}{27} b^2 d e n^2 x^3+\frac {1}{32} b^2 e^2 n^2 x^4 \]
Antiderivative was successfully verified.
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Rule 2304
Rule 2305
Rule 2353
Rubi steps
\begin {align*} \int x (d+e x)^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx &=\int \left (d^2 x \left (a+b \log \left (c x^n\right )\right )^2+2 d e x^2 \left (a+b \log \left (c x^n\right )\right )^2+e^2 x^3 \left (a+b \log \left (c x^n\right )\right )^2\right ) \, dx\\ &=d^2 \int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx+(2 d e) \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx+e^2 \int x^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx\\ &=\frac {1}{2} d^2 x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac {2}{3} d e x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{4} e^2 x^4 \left (a+b \log \left (c x^n\right )\right )^2-\left (b d^2 n\right ) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx-\frac {1}{3} (4 b d e n) \int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx-\frac {1}{2} \left (b e^2 n\right ) \int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx\\ &=\frac {1}{4} b^2 d^2 n^2 x^2+\frac {4}{27} b^2 d e n^2 x^3+\frac {1}{32} b^2 e^2 n^2 x^4-\frac {1}{2} b d^2 n x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {4}{9} b d e n x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{8} b e^2 n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{2} d^2 x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac {2}{3} d e x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{4} e^2 x^4 \left (a+b \log \left (c x^n\right )\right )^2\\ \end {align*}
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Mathematica [A] time = 0.09, size = 134, normalized size = 0.75 \[ \frac {1}{864} x^2 \left (432 d^2 \left (a+b \log \left (c x^n\right )\right )^2+216 b d^2 n \left (-2 a-2 b \log \left (c x^n\right )+b n\right )+576 d e x \left (a+b \log \left (c x^n\right )\right )^2+128 b d e n x \left (-3 a-3 b \log \left (c x^n\right )+b n\right )+216 e^2 x^2 \left (a+b \log \left (c x^n\right )\right )^2+27 b e^2 n x^2 \left (-4 a-4 b \log \left (c x^n\right )+b n\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 363, normalized size = 2.04 \[ \frac {1}{32} \, {\left (b^{2} e^{2} n^{2} - 4 \, a b e^{2} n + 8 \, a^{2} e^{2}\right )} x^{4} + \frac {2}{27} \, {\left (2 \, b^{2} d e n^{2} - 6 \, a b d e n + 9 \, a^{2} d e\right )} x^{3} + \frac {1}{4} \, {\left (b^{2} d^{2} n^{2} - 2 \, a b d^{2} n + 2 \, a^{2} d^{2}\right )} x^{2} + \frac {1}{12} \, {\left (3 \, b^{2} e^{2} x^{4} + 8 \, b^{2} d e x^{3} + 6 \, b^{2} d^{2} x^{2}\right )} \log \relax (c)^{2} + \frac {1}{12} \, {\left (3 \, b^{2} e^{2} n^{2} x^{4} + 8 \, b^{2} d e n^{2} x^{3} + 6 \, b^{2} d^{2} n^{2} x^{2}\right )} \log \relax (x)^{2} - \frac {1}{72} \, {\left (9 \, {\left (b^{2} e^{2} n - 4 \, a b e^{2}\right )} x^{4} + 32 \, {\left (b^{2} d e n - 3 \, a b d e\right )} x^{3} + 36 \, {\left (b^{2} d^{2} n - 2 \, a b d^{2}\right )} x^{2}\right )} \log \relax (c) - \frac {1}{72} \, {\left (9 \, {\left (b^{2} e^{2} n^{2} - 4 \, a b e^{2} n\right )} x^{4} + 32 \, {\left (b^{2} d e n^{2} - 3 \, a b d e n\right )} x^{3} + 36 \, {\left (b^{2} d^{2} n^{2} - 2 \, a b d^{2} n\right )} x^{2} - 12 \, {\left (3 \, b^{2} e^{2} n x^{4} + 8 \, b^{2} d e n x^{3} + 6 \, b^{2} d^{2} n x^{2}\right )} \log \relax (c)\right )} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.32, size = 408, normalized size = 2.29 \[ \frac {1}{4} \, b^{2} n^{2} x^{4} e^{2} \log \relax (x)^{2} + \frac {2}{3} \, b^{2} d n^{2} x^{3} e \log \relax (x)^{2} - \frac {1}{8} \, b^{2} n^{2} x^{4} e^{2} \log \relax (x) - \frac {4}{9} \, b^{2} d n^{2} x^{3} e \log \relax (x) + \frac {1}{2} \, b^{2} n x^{4} e^{2} \log \relax (c) \log \relax (x) + \frac {4}{3} \, b^{2} d n x^{3} e \log \relax (c) \log \relax (x) + \frac {1}{2} \, b^{2} d^{2} n^{2} x^{2} \log \relax (x)^{2} + \frac {1}{32} \, b^{2} n^{2} x^{4} e^{2} + \frac {4}{27} \, b^{2} d n^{2} x^{3} e - \frac {1}{8} \, b^{2} n x^{4} e^{2} \log \relax (c) - \frac {4}{9} \, b^{2} d n x^{3} e \log \relax (c) + \frac {1}{4} \, b^{2} x^{4} e^{2} \log \relax (c)^{2} + \frac {2}{3} \, b^{2} d x^{3} e \log \relax (c)^{2} - \frac {1}{2} \, b^{2} d^{2} n^{2} x^{2} \log \relax (x) + \frac {1}{2} \, a b n x^{4} e^{2} \log \relax (x) + \frac {4}{3} \, a b d n x^{3} e \log \relax (x) + b^{2} d^{2} n x^{2} \log \relax (c) \log \relax (x) + \frac {1}{4} \, b^{2} d^{2} n^{2} x^{2} - \frac {1}{8} \, a b n x^{4} e^{2} - \frac {4}{9} \, a b d n x^{3} e - \frac {1}{2} \, b^{2} d^{2} n x^{2} \log \relax (c) + \frac {1}{2} \, a b x^{4} e^{2} \log \relax (c) + \frac {4}{3} \, a b d x^{3} e \log \relax (c) + \frac {1}{2} \, b^{2} d^{2} x^{2} \log \relax (c)^{2} + a b d^{2} n x^{2} \log \relax (x) - \frac {1}{2} \, a b d^{2} n x^{2} + \frac {1}{4} \, a^{2} x^{4} e^{2} + \frac {2}{3} \, a^{2} d x^{3} e + a b d^{2} x^{2} \log \relax (c) + \frac {1}{2} \, a^{2} d^{2} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.32, size = 2597, normalized size = 14.59 \[ \text {Expression too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 250, normalized size = 1.40 \[ \frac {1}{4} \, b^{2} e^{2} x^{4} \log \left (c x^{n}\right )^{2} - \frac {1}{8} \, a b e^{2} n x^{4} + \frac {1}{2} \, a b e^{2} x^{4} \log \left (c x^{n}\right ) + \frac {2}{3} \, b^{2} d e x^{3} \log \left (c x^{n}\right )^{2} - \frac {4}{9} \, a b d e n x^{3} + \frac {1}{4} \, a^{2} e^{2} x^{4} + \frac {4}{3} \, a b d e x^{3} \log \left (c x^{n}\right ) + \frac {1}{2} \, b^{2} d^{2} x^{2} \log \left (c x^{n}\right )^{2} - \frac {1}{2} \, a b d^{2} n x^{2} + \frac {2}{3} \, a^{2} d e x^{3} + a b d^{2} x^{2} \log \left (c x^{n}\right ) + \frac {1}{2} \, a^{2} d^{2} x^{2} + \frac {1}{4} \, {\left (n^{2} x^{2} - 2 \, n x^{2} \log \left (c x^{n}\right )\right )} b^{2} d^{2} + \frac {4}{27} \, {\left (n^{2} x^{3} - 3 \, n x^{3} \log \left (c x^{n}\right )\right )} b^{2} d e + \frac {1}{32} \, {\left (n^{2} x^{4} - 4 \, n x^{4} \log \left (c x^{n}\right )\right )} b^{2} e^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.62, size = 179, normalized size = 1.01 \[ \ln \left (c\,x^n\right )\,\left (\frac {b\,\left (2\,a-b\,n\right )\,d^2\,x^2}{2}+\frac {4\,b\,\left (3\,a-b\,n\right )\,d\,e\,x^3}{9}+\frac {b\,\left (4\,a-b\,n\right )\,e^2\,x^4}{8}\right )+{\ln \left (c\,x^n\right )}^2\,\left (\frac {b^2\,d^2\,x^2}{2}+\frac {2\,b^2\,d\,e\,x^3}{3}+\frac {b^2\,e^2\,x^4}{4}\right )+\frac {d^2\,x^2\,\left (2\,a^2-2\,a\,b\,n+b^2\,n^2\right )}{4}+\frac {e^2\,x^4\,\left (8\,a^2-4\,a\,b\,n+b^2\,n^2\right )}{32}+\frac {2\,d\,e\,x^3\,\left (9\,a^2-6\,a\,b\,n+2\,b^2\,n^2\right )}{27} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.27, size = 510, normalized size = 2.87 \[ \frac {a^{2} d^{2} x^{2}}{2} + \frac {2 a^{2} d e x^{3}}{3} + \frac {a^{2} e^{2} x^{4}}{4} + a b d^{2} n x^{2} \log {\relax (x )} - \frac {a b d^{2} n x^{2}}{2} + a b d^{2} x^{2} \log {\relax (c )} + \frac {4 a b d e n x^{3} \log {\relax (x )}}{3} - \frac {4 a b d e n x^{3}}{9} + \frac {4 a b d e x^{3} \log {\relax (c )}}{3} + \frac {a b e^{2} n x^{4} \log {\relax (x )}}{2} - \frac {a b e^{2} n x^{4}}{8} + \frac {a b e^{2} x^{4} \log {\relax (c )}}{2} + \frac {b^{2} d^{2} n^{2} x^{2} \log {\relax (x )}^{2}}{2} - \frac {b^{2} d^{2} n^{2} x^{2} \log {\relax (x )}}{2} + \frac {b^{2} d^{2} n^{2} x^{2}}{4} + b^{2} d^{2} n x^{2} \log {\relax (c )} \log {\relax (x )} - \frac {b^{2} d^{2} n x^{2} \log {\relax (c )}}{2} + \frac {b^{2} d^{2} x^{2} \log {\relax (c )}^{2}}{2} + \frac {2 b^{2} d e n^{2} x^{3} \log {\relax (x )}^{2}}{3} - \frac {4 b^{2} d e n^{2} x^{3} \log {\relax (x )}}{9} + \frac {4 b^{2} d e n^{2} x^{3}}{27} + \frac {4 b^{2} d e n x^{3} \log {\relax (c )} \log {\relax (x )}}{3} - \frac {4 b^{2} d e n x^{3} \log {\relax (c )}}{9} + \frac {2 b^{2} d e x^{3} \log {\relax (c )}^{2}}{3} + \frac {b^{2} e^{2} n^{2} x^{4} \log {\relax (x )}^{2}}{4} - \frac {b^{2} e^{2} n^{2} x^{4} \log {\relax (x )}}{8} + \frac {b^{2} e^{2} n^{2} x^{4}}{32} + \frac {b^{2} e^{2} n x^{4} \log {\relax (c )} \log {\relax (x )}}{2} - \frac {b^{2} e^{2} n x^{4} \log {\relax (c )}}{8} + \frac {b^{2} e^{2} x^{4} \log {\relax (c )}^{2}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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