Optimal. Leaf size=109 \[ \frac {1}{2} d x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} b d n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{3} e x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac {2}{9} b e n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{4} b^2 d n^2 x^2+\frac {2}{27} b^2 e n^2 x^3 \]
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Rubi [A] time = 0.11, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2353, 2305, 2304} \[ \frac {1}{2} d x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} b d n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{3} e x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac {2}{9} b e n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{4} b^2 d n^2 x^2+\frac {2}{27} b^2 e n^2 x^3 \]
Antiderivative was successfully verified.
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Rule 2304
Rule 2305
Rule 2353
Rubi steps
\begin {align*} \int x (d+e x) \left (a+b \log \left (c x^n\right )\right )^2 \, dx &=\int \left (d x \left (a+b \log \left (c x^n\right )\right )^2+e x^2 \left (a+b \log \left (c x^n\right )\right )^2\right ) \, dx\\ &=d \int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx+e \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx\\ &=\frac {1}{2} d x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{3} e x^3 \left (a+b \log \left (c x^n\right )\right )^2-(b d n) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx-\frac {1}{3} (2 b e n) \int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx\\ &=\frac {1}{4} b^2 d n^2 x^2+\frac {2}{27} b^2 e n^2 x^3-\frac {1}{2} b d n x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {2}{9} b e n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{2} d x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{3} e x^3 \left (a+b \log \left (c x^n\right )\right )^2\\ \end {align*}
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Mathematica [A] time = 0.06, size = 82, normalized size = 0.75 \[ \frac {1}{108} x^2 \left (54 d \left (a+b \log \left (c x^n\right )\right )^2+27 b d n \left (-2 a-2 b \log \left (c x^n\right )+b n\right )+36 e x \left (a+b \log \left (c x^n\right )\right )^2+8 b e n x \left (-3 a-3 b \log \left (c x^n\right )+b n\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.69, size = 219, normalized size = 2.01 \[ \frac {1}{27} \, {\left (2 \, b^{2} e n^{2} - 6 \, a b e n + 9 \, a^{2} e\right )} x^{3} + \frac {1}{4} \, {\left (b^{2} d n^{2} - 2 \, a b d n + 2 \, a^{2} d\right )} x^{2} + \frac {1}{6} \, {\left (2 \, b^{2} e x^{3} + 3 \, b^{2} d x^{2}\right )} \log \relax (c)^{2} + \frac {1}{6} \, {\left (2 \, b^{2} e n^{2} x^{3} + 3 \, b^{2} d n^{2} x^{2}\right )} \log \relax (x)^{2} - \frac {1}{18} \, {\left (4 \, {\left (b^{2} e n - 3 \, a b e\right )} x^{3} + 9 \, {\left (b^{2} d n - 2 \, a b d\right )} x^{2}\right )} \log \relax (c) - \frac {1}{18} \, {\left (4 \, {\left (b^{2} e n^{2} - 3 \, a b e n\right )} x^{3} + 9 \, {\left (b^{2} d n^{2} - 2 \, a b d n\right )} x^{2} - 6 \, {\left (2 \, b^{2} e n x^{3} + 3 \, b^{2} d n x^{2}\right )} \log \relax (c)\right )} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.26, size = 248, normalized size = 2.28 \[ \frac {1}{3} \, b^{2} n^{2} x^{3} e \log \relax (x)^{2} - \frac {2}{9} \, b^{2} n^{2} x^{3} e \log \relax (x) + \frac {2}{3} \, b^{2} n x^{3} e \log \relax (c) \log \relax (x) + \frac {1}{2} \, b^{2} d n^{2} x^{2} \log \relax (x)^{2} + \frac {2}{27} \, b^{2} n^{2} x^{3} e - \frac {2}{9} \, b^{2} n x^{3} e \log \relax (c) + \frac {1}{3} \, b^{2} x^{3} e \log \relax (c)^{2} - \frac {1}{2} \, b^{2} d n^{2} x^{2} \log \relax (x) + \frac {2}{3} \, a b n x^{3} e \log \relax (x) + b^{2} d n x^{2} \log \relax (c) \log \relax (x) + \frac {1}{4} \, b^{2} d n^{2} x^{2} - \frac {2}{9} \, a b n x^{3} e - \frac {1}{2} \, b^{2} d n x^{2} \log \relax (c) + \frac {2}{3} \, a b x^{3} e \log \relax (c) + \frac {1}{2} \, b^{2} d x^{2} \log \relax (c)^{2} + a b d n x^{2} \log \relax (x) - \frac {1}{2} \, a b d n x^{2} + \frac {1}{3} \, a^{2} x^{3} e + a b d x^{2} \log \relax (c) + \frac {1}{2} \, a^{2} d x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.27, size = 1621, normalized size = 14.87 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 150, normalized size = 1.38 \[ \frac {1}{3} \, b^{2} e x^{3} \log \left (c x^{n}\right )^{2} - \frac {2}{9} \, a b e n x^{3} + \frac {2}{3} \, a b e x^{3} \log \left (c x^{n}\right ) + \frac {1}{2} \, b^{2} d x^{2} \log \left (c x^{n}\right )^{2} - \frac {1}{2} \, a b d n x^{2} + \frac {1}{3} \, a^{2} e x^{3} + a b d x^{2} \log \left (c x^{n}\right ) + \frac {1}{2} \, a^{2} d x^{2} + \frac {1}{4} \, {\left (n^{2} x^{2} - 2 \, n x^{2} \log \left (c x^{n}\right )\right )} b^{2} d + \frac {2}{27} \, {\left (n^{2} x^{3} - 3 \, n x^{3} \log \left (c x^{n}\right )\right )} b^{2} e \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.53, size = 116, normalized size = 1.06 \[ {\ln \left (c\,x^n\right )}^2\,\left (\frac {e\,b^2\,x^3}{3}+\frac {d\,b^2\,x^2}{2}\right )+\ln \left (c\,x^n\right )\,\left (\frac {2\,b\,e\,\left (3\,a-b\,n\right )\,x^3}{9}+\frac {b\,d\,\left (2\,a-b\,n\right )\,x^2}{2}\right )+\frac {d\,x^2\,\left (2\,a^2-2\,a\,b\,n+b^2\,n^2\right )}{4}+\frac {e\,x^3\,\left (9\,a^2-6\,a\,b\,n+2\,b^2\,n^2\right )}{27} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.80, size = 304, normalized size = 2.79 \[ \frac {a^{2} d x^{2}}{2} + \frac {a^{2} e x^{3}}{3} + a b d n x^{2} \log {\relax (x )} - \frac {a b d n x^{2}}{2} + a b d x^{2} \log {\relax (c )} + \frac {2 a b e n x^{3} \log {\relax (x )}}{3} - \frac {2 a b e n x^{3}}{9} + \frac {2 a b e x^{3} \log {\relax (c )}}{3} + \frac {b^{2} d n^{2} x^{2} \log {\relax (x )}^{2}}{2} - \frac {b^{2} d n^{2} x^{2} \log {\relax (x )}}{2} + \frac {b^{2} d n^{2} x^{2}}{4} + b^{2} d n x^{2} \log {\relax (c )} \log {\relax (x )} - \frac {b^{2} d n x^{2} \log {\relax (c )}}{2} + \frac {b^{2} d x^{2} \log {\relax (c )}^{2}}{2} + \frac {b^{2} e n^{2} x^{3} \log {\relax (x )}^{2}}{3} - \frac {2 b^{2} e n^{2} x^{3} \log {\relax (x )}}{9} + \frac {2 b^{2} e n^{2} x^{3}}{27} + \frac {2 b^{2} e n x^{3} \log {\relax (c )} \log {\relax (x )}}{3} - \frac {2 b^{2} e n x^{3} \log {\relax (c )}}{9} + \frac {b^{2} e x^{3} \log {\relax (c )}^{2}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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