Optimal. Leaf size=137 \[ \frac {d^2 \left (a+b \log \left (c x^n\right )\right )^3}{3 b n}+2 d e x \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{2} e^2 x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} b e^2 n x^2 \left (a+b \log \left (c x^n\right )\right )-4 a b d e n x-4 b^2 d e n x \log \left (c x^n\right )+4 b^2 d e n^2 x+\frac {1}{4} b^2 e^2 n^2 x^2 \]
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Rubi [A] time = 0.23, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {2346, 2302, 30, 2296, 2295, 2330, 2305, 2304} \[ \frac {d^2 \left (a+b \log \left (c x^n\right )\right )^3}{3 b n}+2 d e x \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{2} e^2 x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} b e^2 n x^2 \left (a+b \log \left (c x^n\right )\right )-4 a b d e n x-4 b^2 d e n x \log \left (c x^n\right )+4 b^2 d e n^2 x+\frac {1}{4} b^2 e^2 n^2 x^2 \]
Antiderivative was successfully verified.
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Rule 30
Rule 2295
Rule 2296
Rule 2302
Rule 2304
Rule 2305
Rule 2330
Rule 2346
Rubi steps
\begin {align*} \int \frac {(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx &=d \int \frac {(d+e x) \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx+e \int (d+e x) \left (a+b \log \left (c x^n\right )\right )^2 \, dx\\ &=d^2 \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx+e \int \left (d \left (a+b \log \left (c x^n\right )\right )^2+e x \left (a+b \log \left (c x^n\right )\right )^2\right ) \, dx+(d e) \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx\\ &=d e x \left (a+b \log \left (c x^n\right )\right )^2+(d e) \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx+e^2 \int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx+\frac {d^2 \operatorname {Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )}{b n}-(2 b d e n) \int \left (a+b \log \left (c x^n\right )\right ) \, dx\\ &=-2 a b d e n x+2 d e x \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{2} e^2 x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^3}{3 b n}-(2 b d e n) \int \left (a+b \log \left (c x^n\right )\right ) \, dx-\left (2 b^2 d e n\right ) \int \log \left (c x^n\right ) \, dx-\left (b e^2 n\right ) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx\\ &=-4 a b d e n x+2 b^2 d e n^2 x+\frac {1}{4} b^2 e^2 n^2 x^2-2 b^2 d e n x \log \left (c x^n\right )-\frac {1}{2} b e^2 n x^2 \left (a+b \log \left (c x^n\right )\right )+2 d e x \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{2} e^2 x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^3}{3 b n}-\left (2 b^2 d e n\right ) \int \log \left (c x^n\right ) \, dx\\ &=-4 a b d e n x+4 b^2 d e n^2 x+\frac {1}{4} b^2 e^2 n^2 x^2-4 b^2 d e n x \log \left (c x^n\right )-\frac {1}{2} b e^2 n x^2 \left (a+b \log \left (c x^n\right )\right )+2 d e x \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{2} e^2 x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^3}{3 b n}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 114, normalized size = 0.83 \[ \frac {d^2 \left (a+b \log \left (c x^n\right )\right )^3}{3 b n}+2 d e x \left (a+b \log \left (c x^n\right )\right )^2-4 b d e n x \left (a+b \log \left (c x^n\right )-b n\right )+\frac {1}{2} e^2 x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{4} b e^2 n x^2 \left (-2 a-2 b \log \left (c x^n\right )+b n\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.70, size = 293, normalized size = 2.14 \[ \frac {1}{3} \, b^{2} d^{2} n^{2} \log \relax (x)^{3} + \frac {1}{4} \, {\left (b^{2} e^{2} n^{2} - 2 \, a b e^{2} n + 2 \, a^{2} e^{2}\right )} x^{2} + \frac {1}{2} \, {\left (b^{2} e^{2} x^{2} + 4 \, b^{2} d e x\right )} \log \relax (c)^{2} + \frac {1}{2} \, {\left (b^{2} e^{2} n^{2} x^{2} + 4 \, b^{2} d e n^{2} x + 2 \, b^{2} d^{2} n \log \relax (c) + 2 \, a b d^{2} n\right )} \log \relax (x)^{2} + 2 \, {\left (2 \, b^{2} d e n^{2} - 2 \, a b d e n + a^{2} d e\right )} x - \frac {1}{2} \, {\left ({\left (b^{2} e^{2} n - 2 \, a b e^{2}\right )} x^{2} + 8 \, {\left (b^{2} d e n - a b d e\right )} x\right )} \log \relax (c) + \frac {1}{2} \, {\left (2 \, b^{2} d^{2} \log \relax (c)^{2} + 2 \, a^{2} d^{2} - {\left (b^{2} e^{2} n^{2} - 2 \, a b e^{2} n\right )} x^{2} - 8 \, {\left (b^{2} d e n^{2} - a b d e n\right )} x + 2 \, {\left (b^{2} e^{2} n x^{2} + 4 \, b^{2} d e n x + 2 \, a b d^{2}\right )} \log \relax (c)\right )} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.34, size = 321, normalized size = 2.34 \[ \frac {1}{2} \, b^{2} n^{2} x^{2} e^{2} \log \relax (x)^{2} + 2 \, b^{2} d n^{2} x e \log \relax (x)^{2} + \frac {1}{3} \, b^{2} d^{2} n^{2} \log \relax (x)^{3} - \frac {1}{2} \, b^{2} n^{2} x^{2} e^{2} \log \relax (x) - 4 \, b^{2} d n^{2} x e \log \relax (x) + b^{2} n x^{2} e^{2} \log \relax (c) \log \relax (x) + 4 \, b^{2} d n x e \log \relax (c) \log \relax (x) + b^{2} d^{2} n \log \relax (c) \log \relax (x)^{2} + \frac {1}{4} \, b^{2} n^{2} x^{2} e^{2} + 4 \, b^{2} d n^{2} x e - \frac {1}{2} \, b^{2} n x^{2} e^{2} \log \relax (c) - 4 \, b^{2} d n x e \log \relax (c) + \frac {1}{2} \, b^{2} x^{2} e^{2} \log \relax (c)^{2} + 2 \, b^{2} d x e \log \relax (c)^{2} + a b n x^{2} e^{2} \log \relax (x) + 4 \, a b d n x e \log \relax (x) + b^{2} d^{2} \log \relax (c)^{2} \log \relax (x) + a b d^{2} n \log \relax (x)^{2} - \frac {1}{2} \, a b n x^{2} e^{2} - 4 \, a b d n x e + a b x^{2} e^{2} \log \relax (c) + 4 \, a b d x e \log \relax (c) + 2 \, a b d^{2} \log \relax (c) \log \relax (x) + \frac {1}{2} \, a^{2} x^{2} e^{2} + 2 \, a^{2} d x e + a^{2} d^{2} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.51, size = 2543, normalized size = 18.56 \[ \text {Expression too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 198, normalized size = 1.45 \[ \frac {1}{2} \, b^{2} e^{2} x^{2} \log \left (c x^{n}\right )^{2} - \frac {1}{2} \, a b e^{2} n x^{2} + a b e^{2} x^{2} \log \left (c x^{n}\right ) + 2 \, b^{2} d e x \log \left (c x^{n}\right )^{2} - 4 \, a b d e n x + \frac {1}{2} \, a^{2} e^{2} x^{2} + 4 \, a b d e x \log \left (c x^{n}\right ) + \frac {b^{2} d^{2} \log \left (c x^{n}\right )^{3}}{3 \, n} + 4 \, {\left (n^{2} x - n x \log \left (c x^{n}\right )\right )} b^{2} d e + \frac {1}{4} \, {\left (n^{2} x^{2} - 2 \, n x^{2} \log \left (c x^{n}\right )\right )} b^{2} e^{2} + 2 \, a^{2} d e x + \frac {a b d^{2} \log \left (c x^{n}\right )^{2}}{n} + a^{2} d^{2} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.78, size = 152, normalized size = 1.11 \[ {\ln \left (c\,x^n\right )}^2\,\left (\frac {b^2\,e^2\,x^2}{2}+2\,b^2\,d\,e\,x+\frac {a\,b\,d^2}{n}\right )+\ln \left (c\,x^n\right )\,\left (\frac {b\,\left (2\,a-b\,n\right )\,e^2\,x^2}{2}+4\,b\,d\,\left (a-b\,n\right )\,e\,x\right )+a^2\,d^2\,\ln \relax (x)+\frac {e^2\,x^2\,\left (2\,a^2-2\,a\,b\,n+b^2\,n^2\right )}{4}+2\,d\,e\,x\,\left (a^2-2\,a\,b\,n+2\,b^2\,n^2\right )+\frac {b^2\,d^2\,{\ln \left (c\,x^n\right )}^3}{3\,n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.11, size = 398, normalized size = 2.91 \[ a^{2} d^{2} \log {\relax (x )} + 2 a^{2} d e x + \frac {a^{2} e^{2} x^{2}}{2} + a b d^{2} n \log {\relax (x )}^{2} + 2 a b d^{2} \log {\relax (c )} \log {\relax (x )} + 4 a b d e n x \log {\relax (x )} - 4 a b d e n x + 4 a b d e x \log {\relax (c )} + a b e^{2} n x^{2} \log {\relax (x )} - \frac {a b e^{2} n x^{2}}{2} + a b e^{2} x^{2} \log {\relax (c )} + \frac {b^{2} d^{2} n^{2} \log {\relax (x )}^{3}}{3} + b^{2} d^{2} n \log {\relax (c )} \log {\relax (x )}^{2} + b^{2} d^{2} \log {\relax (c )}^{2} \log {\relax (x )} + 2 b^{2} d e n^{2} x \log {\relax (x )}^{2} - 4 b^{2} d e n^{2} x \log {\relax (x )} + 4 b^{2} d e n^{2} x + 4 b^{2} d e n x \log {\relax (c )} \log {\relax (x )} - 4 b^{2} d e n x \log {\relax (c )} + 2 b^{2} d e x \log {\relax (c )}^{2} + \frac {b^{2} e^{2} n^{2} x^{2} \log {\relax (x )}^{2}}{2} - \frac {b^{2} e^{2} n^{2} x^{2} \log {\relax (x )}}{2} + \frac {b^{2} e^{2} n^{2} x^{2}}{4} + b^{2} e^{2} n x^{2} \log {\relax (c )} \log {\relax (x )} - \frac {b^{2} e^{2} n x^{2} \log {\relax (c )}}{2} + \frac {b^{2} e^{2} x^{2} \log {\relax (c )}^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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