Optimal. Leaf size=101 \[ d x \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} b e n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{2} e x^2 \left (a+b \log \left (c x^n\right )\right )^2-2 a b d n x-2 b^2 d n x \log \left (c x^n\right )+2 b^2 d n^2 x+\frac {1}{4} b^2 e n^2 x^2 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {2330, 2296, 2295, 2305, 2304} \[ d x \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} b e n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{2} e x^2 \left (a+b \log \left (c x^n\right )\right )^2-2 a b d n x-2 b^2 d n x \log \left (c x^n\right )+2 b^2 d n^2 x+\frac {1}{4} b^2 e n^2 x^2 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2295
Rule 2296
Rule 2304
Rule 2305
Rule 2330
Rubi steps
\begin {align*} \int (d+e x) \left (a+b \log \left (c x^n\right )\right )^2 \, dx &=\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 \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx+e \int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx\\ &=d x \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{2} e x^2 \left (a+b \log \left (c x^n\right )\right )^2-(2 b d n) \int \left (a+b \log \left (c x^n\right )\right ) \, dx-(b e n) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx\\ &=-2 a b d n x+\frac {1}{4} b^2 e n^2 x^2-\frac {1}{2} b e n x^2 \left (a+b \log \left (c x^n\right )\right )+d x \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{2} e x^2 \left (a+b \log \left (c x^n\right )\right )^2-\left (2 b^2 d n\right ) \int \log \left (c x^n\right ) \, dx\\ &=-2 a b d n x+2 b^2 d n^2 x+\frac {1}{4} b^2 e n^2 x^2-2 b^2 d n x \log \left (c x^n\right )-\frac {1}{2} b e n x^2 \left (a+b \log \left (c x^n\right )\right )+d x \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{2} e x^2 \left (a+b \log \left (c x^n\right )\right )^2\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 77, normalized size = 0.76 \[ \frac {1}{4} x \left (4 d \left (a+b \log \left (c x^n\right )\right )^2-8 b d n \left (a+b \log \left (c x^n\right )-b n\right )+2 e x \left (a+b \log \left (c x^n\right )\right )^2+b e n x \left (-2 a-2 b \log \left (c x^n\right )+b n\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.45, size = 200, normalized size = 1.98 \[ \frac {1}{4} \, {\left (b^{2} e n^{2} - 2 \, a b e n + 2 \, a^{2} e\right )} x^{2} + \frac {1}{2} \, {\left (b^{2} e x^{2} + 2 \, b^{2} d x\right )} \log \relax (c)^{2} + \frac {1}{2} \, {\left (b^{2} e n^{2} x^{2} + 2 \, b^{2} d n^{2} x\right )} \log \relax (x)^{2} + {\left (2 \, b^{2} d n^{2} - 2 \, a b d n + a^{2} d\right )} x - \frac {1}{2} \, {\left ({\left (b^{2} e n - 2 \, a b e\right )} x^{2} + 4 \, {\left (b^{2} d n - a b d\right )} x\right )} \log \relax (c) - \frac {1}{2} \, {\left ({\left (b^{2} e n^{2} - 2 \, a b e n\right )} x^{2} + 4 \, {\left (b^{2} d n^{2} - a b d n\right )} x - 2 \, {\left (b^{2} e n x^{2} + 2 \, b^{2} d n x\right )} \log \relax (c)\right )} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.35, size = 225, normalized size = 2.23 \[ \frac {1}{2} \, b^{2} n^{2} x^{2} e \log \relax (x)^{2} - \frac {1}{2} \, b^{2} n^{2} x^{2} e \log \relax (x) + b^{2} n x^{2} e \log \relax (c) \log \relax (x) + b^{2} d n^{2} x \log \relax (x)^{2} + \frac {1}{4} \, b^{2} n^{2} x^{2} e - \frac {1}{2} \, b^{2} n x^{2} e \log \relax (c) + \frac {1}{2} \, b^{2} x^{2} e \log \relax (c)^{2} - 2 \, b^{2} d n^{2} x \log \relax (x) + a b n x^{2} e \log \relax (x) + 2 \, b^{2} d n x \log \relax (c) \log \relax (x) + 2 \, b^{2} d n^{2} x - \frac {1}{2} \, a b n x^{2} e - 2 \, b^{2} d n x \log \relax (c) + a b x^{2} e \log \relax (c) + b^{2} d x \log \relax (c)^{2} + 2 \, a b d n x \log \relax (x) - 2 \, a b d n x + \frac {1}{2} \, a^{2} x^{2} e + 2 \, a b d x \log \relax (c) + a^{2} d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.28, size = 1548, normalized size = 15.33 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.59, size = 136, normalized size = 1.35 \[ \frac {1}{2} \, b^{2} e x^{2} \log \left (c x^{n}\right )^{2} - \frac {1}{2} \, a b e n x^{2} + a b e x^{2} \log \left (c x^{n}\right ) + b^{2} d x \log \left (c x^{n}\right )^{2} - 2 \, a b d n x + \frac {1}{2} \, a^{2} e x^{2} + 2 \, a b d x \log \left (c x^{n}\right ) + 2 \, {\left (n^{2} x - n x \log \left (c x^{n}\right )\right )} b^{2} d + \frac {1}{4} \, {\left (n^{2} x^{2} - 2 \, n x^{2} \log \left (c x^{n}\right )\right )} b^{2} e + a^{2} d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.68, size = 104, normalized size = 1.03 \[ \ln \left (c\,x^n\right )\,\left (\frac {b\,e\,\left (2\,a-b\,n\right )\,x^2}{2}+2\,b\,d\,\left (a-b\,n\right )\,x\right )+{\ln \left (c\,x^n\right )}^2\,\left (\frac {e\,b^2\,x^2}{2}+d\,b^2\,x\right )+\frac {e\,x^2\,\left (2\,a^2-2\,a\,b\,n+b^2\,n^2\right )}{4}+d\,x\,\left (a^2-2\,a\,b\,n+2\,b^2\,n^2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 1.10, size = 270, normalized size = 2.67 \[ a^{2} d x + \frac {a^{2} e x^{2}}{2} + 2 a b d n x \log {\relax (x )} - 2 a b d n x + 2 a b d x \log {\relax (c )} + a b e n x^{2} \log {\relax (x )} - \frac {a b e n x^{2}}{2} + a b e x^{2} \log {\relax (c )} + b^{2} d n^{2} x \log {\relax (x )}^{2} - 2 b^{2} d n^{2} x \log {\relax (x )} + 2 b^{2} d n^{2} x + 2 b^{2} d n x \log {\relax (c )} \log {\relax (x )} - 2 b^{2} d n x \log {\relax (c )} + b^{2} d x \log {\relax (c )}^{2} + \frac {b^{2} e n^{2} x^{2} \log {\relax (x )}^{2}}{2} - \frac {b^{2} e n^{2} x^{2} \log {\relax (x )}}{2} + \frac {b^{2} e n^{2} x^{2}}{4} + b^{2} e n x^{2} \log {\relax (c )} \log {\relax (x )} - \frac {b^{2} e n x^{2} \log {\relax (c )}}{2} + \frac {b^{2} e x^{2} \log {\relax (c )}^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________