Optimal. Leaf size=20 \[ \frac{1}{3} \left (a x+b \log ^2\left (c x^n\right )\right )^3 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.151753, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051, Rules used = {2561, 2544} \[ \frac{1}{3} \left (a x+b \log ^2\left (c x^n\right )\right )^3 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2561
Rule 2544
Rubi steps
\begin{align*} \int \left (\frac{a}{x^2}+\frac{2 b n \log \left (c x^n\right )}{x^3}\right ) \left (a x^2+b x \log ^2\left (c x^n\right )\right )^2 \, dx &=\int \frac{\left (a x+2 b n \log \left (c x^n\right )\right ) \left (a x^2+b x \log ^2\left (c x^n\right )\right )^2}{x^3} \, dx\\ &=\int \frac{\left (a x+2 b n \log \left (c x^n\right )\right ) \left (a x+b \log ^2\left (c x^n\right )\right )^2}{x} \, dx\\ &=\frac{1}{3} \left (a x+b \log ^2\left (c x^n\right )\right )^3\\ \end{align*}
Mathematica [A] time = 0.0175154, size = 20, normalized size = 1. \[ \frac{1}{3} \left (a x+b \log ^2\left (c x^n\right )\right )^3 \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 1.223, size = 16321, normalized size = 816.1 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.02688, size = 285, normalized size = 14.25 \begin{align*} \frac{1}{3} \, b^{3} \log \left (c x^{n}\right )^{6} + 4 \, a b^{2} n x \log \left (c x^{n}\right )^{3} + a b^{2} x \log \left (c x^{n}\right )^{4} - \frac{1}{2} \, a^{2} b n^{2} x^{2} + a^{2} b n x^{2} \log \left (c x^{n}\right ) + a^{2} b x^{2} \log \left (c x^{n}\right )^{2} + \frac{1}{3} \, a^{3} x^{3} - 12 \,{\left (n x \log \left (c x^{n}\right )^{2} + 2 \,{\left (n^{2} x - n x \log \left (c x^{n}\right )\right )} n\right )} a b^{2} n + \frac{1}{2} \,{\left (n^{2} x^{2} - 2 \, n x^{2} \log \left (c x^{n}\right )\right )} a^{2} b - 4 \,{\left (n x \log \left (c x^{n}\right )^{3} - 3 \,{\left (n x \log \left (c x^{n}\right )^{2} + 2 \,{\left (n^{2} x - n x \log \left (c x^{n}\right )\right )} n\right )} n\right )} a b^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.84464, size = 478, normalized size = 23.9 \begin{align*} \frac{1}{3} \, b^{3} n^{6} \log \left (x\right )^{6} + 2 \, b^{3} n^{5} \log \left (c\right ) \log \left (x\right )^{5} + a b^{2} x \log \left (c\right )^{4} + a^{2} b x^{2} \log \left (c\right )^{2} + \frac{1}{3} \, a^{3} x^{3} +{\left (5 \, b^{3} n^{4} \log \left (c\right )^{2} + a b^{2} n^{4} x\right )} \log \left (x\right )^{4} + \frac{4}{3} \,{\left (5 \, b^{3} n^{3} \log \left (c\right )^{3} + 3 \, a b^{2} n^{3} x \log \left (c\right )\right )} \log \left (x\right )^{3} +{\left (5 \, b^{3} n^{2} \log \left (c\right )^{4} + 6 \, a b^{2} n^{2} x \log \left (c\right )^{2} + a^{2} b n^{2} x^{2}\right )} \log \left (x\right )^{2} + 2 \,{\left (b^{3} n \log \left (c\right )^{5} + 2 \, a b^{2} n x \log \left (c\right )^{3} + a^{2} b n x^{2} \log \left (c\right )\right )} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 13.4414, size = 221, normalized size = 11.05 \begin{align*} \frac{a^{3} x^{3}}{3} + a^{2} b n^{2} x^{2} \log{\left (x \right )}^{2} - a^{2} b n^{2} x^{2} \log{\left (x \right )} + 2 a^{2} b n x^{2} \log{\left (c \right )} \log{\left (x \right )} - a^{2} b n x^{2} \log{\left (c \right )} + a^{2} b n x^{2} \log{\left (c x^{n} \right )} + a^{2} b x^{2} \log{\left (c \right )}^{2} + a b^{2} n^{4} x \log{\left (x \right )}^{4} + 4 a b^{2} n^{3} x \log{\left (c \right )} \log{\left (x \right )}^{3} + 6 a b^{2} n^{2} x \log{\left (c \right )}^{2} \log{\left (x \right )}^{2} + 4 a b^{2} n x \log{\left (c \right )}^{3} \log{\left (x \right )} + a b^{2} x \log{\left (c \right )}^{4} - 2 b^{3} n \left (\begin{cases} - \log{\left (c \right )}^{5} \log{\left (x \right )} & \text{for}\: n = 0 \\- \frac{\log{\left (c x^{n} \right )}^{6}}{6 n} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.29055, size = 267, normalized size = 13.35 \begin{align*} \frac{1}{3} \, b^{3} n^{6} \log \left (x\right )^{6} + 2 \, b^{3} n^{5} \log \left (c\right ) \log \left (x\right )^{5} + 2 \, b^{3} n \log \left (c\right )^{5} \log \left (x\right ) + a b^{2} x \log \left (c\right )^{4} + a^{2} b x^{2} \log \left (c\right )^{2} + \frac{1}{3} \, a^{3} x^{3} +{\left (5 \, b^{3} n^{4} \log \left (c\right )^{2} + a b^{2} n^{4} x\right )} \log \left (x\right )^{4} + \frac{4}{3} \,{\left (5 \, b^{3} n^{3} \log \left (c\right )^{3} + 3 \, a b^{2} n^{3} x \log \left (c\right )\right )} \log \left (x\right )^{3} +{\left (5 \, b^{3} n^{2} \log \left (c\right )^{4} + 6 \, a b^{2} n^{2} x \log \left (c\right )^{2} + a^{2} b n^{2} x^{2}\right )} \log \left (x\right )^{2} + 2 \,{\left (2 \, a b^{2} n x \log \left (c\right )^{3} + a^{2} b n x^{2} \log \left (c\right )\right )} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]