Optimal. Leaf size=118 \[ 3 d^2 e x \left (a+b \log \left (c x^n\right )\right )-\frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{x}+d e^2 x^3 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{5} e^3 x^5 \left (a+b \log \left (c x^n\right )\right )-3 b d^2 e n x-\frac{b d^3 n}{x}-\frac{1}{3} b d e^2 n x^3-\frac{1}{25} b e^3 n x^5 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0823846, antiderivative size = 92, normalized size of antiderivative = 0.78, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {270, 2334} \[ -\frac{1}{5} \left (-15 d^2 e x+\frac{5 d^3}{x}-5 d e^2 x^3-e^3 x^5\right ) \left (a+b \log \left (c x^n\right )\right )-3 b d^2 e n x-\frac{b d^3 n}{x}-\frac{1}{3} b d e^2 n x^3-\frac{1}{25} b e^3 n x^5 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 270
Rule 2334
Rubi steps
\begin{align*} \int \frac{\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx &=-\frac{1}{5} \left (\frac{5 d^3}{x}-15 d^2 e x-5 d e^2 x^3-e^3 x^5\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (3 d^2 e-\frac{d^3}{x^2}+d e^2 x^2+\frac{e^3 x^4}{5}\right ) \, dx\\ &=-\frac{b d^3 n}{x}-3 b d^2 e n x-\frac{1}{3} b d e^2 n x^3-\frac{1}{25} b e^3 n x^5-\frac{1}{5} \left (\frac{5 d^3}{x}-15 d^2 e x-5 d e^2 x^3-e^3 x^5\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}
Mathematica [A] time = 0.055613, size = 123, normalized size = 1.04 \[ -\frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{x}+d e^2 x^3 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{5} e^3 x^5 \left (a+b \log \left (c x^n\right )\right )+3 a d^2 e x+3 b d^2 e x \log \left (c x^n\right )-3 b d^2 e n x-\frac{b d^3 n}{x}-\frac{1}{3} b d e^2 n x^3-\frac{1}{25} b e^3 n x^5 \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.238, size = 587, normalized size = 5. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.00805, size = 182, normalized size = 1.54 \begin{align*} -\frac{1}{25} \, b e^{3} n x^{5} + \frac{1}{5} \, b e^{3} x^{5} \log \left (c x^{n}\right ) + \frac{1}{5} \, a e^{3} x^{5} - \frac{1}{3} \, b d e^{2} n x^{3} + b d e^{2} x^{3} \log \left (c x^{n}\right ) + a d e^{2} x^{3} - 3 \, b d^{2} e n x + 3 \, b d^{2} e x \log \left (c x^{n}\right ) + 3 \, a d^{2} e x - \frac{b d^{3} n}{x} - \frac{b d^{3} \log \left (c x^{n}\right )}{x} - \frac{a d^{3}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.28806, size = 362, normalized size = 3.07 \begin{align*} -\frac{3 \,{\left (b e^{3} n - 5 \, a e^{3}\right )} x^{6} + 75 \, b d^{3} n + 25 \,{\left (b d e^{2} n - 3 \, a d e^{2}\right )} x^{4} + 75 \, a d^{3} + 225 \,{\left (b d^{2} e n - a d^{2} e\right )} x^{2} - 15 \,{\left (b e^{3} x^{6} + 5 \, b d e^{2} x^{4} + 15 \, b d^{2} e x^{2} - 5 \, b d^{3}\right )} \log \left (c\right ) - 15 \,{\left (b e^{3} n x^{6} + 5 \, b d e^{2} n x^{4} + 15 \, b d^{2} e n x^{2} - 5 \, b d^{3} n\right )} \log \left (x\right )}{75 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 9.0539, size = 190, normalized size = 1.61 \begin{align*} - \frac{a d^{3}}{x} + 3 a d^{2} e x + a d e^{2} x^{3} + \frac{a e^{3} x^{5}}{5} - \frac{b d^{3} n \log{\left (x \right )}}{x} - \frac{b d^{3} n}{x} - \frac{b d^{3} \log{\left (c \right )}}{x} + 3 b d^{2} e n x \log{\left (x \right )} - 3 b d^{2} e n x + 3 b d^{2} e x \log{\left (c \right )} + b d e^{2} n x^{3} \log{\left (x \right )} - \frac{b d e^{2} n x^{3}}{3} + b d e^{2} x^{3} \log{\left (c \right )} + \frac{b e^{3} n x^{5} \log{\left (x \right )}}{5} - \frac{b e^{3} n x^{5}}{25} + \frac{b e^{3} x^{5} \log{\left (c \right )}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.31192, size = 224, normalized size = 1.9 \begin{align*} \frac{15 \, b n x^{6} e^{3} \log \left (x\right ) - 3 \, b n x^{6} e^{3} + 15 \, b x^{6} e^{3} \log \left (c\right ) + 75 \, b d n x^{4} e^{2} \log \left (x\right ) + 15 \, a x^{6} e^{3} - 25 \, b d n x^{4} e^{2} + 75 \, b d x^{4} e^{2} \log \left (c\right ) + 225 \, b d^{2} n x^{2} e \log \left (x\right ) + 75 \, a d x^{4} e^{2} - 225 \, b d^{2} n x^{2} e + 225 \, b d^{2} x^{2} e \log \left (c\right ) + 225 \, a d^{2} x^{2} e - 75 \, b d^{3} n \log \left (x\right ) - 75 \, b d^{3} n - 75 \, b d^{3} \log \left (c\right ) - 75 \, a d^{3}}{75 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]