Optimal. Leaf size=51 \[ -\frac{b^2 p \log \left (a x^2+b\right )}{4 a^2}+\frac{1}{4} x^4 \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )+\frac{b p x^2}{4 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0335074, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2455, 263, 266, 43} \[ -\frac{b^2 p \log \left (a x^2+b\right )}{4 a^2}+\frac{1}{4} x^4 \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )+\frac{b p x^2}{4 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2455
Rule 263
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^3 \log \left (c \left (a+\frac{b}{x^2}\right )^p\right ) \, dx &=\frac{1}{4} x^4 \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )+\frac{1}{2} (b p) \int \frac{x}{a+\frac{b}{x^2}} \, dx\\ &=\frac{1}{4} x^4 \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )+\frac{1}{2} (b p) \int \frac{x^3}{b+a x^2} \, dx\\ &=\frac{1}{4} x^4 \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )+\frac{1}{4} (b p) \operatorname{Subst}\left (\int \frac{x}{b+a x} \, dx,x,x^2\right )\\ &=\frac{1}{4} x^4 \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )+\frac{1}{4} (b p) \operatorname{Subst}\left (\int \left (\frac{1}{a}-\frac{b}{a (b+a x)}\right ) \, dx,x,x^2\right )\\ &=\frac{b p x^2}{4 a}+\frac{1}{4} x^4 \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )-\frac{b^2 p \log \left (b+a x^2\right )}{4 a^2}\\ \end{align*}
Mathematica [A] time = 0.0202259, size = 56, normalized size = 1.1 \[ \frac{1}{4} b p \left (-\frac{b \log \left (a+\frac{b}{x^2}\right )}{a^2}-\frac{2 b \log (x)}{a^2}+\frac{x^2}{a}\right )+\frac{1}{4} x^4 \log \left (c \left (a+\frac{b}{x^2}\right )^p\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.245, size = 0, normalized size = 0. \begin{align*} \int{x}^{3}\ln \left ( c \left ( a+{\frac{b}{{x}^{2}}} \right ) ^{p} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.08037, size = 59, normalized size = 1.16 \begin{align*} \frac{1}{4} \, x^{4} \log \left ({\left (a + \frac{b}{x^{2}}\right )}^{p} c\right ) + \frac{1}{4} \, b p{\left (\frac{x^{2}}{a} - \frac{b \log \left (a x^{2} + b\right )}{a^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.32064, size = 127, normalized size = 2.49 \begin{align*} \frac{a^{2} p x^{4} \log \left (\frac{a x^{2} + b}{x^{2}}\right ) + a^{2} x^{4} \log \left (c\right ) + a b p x^{2} - b^{2} p \log \left (a x^{2} + b\right )}{4 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 29.7322, size = 87, normalized size = 1.71 \begin{align*} \begin{cases} \frac{p x^{4} \log{\left (a + \frac{b}{x^{2}} \right )}}{4} + \frac{x^{4} \log{\left (c \right )}}{4} + \frac{b p x^{2}}{4 a} - \frac{b^{2} p \log{\left (a x^{2} + b \right )}}{4 a^{2}} & \text{for}\: a \neq 0 \\\frac{p x^{4} \log{\left (b \right )}}{4} - \frac{p x^{4} \log{\left (x \right )}}{2} + \frac{p x^{4}}{8} + \frac{x^{4} \log{\left (c \right )}}{4} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.21072, size = 80, normalized size = 1.57 \begin{align*} \frac{1}{4} \, p x^{4} \log \left (a x^{2} + b\right ) - \frac{1}{4} \, p x^{4} \log \left (x^{2}\right ) + \frac{1}{4} \, x^{4} \log \left (c\right ) + \frac{b p x^{2}}{4 \, a} - \frac{b^{2} p \log \left (a x^{2} + b\right )}{4 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]