Optimal. Leaf size=105 \[ \frac{(e x)^{m+1} \left (a+\frac{b}{x^2}\right )^p \left (\frac{b}{a x^2}+1\right )^{-p} \left (c+\frac{d}{x^2}\right )^q \left (\frac{d}{c x^2}+1\right )^{-q} F_1\left (\frac{1}{2} (-m-1);-p,-q;\frac{1-m}{2};-\frac{b}{a x^2},-\frac{d}{c x^2}\right )}{e (m+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11591, antiderivative size = 101, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {497, 511, 510} \[ \frac{x (e x)^m \left (a+\frac{b}{x^2}\right )^p \left (\frac{b}{a x^2}+1\right )^{-p} \left (c+\frac{d}{x^2}\right )^q \left (\frac{d}{c x^2}+1\right )^{-q} F_1\left (\frac{1}{2} (-m-1);-p,-q;\frac{1-m}{2};-\frac{b}{a x^2},-\frac{d}{c x^2}\right )}{m+1} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 497
Rule 511
Rule 510
Rubi steps
\begin{align*} \int \left (a+\frac{b}{x^2}\right )^p \left (c+\frac{d}{x^2}\right )^q (e x)^m \, dx &=-\left (\left (\left (\frac{1}{x}\right )^m (e x)^m\right ) \operatorname{Subst}\left (\int x^{-2-m} \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx,x,\frac{1}{x}\right )\right )\\ &=-\left (\left (\left (a+\frac{b}{x^2}\right )^p \left (1+\frac{b}{a x^2}\right )^{-p} \left (\frac{1}{x}\right )^m (e x)^m\right ) \operatorname{Subst}\left (\int x^{-2-m} \left (1+\frac{b x^2}{a}\right )^p \left (c+d x^2\right )^q \, dx,x,\frac{1}{x}\right )\right )\\ &=-\left (\left (\left (a+\frac{b}{x^2}\right )^p \left (1+\frac{b}{a x^2}\right )^{-p} \left (c+\frac{d}{x^2}\right )^q \left (1+\frac{d}{c x^2}\right )^{-q} \left (\frac{1}{x}\right )^m (e x)^m\right ) \operatorname{Subst}\left (\int x^{-2-m} \left (1+\frac{b x^2}{a}\right )^p \left (1+\frac{d x^2}{c}\right )^q \, dx,x,\frac{1}{x}\right )\right )\\ &=\frac{\left (a+\frac{b}{x^2}\right )^p \left (1+\frac{b}{a x^2}\right )^{-p} \left (c+\frac{d}{x^2}\right )^q \left (1+\frac{d}{c x^2}\right )^{-q} x (e x)^m F_1\left (\frac{1}{2} (-1-m);-p,-q;\frac{1-m}{2};-\frac{b}{a x^2},-\frac{d}{c x^2}\right )}{1+m}\\ \end{align*}
Mathematica [A] time = 0.128659, size = 115, normalized size = 1.1 \[ \frac{x (e x)^m \left (a+\frac{b}{x^2}\right )^p \left (\frac{a x^2}{b}+1\right )^{-p} \left (c+\frac{d}{x^2}\right )^q \left (\frac{c x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2} (m-2 p-2 q+1);-p,-q;\frac{1}{2} (m-2 p-2 q+3);-\frac{a x^2}{b},-\frac{c x^2}{d}\right )}{m-2 p-2 q+1} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.227, size = 0, normalized size = 0. \begin{align*} \int \left ( a+{\frac{b}{{x}^{2}}} \right ) ^{p} \left ( c+{\frac{d}{{x}^{2}}} \right ) ^{q} \left ( ex \right ) ^{m}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{m}{\left (a + \frac{b}{x^{2}}\right )}^{p}{\left (c + \frac{d}{x^{2}}\right )}^{q}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (e x\right )^{m} \left (\frac{a x^{2} + b}{x^{2}}\right )^{p} \left (\frac{c x^{2} + d}{x^{2}}\right )^{q}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{m}{\left (a + \frac{b}{x^{2}}\right )}^{p}{\left (c + \frac{d}{x^{2}}\right )}^{q}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]