Optimal. Leaf size=209 \[ -\frac{\left (\sqrt{-a}-\sqrt{c} x\right ) \left (a+c x^2\right )^p (d+e x)^{-2 p-1} \left (-\frac{\left (\sqrt{-a}+\sqrt{c} x\right ) \left (\sqrt{-a} e+\sqrt{c} d\right )}{\left (\sqrt{-a}-\sqrt{c} x\right ) \left (\sqrt{c} d-\sqrt{-a} e\right )}\right )^{-p} \, _2F_1\left (-2 p-1,-p;-2 p;\frac{2 \sqrt{-a} \sqrt{c} (d+e x)}{\left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{-a}-\sqrt{c} x\right )}\right )}{(2 p+1) \left (\sqrt{-a} e+\sqrt{c} d\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0839888, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {727} \[ -\frac{\left (\sqrt{-a}-\sqrt{c} x\right ) \left (a+c x^2\right )^p (d+e x)^{-2 p-1} \left (-\frac{\left (\sqrt{-a}+\sqrt{c} x\right ) \left (\sqrt{-a} e+\sqrt{c} d\right )}{\left (\sqrt{-a}-\sqrt{c} x\right ) \left (\sqrt{c} d-\sqrt{-a} e\right )}\right )^{-p} \, _2F_1\left (-2 p-1,-p;-2 p;\frac{2 \sqrt{-a} \sqrt{c} (d+e x)}{\left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{-a}-\sqrt{c} x\right )}\right )}{(2 p+1) \left (\sqrt{-a} e+\sqrt{c} d\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 727
Rubi steps
\begin{align*} \int (d+e x)^{-2-2 p} \left (a+c x^2\right )^p \, dx &=-\frac{\left (\sqrt{-a}-\sqrt{c} x\right ) \left (-\frac{\left (\sqrt{c} d+\sqrt{-a} e\right ) \left (\sqrt{-a}+\sqrt{c} x\right )}{\left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{-a}-\sqrt{c} x\right )}\right )^{-p} (d+e x)^{-1-2 p} \left (a+c x^2\right )^p \, _2F_1\left (-1-2 p,-p;-2 p;\frac{2 \sqrt{-a} \sqrt{c} (d+e x)}{\left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{-a}-\sqrt{c} x\right )}\right )}{\left (\sqrt{c} d+\sqrt{-a} e\right ) (1+2 p)}\\ \end{align*}
Mathematica [A] time = 0.13412, size = 200, normalized size = 0.96 \[ \frac{\left (\sqrt{c} x-\sqrt{-a}\right ) \left (a+c x^2\right )^p (d+e x)^{-2 p-1} \left (1-\frac{c (d+e x)}{c d-\sqrt{-a} \sqrt{c} e}\right )^{-p} \left (1-\frac{c (d+e x)}{\sqrt{-a} \sqrt{c} e+c d}\right )^p \, _2F_1\left (-2 p-1,-p;-2 p;\frac{2 \sqrt{-a} \sqrt{c} (d+e x)}{\left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{-a}-\sqrt{c} x\right )}\right )}{(2 p+1) \left (\sqrt{-a} e+\sqrt{c} d\right )} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.583, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) ^{-2-2\,p} \left ( c{x}^{2}+a \right ) ^{p}\, 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 (c x^{2} + a\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 2}\,{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 (c x^{2} + a\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 2}, 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 (c x^{2} + a\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]