Optimal. Leaf size=92 \[ \frac{c d \left (a+c x^2\right )^{p+1} \, _2F_1\left (2,p+1;p+2;\frac{c x^2}{a}+1\right )}{2 a^2 (p+1)}-\frac{e \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \, _2F_1\left (-\frac{1}{2},-p;\frac{1}{2};-\frac{c x^2}{a}\right )}{x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0503845, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {764, 266, 65, 365, 364} \[ \frac{c d \left (a+c x^2\right )^{p+1} \, _2F_1\left (2,p+1;p+2;\frac{c x^2}{a}+1\right )}{2 a^2 (p+1)}-\frac{e \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \, _2F_1\left (-\frac{1}{2},-p;\frac{1}{2};-\frac{c x^2}{a}\right )}{x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 764
Rule 266
Rule 65
Rule 365
Rule 364
Rubi steps
\begin{align*} \int \frac{(d+e x) \left (a+c x^2\right )^p}{x^3} \, dx &=d \int \frac{\left (a+c x^2\right )^p}{x^3} \, dx+e \int \frac{\left (a+c x^2\right )^p}{x^2} \, dx\\ &=\frac{1}{2} d \operatorname{Subst}\left (\int \frac{(a+c x)^p}{x^2} \, dx,x,x^2\right )+\left (e \left (a+c x^2\right )^p \left (1+\frac{c x^2}{a}\right )^{-p}\right ) \int \frac{\left (1+\frac{c x^2}{a}\right )^p}{x^2} \, dx\\ &=-\frac{e \left (a+c x^2\right )^p \left (1+\frac{c x^2}{a}\right )^{-p} \, _2F_1\left (-\frac{1}{2},-p;\frac{1}{2};-\frac{c x^2}{a}\right )}{x}+\frac{c d \left (a+c x^2\right )^{1+p} \, _2F_1\left (2,1+p;2+p;1+\frac{c x^2}{a}\right )}{2 a^2 (1+p)}\\ \end{align*}
Mathematica [A] time = 0.0488155, size = 89, normalized size = 0.97 \[ \frac{1}{2} \left (a+c x^2\right )^p \left (\frac{c d \left (a+c x^2\right ) \, _2F_1\left (2,p+1;p+2;\frac{c x^2}{a}+1\right )}{a^2 (p+1)}-\frac{2 e \left (\frac{c x^2}{a}+1\right )^{-p} \, _2F_1\left (-\frac{1}{2},-p;\frac{1}{2};-\frac{c x^2}{a}\right )}{x}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.035, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex+d \right ) \left ( c{x}^{2}+a \right ) ^{p}}{{x}^{3}}}\, 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 \frac{{\left (e x + d\right )}{\left (c x^{2} + a\right )}^{p}}{x^{3}}\,{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 (\frac{{\left (e x + d\right )}{\left (c x^{2} + a\right )}^{p}}{x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 19.8266, size = 71, normalized size = 0.77 \begin{align*} - \frac{a^{p} e{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, - p \\ \frac{1}{2} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{x} - \frac{c^{p} d x^{2 p} \Gamma \left (1 - p\right ){{}_{2}F_{1}\left (\begin{matrix} - p, 1 - p \\ 2 - p \end{matrix}\middle |{\frac{a e^{i \pi }}{c x^{2}}} \right )}}{2 x^{2} \Gamma \left (2 - p\right )} \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 \frac{{\left (e x + d\right )}{\left (c x^{2} + a\right )}^{p}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]