Optimal. Leaf size=38 \[ \frac{(c x)^m \, _2F_1\left (1,\frac{m}{2};\frac{m+2}{2};-\frac{b x^2}{a}\right )}{a c m} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0085645, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {364} \[ \frac{(c x)^m \, _2F_1\left (1,\frac{m}{2};\frac{m+2}{2};-\frac{b x^2}{a}\right )}{a c m} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 364
Rubi steps
\begin{align*} \int \frac{(c x)^{-1+m}}{a+b x^2} \, dx &=\frac{(c x)^m \, _2F_1\left (1,\frac{m}{2};\frac{2+m}{2};-\frac{b x^2}{a}\right )}{a c m}\\ \end{align*}
Mathematica [A] time = 0.0073654, size = 38, normalized size = 1. \[ \frac{x (c x)^{m-1} \, _2F_1\left (1,\frac{m}{2};\frac{m}{2}+1;-\frac{b x^2}{a}\right )}{a m} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.028, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( cx \right ) ^{-1+m}}{b{x}^{2}+a}}\, 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 (c x\right )^{m - 1}}{b x^{2} + a}\,{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 (c x\right )^{m - 1}}{b x^{2} + a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 18.7562, size = 39, normalized size = 1.03 \begin{align*} \frac{c^{m} m x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2}\right ) \Gamma \left (\frac{m}{2}\right )}{4 a c \Gamma \left (\frac{m}{2} + 1\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 (c x\right )^{m - 1}}{b x^{2} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]