Optimal. Leaf size=91 \[ \frac{A (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{c x^2}{a}\right )}{a e (m+1)}+\frac{B (e x)^{m+2} \, _2F_1\left (1,\frac{m+2}{2};\frac{m+4}{2};-\frac{c x^2}{a}\right )}{a e^2 (m+2)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0421989, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {808, 364} \[ \frac{A (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{c x^2}{a}\right )}{a e (m+1)}+\frac{B (e x)^{m+2} \, _2F_1\left (1,\frac{m+2}{2};\frac{m+4}{2};-\frac{c x^2}{a}\right )}{a e^2 (m+2)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 808
Rule 364
Rubi steps
\begin{align*} \int \frac{(e x)^m (A+B x)}{a+c x^2} \, dx &=A \int \frac{(e x)^m}{a+c x^2} \, dx+\frac{B \int \frac{(e x)^{1+m}}{a+c x^2} \, dx}{e}\\ &=\frac{A (e x)^{1+m} \, _2F_1\left (1,\frac{1+m}{2};\frac{3+m}{2};-\frac{c x^2}{a}\right )}{a e (1+m)}+\frac{B (e x)^{2+m} \, _2F_1\left (1,\frac{2+m}{2};\frac{4+m}{2};-\frac{c x^2}{a}\right )}{a e^2 (2+m)}\\ \end{align*}
Mathematica [A] time = 0.0304526, size = 82, normalized size = 0.9 \[ \frac{x (e x)^m \left (A (m+2) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{c x^2}{a}\right )+B (m+1) x \, _2F_1\left (1,\frac{m}{2}+1;\frac{m}{2}+2;-\frac{c x^2}{a}\right )\right )}{a (m+1) (m+2)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.062, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex \right ) ^{m} \left ( Bx+A \right ) }{c{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 (B x + A\right )} \left (e x\right )^{m}}{c 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 (B x + A\right )} \left (e x\right )^{m}}{c x^{2} + a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 4.47315, size = 192, normalized size = 2.11 \begin{align*} \frac{A e^{m} m x x^{m} \Phi \left (\frac{c x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + \frac{1}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{1}{2}\right )}{4 a \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )} + \frac{A e^{m} x x^{m} \Phi \left (\frac{c x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + \frac{1}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{1}{2}\right )}{4 a \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )} + \frac{B e^{m} m x^{2} x^{m} \Phi \left (\frac{c x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + 1\right ) \Gamma \left (\frac{m}{2} + 1\right )}{4 a \Gamma \left (\frac{m}{2} + 2\right )} + \frac{B e^{m} x^{2} x^{m} \Phi \left (\frac{c x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + 1\right ) \Gamma \left (\frac{m}{2} + 1\right )}{2 a \Gamma \left (\frac{m}{2} + 2\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 (B x + A\right )} \left (e x\right )^{m}}{c x^{2} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]