Optimal. Leaf size=70 \[ \frac{x^{m+1} \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};a^2 x^2\right )}{m+1}+\frac{a x^{m+2} \, _2F_1\left (2,\frac{m+2}{2};\frac{m+4}{2};a^2 x^2\right )}{m+2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0343823, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {848, 82, 73, 364} \[ \frac{x^{m+1} \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};a^2 x^2\right )}{m+1}+\frac{a x^{m+2} \, _2F_1\left (2,\frac{m+2}{2};\frac{m+4}{2};a^2 x^2\right )}{m+2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 848
Rule 82
Rule 73
Rule 364
Rubi steps
\begin{align*} \int \frac{x^m}{(1-a x) \left (1-a^2 x^2\right )} \, dx &=\int \frac{x^m}{(1-a x)^2 (1+a x)} \, dx\\ &=a \int \frac{x^{1+m}}{(1-a x)^2 (1+a x)^2} \, dx+\int \frac{x^m}{(1-a x)^2 (1+a x)^2} \, dx\\ &=a \int \frac{x^{1+m}}{\left (1-a^2 x^2\right )^2} \, dx+\int \frac{x^m}{\left (1-a^2 x^2\right )^2} \, dx\\ &=\frac{x^{1+m} \, _2F_1\left (2,\frac{1+m}{2};\frac{3+m}{2};a^2 x^2\right )}{1+m}+\frac{a x^{2+m} \, _2F_1\left (2,\frac{2+m}{2};\frac{4+m}{2};a^2 x^2\right )}{2+m}\\ \end{align*}
Mathematica [A] time = 0.0057891, size = 67, normalized size = 0.96 \[ x^{m+1} \left (\frac{a x \, _2F_1\left (2,\frac{m}{2}+1;\frac{m}{2}+2;a^2 x^2\right )}{m+2}+\frac{\, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};a^2 x^2\right )}{m+1}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.043, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{m}}{ \left ( -ax+1 \right ) \left ( -{a}^{2}{x}^{2}+1 \right ) }}\, 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{x^{m}}{{\left (a^{2} x^{2} - 1\right )}{\left (a x - 1\right )}}\,{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{x^{m}}{a^{3} x^{3} - a^{2} x^{2} - a x + 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m}}{\left (a x - 1\right )^{2} \left (a x + 1\right )}\, dx \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{x^{m}}{{\left (a^{2} x^{2} - 1\right )}{\left (a x - 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]