Optimal. Leaf size=62 \[ \frac{a x^{m+1} \text{Hypergeometric2F1}\left (1,\frac{m+1}{2},\frac{m+3}{2},-\frac{e x^2}{d}\right )}{d (m+1)}+b \text{Unintegrable}\left (\frac{x^m \tan ^{-1}(c x)}{d+e x^2},x\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.120963, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{x^m \left (a+b \tan ^{-1}(c x)\right )}{d+e x^2} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin{align*} \int \frac{x^m \left (a+b \tan ^{-1}(c x)\right )}{d+e x^2} \, dx &=a \int \frac{x^m}{d+e x^2} \, dx+b \int \frac{x^m \tan ^{-1}(c x)}{d+e x^2} \, dx\\ &=\frac{a x^{1+m} \, _2F_1\left (1,\frac{1+m}{2};\frac{3+m}{2};-\frac{e x^2}{d}\right )}{d (1+m)}+b \int \frac{x^m \tan ^{-1}(c x)}{d+e x^2} \, dx\\ \end{align*}
Mathematica [A] time = 2.22537, size = 0, normalized size = 0. \[ \int \frac{x^m \left (a+b \tan ^{-1}(c x)\right )}{d+e x^2} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 1.5, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{m} \left ( a+b\arctan \left ( cx \right ) \right ) }{e{x}^{2}+d}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \arctan \left (c x\right ) + a\right )} x^{m}}{e x^{2} + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b \arctan \left (c x\right ) + a\right )} x^{m}}{e x^{2} + d}, 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 [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \arctan \left (c x\right ) + a\right )} x^{m}}{e x^{2} + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]