Optimal. Leaf size=39 \[ -\frac{e (d+e x)^{m-1} \, _2F_1\left (2,m-1;m;\frac{e x}{d}+1\right )}{c^2 d^2 (1-m)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02125, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {626, 12, 65} \[ -\frac{e (d+e x)^{m-1} \, _2F_1\left (2,m-1;m;\frac{e x}{d}+1\right )}{c^2 d^2 (1-m)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 626
Rule 12
Rule 65
Rubi steps
\begin{align*} \int \frac{(d+e x)^m}{\left (c d x+c e x^2\right )^2} \, dx &=\int \frac{(d+e x)^{-2+m}}{c^2 x^2} \, dx\\ &=\frac{\int \frac{(d+e x)^{-2+m}}{x^2} \, dx}{c^2}\\ &=-\frac{e (d+e x)^{-1+m} \, _2F_1\left (2,-1+m;m;1+\frac{e x}{d}\right )}{c^2 d^2 (1-m)}\\ \end{align*}
Mathematica [A] time = 0.0126374, size = 36, normalized size = 0.92 \[ \frac{e (d+e x)^{m-1} \, _2F_1\left (2,m-1;m;\frac{e x}{d}+1\right )}{c^2 d^2 (m-1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.563, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex+d \right ) ^{m}}{ \left ( ce{x}^{2}+cdx \right ) ^{2}}}\, 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 )}^{m}}{{\left (c e x^{2} + c d x\right )}^{2}}\,{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 )}^{m}}{c^{2} e^{2} x^{4} + 2 \, c^{2} d e x^{3} + c^{2} d^{2} x^{2}}, 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*} \frac{\int \frac{\left (d + e x\right )^{m}}{d^{2} x^{2} + 2 d e x^{3} + e^{2} x^{4}}\, dx}{c^{2}} \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 )}^{m}}{{\left (c e x^{2} + c d x\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]