Optimal. Leaf size=51 \[ -\frac{(d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{b (d+e x)}{b d-a e}\right )}{(m+1) (b d-a e)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0145051, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {27, 68} \[ -\frac{(d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{b (d+e x)}{b d-a e}\right )}{(m+1) (b d-a e)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 27
Rule 68
Rubi steps
\begin{align*} \int \frac{(a+b x) (d+e x)^m}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac{(d+e x)^m}{a+b x} \, dx\\ &=-\frac{(d+e x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac{b (d+e x)}{b d-a e}\right )}{(b d-a e) (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0093286, size = 51, normalized size = 1. \[ -\frac{(d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{b (d+e x)}{b d-a e}\right )}{(m+1) (b d-a e)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.078, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) \left ( ex+d \right ) ^{m}}{{b}^{2}{x}^{2}+2\,abx+{a}^{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 (b x + a\right )}{\left (e x + d\right )}^{m}}{b^{2} x^{2} + 2 \, a b x + a^{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}}{b x + a}, 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{\left (d + e x\right )^{m}}{a + b x}\, 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{{\left (b x + a\right )}{\left (e x + d\right )}^{m}}{b^{2} x^{2} + 2 \, a b x + a^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]