Optimal. Leaf size=62 \[ \frac{(b c-a d) (a+b x)^{n+1} (c+d x)^{-n-1} \, _2F_1\left (2,n+1;n+2;\frac{c (a+b x)}{a (c+d x)}\right )}{a^2 (n+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0143239, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {131} \[ \frac{(b c-a d) (a+b x)^{n+1} (c+d x)^{-n-1} \, _2F_1\left (2,n+1;n+2;\frac{c (a+b x)}{a (c+d x)}\right )}{a^2 (n+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 131
Rubi steps
\begin{align*} \int \frac{(a+b x)^n (c+d x)^{-n}}{x^2} \, dx &=\frac{(b c-a d) (a+b x)^{1+n} (c+d x)^{-1-n} \, _2F_1\left (2,1+n;2+n;\frac{c (a+b x)}{a (c+d x)}\right )}{a^2 (1+n)}\\ \end{align*}
Mathematica [A] time = 0.0157296, size = 62, normalized size = 1. \[ \frac{(b c-a d) (a+b x)^{n+1} (c+d x)^{-n-1} \, _2F_1\left (2,n+1;n+2;\frac{c (a+b x)}{a (c+d x)}\right )}{a^2 (n+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.059, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{n}}{{x}^{2} \left ( dx+c \right ) ^{n}}}\, 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 )}^{n}}{{\left (d x + c\right )}^{n} x^{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 (b x + a\right )}^{n}}{{\left (d x + c\right )}^{n} x^{2}}, 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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{n}}{{\left (d x + c\right )}^{n} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]