Optimal. Leaf size=52 \[ -\frac{(a c-b c x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{a-b x}{2 a}\right )}{2 a b c (n+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0145496, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {68} \[ -\frac{(a c-b c x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{a-b x}{2 a}\right )}{2 a b c (n+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 68
Rubi steps
\begin{align*} \int \frac{(a c-b c x)^n}{a+b x} \, dx &=-\frac{(a c-b c x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac{a-b x}{2 a}\right )}{2 a b c (1+n)}\\ \end{align*}
Mathematica [A] time = 0.0128123, size = 52, normalized size = 1. \[ -\frac{(a-b x) (c (a-b x))^n \, _2F_1\left (1,n+1;n+2;\frac{a-b x}{2 a}\right )}{2 a b (n+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.04, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( -bcx+ac \right ) ^{n}}{bx+a}}\, 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 c x + a c\right )}^{n}}{b x + a}\,{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 c x + a c\right )}^{n}}{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 (- c \left (- a + b x\right )\right )^{n}}{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 c x + a c\right )}^{n}}{b x + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]