Optimal. Leaf size=151 \[ \frac{(a+b x)^{-n} (e+f x)^{n+1} \left (-\frac{f (a+b x)}{b e-a f}\right )^n (b (c f-d e (1-n))-a d f n) \, _2F_1\left (n,n+1;n+2;\frac{b (e+f x)}{b e-a f}\right )}{f^2 n (n+1) (b e-a f)}+\frac{(a+b x)^{1-n} (d e-c f) (e+f x)^n}{f n (b e-a f)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0701446, antiderivative size = 150, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {79, 70, 69} \[ \frac{(a+b x)^{-n} (e+f x)^{n+1} \left (-\frac{f (a+b x)}{b e-a f}\right )^n (-a d f n+b c f-b d e (1-n)) \, _2F_1\left (n,n+1;n+2;\frac{b (e+f x)}{b e-a f}\right )}{f^2 n (n+1) (b e-a f)}+\frac{(a+b x)^{1-n} (d e-c f) (e+f x)^n}{f n (b e-a f)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 79
Rule 70
Rule 69
Rubi steps
\begin{align*} \int (a+b x)^{-n} (c+d x) (e+f x)^{-1+n} \, dx &=\frac{(d e-c f) (a+b x)^{1-n} (e+f x)^n}{f (b e-a f) n}-\frac{(b c f-d (b e (1-n)+a f n)) \int (a+b x)^{-n} (e+f x)^n \, dx}{f (-b e+a f) n}\\ &=\frac{(d e-c f) (a+b x)^{1-n} (e+f x)^n}{f (b e-a f) n}-\frac{\left ((b c f-d (b e (1-n)+a f n)) (a+b x)^{-n} \left (\frac{f (a+b x)}{-b e+a f}\right )^n\right ) \int (e+f x)^n \left (-\frac{a f}{b e-a f}-\frac{b f x}{b e-a f}\right )^{-n} \, dx}{f (-b e+a f) n}\\ &=\frac{(d e-c f) (a+b x)^{1-n} (e+f x)^n}{f (b e-a f) n}+\frac{(b c f-b d e (1-n)-a d f n) (a+b x)^{-n} \left (-\frac{f (a+b x)}{b e-a f}\right )^n (e+f x)^{1+n} \, _2F_1\left (n,1+n;2+n;\frac{b (e+f x)}{b e-a f}\right )}{f^2 (b e-a f) n (1+n)}\\ \end{align*}
Mathematica [A] time = 0.0919826, size = 123, normalized size = 0.81 \[ \frac{(a+b x)^{-n} (e+f x)^n \left (\frac{(e+f x) \left (\frac{f (a+b x)}{a f-b e}\right )^n (a d f n-b (c f+d e (n-1))) \, _2F_1\left (n,n+1;n+2;\frac{b (e+f x)}{b e-a f}\right )}{n+1}+f (a+b x) (c f-d e)\right )}{f^2 n (a f-b e)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.052, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dx+c \right ) \left ( fx+e \right ) ^{-1+n}}{ \left ( bx+a \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 (d x + c\right )}{\left (f x + e\right )}^{n - 1}}{{\left (b x + a\right )}^{n}}\,{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 (d x + c\right )}{\left (f x + e\right )}^{n - 1}}{{\left (b x + a\right )}^{n}}, 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 (d x + c\right )}{\left (f x + e\right )}^{n - 1}}{{\left (b x + a\right )}^{n}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]