Optimal. Leaf size=230 \[ -\frac{(a+b x)^m (d e-c f) (c+d x)^{-m} \, _2F_1\left (1,-m;1-m;\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{f^2 m}+\frac{d (a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m (b (d e-c f (1-m))-a d f m) \, _2F_1\left (m,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{b f^2 m (m+1) (b c-a d)}-\frac{d (a+b x)^{m+1} (d e-c f) (c+d x)^{-m}}{f^2 m (b c-a d)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.118948, antiderivative size = 220, normalized size of antiderivative = 0.96, number of steps used = 7, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {105, 70, 69, 131} \[ \frac{(a+b x)^m (d e-c f) (c+d x)^{-m} \, _2F_1\left (1,m;m+1;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{f^2 m}-\frac{(a+b x)^m (d e-c f) (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,m;m+1;-\frac{d (a+b x)}{b c-a d}\right )}{f^2 m}+\frac{d (a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{b f (m+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 105
Rule 70
Rule 69
Rule 131
Rubi steps
\begin{align*} \int \frac{(a+b x)^m (c+d x)^{1-m}}{e+f x} \, dx &=\frac{d \int (a+b x)^m (c+d x)^{-m} \, dx}{f}-\frac{(d e-c f) \int \frac{(a+b x)^m (c+d x)^{-m}}{e+f x} \, dx}{f}\\ &=-\frac{(b (d e-c f)) \int (a+b x)^{-1+m} (c+d x)^{-m} \, dx}{f^2}+\frac{((b e-a f) (d e-c f)) \int \frac{(a+b x)^{-1+m} (c+d x)^{-m}}{e+f x} \, dx}{f^2}+\frac{\left (d (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m\right ) \int (a+b x)^m \left (\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}\right )^{-m} \, dx}{f}\\ &=\frac{(d e-c f) (a+b x)^m (c+d x)^{-m} \, _2F_1\left (1,m;1+m;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{f^2 m}+\frac{d (a+b x)^{1+m} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,1+m;2+m;-\frac{d (a+b x)}{b c-a d}\right )}{b f (1+m)}-\frac{\left (b (d e-c f) (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m\right ) \int (a+b x)^{-1+m} \left (\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}\right )^{-m} \, dx}{f^2}\\ &=\frac{(d e-c f) (a+b x)^m (c+d x)^{-m} \, _2F_1\left (1,m;1+m;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{f^2 m}-\frac{(d e-c f) (a+b x)^m (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,m;1+m;-\frac{d (a+b x)}{b c-a d}\right )}{f^2 m}+\frac{d (a+b x)^{1+m} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,1+m;2+m;-\frac{d (a+b x)}{b c-a d}\right )}{b f (1+m)}\\ \end{align*}
Mathematica [A] time = 0.179883, size = 177, normalized size = 0.77 \[ \frac{(a+b x)^m (c+d x)^{-m} \left (\frac{(d e-c f) \left (\, _2F_1\left (1,m;m+1;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )-\left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,m;m+1;\frac{d (a+b x)}{a d-b c}\right )\right )}{f m}+\frac{d (a+b x) \left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,m+1;m+2;\frac{d (a+b x)}{a d-b c}\right )}{b (m+1)}\right )}{f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.07, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{1-m}}{fx+e}}\, 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 )}^{m}{\left (d x + c\right )}^{-m + 1}}{f x + e}\,{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 )}^{m}{\left (d x + c\right )}^{-m + 1}}{f x + e}, 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 )}^{m}{\left (d x + c\right )}^{-m + 1}}{f x + e}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]