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 = 0.12, 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 69
Rule 70
Rule 105
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 = 0.18, 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]
________________________________________________________________________________________
fricas [F] time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m + 1}}{f x + e}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m + 1}}{f x + e}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.24, size = 0, normalized size = 0.00 \[ \int \frac {\left (b x +a \right )^{m} \left (d x +c \right )^{-m +1}}{f x +e}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m + 1}}{f x + e}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,x\right )}^m\,{\left (c+d\,x\right )}^{1-m}}{e+f\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________