Integrand size = 29, antiderivative size = 233 \[ \int \frac {(a+b x)^m (A+B x) (c+d x)^{-m}}{e+f x} \, dx=-\frac {d (B e-A f) (a+b x)^{1+m} (c+d x)^{-m}}{(b c-a d) f^2 m}-\frac {(B e-A f) (a+b x)^m (c+d x)^{-m} \operatorname {Hypergeometric2F1}\left (1,-m,1-m,\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{f^2 m}-\frac {(a B d f m-b (B d e-A d f+B c f m)) (a+b x)^{1+m} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \operatorname {Hypergeometric2F1}\left (m,1+m,2+m,-\frac {d (a+b x)}{b c-a d}\right )}{b (b c-a d) f^2 m (1+m)} \]
[Out]
Time = 0.15 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {161, 133, 80, 72, 71} \[ \int \frac {(a+b x)^m (A+B x) (c+d x)^{-m}}{e+f x} \, dx=-\frac {(a+b x)^m (B e-A f) (c+d x)^{-m} \operatorname {Hypergeometric2F1}\left (1,-m,1-m,\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{f^2 m}-\frac {(a+b x)^{m+1} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \operatorname {Hypergeometric2F1}\left (m,m+1,m+2,-\frac {d (a+b x)}{b c-a d}\right ) (a B d f m-b (-A d f+B c f m+B d e))}{b f^2 m (m+1) (b c-a d)}-\frac {d (a+b x)^{m+1} (B e-A f) (c+d x)^{-m}}{f^2 m (b c-a d)} \]
[In]
[Out]
Rule 71
Rule 72
Rule 80
Rule 133
Rule 161
Rubi steps \begin{align*} \text {integral}& = \frac {\int (a+b x)^m (c+d x)^{-1-m} (-B d e+B c f+A d f+B d f x) \, dx}{f^2}+\frac {((B e-A f) (d e-c f)) \int \frac {(a+b x)^m (c+d x)^{-1-m}}{e+f x} \, dx}{f^2} \\ & = -\frac {d (B e-A f) (a+b x)^{1+m} (c+d x)^{-m}}{(b c-a d) f^2 m}-\frac {(B e-A f) (a+b x)^m (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 {(b d (-B d e+B c f+A d f)-B d f (-a d m+b c (1+m))) \int (a+b x)^m (c+d x)^{-m} \, dx}{d (-b c+a d) f^2 m} \\ & = -\frac {d (B e-A f) (a+b x)^{1+m} (c+d x)^{-m}}{(b c-a d) f^2 m}-\frac {(B e-A f) (a+b x)^m (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 {\left ((b d (-B d e+B c f+A d f)-B d f (-a d m+b c (1+m))) (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}{d (-b c+a d) f^2 m} \\ & = -\frac {d (B e-A f) (a+b x)^{1+m} (c+d x)^{-m}}{(b c-a d) f^2 m}-\frac {(B e-A f) (a+b x)^m (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 {(a B d f m-b (B d e-A d f+B c f m)) (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 (b c-a d) f^2 m (1+m)} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.75 \[ \int \frac {(a+b x)^m (A+B x) (c+d x)^{-m}}{e+f x} \, dx=\frac {(a+b x)^m (c+d x)^{-m} \left (b (B e-A f) (1+m) \operatorname {Hypergeometric2F1}\left (1,m,1+m,\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )+\left (\frac {b (c+d x)}{b c-a d}\right )^m \left (-b (B e-A f) (1+m) \operatorname {Hypergeometric2F1}\left (m,m,1+m,\frac {d (a+b x)}{-b c+a d}\right )+B f m (a+b x) \operatorname {Hypergeometric2F1}\left (m,1+m,2+m,\frac {d (a+b x)}{-b c+a d}\right )\right )\right )}{b f^2 m (1+m)} \]
[In]
[Out]
\[\int \frac {\left (b x +a \right )^{m} \left (B x +A \right ) \left (d x +c \right )^{-m}}{f x +e}d x\]
[In]
[Out]
\[ \int \frac {(a+b x)^m (A+B x) (c+d x)^{-m}}{e+f x} \, dx=\int { \frac {{\left (B x + A\right )} {\left (b x + a\right )}^{m}}{{\left (f x + e\right )} {\left (d x + c\right )}^{m}} \,d x } \]
[In]
[Out]
Exception generated. \[ \int \frac {(a+b x)^m (A+B x) (c+d x)^{-m}}{e+f x} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
[In]
[Out]
\[ \int \frac {(a+b x)^m (A+B x) (c+d x)^{-m}}{e+f x} \, dx=\int { \frac {{\left (B x + A\right )} {\left (b x + a\right )}^{m}}{{\left (f x + e\right )} {\left (d x + c\right )}^{m}} \,d x } \]
[In]
[Out]
\[ \int \frac {(a+b x)^m (A+B x) (c+d x)^{-m}}{e+f x} \, dx=\int { \frac {{\left (B x + A\right )} {\left (b x + a\right )}^{m}}{{\left (f x + e\right )} {\left (d x + c\right )}^{m}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(a+b x)^m (A+B x) (c+d x)^{-m}}{e+f x} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (a+b\,x\right )}^m}{\left (e+f\,x\right )\,{\left (c+d\,x\right )}^m} \,d x \]
[In]
[Out]