Optimal. Leaf size=233 \[ -\frac {(a+b x)^m (B e-A f) (c+d x)^{-m} \, _2F_1\left (1,-m;1-m;\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{f^2 m}-\frac {(a+b x)^{m+1} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,m+1;m+2;-\frac {d (a+b x)}{b c-a d}\right ) (a B d f m-b (-A d f+B c f m+B d e))}{b f^2 m (m+1) (b c-a d)}-\frac {d (a+b x)^{m+1} (B e-A f) (c+d x)^{-m}}{f^2 m (b c-a d)} \]
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Rubi [A] time = 0.13, antiderivative size = 220, normalized size of antiderivative = 0.94, number of steps used = 7, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {157, 70, 69, 105, 131} \[ \frac {(a+b x)^m (B e-A f) (c+d x)^{-m} \, _2F_1\left (1,m;m+1;\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{f^2 m}-\frac {(a+b x)^m (B e-A f) (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,m;m+1;-\frac {d (a+b x)}{b c-a d}\right )}{f^2 m}+\frac {B (a+b x)^{m+1} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,m+1;m+2;-\frac {d (a+b x)}{b c-a d}\right )}{b f (m+1)} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 105
Rule 131
Rule 157
Rubi steps
\begin {align*} \int \frac {(a+b x)^m (A+B x) (c+d x)^{-m}}{e+f x} \, dx &=\frac {B \int (a+b x)^m (c+d x)^{-m} \, dx}{f}+\frac {(-B e+A f) \int \frac {(a+b x)^m (c+d x)^{-m}}{e+f x} \, dx}{f}\\ &=-\frac {(b (B e-A f)) \int (a+b x)^{-1+m} (c+d x)^{-m} \, dx}{f^2}+\frac {((b e-a f) (B e-A f)) \int \frac {(a+b x)^{-1+m} (c+d x)^{-m}}{e+f x} \, dx}{f^2}+\frac {\left (B (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 {(B e-A 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 {B (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 (B e-A 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 {(B e-A 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 {(B e-A 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 {B (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*}
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Mathematica [A] time = 0.14, size = 174, normalized size = 0.75 \[ \frac {(a+b x)^m (c+d x)^{-m} \left (\left (\frac {b (c+d x)}{b c-a d}\right )^m \left (B f m (a+b x) \, _2F_1\left (m,m+1;m+2;\frac {d (a+b x)}{a d-b c}\right )-b (m+1) (B e-A f) \, _2F_1\left (m,m;m+1;\frac {d (a+b x)}{a d-b c}\right )\right )+b (m+1) (B e-A f) \, _2F_1\left (1,m;m+1;\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )\right )}{b f^2 m (m+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B x + A\right )} {\left (b x + a\right )}^{m}}{{\left (f x + e\right )} {\left (d x + c\right )}^{m}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B x + A\right )} {\left (b x + a\right )}^{m}}{{\left (f x + e\right )} {\left (d x + c\right )}^{m}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.25, size = 0, normalized size = 0.00 \[ \int \frac {\left (B x +A \right ) \left (b x +a \right )^{m} \left (d x +c \right )^{-m}}{f x +e}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B x + A\right )} {\left (b x + a\right )}^{m}}{{\left (f x + e\right )} {\left (d x + c\right )}^{m}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (A+B\,x\right )\,{\left (a+b\,x\right )}^m}{\left (e+f\,x\right )\,{\left (c+d\,x\right )}^m} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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