Optimal. Leaf size=135 \[ \frac {f (a+b x)^{m+1} (c+d x)^{1-m}}{2 b d}-\frac {(a+b x)^{m+1} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m (a d f (1-m)-b (2 d e-c f (m+1))) \, _2F_1\left (m,m+1;m+2;-\frac {d (a+b x)}{b c-a d}\right )}{2 b^2 d (m+1)} \]
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Rubi [A] time = 0.06, antiderivative size = 134, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {80, 70, 69} \[ \frac {(a+b x)^{m+1} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m (-a d f (1-m)-b c f (m+1)+2 b d e) \, _2F_1\left (m,m+1;m+2;-\frac {d (a+b x)}{b c-a d}\right )}{2 b^2 d (m+1)}+\frac {f (a+b x)^{m+1} (c+d x)^{1-m}}{2 b d} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 80
Rubi steps
\begin {align*} \int (a+b x)^m (c+d x)^{-m} (e+f x) \, dx &=\frac {f (a+b x)^{1+m} (c+d x)^{1-m}}{2 b d}+\frac {(2 b d e-f (a d (1-m)+b c (1+m))) \int (a+b x)^m (c+d x)^{-m} \, dx}{2 b d}\\ &=\frac {f (a+b x)^{1+m} (c+d x)^{1-m}}{2 b d}+\frac {\left ((2 b d e-f (a d (1-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}{2 b d}\\ &=\frac {f (a+b x)^{1+m} (c+d x)^{1-m}}{2 b d}+\frac {(2 b d e-a d f (1-m)-b c f (1+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 )}{2 b^2 d (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 109, normalized size = 0.81 \[ \frac {(a+b x)^{m+1} (c+d x)^{-m} \left (b f (c+d x)-\frac {\left (\frac {b (c+d x)}{b c-a d}\right )^m (-a d f (m-1)+b c f (m+1)-2 b d e) \, _2F_1\left (m,m+1;m+2;\frac {d (a+b x)}{a d-b c}\right )}{m+1}\right )}{2 b^2 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.00, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (f x + e\right )} {\left (b x + a\right )}^{m}}{{\left (d x + c\right )}^{m}}, x\right ) \]
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 \[ \int \frac {{\left (f x + e\right )} {\left (b x + a\right )}^{m}}{{\left (d x + c\right )}^{m}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.23, size = 0, normalized size = 0.00 \[ \int \left (f x +e \right ) \left (b x +a \right )^{m} \left (d x +c \right )^{-m}\, dx \]
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 \[ \int \frac {{\left (f x + e\right )} {\left (b x + a\right )}^{m}}{{\left (d x + c\right )}^{m}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (e+f\,x\right )\,{\left (a+b\,x\right )}^m}{{\left (c+d\,x\right )}^m} \,d x \]
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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