Optimal. Leaf size=145 \[ \frac {f (a+b x)^{m+1} (c+d x)^{2-m}}{3 b d}-\frac {(b c-a d) (a+b x)^{m+1} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m (a d f (2-m)-b (3 d e-c f (m+1))) \, _2F_1\left (m-1,m+1;m+2;-\frac {d (a+b x)}{b c-a d}\right )}{3 b^3 d (m+1)} \]
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Rubi [A] time = 0.08, antiderivative size = 144, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {80, 70, 69} \[ \frac {(b c-a d) (a+b x)^{m+1} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m (-a d f (2-m)-b c f (m+1)+3 b d e) \, _2F_1\left (m-1,m+1;m+2;-\frac {d (a+b x)}{b c-a d}\right )}{3 b^3 d (m+1)}+\frac {f (a+b x)^{m+1} (c+d x)^{2-m}}{3 b d} \]
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)^{1-m} (e+f x) \, dx &=\frac {f (a+b x)^{1+m} (c+d x)^{2-m}}{3 b d}+\frac {(3 b d e-f (a d (2-m)+b c (1+m))) \int (a+b x)^m (c+d x)^{1-m} \, dx}{3 b d}\\ &=\frac {f (a+b x)^{1+m} (c+d x)^{2-m}}{3 b d}+\frac {\left ((b c-a d) (3 b d e-f (a d (2-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 )^{1-m} \, dx}{3 b^2 d}\\ &=\frac {f (a+b x)^{1+m} (c+d x)^{2-m}}{3 b d}+\frac {(b c-a d) (3 b d e-a d f (2-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 (-1+m,1+m;2+m;-\frac {d (a+b x)}{b c-a d}\right )}{3 b^3 d (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 126, normalized size = 0.87 \[ \frac {(a+b x)^{m+1} (c+d x)^{-m} \left (b^2 f (m+1) (c+d x)^2-(b c-a d) \left (\frac {b (c+d x)}{b c-a d}\right )^m (-a d f (m-2)+b c f (m+1)-3 b d e) \, _2F_1\left (m-1,m+1;m+2;\frac {d (a+b x)}{a d-b c}\right )\right )}{3 b^3 d (m+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.08, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (f x + e\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m + 1}, x\right ) \]
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 \[ \int {\left (f x + e\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m + 1}\,{d x} \]
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 \[ \int \left (f x +e \right ) \left (b x +a \right )^{m} \left (d x +c \right )^{-m +1}\, dx \]
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 \[ \int {\left (f x + e\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m + 1}\,{d x} \]
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 \[ \int \left (e+f\,x\right )\,{\left (a+b\,x\right )}^m\,{\left (c+d\,x\right )}^{1-m} \,d x \]
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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