Optimal. Leaf size=147 \[ \frac{f (a+b x)^{m+1} (c+d x)^{3-m}}{4 b d}-\frac{(b c-a d)^2 (a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m (a d f (3-m)-b (4 d e-c f (m+1))) \, _2F_1\left (m-2,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{4 b^4 d (m+1)} \]
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Rubi [A] time = 0.0730053, antiderivative size = 146, 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} \[ \frac{(b c-a d)^2 (a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m (-a d f (3-m)-b c f (m+1)+4 b d e) \, _2F_1\left (m-2,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{4 b^4 d (m+1)}+\frac{f (a+b x)^{m+1} (c+d x)^{3-m}}{4 b d} \]
Antiderivative was successfully verified.
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Rule 80
Rule 70
Rule 69
Rubi steps
\begin{align*} \int (a+b x)^m (c+d x)^{2-m} (e+f x) \, dx &=\frac{f (a+b x)^{1+m} (c+d x)^{3-m}}{4 b d}+\frac{(4 b d e-f (a d (3-m)+b c (1+m))) \int (a+b x)^m (c+d x)^{2-m} \, dx}{4 b d}\\ &=\frac{f (a+b x)^{1+m} (c+d x)^{3-m}}{4 b d}+\frac{\left ((b c-a d)^2 (4 b d e-f (a d (3-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 )^{2-m} \, dx}{4 b^3 d}\\ &=\frac{f (a+b x)^{1+m} (c+d x)^{3-m}}{4 b d}+\frac{(b c-a d)^2 (4 b d e-a d f (3-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 (-2+m,1+m;2+m;-\frac{d (a+b x)}{b c-a d}\right )}{4 b^4 d (1+m)}\\ \end{align*}
Mathematica [A] time = 0.139492, size = 128, normalized size = 0.87 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m} \left (b^3 f (m+1) (c+d x)^3-(b c-a d)^2 \left (\frac{b (c+d x)}{b c-a d}\right )^m (-a d f (m-3)+b c f (m+1)-4 b d e) \, _2F_1\left (m-2,m+1;m+2;\frac{d (a+b x)}{a d-b c}\right )\right )}{4 b^4 d (m+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.057, size = 0, normalized size = 0. \begin{align*} \int \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{2-m} \left ( fx+e \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (f x + e\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (f x + e\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (f x + e\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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