Optimal. Leaf size=85 \[ \frac{(b c-a d)^3 (a+b x)^{m+1} (c+d x)^{-m-1} \, _2F_1\left (4,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(m+1) (b e-a f)^4} \]
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Rubi [A] time = 0.0205919, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {131} \[ \frac{(b c-a d)^3 (a+b x)^{m+1} (c+d x)^{-m-1} \, _2F_1\left (4,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(m+1) (b e-a f)^4} \]
Antiderivative was successfully verified.
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Rule 131
Rubi steps
\begin{align*} \int \frac{(a+b x)^m (c+d x)^{2-m}}{(e+f x)^4} \, dx &=\frac{(b c-a d)^3 (a+b x)^{1+m} (c+d x)^{-1-m} \, _2F_1\left (4,1+m;2+m;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(b e-a f)^4 (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0379231, size = 86, normalized size = 1.01 \[ \frac{(b c-a d)^3 (a+b x)^{m+1} (c+d x)^{-m-1} \, _2F_1\left (4,m+1;m+2;-\frac{(c f-d e) (a+b x)}{(b e-a f) (c+d x)}\right )}{(m+1) (b e-a f)^4} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.119, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{2-m}}{ \left ( fx+e \right ) ^{4}}}\, 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 \frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 2}}{{\left (f x + e\right )}^{4}}\,{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 (\frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 2}}{f^{4} x^{4} + 4 \, e f^{3} x^{3} + 6 \, e^{2} f^{2} x^{2} + 4 \, e^{3} f x + e^{4}}, 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 \frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 2}}{{\left (f x + e\right )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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