Optimal. Leaf size=174 \[ \frac{(b c-a d) (a+b x)^{m+1} (c+d x)^{-m-1} (b (2 d e-c f (1-m))-a d f (m+1)) \, _2F_1\left (2,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{2 (m+1) (b e-a f)^3 (d e-c f)}-\frac{f (a+b x)^{m+1} (c+d x)^{1-m}}{2 (e+f x)^2 (b e-a f) (d e-c f)} \]
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Rubi [A] time = 0.0770256, antiderivative size = 173, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {96, 131} \[ \frac{(b c-a d) (a+b x)^{m+1} (c+d x)^{-m-1} (-a d f (m+1)-b c f (1-m)+2 b d e) \, _2F_1\left (2,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{2 (m+1) (b e-a f)^3 (d e-c f)}-\frac{f (a+b x)^{m+1} (c+d x)^{1-m}}{2 (e+f x)^2 (b e-a f) (d e-c f)} \]
Antiderivative was successfully verified.
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Rule 96
Rule 131
Rubi steps
\begin{align*} \int \frac{(a+b x)^m (c+d x)^{-m}}{(e+f x)^3} \, dx &=-\frac{f (a+b x)^{1+m} (c+d x)^{1-m}}{2 (b e-a f) (d e-c f) (e+f x)^2}+\frac{(2 b d e-b c f (1-m)-a d f (1+m)) \int \frac{(a+b x)^m (c+d x)^{-m}}{(e+f x)^2} \, dx}{2 (b e-a f) (d e-c f)}\\ &=-\frac{f (a+b x)^{1+m} (c+d x)^{1-m}}{2 (b e-a f) (d e-c f) (e+f x)^2}+\frac{(b c-a d) (2 b d e-b c f (1-m)-a d f (1+m)) (a+b x)^{1+m} (c+d x)^{-1-m} \, _2F_1\left (2,1+m;2+m;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{2 (b e-a f)^3 (d e-c f) (1+m)}\\ \end{align*}
Mathematica [A] time = 0.101075, size = 146, normalized size = 0.84 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m-1} \left (\frac{(b c-a d) (-a d f (m+1)+b c f (m-1)+2 b d e) \, _2F_1\left (2,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(m+1) (b e-a f)^2}-\frac{f (c+d x)^2}{(e+f x)^2}\right )}{2 (b e-a f) (d e-c f)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.084, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{m}}{ \left ( dx+c \right ) ^{m} \left ( fx+e \right ) ^{3}}}\, 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 (f x + e\right )}^{3}{\left (d x + c\right )}^{m}}\,{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 (f^{3} x^{3} + 3 \, e f^{2} x^{2} + 3 \, e^{2} f x + e^{3}\right )}{\left (d x + c\right )}^{m}}, 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 (f x + e\right )}^{3}{\left (d x + c\right )}^{m}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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