Optimal. Leaf size=283 \[ \frac{(a+b x)^m (c+d x)^{-m} \left (-a^2 d^2 f^2 \left (m^2+3 m+2\right )+2 a b d f (m+1) (c f m+2 d e)+b^2 \left (-\left (-c^2 f^2 (1-m) m+4 c d e f m+2 d^2 e^2\right )\right )\right ) \, _2F_1\left (1,-m;1-m;\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{2 m (b e-a f)^2 (d e-c f)^3}-\frac{f (a+b x)^{m+1} (c+d x)^{-m} (b (3 d e-c f (1-m))-a d f (m+2))}{2 (e+f x) (b e-a f)^2 (d e-c f)^2}-\frac{f (a+b x)^{m+1} (c+d x)^{-m}}{2 (e+f x)^2 (b e-a f) (d e-c f)} \]
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Rubi [A] time = 0.285844, antiderivative size = 300, normalized size of antiderivative = 1.06, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {129, 151, 12, 131} \[ \frac{(a+b x)^{m+1} (c+d x)^{-m-1} \left (-a^2 d^2 f^2 \left (m^2+3 m+2\right )+2 a b d f (m+1) (c f m+2 d e)+b^2 \left (-\left (-c^2 f^2 (1-m) m+4 c d e f m+2 d^2 e^2\right )\right )\right ) \, _2F_1\left (2,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{2 m (m+1) (b e-a f)^3 (d e-c f)^2}-\frac{f (a+b x)^{m+1} (c+d x)^{1-m} (a d f (m+2)-b (c f m+2 d e))}{2 m (e+f x)^2 (b c-a d) (b e-a f) (d e-c f)^2}+\frac{d (a+b x)^{m+1} (c+d x)^{-m}}{m (e+f x)^2 (b c-a d) (d e-c f)} \]
Antiderivative was successfully verified.
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Rule 129
Rule 151
Rule 12
Rule 131
Rubi steps
\begin{align*} \int \frac{(a+b x)^m (c+d x)^{-1-m}}{(e+f x)^3} \, dx &=\frac{d (a+b x)^{1+m} (c+d x)^{-m}}{(b c-a d) (d e-c f) m (e+f x)^2}+\frac{\int \frac{(a+b x)^m (c+d x)^{-m} (a d f (2+m)-b (d e+c f m)+b d f x)}{(e+f x)^3} \, dx}{(b c-a d) (d e-c f) m}\\ &=-\frac{f (a d f (2+m)-b (2 d e+c f m)) (a+b x)^{1+m} (c+d x)^{1-m}}{2 (b c-a d) (b e-a f) (d e-c f)^2 m (e+f x)^2}+\frac{d (a+b x)^{1+m} (c+d x)^{-m}}{(b c-a d) (d e-c f) m (e+f x)^2}-\frac{\int \frac{\left (-2 a b d f (1+m) (2 d e+c f m)+b^2 \left (2 d^2 e^2+4 c d e f m-c^2 f^2 (1-m) m\right )+a^2 d^2 f^2 \left (2+3 m+m^2\right )\right ) (a+b x)^m (c+d x)^{-m}}{(e+f x)^2} \, dx}{2 (b c-a d) (b e-a f) (d e-c f)^2 m}\\ &=-\frac{f (a d f (2+m)-b (2 d e+c f m)) (a+b x)^{1+m} (c+d x)^{1-m}}{2 (b c-a d) (b e-a f) (d e-c f)^2 m (e+f x)^2}+\frac{d (a+b x)^{1+m} (c+d x)^{-m}}{(b c-a d) (d e-c f) m (e+f x)^2}+\frac{\left (2 a b d f (1+m) (2 d e+c f m)-b^2 \left (2 d^2 e^2+4 c d e f m-c^2 f^2 (1-m) m\right )-a^2 d^2 f^2 \left (2+3 m+m^2\right )\right ) \int \frac{(a+b x)^m (c+d x)^{-m}}{(e+f x)^2} \, dx}{2 (b c-a d) (b e-a f) (d e-c f)^2 m}\\ &=-\frac{f (a d f (2+m)-b (2 d e+c f m)) (a+b x)^{1+m} (c+d x)^{1-m}}{2 (b c-a d) (b e-a f) (d e-c f)^2 m (e+f x)^2}+\frac{d (a+b x)^{1+m} (c+d x)^{-m}}{(b c-a d) (d e-c f) m (e+f x)^2}+\frac{\left (2 a b d f (1+m) (2 d e+c f m)-b^2 \left (2 d^2 e^2+4 c d e f m-c^2 f^2 (1-m) m\right )-a^2 d^2 f^2 \left (2+3 m+m^2\right )\right ) (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)^2 m (1+m)}\\ \end{align*}
Mathematica [A] time = 0.401472, size = 260, normalized size = 0.92 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m} \left (-\frac{(b c-a d) \left (a^2 d^2 f^2 \left (m^2+3 m+2\right )-2 a b d f (m+1) (c f m+2 d e)+b^2 \left (c^2 f^2 (m-1) m+4 c d e f m+2 d^2 e^2\right )\right ) \, _2F_1\left (2,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(m+1) (c+d x) (b e-a f)^3 (c f-d e)}-\frac{f (c+d x) (b (c f m+2 d e)-a d f (m+2))}{(e+f x)^2 (b e-a f) (d e-c f)}-\frac{2 d}{(e+f x)^2}\right )}{2 m (b c-a d) (c f-d e)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.089, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{-1-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 (d x + c\right )}^{-m - 1}}{{\left (f x + e\right )}^{3}}\,{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 - 1}}{f^{3} x^{3} + 3 \, e f^{2} x^{2} + 3 \, e^{2} f x + e^{3}}, 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 - 1}}{{\left (f x + e\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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