Optimal. Leaf size=233 \[ -\frac {f (a+b x)^m (c+d x)^{-m} (a d f (m+2)-b (c f m+2 d e)) \, _2F_1\left (1,-m;1-m;\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{m (b e-a f) (d e-c f)^3}-\frac {d (a+b x)^{m+1} (c+d x)^{-m-1} (a d f (m+2)-b (c f (m+1)+d e))}{(m+1) (b c-a d) (b e-a f) (d e-c f)^2}-\frac {f (a+b x)^{m+1} (c+d x)^{-m-1}}{(e+f x) (b e-a f) (d e-c f)} \]
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Rubi [A] time = 0.20, antiderivative size = 243, normalized size of antiderivative = 1.04, 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} \[ \frac {f (a+b x)^{m+1} (c+d x)^{-m-1} (a d f (m+2)-b (c f m+2 d e)) \, _2F_1\left (1,m+1;m+2;\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(m+1) (b e-a f)^2 (d e-c f)^2}+\frac {d (a+b x)^{m+1} (c+d x)^{-m-1}}{(m+1) (e+f x) (b c-a d) (d e-c f)}+\frac {f (a+b x)^{m+1} (c+d x)^{-m} (-a d f (m+2)+b c f (m+1)+b d e)}{(m+1) (e+f x) (b c-a d) (b e-a f) (d e-c f)^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 129
Rule 131
Rule 151
Rubi steps
\begin {align*} \int \frac {(a+b x)^m (c+d x)^{-2-m}}{(e+f x)^2} \, dx &=\frac {d (a+b x)^{1+m} (c+d x)^{-1-m}}{(b c-a d) (d e-c f) (1+m) (e+f x)}+\frac {\int \frac {(a+b x)^m (c+d x)^{-1-m} (-f (b c (1+m)-a d (2+m))+b d f x)}{(e+f x)^2} \, dx}{(b c-a d) (d e-c f) (1+m)}\\ &=\frac {d (a+b x)^{1+m} (c+d x)^{-1-m}}{(b c-a d) (d e-c f) (1+m) (e+f x)}+\frac {f (b d e+b c f (1+m)-a d f (2+m)) (a+b x)^{1+m} (c+d x)^{-m}}{(b c-a d) (b e-a f) (d e-c f)^2 (1+m) (e+f x)}+\frac {\int \frac {(b c-a d) f (1+m) (a d f (2+m)-b (2 d e+c f m)) (a+b x)^m (c+d x)^{-1-m}}{e+f x} \, dx}{(b c-a d) (b e-a f) (d e-c f)^2 (1+m)}\\ &=\frac {d (a+b x)^{1+m} (c+d x)^{-1-m}}{(b c-a d) (d e-c f) (1+m) (e+f x)}+\frac {f (b d e+b c f (1+m)-a d f (2+m)) (a+b x)^{1+m} (c+d x)^{-m}}{(b c-a d) (b e-a f) (d e-c f)^2 (1+m) (e+f x)}+\frac {(f (a d f (2+m)-b (2 d e+c f m))) \int \frac {(a+b x)^m (c+d x)^{-1-m}}{e+f x} \, dx}{(b e-a f) (d e-c f)^2}\\ &=\frac {d (a+b x)^{1+m} (c+d x)^{-1-m}}{(b c-a d) (d e-c f) (1+m) (e+f x)}+\frac {f (b d e+b c f (1+m)-a d f (2+m)) (a+b x)^{1+m} (c+d x)^{-m}}{(b c-a d) (b e-a f) (d e-c f)^2 (1+m) (e+f x)}+\frac {f (a d f (2+m)-b (2 d e+c f m)) (a+b x)^{1+m} (c+d x)^{-1-m} \, _2F_1\left (1,1+m;2+m;\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(b e-a f)^2 (d e-c f)^2 (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 193, normalized size = 0.83 \[ \frac {(a+b x)^{m+1} (c+d x)^{-m-1} \left (f (e+f x) (b c-a d) (a d f (m+2)-b (c f m+2 d e)) \, _2F_1\left (1,m+1;m+2;\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )+f (c+d x) (b e-a f) (-a d f (m+2)+b c f (m+1)+b d e)+d (b e-a f)^2 (d e-c f)\right )}{(m+1) (e+f x) (b c-a d) (b e-a f)^2 (d e-c f)^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.03, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 2}}{f^{2} x^{2} + 2 \, e f x + e^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 2}}{{\left (f x + e\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.24, size = 0, normalized size = 0.00 \[ \int \frac {\left (b x +a \right )^{m} \left (d x +c \right )^{-m -2}}{\left (f x +e \right )^{2}}\, dx \]
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 \[ \int \frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 2}}{{\left (f x + e\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,x\right )}^m}{{\left (e+f\,x\right )}^2\,{\left (c+d\,x\right )}^{m+2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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