Optimal. Leaf size=634 \[ -\frac{d (a+b x)^{m+1} (c+d x)^{-m-2} \left (a^2 d^2 f^2 \left (m^2+7 m+12\right )-2 a b d f \left (c f \left (m^2+6 m+10\right )+d e (m+2)\right )+b^2 \left (-\left (-c^2 f^2 \left (m^2+5 m+6\right )-2 c d e f (m+4)+2 d^2 e^2\right )\right )\right )}{(m+2) (m+3) (b c-a d)^2 (b e-a f) (d e-c f)^3}-\frac{d (a+b x)^{m+1} (c+d x)^{-m-1} \left (-a^2 b d^2 f^2 (m+3) \left (c f \left (3 m^2+15 m+20\right )+d e (3 m+4)\right )+a^3 d^3 f^3 \left (m^3+9 m^2+26 m+24\right )-a b^2 d f \left (-c^2 f^2 \left (3 m^3+21 m^2+50 m+44\right )-2 c d e f \left (3 m^2+15 m+16\right )+2 d^2 e^2 (m+2)\right )+b^3 \left (-\left (c^2 d e f^2 \left (3 m^2+17 m+26\right )+c^3 f^3 \left (m^3+6 m^2+11 m+6\right )-2 c d^2 e^2 f (m+5)+2 d^3 e^3\right )\right )\right )}{(m+1) (m+2) (m+3) (b c-a d)^3 (b e-a f) (d e-c f)^4}-\frac{f^3 (a+b x)^m (c+d x)^{-m} (a d f (m+4)-b (c f m+4 d e)) \, _2F_1\left (1,-m;1-m;\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{m (b e-a f) (d e-c f)^5}-\frac{d (a+b x)^{m+1} (c+d x)^{-m-3} (a d f (m+4)-b (c f (m+3)+d e))}{(m+3) (b c-a d) (b e-a f) (d e-c f)^2}-\frac{f (a+b x)^{m+1} (c+d x)^{-m-3}}{(e+f x) (b e-a f) (d e-c f)} \]
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Rubi [A] time = 1.41366, antiderivative size = 646, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {129, 151, 155, 12, 131} \[ -\frac{d (a+b x)^{m+1} (c+d x)^{-m-2} \left (a^2 d^2 f^2 \left (m^2+7 m+12\right )-2 a b d f \left (c f \left (m^2+6 m+10\right )+d e (m+2)\right )+b^2 \left (-\left (-c^2 f^2 \left (m^2+5 m+6\right )-2 c d e f (m+4)+2 d^2 e^2\right )\right )\right )}{(m+2) (m+3) (b c-a d)^2 (b e-a f) (d e-c f)^3}-\frac{d (a+b x)^{m+1} (c+d x)^{-m-1} \left (-a^2 b d^2 f^2 (m+3) \left (c f \left (3 m^2+15 m+20\right )+d e (3 m+4)\right )+a^3 d^3 f^3 \left (m^3+9 m^2+26 m+24\right )-a b^2 d f \left (-c^2 f^2 \left (3 m^3+21 m^2+50 m+44\right )-2 c d e f \left (3 m^2+15 m+16\right )+2 d^2 e^2 (m+2)\right )+b^3 \left (-\left (c^2 d e f^2 \left (3 m^2+17 m+26\right )+c^3 f^3 \left (m^3+6 m^2+11 m+6\right )-2 c d^2 e^2 f (m+5)+2 d^3 e^3\right )\right )\right )}{(m+1) (m+2) (m+3) (b c-a d)^3 (b e-a f) (d e-c f)^4}+\frac{f^3 (a+b x)^{m+1} (c+d x)^{-m-1} (a d f (m+4)-b (c f m+4 d e)) \, _2F_1\left (1,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(m+1) (b e-a f)^2 (d e-c f)^4}+\frac{d (a+b x)^{m+1} (c+d x)^{-m-3}}{(m+3) (e+f x) (b c-a d) (d e-c f)}+\frac{f (a+b x)^{m+1} (c+d x)^{-m-2} (-a d f (m+4)+b c f (m+3)+b d e)}{(m+3) (e+f x) (b c-a d) (b e-a f) (d e-c f)^2} \]
Antiderivative was successfully verified.
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Rule 129
Rule 151
Rule 155
Rule 12
Rule 131
Rubi steps
\begin{align*} \int \frac{(a+b x)^m (c+d x)^{-4-m}}{(e+f x)^2} \, dx &=\frac{d (a+b x)^{1+m} (c+d x)^{-3-m}}{(b c-a d) (d e-c f) (3+m) (e+f x)}+\frac{\int \frac{(a+b x)^m (c+d x)^{-3-m} (2 b d e-b c f (3+m)+a d f (4+m)+3 b d f x)}{(e+f x)^2} \, dx}{(b c-a d) (d e-c f) (3+m)}\\ &=\frac{d (a+b x)^{1+m} (c+d x)^{-3-m}}{(b c-a d) (d e-c f) (3+m) (e+f x)}-\frac{f (a d f (4+m)-b (d e+c f (3+m))) (a+b x)^{1+m} (c+d x)^{-2-m}}{(b c-a d) (b e-a f) (d e-c f)^2 (3+m) (e+f x)}-\frac{\int \frac{(a+b x)^m (c+d x)^{-3-m} \left (a^2 d^2 f^2 \left (12+7 m+m^2\right )-2 a b d f (2+m) (d e+c f (3+m))-b^2 \left (2 d^2 e^2-2 c d e f (3+m)-c^2 f^2 m (3+m)\right )+2 b d f (a d f (4+m)-b (d e+c f (3+m))) x\right )}{e+f x} \, dx}{(b c-a d) (b e-a f) (d e-c f)^2 (3+m)}\\ &=-\frac{d \left (a^2 d^2 f^2 \left (12+7 m+m^2\right )-b^2 \left (2 d^2 e^2-2 c d e f (4+m)-c^2 f^2 \left (6+5 m+m^2\right )\right )-2 a b d f \left (d e (2+m)+c f \left (10+6 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-2-m}}{(b c-a d)^2 (b e-a f) (d e-c f)^3 (2+m) (3+m)}+\frac{d (a+b x)^{1+m} (c+d x)^{-3-m}}{(b c-a d) (d e-c f) (3+m) (e+f x)}-\frac{f (a d f (4+m)-b (d e+c f (3+m))) (a+b x)^{1+m} (c+d x)^{-2-m}}{(b c-a d) (b e-a f) (d e-c f)^2 (3+m) (e+f x)}-\frac{\int \frac{(a+b x)^m (c+d x)^{-2-m} \left (a^3 d^3 f^3 \left (24+26 m+9 m^2+m^3\right )-b^3 \left (2 d^3 e^3-2 c d^2 e^2 f (4+m)+3 c^2 d e f^2 \left (6+5 m+m^2\right )+c^3 f^3 m \left (6+5 m+m^2\right )\right )-a^2 b d^2 f^2 (3+m) \left (d e (4+3 m)+c f \left (16+14 m+3 m^2\right )\right )-a b^2 d f \left (2 d^2 e^2 (2+m)-2 c d e f \left (14+14 m+3 m^2\right )-c^2 f^2 \left (24+38 m+19 m^2+3 m^3\right )\right )+b d f \left (a^2 d^2 f^2 \left (12+7 m+m^2\right )-b^2 \left (2 d^2 e^2-2 c d e f (4+m)-c^2 f^2 \left (6+5 m+m^2\right )\right )-2 a b d f \left (d e (2+m)+c f \left (10+6 m+m^2\right )\right )\right ) x\right )}{e+f x} \, dx}{(b c-a d)^2 (b e-a f) (d e-c f)^3 (2+m) (3+m)}\\ &=-\frac{d \left (a^2 d^2 f^2 \left (12+7 m+m^2\right )-b^2 \left (2 d^2 e^2-2 c d e f (4+m)-c^2 f^2 \left (6+5 m+m^2\right )\right )-2 a b d f \left (d e (2+m)+c f \left (10+6 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-2-m}}{(b c-a d)^2 (b e-a f) (d e-c f)^3 (2+m) (3+m)}-\frac{d \left (a^3 d^3 f^3 \left (24+26 m+9 m^2+m^3\right )-a^2 b d^2 f^2 (3+m) \left (d e (4+3 m)+c f \left (20+15 m+3 m^2\right )\right )-b^3 \left (2 d^3 e^3-2 c d^2 e^2 f (5+m)+c^2 d e f^2 \left (26+17 m+3 m^2\right )+c^3 f^3 \left (6+11 m+6 m^2+m^3\right )\right )-a b^2 d f \left (2 d^2 e^2 (2+m)-2 c d e f \left (16+15 m+3 m^2\right )-c^2 f^2 \left (44+50 m+21 m^2+3 m^3\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-1-m}}{(b c-a d)^3 (b e-a f) (d e-c f)^4 (1+m) (2+m) (3+m)}+\frac{d (a+b x)^{1+m} (c+d x)^{-3-m}}{(b c-a d) (d e-c f) (3+m) (e+f x)}-\frac{f (a d f (4+m)-b (d e+c f (3+m))) (a+b x)^{1+m} (c+d x)^{-2-m}}{(b c-a d) (b e-a f) (d e-c f)^2 (3+m) (e+f x)}-\frac{\int \frac{(b c-a d)^3 f^3 (1+m) (2+m) (3+m) (4 b d e-4 a d f+b c f m-a d f m) (a+b x)^m (c+d x)^{-1-m}}{e+f x} \, dx}{(b c-a d)^3 (b e-a f) (d e-c f)^4 (1+m) (2+m) (3+m)}\\ &=-\frac{d \left (a^2 d^2 f^2 \left (12+7 m+m^2\right )-b^2 \left (2 d^2 e^2-2 c d e f (4+m)-c^2 f^2 \left (6+5 m+m^2\right )\right )-2 a b d f \left (d e (2+m)+c f \left (10+6 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-2-m}}{(b c-a d)^2 (b e-a f) (d e-c f)^3 (2+m) (3+m)}-\frac{d \left (a^3 d^3 f^3 \left (24+26 m+9 m^2+m^3\right )-a^2 b d^2 f^2 (3+m) \left (d e (4+3 m)+c f \left (20+15 m+3 m^2\right )\right )-b^3 \left (2 d^3 e^3-2 c d^2 e^2 f (5+m)+c^2 d e f^2 \left (26+17 m+3 m^2\right )+c^3 f^3 \left (6+11 m+6 m^2+m^3\right )\right )-a b^2 d f \left (2 d^2 e^2 (2+m)-2 c d e f \left (16+15 m+3 m^2\right )-c^2 f^2 \left (44+50 m+21 m^2+3 m^3\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-1-m}}{(b c-a d)^3 (b e-a f) (d e-c f)^4 (1+m) (2+m) (3+m)}+\frac{d (a+b x)^{1+m} (c+d x)^{-3-m}}{(b c-a d) (d e-c f) (3+m) (e+f x)}-\frac{f (a d f (4+m)-b (d e+c f (3+m))) (a+b x)^{1+m} (c+d x)^{-2-m}}{(b c-a d) (b e-a f) (d e-c f)^2 (3+m) (e+f x)}+\frac{\left (f^3 (a d f (4+m)-b (4 d e+c f m))\right ) \int \frac{(a+b x)^m (c+d x)^{-1-m}}{e+f x} \, dx}{(b e-a f) (d e-c f)^4}\\ &=-\frac{d \left (a^2 d^2 f^2 \left (12+7 m+m^2\right )-b^2 \left (2 d^2 e^2-2 c d e f (4+m)-c^2 f^2 \left (6+5 m+m^2\right )\right )-2 a b d f \left (d e (2+m)+c f \left (10+6 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-2-m}}{(b c-a d)^2 (b e-a f) (d e-c f)^3 (2+m) (3+m)}-\frac{d \left (a^3 d^3 f^3 \left (24+26 m+9 m^2+m^3\right )-a^2 b d^2 f^2 (3+m) \left (d e (4+3 m)+c f \left (20+15 m+3 m^2\right )\right )-b^3 \left (2 d^3 e^3-2 c d^2 e^2 f (5+m)+c^2 d e f^2 \left (26+17 m+3 m^2\right )+c^3 f^3 \left (6+11 m+6 m^2+m^3\right )\right )-a b^2 d f \left (2 d^2 e^2 (2+m)-2 c d e f \left (16+15 m+3 m^2\right )-c^2 f^2 \left (44+50 m+21 m^2+3 m^3\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-1-m}}{(b c-a d)^3 (b e-a f) (d e-c f)^4 (1+m) (2+m) (3+m)}+\frac{d (a+b x)^{1+m} (c+d x)^{-3-m}}{(b c-a d) (d e-c f) (3+m) (e+f x)}-\frac{f (a d f (4+m)-b (d e+c f (3+m))) (a+b x)^{1+m} (c+d x)^{-2-m}}{(b c-a d) (b e-a f) (d e-c f)^2 (3+m) (e+f x)}+\frac{f^3 (a d f (4+m)-b (4 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)^4 (1+m)}\\ \end{align*}
Mathematica [A] time = 2.71012, size = 562, normalized size = 0.89 \[ -\frac{(a+b x)^{m+1} (c+d x)^{-m-3} \left (\frac{(c+d x) \left ((c+d x) \left (d (m+1) (b e-a f) \left (a^2 b d^2 f^2 (m+3) \left (c f \left (3 m^2+15 m+20\right )+d e (3 m+4)\right )-a^3 d^3 f^3 \left (m^3+9 m^2+26 m+24\right )+a b^2 d f \left (-c^2 f^2 \left (3 m^3+21 m^2+50 m+44\right )-2 c d e f \left (3 m^2+15 m+16\right )+2 d^2 e^2 (m+2)\right )+b^3 \left (c^2 d e f^2 \left (3 m^2+17 m+26\right )+c^3 f^3 \left (m^3+6 m^2+11 m+6\right )-2 c d^2 e^2 f (m+5)+2 d^3 e^3\right )\right )-f^3 \left (m^3+6 m^2+11 m+6\right ) (b c-a d)^3 (b (c f m+4 d e)-a d f (m+4)) \, _2F_1\left (1,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )\right )+d (m+1)^2 (b c-a d) (b e-a f) (d e-c f) \left (-a^2 d^2 f^2 \left (m^2+7 m+12\right )+2 a b d f \left (c f \left (m^2+6 m+10\right )+d e (m+2)\right )+b^2 \left (-c^2 f^2 \left (m^2+5 m+6\right )-2 c d e f (m+4)+2 d^2 e^2\right )\right )\right )}{(m+1)^2 (m+2) (b c-a d)^2 (b e-a f)^2 (d e-c f)^3}+\frac{f (c+d x) (-a d f (m+4)+b c f (m+3)+b d e)}{(e+f x) (b e-a f) (d e-c f)}+\frac{d}{e+f x}\right )}{(m+3) (b c-a d) (c f-d e)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.08, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{-4-m}}{ \left ( fx+e \right ) ^{2}}}\, 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 - 4}}{{\left (f x + e\right )}^{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 (\frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 4}}{f^{2} x^{2} + 2 \, e f x + e^{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 \frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 4}}{{\left (f x + e\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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