Optimal. Leaf size=634 \[ -\frac {d (a d f (4+m)-b (d e+c f (3+m))) (a+b x)^{1+m} (c+d x)^{-3-m}}{(b c-a d) (b e-a f) (d e-c f)^2 (3+m)}-\frac {d \left (a^2 d^2 f^2 \left (12+7 m+m^2\right )-b^2 \left (2 d^2 e^2-2 c d e f (4+m)-c^2 f^2 \left (6+5 m+m^2\right )\right )-2 a b d f \left (d e (2+m)+c f \left (10+6 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-2-m}}{(b c-a d)^2 (b e-a f) (d e-c f)^3 (2+m) (3+m)}-\frac {d \left (a^3 d^3 f^3 \left (24+26 m+9 m^2+m^3\right )-a^2 b d^2 f^2 (3+m) \left (d e (4+3 m)+c f \left (20+15 m+3 m^2\right )\right )-b^3 \left (2 d^3 e^3-2 c d^2 e^2 f (5+m)+c^2 d e f^2 \left (26+17 m+3 m^2\right )+c^3 f^3 \left (6+11 m+6 m^2+m^3\right )\right )-a b^2 d f \left (2 d^2 e^2 (2+m)-2 c d e f \left (16+15 m+3 m^2\right )-c^2 f^2 \left (44+50 m+21 m^2+3 m^3\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-1-m}}{(b c-a d)^3 (b e-a f) (d e-c f)^4 (1+m) (2+m) (3+m)}-\frac {f (a+b x)^{1+m} (c+d x)^{-3-m}}{(b e-a f) (d e-c f) (e+f x)}-\frac {f^3 (a d f (4+m)-b (4 d e+c f m)) (a+b x)^m (c+d x)^{-m} \, _2F_1\left (1,-m;1-m;\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{(b e-a f) (d e-c f)^5 m} \]
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Rubi [A]
time = 0.92, antiderivative size = 632, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {105, 160, 12,
133} \begin {gather*} -\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 )-\left (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+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^3 d^3 f^3 \left (m^3+9 m^2+26 m+24\right )-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 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 )-\left (b^3 \left (c^3 f^3 \left (m^3+6 m^2+11 m+6\right )+c^2 d e f^2 \left (3 m^2+17 m+26\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)+b 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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 105
Rule 133
Rule 160
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*}
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Mathematica [A]
time = 2.08, size = 562, normalized size = 0.89 \begin {gather*} -\frac {(a+b x)^{1+m} (c+d x)^{-3-m} \left (\frac {d}{e+f x}+\frac {f (b d e+b c f (3+m)-a d f (4+m)) (c+d x)}{(b e-a f) (d e-c f) (e+f x)}+\frac {(c+d x) \left (d (b c-a d) (b e-a f) (d e-c f) (1+m)^2 \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 )+(c+d x) \left (d (b e-a f) (1+m) \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 )-(b c-a d)^3 f^3 \left (6+11 m+6 m^2+m^3\right ) (-a d f (4+m)+b (4 d e+c f 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 )\right )\right )}{(b c-a d)^2 (b e-a f)^2 (d e-c f)^3 (1+m)^2 (2+m)}\right )}{(b c-a d) (-d e+c f) (3+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{m} \left (d x +c \right )^{-4-m}}{\left (f x +e \right )^{2}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^m}{{\left (e+f\,x\right )}^2\,{\left (c+d\,x\right )}^{m+4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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