Optimal. Leaf size=397 \[ \frac {3 b d (d e-c f)^2 (a+b x)^m (c+d x)^{1-m}}{(b c-a d) f^4 m}+\frac {d^2 (a+b x)^{1+m} (c+d x)^{1-m}}{2 b f^2}-\frac {(d e-c f)^2 (a+b x)^m (c+d x)^{1-m}}{f^3 (e+f x)}+\frac {(d e-c f)^2 (a d f (3-m)-b (3 d e-c f m)) (a+b x)^m (c+d x)^{-m} \, _2F_1\left (1,m;1+m;\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{f^4 (b e-a f) m}+\frac {d^2 \left (2 a b d f (2 d e-c f (3-m)) m+a^2 d^2 f^2 (1-m) m-b^2 \left (6 d^2 e^2-4 c d e f (3-m)+c^2 f^2 \left (6-5 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,1+m;2+m;-\frac {d (a+b x)}{b c-a d}\right )}{2 b^2 (b c-a d) f^4 m (1+m)} \]
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Rubi [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in
optimal.
time = 0.04, antiderivative size = 113, normalized size of antiderivative = 0.28, number of steps
used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {142, 141}
\begin {gather*} \frac {(b c-a d)^3 (a+b x)^{m+1} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m F_1\left (m+1;m-3,2;m+2;-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^2 (m+1) (b e-a f)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 141
Rule 142
Rubi steps
\begin {align*} \int \frac {(a+b x)^m (c+d x)^{3-m}}{(e+f x)^2} \, dx &=\frac {\left ((b c-a d)^3 (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m\right ) \int \frac {(a+b x)^m \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{3-m}}{(e+f x)^2} \, dx}{b^3}\\ &=\frac {(b c-a d)^3 (a+b x)^{1+m} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m F_1\left (1+m;-3+m,2;2+m;-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^2 (b e-a f)^2 (1+m)}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in
optimal.
time = 0.22, size = 111, normalized size = 0.28 \begin {gather*} \frac {(b c-a d)^3 (a+b x)^{1+m} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m F_1\left (1+m;-3+m,2;2+m;\frac {d (a+b x)}{-b c+a d},\frac {f (a+b x)}{-b e+a f}\right )}{b^2 (b e-a f)^2 (1+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{m} \left (d x +c \right )^{3-m}}{\left (f x +e \right )^{2}}\, dx\]
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 (c+d\,x\right )}^{3-m}}{{\left (e+f\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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