Optimal. Leaf size=190 \[ -\frac {(a+b x)^m (c+d x)^{1-m}}{f (e+f x)}+\frac {(a d f (1-m)-b (d e-c f m)) (a+b x)^m (c+d x)^{-m} \, _2F_1\left (1,m;1+m;\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{f^2 (b e-a f) m}+\frac {d (a+b x)^m (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,m;1+m;-\frac {d (a+b x)}{b c-a d}\right )}{f^2 m} \]
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Rubi [A]
time = 0.12, antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {132, 72, 71,
156, 12, 133} \begin {gather*} \frac {(a+b x)^m (c+d x)^{-m} (a d f (1-m)-b (d e-c f m)) \, _2F_1\left (1,m;m+1;\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{f^2 m (b e-a f)}-\frac {(a+b x)^m (c+d x)^{1-m}}{f (e+f x)}+\frac {d (a+b x)^m (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,m;m+1;-\frac {d (a+b x)}{b c-a d}\right )}{f^2 m} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 71
Rule 72
Rule 132
Rule 133
Rule 156
Rubi steps
\begin {align*} \int \frac {(a+b x)^m (c+d x)^{1-m}}{(e+f x)^2} \, dx &=\frac {\left ((b c-a d) (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 )^{1-m}}{(e+f x)^2} \, dx}{b}\\ &=\frac {(b c-a d) (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;-1+m,2;2+m;-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{(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 = 0.24, size = 203, normalized size = 1.07 \begin {gather*} -\frac {(a+b x)^{1+m} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \left (\frac {b (e+f x)}{b e-a f}\right )^{-m} \left (d (e+f x) \left (\frac {b (e+f x)}{b e-a f}\right )^m F_1\left (1+m;m,1;2+m;\frac {d (a+b x)}{-b c+a d},\frac {f (a+b x)}{-b e+a f}\right )+(-d e+c f) \, _2F_1\left (m,1+m;2+m;\frac {(-d e+c f) (a+b x)}{(b c-a d) (e+f x)}\right )\right )}{f (-b e+a f) (1+m) (e+f x)} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{m} \left (d x +c \right )^{1-m}}{\left (f x +e \right )^{2}}\, dx\]
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 = -0.01 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^m\,{\left (c+d\,x\right )}^{1-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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