Optimal. Leaf size=120 \[ \frac {d (a+b x)^{1+m} (c+d x)^{-1-m}}{(b c-a d) (d e-c f) (1+m)}+\frac {f (a+b x)^m (c+d x)^{-m} \, _2F_1\left (1,-m;1-m;\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{(d e-c f)^2 m} \]
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Rubi [A]
time = 0.04, antiderivative size = 120, normalized size of antiderivative = 1.00, 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 = {98, 133}
\begin {gather*} \frac {f (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 )}{m (d e-c f)^2}+\frac {d (a+b x)^{m+1} (c+d x)^{-m-1}}{(m+1) (b c-a d) (d e-c f)} \end {gather*}
Antiderivative was successfully verified.
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Rule 98
Rule 133
Rubi steps
\begin {align*} \int \frac {(a+b x)^m (c+d x)^{-2-m}}{e+f x} \, dx &=\frac {d (a+b x)^{1+m} (c+d x)^{-1-m}}{(b c-a d) (d e-c f) (1+m)}-\frac {f \int \frac {(a+b x)^m (c+d x)^{-1-m}}{e+f x} \, dx}{d e-c f}\\ &=\frac {d (a+b x)^{1+m} (c+d x)^{-1-m}}{(b c-a d) (d e-c f) (1+m)}-\frac {f (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) (d e-c f) (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 115, normalized size = 0.96 \begin {gather*} \frac {(a+b x)^{1+m} (c+d x)^{-1-m} \left (-b d e+a d f+(b c-a d) f \, _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 )}{(b c-a d) (b e-a f) (-d e+c f) (1+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{m} \left (d x +c \right )^{-2-m}}{f x +e}\, dx\]
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 (e+f\,x\right )\,{\left (c+d\,x\right )}^{m+2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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