Optimal. Leaf size=124 \[ \frac {(d e-c f) (a+b x)^{1+m} (c+d x)^{-1-m}}{d (b c-a d) (1+m)}-\frac {f (a+b x)^m \left (-\frac {d (a+b x)}{b c-a d}\right )^{-m} (c+d x)^{-m} \, _2F_1\left (-m,-m;1-m;\frac {b (c+d x)}{b c-a d}\right )}{d^2 m} \]
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Rubi [A]
time = 0.04, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {80, 72, 71}
\begin {gather*} \frac {(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-1}}{d (m+1) (b c-a d)}-\frac {f (a+b x)^m (c+d x)^{-m} \left (-\frac {d (a+b x)}{b c-a d}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac {b (c+d x)}{b c-a d}\right )}{d^2 m} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 72
Rule 80
Rubi steps
\begin {align*} \int (a+b x)^m (c+d x)^{-2-m} (e+f x) \, dx &=\frac {(d e-c f) (a+b x)^{1+m} (c+d x)^{-1-m}}{d (b c-a d) (1+m)}+\frac {f \int (a+b x)^m (c+d x)^{-1-m} \, dx}{d}\\ &=\frac {(d e-c f) (a+b x)^{1+m} (c+d x)^{-1-m}}{d (b c-a d) (1+m)}+\frac {\left (f (a+b x)^m \left (\frac {d (a+b x)}{-b c+a d}\right )^{-m}\right ) \int (c+d x)^{-1-m} \left (-\frac {a d}{b c-a d}-\frac {b d x}{b c-a d}\right )^m \, dx}{d}\\ &=\frac {(d e-c f) (a+b x)^{1+m} (c+d x)^{-1-m}}{d (b c-a d) (1+m)}-\frac {f (a+b x)^m \left (-\frac {d (a+b x)}{b c-a d}\right )^{-m} (c+d x)^{-m} \, _2F_1\left (-m,-m;1-m;\frac {b (c+d x)}{b c-a d}\right )}{d^2 m}\\ \end {align*}
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Mathematica [A]
time = 0.22, size = 132, normalized size = 1.06 \begin {gather*} \frac {(a+b x)^m (c+d x)^{-1-m} \left (d^2 e m (a+b x)-c d f m (a+b x)-(b c-a d) f (1+m) \left (\frac {d (a+b x)}{-b c+a d}\right )^{-m} (c+d x) \, _2F_1\left (-m,-m;1-m;\frac {b (c+d x)}{b c-a d}\right )\right )}{d^2 (b c-a d) m (1+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \left (b x +a \right )^{m} \left (d x +c \right )^{-2-m} \left (f x +e \right )\, dx\]
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 (e+f\,x\right )\,{\left (a+b\,x\right )}^m}{{\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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