Optimal. Leaf size=135 \[ \frac {f (a+b x)^{1+m} (c+d x)^{1-m}}{2 b d}-\frac {(a d f (1-m)-b (2 d e-c f (1+m))) (a+b x)^{1+m} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,1+m;2+m;-\frac {d (a+b x)}{b c-a d}\right )}{2 b^2 d (1+m)} \]
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Rubi [A]
time = 0.04, antiderivative size = 134, normalized size of antiderivative = 0.99, number of steps
used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {81, 72, 71}
\begin {gather*} \frac {(a+b x)^{m+1} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m (-a d f (1-m)-b c f (m+1)+2 b d e) \, _2F_1\left (m,m+1;m+2;-\frac {d (a+b x)}{b c-a d}\right )}{2 b^2 d (m+1)}+\frac {f (a+b x)^{m+1} (c+d x)^{1-m}}{2 b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 72
Rule 81
Rubi steps
\begin {align*} \int (a+b x)^m (c+d x)^{-m} (e+f x) \, dx &=\frac {f (a+b x)^{1+m} (c+d x)^{1-m}}{2 b d}+\frac {(2 b d e-f (a d (1-m)+b c (1+m))) \int (a+b x)^m (c+d x)^{-m} \, dx}{2 b d}\\ &=\frac {f (a+b x)^{1+m} (c+d x)^{1-m}}{2 b d}+\frac {\left ((2 b d e-f (a d (1-m)+b c (1+m))) (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m\right ) \int (a+b x)^m \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{-m} \, dx}{2 b d}\\ &=\frac {f (a+b x)^{1+m} (c+d x)^{1-m}}{2 b d}+\frac {(2 b d e-a d f (1-m)-b c f (1+m)) (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 d (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 109, normalized size = 0.81 \begin {gather*} \frac {(a+b x)^{1+m} (c+d x)^{-m} \left (b f (c+d x)-\frac {(-2 b d e-a d f (-1+m)+b c f (1+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 )}{1+m}\right )}{2 b^2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (b x +a \right )^{m} \left (f x +e \right ) \left (d x +c \right )^{-m}\, dx\]
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \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} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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