Optimal. Leaf size=147 \[ \frac {f (a+b x)^{1+m} (c+d x)^{3-m}}{4 b d}-\frac {(b c-a d)^2 (a d f (3-m)-b (4 d e-c f (1+m))) (a+b x)^{1+m} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (-2+m,1+m;2+m;-\frac {d (a+b x)}{b c-a d}\right )}{4 b^4 d (1+m)} \]
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Rubi [A]
time = 0.05, antiderivative size = 146, normalized size of antiderivative = 0.99, 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 = {81, 72, 71}
\begin {gather*} \frac {(b c-a d)^2 (a+b x)^{m+1} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m (-a d f (3-m)-b c f (m+1)+4 b d e) \, _2F_1\left (m-2,m+1;m+2;-\frac {d (a+b x)}{b c-a d}\right )}{4 b^4 d (m+1)}+\frac {f (a+b x)^{m+1} (c+d x)^{3-m}}{4 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)^{2-m} (e+f x) \, dx &=\frac {f (a+b x)^{1+m} (c+d x)^{3-m}}{4 b d}+\frac {(4 b d e-f (a d (3-m)+b c (1+m))) \int (a+b x)^m (c+d x)^{2-m} \, dx}{4 b d}\\ &=\frac {f (a+b x)^{1+m} (c+d x)^{3-m}}{4 b d}+\frac {\left ((b c-a d)^2 (4 b d e-f (a d (3-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 )^{2-m} \, dx}{4 b^3 d}\\ &=\frac {f (a+b x)^{1+m} (c+d x)^{3-m}}{4 b d}+\frac {(b c-a d)^2 (4 b d e-a d f (3-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 (-2+m,1+m;2+m;-\frac {d (a+b x)}{b c-a d}\right )}{4 b^4 d (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 128, normalized size = 0.87 \begin {gather*} \frac {(a+b x)^{1+m} (c+d x)^{-m} \left (b^3 f (1+m) (c+d x)^3-(b c-a d)^2 (-4 b d e-a d f (-3+m)+b c f (1+m)) \left (\frac {b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (-2+m,1+m;2+m;\frac {d (a+b x)}{-b c+a d}\right )\right )}{4 b^4 d (1+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (b x +a \right )^{m} \left (d x +c \right )^{2-m} \left (f x +e \right )\, dx\]
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(-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 \left (e+f\,x\right )\,{\left (a+b\,x\right )}^m\,{\left (c+d\,x\right )}^{2-m} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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