Optimal. Leaf size=121 \[ -\frac {7 (21-m) (1+2 x)^{-m} (2+3 x)^{1+m}}{3 m}-\frac {1}{3} (5-4 x) (1+2 x)^{-m} (2+3 x)^{1+m}+\frac {2^{-1-m} \left (441-86 m+2 m^2\right ) (1+2 x)^{1-m} \, _2F_1(1-m,-m;2-m;-3 (1+2 x))}{3 (1-m) m} \]
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Rubi [A]
time = 0.07, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {92, 80, 71}
\begin {gather*} \frac {2^{-m-1} \left (2 m^2-86 m+441\right ) (2 x+1)^{1-m} \, _2F_1(1-m,-m;2-m;-3 (2 x+1))}{3 (1-m) m}-\frac {1}{3} (5-4 x) (3 x+2)^{m+1} (2 x+1)^{-m}-\frac {7 (21-m) (3 x+2)^{m+1} (2 x+1)^{-m}}{3 m} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 80
Rule 92
Rubi steps
\begin {align*} \int (5-4 x)^2 (1+2 x)^{-1-m} (2+3 x)^m \, dx &=-\frac {1}{3} (5-4 x) (1+2 x)^{-m} (2+3 x)^{1+m}+\frac {1}{12} \int (1+2 x)^{-1-m} (2+3 x)^m (4 (82-5 m)-8 (65-2 m) x) \, dx\\ &=-\frac {7 (21-m) (1+2 x)^{-m} (2+3 x)^{1+m}}{3 m}-\frac {1}{3} (5-4 x) (1+2 x)^{-m} (2+3 x)^{1+m}+\frac {(24 (82-5 m)+8 (65-2 m) (-4 m+3 (1+m))) \int (1+2 x)^{-m} (2+3 x)^m \, dx}{24 m}\\ &=-\frac {7 (21-m) (1+2 x)^{-m} (2+3 x)^{1+m}}{3 m}-\frac {1}{3} (5-4 x) (1+2 x)^{-m} (2+3 x)^{1+m}+\frac {2^{-1-m} \left (441-86 m+2 m^2\right ) (1+2 x)^{1-m} \, _2F_1(1-m,-m;2-m;-3 (1+2 x))}{3 (1-m) m}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 96, normalized size = 0.79 \begin {gather*} \frac {1}{6} (1+2 x)^{-m} \left (\frac {14 (-21+m) (2+3 x)^{1+m}}{m}+2 (2+3 x)^{1+m} (-5+4 x)-\frac {2^{-m} \left (441-86 m+2 m^2\right ) (1+2 x) \, _2F_1(1-m,-m;2-m;-3-6 x)}{(-1+m) m}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \left (5-4 x \right )^{2} \left (1+2 x \right )^{-1-m} \left (2+3 x \right )^{m}\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (2 x + 1\right )^{- m - 1} \left (3 x + 2\right )^{m} \left (4 x - 5\right )^{2}\, dx \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 (3\,x+2\right )}^m\,{\left (4\,x-5\right )}^2}{{\left (2\,x+1\right )}^{m+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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