Optimal. Leaf size=142 \[ -\frac {2}{9} (5-4 x)^2 (1+2 x)^{-m} (2+3 x)^{1+m}-\frac {(1+2 x)^{-m} (2+3 x)^{1+m} \left (9261-512 m+4 m^2-4 (109-2 m) m x\right )}{27 m}+\frac {2^{-1-m} \left (27783-8324 m+390 m^2-4 m^3\right ) (1+2 x)^{1-m} \, _2F_1(1-m,-m;2-m;-3 (1+2 x))}{27 (1-m) m} \]
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Rubi [A]
time = 0.12, antiderivative size = 142, 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 = {102, 151, 71}
\begin {gather*} \frac {2^{-m-1} \left (-4 m^3+390 m^2-8324 m+27783\right ) (2 x+1)^{1-m} \, _2F_1(1-m,-m;2-m;-3 (2 x+1))}{27 (1-m) m}-\frac {(3 x+2)^{m+1} \left (4 m^2-4 (109-2 m) m x-512 m+9261\right ) (2 x+1)^{-m}}{27 m}-\frac {2}{9} (5-4 x)^2 (3 x+2)^{m+1} (2 x+1)^{-m} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 102
Rule 151
Rubi steps
\begin {align*} \int (5-4 x)^3 (1+2 x)^{-1-m} (2+3 x)^m \, dx &=-\frac {2}{9} (5-4 x)^2 (1+2 x)^{-m} (2+3 x)^{1+m}+\frac {1}{18} \int (5-4 x) (1+2 x)^{-1-m} (2+3 x)^m (2 (223-10 m)-8 (109-2 m) x) \, dx\\ &=-\frac {2}{9} (5-4 x)^2 (1+2 x)^{-m} (2+3 x)^{1+m}-\frac {(1+2 x)^{-m} (2+3 x)^{1+m} \left (9261-512 m+4 m^2-4 (109-2 m) m x\right )}{27 m}+\frac {(288 (109-2 m) (1+m) (2+m)+6 (1+m) (-6 (-8 (223-10 m)-40 (109-2 m))-128 (109-2 m) m)+4 (180 (223-10 m)+12 (-8 (223-10 m)-40 (109-2 m)) m-128 (109-2 m) (1-m) m)) \int (1+2 x)^{-m} (2+3 x)^m \, dx}{432 m}\\ &=-\frac {2}{9} (5-4 x)^2 (1+2 x)^{-m} (2+3 x)^{1+m}-\frac {(1+2 x)^{-m} (2+3 x)^{1+m} \left (9261-512 m+4 m^2-4 (109-2 m) m x\right )}{27 m}+\frac {2^{-1-m} \left (27783-8324 m+390 m^2-4 m^3\right ) (1+2 x)^{1-m} \, _2F_1(1-m,-m;2-m;-3 (1+2 x))}{27 (1-m) m}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 106, normalized size = 0.75 \begin {gather*} \frac {2^{-1-m} (1+2 x)^{-m} \left (-\left ((-1+m) (4+6 x)^{1+m} \left (9261+m^2 (4+8 x)+m \left (-362-676 x+96 x^2\right )\right )\right )+\left (-27783+8324 m-390 m^2+4 m^3\right ) (1+2 x) \, _2F_1(1-m,-m;2-m;-3-6 x)\right )}{27 (-1+m) m} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \left (5-4 x \right )^{3} \left (1+2 x \right )^{-1-m} \left (2+3 x \right )^{m}\, dx\]
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(-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 (3\,x+2\right )}^m\,{\left (4\,x-5\right )}^3}{{\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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