Optimal. Leaf size=188 \[ -\frac {1}{9} (107-2 m) (5-4 x)^2 (1+2 x)^{-2-m} (2+3 x)^{1+m}-\frac {1}{3} (5-4 x)^3 (1+2 x)^{-2-m} (2+3 x)^{1+m}+\frac {7 (1+2 x)^{-2-m} (2+3 x)^{1+m} \left (3 \left (4638+485 m+108 m^2-2 m^3\right )+2 \left (15209+1882 m-530 m^2+8 m^3\right ) x\right )}{9 \left (2+3 m+m^2\right )}-\frac {2^{2-m} \left (1323-85 m+m^2\right ) (1+2 x)^{-m} \, _2F_1(-m,-m;1-m;-3 (1+2 x))}{9 m} \]
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Rubi [A]
time = 0.13, antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {102, 158, 150,
71} \begin {gather*} -\frac {2^{2-m} \left (m^2-85 m+1323\right ) (2 x+1)^{-m} \, _2F_1(-m,-m;1-m;-3 (2 x+1))}{9 m}+\frac {7 (3 x+2)^{m+1} \left (2 \left (8 m^3-530 m^2+1882 m+15209\right ) x+3 \left (-2 m^3+108 m^2+485 m+4638\right )\right ) (2 x+1)^{-m-2}}{9 \left (m^2+3 m+2\right )}-\frac {1}{3} (5-4 x)^3 (3 x+2)^{m+1} (2 x+1)^{-m-2}-\frac {1}{9} (107-2 m) (5-4 x)^2 (3 x+2)^{m+1} (2 x+1)^{-m-2} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 102
Rule 150
Rule 158
Rubi steps
\begin {align*} \int (5-4 x)^4 (1+2 x)^{-3-m} (2+3 x)^m \, dx &=-\frac {1}{3} (5-4 x)^3 (1+2 x)^{-2-m} (2+3 x)^{1+m}+\frac {1}{12} \int (5-4 x)^2 (1+2 x)^{-3-m} (2+3 x)^m (4 (26-5 m)-8 (107-2 m) x) \, dx\\ &=-\frac {1}{9} (107-2 m) (5-4 x)^2 (1+2 x)^{-2-m} (2+3 x)^{1+m}-\frac {1}{3} (5-4 x)^3 (1+2 x)^{-2-m} (2+3 x)^{1+m}+\frac {1}{72} \int (5-4 x) (1+2 x)^{-3-m} (2+3 x)^m \left (-8 \left (3997+528 m-10 m^2\right )-64 \left (1323-85 m+m^2\right ) x\right ) \, dx\\ &=-\frac {1}{9} (107-2 m) (5-4 x)^2 (1+2 x)^{-2-m} (2+3 x)^{1+m}-\frac {1}{3} (5-4 x)^3 (1+2 x)^{-2-m} (2+3 x)^{1+m}+\frac {7 (1+2 x)^{-2-m} (2+3 x)^{1+m} \left (3 \left (4638+485 m+108 m^2-2 m^3\right )+2 \left (15209+1882 m-530 m^2+8 m^3\right ) x\right )}{9 \left (2+3 m+m^2\right )}+\frac {1}{9} \left (8 \left (1323-85 m+m^2\right )\right ) \int (1+2 x)^{-1-m} (2+3 x)^m \, dx\\ &=-\frac {1}{9} (107-2 m) (5-4 x)^2 (1+2 x)^{-2-m} (2+3 x)^{1+m}-\frac {1}{3} (5-4 x)^3 (1+2 x)^{-2-m} (2+3 x)^{1+m}+\frac {7 (1+2 x)^{-2-m} (2+3 x)^{1+m} \left (3 \left (4638+485 m+108 m^2-2 m^3\right )+2 \left (15209+1882 m-530 m^2+8 m^3\right ) x\right )}{9 \left (2+3 m+m^2\right )}-\frac {2^{2-m} \left (1323-85 m+m^2\right ) (1+2 x)^{-m} \, _2F_1(-m,-m;1-m;-3 (1+2 x))}{9 m}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in
optimal.
time = 0.87, size = 214, normalized size = 1.14 \begin {gather*} \frac {2744 (1+2 x)^{-1-m} (2+3 x)^{1+m}}{1+m}+\left (\frac {23}{14}\right )^m (5-4 x)^2 F_1\left (2;-m,m;3;\frac {3}{23} (5-4 x),\frac {1}{7} (5-4 x)\right )+\frac {21\ 2^{2-m} (1+2 x)^{1-m} \, _2F_1(1-m,-m;2-m;-3-6 x)}{-1+m}-\frac {49\ 2^{3+m} (-1-2 x)^m (2+4 x)^{-m} (6+9 x)^{1+m} \, _2F_1(1+m,1+m;2+m;4+6 x)}{1+m}-\frac {2401\ 3^{2+m} (-1-2 x)^m (1+2 x)^{-m} (2+3 x)^{1+m} \, _2F_1(1+m,3+m;2+m;4+6 x)}{1+m} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \left (5-4 x \right )^{4} \left (1+2 x \right )^{-3-m} \left (2+3 x \right )^{m}\, dx\]
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 )}^4}{{\left (2\,x+1\right )}^{m+3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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