Optimal. Leaf size=132 \[ \frac {2^{-m} \left (2 m^2-128 m+1323\right ) (2 x+1)^{-m} \, _2F_1(-m,-m;1-m;-3 (2 x+1))}{9 m}-\frac {(3 x+2)^{m+1} \left (4 m^2-8 (43-m) (m+1) x-315 m+2768\right ) (2 x+1)^{-m-1}}{9 (m+1)}-\frac {1}{3} (5-4 x)^2 (3 x+2)^{m+1} (2 x+1)^{-m-1} \]
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Rubi [A] time = 0.14, antiderivative size = 132, 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 = {100, 143, 69} \[ \frac {2^{-m} \left (2 m^2-128 m+1323\right ) (2 x+1)^{-m} \, _2F_1(-m,-m;1-m;-3 (2 x+1))}{9 m}-\frac {(3 x+2)^{m+1} \left (4 m^2-8 (43-m) (m+1) x-315 m+2768\right ) (2 x+1)^{-m-1}}{9 (m+1)}-\frac {1}{3} (5-4 x)^2 (3 x+2)^{m+1} (2 x+1)^{-m-1} \]
Antiderivative was successfully verified.
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Rule 69
Rule 100
Rule 143
Rubi steps
\begin {align*} \int (5-4 x)^3 (1+2 x)^{-2-m} (2+3 x)^m \, dx &=-\frac {1}{3} (5-4 x)^2 (1+2 x)^{-1-m} (2+3 x)^{1+m}+\frac {1}{12} \int (5-4 x) (1+2 x)^{-2-m} (2+3 x)^m (4 (54-5 m)-16 (43-m) x) \, dx\\ &=-\frac {1}{3} (5-4 x)^2 (1+2 x)^{-1-m} (2+3 x)^{1+m}-\frac {(1+2 x)^{-1-m} (2+3 x)^{1+m} \left (2768-315 m+4 m^2-8 (43-m) (1+m) x\right )}{9 (1+m)}-\frac {1}{9} \left (2 \left (1323-128 m+2 m^2\right )\right ) \int (1+2 x)^{-1-m} (2+3 x)^m \, dx\\ &=-\frac {1}{3} (5-4 x)^2 (1+2 x)^{-1-m} (2+3 x)^{1+m}-\frac {(1+2 x)^{-1-m} (2+3 x)^{1+m} \left (2768-315 m+4 m^2-8 (43-m) (1+m) x\right )}{9 (1+m)}+\frac {2^{-m} \left (1323-128 m+2 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 [A] time = 0.10, size = 114, normalized size = 0.86 \[ \frac {2^{-m} (2 x+1)^{-m-1} \left (\left (2 m^3-126 m^2+1195 m+1323\right ) (2 x+1) \, _2F_1(-m,-m;1-m;-6 x-3)-2^m m (3 x+2)^{m+1} \left (m^2 (8 x+4)+24 m \left (2 x^2-19 x-10\right )+48 x^2-464 x+2843\right )\right )}{9 m (m+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (64 \, x^{3} - 240 \, x^{2} + 300 \, x - 125\right )} {\left (3 \, x + 2\right )}^{m} {\left (2 \, x + 1\right )}^{-m - 2}, x\right ) \]
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 \[ \int -{\left (3 \, x + 2\right )}^{m} {\left (2 \, x + 1\right )}^{-m - 2} {\left (4 \, x - 5\right )}^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.23, size = 0, normalized size = 0.00 \[ \int \left (-4 x +5\right )^{3} \left (2 x +1\right )^{-m -2} \left (3 x +2\right )^{m}\, dx \]
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 \[ -\int {\left (3 \, x + 2\right )}^{m} {\left (2 \, x + 1\right )}^{-m - 2} {\left (4 \, x - 5\right )}^{3}\,{d x} \]
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 \[ -\int \frac {{\left (3\,x+2\right )}^m\,{\left (4\,x-5\right )}^3}{{\left (2\,x+1\right )}^{m+2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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