Optimal. Leaf size=188 \[ -\frac {2^{2-m} \left (m^2-85 m+1323\right ) (2 x+1)^{-m} \, _2F_1(-m,-m;1-m;-3 (2 x+1))}{9 m}+\frac {7 (3 x+2)^{m+1} \left (2 \left (8 m^3-530 m^2+1882 m+15209\right ) x+3 \left (-2 m^3+108 m^2+485 m+4638\right )\right ) (2 x+1)^{-m-2}}{9 \left (m^2+3 m+2\right )}-\frac {1}{3} (5-4 x)^3 (3 x+2)^{m+1} (2 x+1)^{-m-2}-\frac {1}{9} (107-2 m) (5-4 x)^2 (3 x+2)^{m+1} (2 x+1)^{-m-2} \]
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Rubi [A] time = 0.19, 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 = {100, 153, 145, 69} \[ -\frac {2^{2-m} \left (m^2-85 m+1323\right ) (2 x+1)^{-m} \, _2F_1(-m,-m;1-m;-3 (2 x+1))}{9 m}+\frac {7 (3 x+2)^{m+1} \left (2 \left (8 m^3-530 m^2+1882 m+15209\right ) x+3 \left (-2 m^3+108 m^2+485 m+4638\right )\right ) (2 x+1)^{-m-2}}{9 \left (m^2+3 m+2\right )}-\frac {1}{3} (5-4 x)^3 (3 x+2)^{m+1} (2 x+1)^{-m-2}-\frac {1}{9} (107-2 m) (5-4 x)^2 (3 x+2)^{m+1} (2 x+1)^{-m-2} \]
Antiderivative was successfully verified.
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Rule 69
Rule 100
Rule 145
Rule 153
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 [A] time = 0.25, size = 153, normalized size = 0.81 \[ \frac {2^{-m} (2 x+1)^{-m-2} \left (2^m (3 x+2)^{m+1} \left (16 m^3 \left (6 x^2+7 x+2\right )+32 m^2 \left (18 x^3-219 x^2-274 x-80\right )+m \left (1728 x^3-21696 x^2+143632 x+12629\right )+6 \left (192 x^3-2432 x^2+118699 x+49177\right )\right )-4 \left (m^3-83 m^2+1153 m+2646\right ) (2 x+1) \, _2F_1(-m-1,-m-1;-m;-6 x-3)\right )}{27 (m+1) (m+2)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (256 \, x^{4} - 1280 \, x^{3} + 2400 \, x^{2} - 2000 \, x + 625\right )} {\left (3 \, x + 2\right )}^{m} {\left (2 \, x + 1\right )}^{-m - 3}, x\right ) \]
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 \[ \int {\left (3 \, x + 2\right )}^{m} {\left (2 \, x + 1\right )}^{-m - 3} {\left (4 \, x - 5\right )}^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.24, size = 0, normalized size = 0.00 \[ \int \left (-4 x +5\right )^{4} \left (2 x +1\right )^{-m -3} \left (3 x +2\right )^{m}\, dx \]
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 \[ \int {\left (3 \, x + 2\right )}^{m} {\left (2 \, x + 1\right )}^{-m - 3} {\left (4 \, x - 5\right )}^{4}\,{d x} \]
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 \[ \int \frac {{\left (3\,x+2\right )}^m\,{\left (4\,x-5\right )}^4}{{\left (2\,x+1\right )}^{m+3}} \,d x \]
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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