Optimal. Leaf size=139 \[ -\frac {2^{1-m} (63-2 m) (2 x+1)^{-m} \, _2F_1(-m,-m;1-m;-3 (2 x+1))}{3 m}+\frac {7 (3 x+2)^{m+1} \left (2 \left (-8 m^2+102 m+677\right ) x+3 \left (2 m^2-m+186\right )\right ) (2 x+1)^{-m-2}}{3 \left (m^2+3 m+2\right )}-\frac {2}{3} (5-4 x)^2 (3 x+2)^{m+1} (2 x+1)^{-m-2} \]
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Rubi [A] time = 0.10, antiderivative size = 139, 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, 145, 69} \[ -\frac {2^{1-m} (63-2 m) (2 x+1)^{-m} \, _2F_1(-m,-m;1-m;-3 (2 x+1))}{3 m}+\frac {7 (3 x+2)^{m+1} \left (2 \left (-8 m^2+102 m+677\right ) x+3 \left (2 m^2-m+186\right )\right ) (2 x+1)^{-m-2}}{3 \left (m^2+3 m+2\right )}-\frac {2}{3} (5-4 x)^2 (3 x+2)^{m+1} (2 x+1)^{-m-2} \]
Antiderivative was successfully verified.
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Rule 69
Rule 100
Rule 145
Rubi steps
\begin {align*} \int (5-4 x)^3 (1+2 x)^{-3-m} (2+3 x)^m \, dx &=-\frac {2}{3} (5-4 x)^2 (1+2 x)^{-2-m} (2+3 x)^{1+m}+\frac {1}{6} \int (5-4 x) (1+2 x)^{-3-m} (2+3 x)^m (-2 (7+10 m)-8 (63-2 m) x) \, dx\\ &=-\frac {2}{3} (5-4 x)^2 (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 (186-m+2 m^2\right )+2 \left (677+102 m-8 m^2\right ) x\right )}{3 \left (2+3 m+m^2\right )}+\frac {1}{3} (4 (63-2 m)) \int (1+2 x)^{-1-m} (2+3 x)^m \, dx\\ &=-\frac {2}{3} (5-4 x)^2 (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 (186-m+2 m^2\right )+2 \left (677+102 m-8 m^2\right ) x\right )}{3 \left (2+3 m+m^2\right )}-\frac {2^{1-m} (63-2 m) (1+2 x)^{-m} \, _2F_1(-m,-m;1-m;-3 (1+2 x))}{3 m}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 123, normalized size = 0.88 \[ \frac {2^{-m} (2 x+1)^{-m-2} \left (2 \left (2 m^3-57 m^2-185 m-126\right ) (2 x+1)^2 \, _2F_1(-m,-m;1-m;-6 x-3)-2^m m (3 x+2)^{m+1} \left (3 m \left (32 x^2-556 x+57\right )+8 (2 m x+m)^2+64 x^2-9638 x-3806\right )\right )}{3 m (m+1) (m+2)} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.08, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (64 \, x^{3} - 240 \, x^{2} + 300 \, x - 125\right )} {\left (3 \, x + 2\right )}^{m} {\left (2 \, x + 1\right )}^{-m - 3}, x\right ) \]
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 \[ \int -{\left (3 \, x + 2\right )}^{m} {\left (2 \, x + 1\right )}^{-m - 3} {\left (4 \, x - 5\right )}^{3}\,{d x} \]
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 \[ \int \left (-4 x +5\right )^{3} \left (2 x +1\right )^{-m -3} \left (3 x +2\right )^{m}\, dx \]
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 \[ -\int {\left (3 \, x + 2\right )}^{m} {\left (2 \, x + 1\right )}^{-m - 3} {\left (4 \, x - 5\right )}^{3}\,{d x} \]
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 \[ -\int \frac {{\left (3\,x+2\right )}^m\,{\left (4\,x-5\right )}^3}{{\left (2\,x+1\right )}^{m+3}} \,d x \]
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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