Optimal. Leaf size=188 \[ \frac{2^{-m-1} \left (2 m^4-440 m^3+29050 m^2-639760 m+3528363\right ) (2 x+1)^{1-m} \, _2F_1(1-m,-m;2-m;-3 (2 x+1))}{1215 (1-m)}-\frac{(3 x+2)^{m+1} \left (-24 \left (m^2-154 m+4359\right ) x-2 m^3+426 m^2-25441 m+386850\right ) (2 x+1)^{1-m}}{1215}-\frac{2}{15} (5-4 x)^3 (3 x+2)^{m+1} (2 x+1)^{1-m}-\frac{1}{45} (88-m) (5-4 x)^2 (3 x+2)^{m+1} (2 x+1)^{1-m} \]
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Rubi [A] time = 0.225622, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {100, 153, 147, 69} \[ \frac{2^{-m-1} \left (2 m^4-440 m^3+29050 m^2-639760 m+3528363\right ) (2 x+1)^{1-m} \, _2F_1(1-m,-m;2-m;-3 (2 x+1))}{1215 (1-m)}-\frac{(3 x+2)^{m+1} \left (-24 \left (m^2-154 m+4359\right ) x-2 m^3+426 m^2-25441 m+386850\right ) (2 x+1)^{1-m}}{1215}-\frac{2}{15} (5-4 x)^3 (3 x+2)^{m+1} (2 x+1)^{1-m}-\frac{1}{45} (88-m) (5-4 x)^2 (3 x+2)^{m+1} (2 x+1)^{1-m} \]
Antiderivative was successfully verified.
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Rule 100
Rule 153
Rule 147
Rule 69
Rubi steps
\begin{align*} \int (5-4 x)^4 (1+2 x)^{-m} (2+3 x)^m \, dx &=-\frac{2}{15} (5-4 x)^3 (1+2 x)^{1-m} (2+3 x)^{1+m}+\frac{1}{30} \int (5-4 x)^2 (1+2 x)^{-m} (2+3 x)^m (2 (397-10 m)-16 (88-m) x) \, dx\\ &=-\frac{1}{45} (88-m) (5-4 x)^2 (1+2 x)^{1-m} (2+3 x)^{1+m}-\frac{2}{15} (5-4 x)^3 (1+2 x)^{1-m} (2+3 x)^{1+m}+\frac{1}{720} \int (5-4 x) (1+2 x)^{-m} (2+3 x)^m \left (16 \left (7627-609 m+5 m^2\right )-64 \left (4359-154 m+m^2\right ) x\right ) \, dx\\ &=-\frac{1}{45} (88-m) (5-4 x)^2 (1+2 x)^{1-m} (2+3 x)^{1+m}-\frac{2}{15} (5-4 x)^3 (1+2 x)^{1-m} (2+3 x)^{1+m}-\frac{(1+2 x)^{1-m} (2+3 x)^{1+m} \left (386850-25441 m+426 m^2-2 m^3-24 \left (4359-154 m+m^2\right ) x\right )}{1215}+\frac{\left (3528363-639760 m+29050 m^2-440 m^3+2 m^4\right ) \int (1+2 x)^{-m} (2+3 x)^m \, dx}{1215}\\ &=-\frac{1}{45} (88-m) (5-4 x)^2 (1+2 x)^{1-m} (2+3 x)^{1+m}-\frac{2}{15} (5-4 x)^3 (1+2 x)^{1-m} (2+3 x)^{1+m}-\frac{(1+2 x)^{1-m} (2+3 x)^{1+m} \left (386850-25441 m+426 m^2-2 m^3-24 \left (4359-154 m+m^2\right ) x\right )}{1215}+\frac{2^{-1-m} \left (3528363-639760 m+29050 m^2-440 m^3+2 m^4\right ) (1+2 x)^{1-m} \, _2F_1(1-m,-m;2-m;-3 (1+2 x))}{1215 (1-m)}\\ \end{align*}
Mathematica [A] time = 0.388345, size = 227, normalized size = 1.21 \[ \frac{(2 x+1)^{1-m} \left (483 \left (2^{-m} \left (-2 m^2+132 m-1453\right ) \, _2F_1(1-m,-m;2-m;-6 x-3)+4 (m-1) (m+12 x-59) (3 x+2)^{m+1}\right )-(88-m) \left (2^{2-m} (m-66) \, _2F_1(-m-2,1-m;2-m;-6 x-3)+23\ 2^{1-m} (111-2 m) \, _2F_1(-m-1,1-m;2-m;-6 x-3)+529\ 2^{-m} (m-45) \, _2F_1(1-m,-m;2-m;-6 x-3)-18 (m-1) (5-4 x)^2 (3 x+2)^{m+1}\right )-108 (m-1) (4 x-5)^3 (3 x+2)^{m+1}\right )}{810 (1-m)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.066, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( 5-4\,x \right ) ^{4} \left ( 2+3\,x \right ) ^{m}}{ \left ( 1+2\,x \right ) ^{m}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (3 \, x + 2\right )}^{m}{\left (4 \, x - 5\right )}^{4}}{{\left (2 \, x + 1\right )}^{m}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\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}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (3 \, x + 2\right )}^{m}{\left (4 \, x - 5\right )}^{4}}{{\left (2 \, x + 1\right )}^{m}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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