Optimal. Leaf size=333 \[ \frac{2^{3-m} (42-m) (2 x+1)^{-m} \, _2F_1(-m,-m;1-m;-3 (2 x+1))}{3 m}+\frac{14 (15-2 m) \left (2 m^2+52 m+579\right ) (3 x+2)^{m+1} (2 x+1)^{-m-2}}{9 (m+2) (m+3)}-\frac{14 (15-2 m) \left (2 m^2+52 m+579\right ) (3 x+2)^{m+1} (2 x+1)^{-m-1}}{3 (m+3) \left (m^2+3 m+2\right )}-\frac{2}{3} (5-4 x)^3 (3 x+2)^{m+1} (2 x+1)^{-m-3}-\frac{49 (15-2 m) (2 m+27) (3 x+2)^{m+1} (2 x+1)^{-m-3}}{9 (m+3)}+\frac{14}{9} (15-2 m) (5-4 x) (3 x+2)^{m+1} (2 x+1)^{-m-3}+\frac{196 (42-m) (3 x+2)^{m+1} (2 x+1)^{-m-2}}{3 (m+2)}-\frac{28 (42-m) (4 m+29) (3 x+2)^{m+1} (2 x+1)^{-m-1}}{3 (m+1) (m+2)} \]
[Out]
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Rubi [A] time = 0.816337, antiderivative size = 333, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{2^{3-m} (42-m) (2 x+1)^{-m} \, _2F_1(-m,-m;1-m;-3 (2 x+1))}{3 m}+\frac{14 (15-2 m) \left (2 m^2+52 m+579\right ) (3 x+2)^{m+1} (2 x+1)^{-m-2}}{9 (m+2) (m+3)}-\frac{14 (15-2 m) \left (2 m^2+52 m+579\right ) (3 x+2)^{m+1} (2 x+1)^{-m-1}}{3 (m+3) \left (m^2+3 m+2\right )}-\frac{2}{3} (5-4 x)^3 (3 x+2)^{m+1} (2 x+1)^{-m-3}-\frac{49 (15-2 m) (2 m+27) (3 x+2)^{m+1} (2 x+1)^{-m-3}}{9 (m+3)}+\frac{14}{9} (15-2 m) (5-4 x) (3 x+2)^{m+1} (2 x+1)^{-m-3}+\frac{196 (42-m) (3 x+2)^{m+1} (2 x+1)^{-m-2}}{3 (m+2)}-\frac{28 (42-m) (4 m+29) (3 x+2)^{m+1} (2 x+1)^{-m-1}}{3 (m+1) (m+2)} \]
Antiderivative was successfully verified.
[In] Int[(5 - 4*x)^4*(1 + 2*x)^(-4 - m)*(2 + 3*x)^m,x]
[Out]
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Rubi in Sympy [A] time = 79.323, size = 286, normalized size = 0.86 \[ - \frac{49 \left (- 2 m + 15\right ) \left (2 m + 27\right ) \left (2 x + 1\right )^{- m - 3} \left (3 x + 2\right )^{m + 1}}{9 \left (m + 3\right )} + \frac{14 \left (- 2 m + 15\right ) \left (2 x + 1\right )^{- m - 2} \left (3 x + 2\right )^{m + 1} \left (2 m^{2} + 52 m + 579\right )}{9 \left (m + 2\right ) \left (m + 3\right )} - \frac{14 \left (- 2 m + 15\right ) \left (2 x + 1\right )^{- m - 1} \left (3 x + 2\right )^{m + 1} \left (2 m^{2} + 52 m + 579\right )}{3 \left (m + 1\right ) \left (m + 2\right ) \left (m + 3\right )} + \frac{196 \left (- m + 42\right ) \left (2 x + 1\right )^{- m - 2} \left (3 x + 2\right )^{m + 1}}{3 \left (m + 2\right )} - \frac{28 \left (- m + 42\right ) \left (4 m + 29\right ) \left (2 x + 1\right )^{- m - 1} \left (3 x + 2\right )^{m + 1}}{3 \left (m + 1\right ) \left (m + 2\right )} + \left (- \frac{7 m}{9} + \frac{35}{6}\right ) \left (- 16 x + 20\right ) \left (2 x + 1\right )^{- m - 3} \left (3 x + 2\right )^{m + 1} - \frac{2 \left (- 4 x + 5\right )^{3} \left (2 x + 1\right )^{- m - 3} \left (3 x + 2\right )^{m + 1}}{3} + \frac{8 \cdot 2^{- m} \left (- m + 42\right ) \left (2 x + 1\right )^{- m}{{}_{2}F_{1}\left (\begin{matrix} - m, - m \\ - m + 1 \end{matrix}\middle |{- 6 x - 3} \right )}}{3 m} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((5-4*x)**4*(1+2*x)**(-4-m)*(2+3*x)**m,x)
[Out]
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Mathematica [A] time = 0.742909, size = 166, normalized size = 0.5 \[ (2 x+1)^{-m} \left (\frac{7 (3 x+2)^{m+1} \left (32 (2 x+1) (-6 x-3)^m \, _2F_1(m+1,m+1;m+2;6 x+4)+21 \left (168 (2 x+1) (-6 x-3)^m \, _2F_1(m+1,m+3;m+2;6 x+4)-49\ 3^{m+2} (-2 x-1)^{m+1} \, _2F_1(m+1,m+4;m+2;6 x+4)-8\right )\right )}{(m+1) (2 x+1)}-\frac{2^{3-m} (2 x+1) \, _2F_1(1-m,-m;2-m;-6 x-3)}{m-1}\right ) \]
Warning: Unable to verify antiderivative.
[In] Integrate[(5 - 4*x)^4*(1 + 2*x)^(-4 - m)*(2 + 3*x)^m,x]
[Out]
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Maple [F] time = 0.094, size = 0, normalized size = 0. \[ \int \left ( 5-4\,x \right ) ^{4} \left ( 1+2\,x \right ) ^{-4-m} \left ( 2+3\,x \right ) ^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((5-4*x)^4*(1+2*x)^(-4-m)*(2+3*x)^m,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (3 \, x + 2\right )}^{m}{\left (2 \, x + 1\right )}^{-m - 4}{\left (4 \, x - 5\right )}^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^m*(2*x + 1)^(-m - 4)*(4*x - 5)^4,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (256 \, x^{4} - 1280 \, x^{3} + 2400 \, x^{2} - 2000 \, x + 625\right )}{\left (3 \, x + 2\right )}^{m}{\left (2 \, x + 1\right )}^{-m - 4}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^m*(2*x + 1)^(-m - 4)*(4*x - 5)^4,x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5-4*x)**4*(1+2*x)**(-4-m)*(2+3*x)**m,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (3 \, x + 2\right )}^{m}{\left (2 \, x + 1\right )}^{-m - 4}{\left (4 \, x - 5\right )}^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^m*(2*x + 1)^(-m - 4)*(4*x - 5)^4,x, algorithm="giac")
[Out]