3.3071 \(\int (5-4 x)^4 (1+2 x)^{-3-m} (2+3 x)^m \, dx\)

Optimal. Leaf size=188 \[ -\frac{2^{2-m} \left (m^2-85 m+1323\right ) (2 x+1)^{-m} \, _2F_1(-m,-m;1-m;-3 (2 x+1))}{9 m}+\frac{7 (3 x+2)^{m+1} \left (2 \left (8 m^3-530 m^2+1882 m+15209\right ) x+3 \left (-2 m^3+108 m^2+485 m+4638\right )\right ) (2 x+1)^{-m-2}}{9 \left (m^2+3 m+2\right )}-\frac{1}{3} (5-4 x)^3 (3 x+2)^{m+1} (2 x+1)^{-m-2}-\frac{1}{9} (107-2 m) (5-4 x)^2 (3 x+2)^{m+1} (2 x+1)^{-m-2} \]

[Out]

-((107 - 2*m)*(5 - 4*x)^2*(1 + 2*x)^(-2 - m)*(2 + 3*x)^(1 + m))/9 - ((5 - 4*x)^3
*(1 + 2*x)^(-2 - m)*(2 + 3*x)^(1 + m))/3 + (7*(1 + 2*x)^(-2 - m)*(2 + 3*x)^(1 +
m)*(3*(4638 + 485*m + 108*m^2 - 2*m^3) + 2*(15209 + 1882*m - 530*m^2 + 8*m^3)*x)
)/(9*(2 + 3*m + m^2)) - (2^(2 - m)*(1323 - 85*m + m^2)*Hypergeometric2F1[-m, -m,
1 - m, -3*(1 + 2*x)])/(9*m*(1 + 2*x)^m)

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Rubi [A]  time = 0.508245, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{2^{2-m} \left (m^2-85 m+1323\right ) (2 x+1)^{-m} \, _2F_1(-m,-m;1-m;-3 (2 x+1))}{9 m}+\frac{7 (3 x+2)^{m+1} \left (2 \left (8 m^3-530 m^2+1882 m+15209\right ) x+3 \left (-2 m^3+108 m^2+485 m+4638\right )\right ) (2 x+1)^{-m-2}}{9 \left (m^2+3 m+2\right )}-\frac{1}{3} (5-4 x)^3 (3 x+2)^{m+1} (2 x+1)^{-m-2}-\frac{1}{9} (107-2 m) (5-4 x)^2 (3 x+2)^{m+1} (2 x+1)^{-m-2} \]

Antiderivative was successfully verified.

[In]  Int[(5 - 4*x)^4*(1 + 2*x)^(-3 - m)*(2 + 3*x)^m,x]

[Out]

-((107 - 2*m)*(5 - 4*x)^2*(1 + 2*x)^(-2 - m)*(2 + 3*x)^(1 + m))/9 - ((5 - 4*x)^3
*(1 + 2*x)^(-2 - m)*(2 + 3*x)^(1 + m))/3 + (7*(1 + 2*x)^(-2 - m)*(2 + 3*x)^(1 +
m)*(3*(4638 + 485*m + 108*m^2 - 2*m^3) + 2*(15209 + 1882*m - 530*m^2 + 8*m^3)*x)
)/(9*(2 + 3*m + m^2)) - (2^(2 - m)*(1323 - 85*m + m^2)*Hypergeometric2F1[-m, -m,
1 - m, -3*(1 + 2*x)])/(9*m*(1 + 2*x)^m)

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Rubi in Sympy [A]  time = 38.1065, size = 156, normalized size = 0.83 \[ - \left (- \frac{2 m}{9} + \frac{107}{9}\right ) \left (- 4 x + 5\right )^{2} \left (2 x + 1\right )^{- m - 2} \left (3 x + 2\right )^{m + 1} - \frac{\left (- 4 x + 5\right )^{3} \left (2 x + 1\right )^{- m - 2} \left (3 x + 2\right )^{m + 1}}{3} + \frac{\left (2 x + 1\right )^{- m - 2} \left (3 x + 2\right )^{m + 1} \left (- 1344 m^{3} + 72576 m^{2} + 325920 m + x \left (3584 m^{3} - 237440 m^{2} + 843136 m + 6813632\right ) + 3116736\right )}{288 \left (m + 1\right ) \left (m + 2\right )} - \frac{4 \cdot 2^{- m} \left (2 x + 1\right )^{- m} \left (m^{2} - 85 m + 1323\right ){{}_{2}F_{1}\left (\begin{matrix} - m, - m \\ - m + 1 \end{matrix}\middle |{- 6 x - 3} \right )}}{9 m} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-4*x)**4*(1+2*x)**(-3-m)*(2+3*x)**m,x)

[Out]

-(-2*m/9 + 107/9)*(-4*x + 5)**2*(2*x + 1)**(-m - 2)*(3*x + 2)**(m + 1) - (-4*x +
5)**3*(2*x + 1)**(-m - 2)*(3*x + 2)**(m + 1)/3 + (2*x + 1)**(-m - 2)*(3*x + 2)*
*(m + 1)*(-1344*m**3 + 72576*m**2 + 325920*m + x*(3584*m**3 - 237440*m**2 + 8431
36*m + 6813632) + 3116736)/(288*(m + 1)*(m + 2)) - 4*2**(-m)*(2*x + 1)**(-m)*(m*
*2 - 85*m + 1323)*hyper((-m, -m), (-m + 1,), -6*x - 3)/(9*m)

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Mathematica [C]  time = 0.964708, size = 318, normalized size = 1.69 \[ 21 \left (\frac{23 (5-4 x)^2 (4 x+2)^{-m} (6 x+4)^m F_1\left (2;-m,m;3;-\frac{3}{23} (4 x-5),\frac{1}{7} (5-4 x)\right )}{483 F_1\left (2;-m,m;3;\frac{3}{23} (5-4 x),\frac{1}{7} (5-4 x)\right )+m (4 x-5) \left (21 F_1\left (3;1-m,m;4;\frac{3}{23} (5-4 x),\frac{1}{7} (5-4 x)\right )-23 F_1\left (3;-m,m+1;4;\frac{3}{23} (5-4 x),\frac{1}{7} (5-4 x)\right )\right )}+\frac{2^{2-m} (2 x+1)^{1-m} \, _2F_1(1-m,-m;2-m;-6 x-3)}{m-1}-\frac{56 (-6 x-3)^m (3 x+2)^{m+1} (2 x+1)^{-m} \, _2F_1(m+1,m+1;m+2;6 x+4)}{m+1}-\frac{1029 (3 x+2) (-2 x-1)^m (9 x+6)^m (2 x+1)^{-m} \, _2F_1(m+1,m+3;m+2;6 x+4)}{m+1}+\frac{392 (3 x+2)^{m+1} (2 x+1)^{-m-1}}{3 m+3}\right ) \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(5 - 4*x)^4*(1 + 2*x)^(-3 - m)*(2 + 3*x)^m,x]

[Out]

21*((392*(1 + 2*x)^(-1 - m)*(2 + 3*x)^(1 + m))/(3 + 3*m) + (23*(5 - 4*x)^2*(4 +
6*x)^m*AppellF1[2, -m, m, 3, (-3*(-5 + 4*x))/23, (5 - 4*x)/7])/((2 + 4*x)^m*(483
*AppellF1[2, -m, m, 3, (3*(5 - 4*x))/23, (5 - 4*x)/7] + m*(-5 + 4*x)*(21*AppellF
1[3, 1 - m, m, 4, (3*(5 - 4*x))/23, (5 - 4*x)/7] - 23*AppellF1[3, -m, 1 + m, 4,
(3*(5 - 4*x))/23, (5 - 4*x)/7]))) + (2^(2 - m)*(1 + 2*x)^(1 - m)*Hypergeometric2
F1[1 - m, -m, 2 - m, -3 - 6*x])/(-1 + m) - (56*(-3 - 6*x)^m*(2 + 3*x)^(1 + m)*Hy
pergeometric2F1[1 + m, 1 + m, 2 + m, 4 + 6*x])/((1 + m)*(1 + 2*x)^m) - (1029*(-1
- 2*x)^m*(2 + 3*x)*(6 + 9*x)^m*Hypergeometric2F1[1 + m, 3 + m, 2 + m, 4 + 6*x])
/((1 + m)*(1 + 2*x)^m))

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Maple [F]  time = 0.084, size = 0, normalized size = 0. \[ \int \left ( 5-4\,x \right ) ^{4} \left ( 1+2\,x \right ) ^{-3-m} \left ( 2+3\,x \right ) ^{m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-4*x)^4*(1+2*x)^(-3-m)*(2+3*x)^m,x)

[Out]

int((5-4*x)^4*(1+2*x)^(-3-m)*(2+3*x)^m,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (3 \, x + 2\right )}^{m}{\left (2 \, x + 1\right )}^{-m - 3}{\left (4 \, x - 5\right )}^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^m*(2*x + 1)^(-m - 3)*(4*x - 5)^4,x, algorithm="maxima")

[Out]

integrate((3*x + 2)^m*(2*x + 1)^(-m - 3)*(4*x - 5)^4, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (256 \, x^{4} - 1280 \, x^{3} + 2400 \, x^{2} - 2000 \, x + 625\right )}{\left (3 \, x + 2\right )}^{m}{\left (2 \, x + 1\right )}^{-m - 3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^m*(2*x + 1)^(-m - 3)*(4*x - 5)^4,x, algorithm="fricas")

[Out]

integral((256*x^4 - 1280*x^3 + 2400*x^2 - 2000*x + 625)*(3*x + 2)^m*(2*x + 1)^(-
m - 3), x)

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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)**(-3-m)*(2+3*x)**m,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (3 \, x + 2\right )}^{m}{\left (2 \, x + 1\right )}^{-m - 3}{\left (4 \, x - 5\right )}^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^m*(2*x + 1)^(-m - 3)*(4*x - 5)^4,x, algorithm="giac")

[Out]

integrate((3*x + 2)^m*(2*x + 1)^(-m - 3)*(4*x - 5)^4, x)