3.3054 \(\int (5-4 x)^3 (1+2 x)^{-1-m} (2+3 x)^m \, dx\)

Optimal. Leaf size=142 \[ \frac{2^{-m-1} \left (-4 m^3+390 m^2-8324 m+27783\right ) (2 x+1)^{1-m} \, _2F_1(1-m,-m;2-m;-3 (2 x+1))}{27 (1-m) m}-\frac{(3 x+2)^{m+1} \left (4 m^2-4 (109-2 m) m x-512 m+9261\right ) (2 x+1)^{-m}}{27 m}-\frac{2}{9} (5-4 x)^2 (3 x+2)^{m+1} (2 x+1)^{-m} \]

[Out]

(-2*(5 - 4*x)^2*(2 + 3*x)^(1 + m))/(9*(1 + 2*x)^m) - ((2 + 3*x)^(1 + m)*(9261 -
512*m + 4*m^2 - 4*(109 - 2*m)*m*x))/(27*m*(1 + 2*x)^m) + (2^(-1 - m)*(27783 - 83
24*m + 390*m^2 - 4*m^3)*(1 + 2*x)^(1 - m)*Hypergeometric2F1[1 - m, -m, 2 - m, -3
*(1 + 2*x)])/(27*(1 - m)*m)

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Rubi [A]  time = 0.434518, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{2^{-m-1} \left (-4 m^3+390 m^2-8324 m+27783\right ) (2 x+1)^{1-m} \, _2F_1(1-m,-m;2-m;-3 (2 x+1))}{27 (1-m) m}-\frac{(3 x+2)^{m+1} \left (4 m^2-4 (109-2 m) m x-512 m+9261\right ) (2 x+1)^{-m}}{27 m}-\frac{2}{9} (5-4 x)^2 (3 x+2)^{m+1} (2 x+1)^{-m} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(5 - 4*x)^2*(2 + 3*x)^(1 + m))/(9*(1 + 2*x)^m) - ((2 + 3*x)^(1 + m)*(9261 -
512*m + 4*m^2 - 4*(109 - 2*m)*m*x))/(27*m*(1 + 2*x)^m) + (2^(-1 - m)*(27783 - 83
24*m + 390*m^2 - 4*m^3)*(1 + 2*x)^(1 - m)*Hypergeometric2F1[1 - m, -m, 2 - m, -3
*(1 + 2*x)])/(27*(1 - m)*m)

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Rubi in Sympy [A]  time = 27.1491, size = 110, normalized size = 0.77 \[ - \frac{2 \left (- 4 x + 5\right )^{2} \left (2 x + 1\right )^{- m} \left (3 x + 2\right )^{m + 1}}{9} - \frac{\left (2 x + 1\right )^{- m} \left (3 x + 2\right )^{m + 1} \left (64 m^{2} - 64 m x \left (- 2 m + 109\right ) - 8192 m + 148176\right )}{432 m} + \frac{2^{- m} \left (2 x + 1\right )^{- m + 1} \left (- 4 m^{3} + 390 m^{2} - 8324 m + 27783\right ){{}_{2}F_{1}\left (\begin{matrix} - m, - m + 1 \\ - m + 2 \end{matrix}\middle |{- 6 x - 3} \right )}}{54 m \left (- m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(-4*x + 5)**2*(2*x + 1)**(-m)*(3*x + 2)**(m + 1)/9 - (2*x + 1)**(-m)*(3*x + 2
)**(m + 1)*(64*m**2 - 64*m*x*(-2*m + 109) - 8192*m + 148176)/(432*m) + 2**(-m)*(
2*x + 1)**(-m + 1)*(-4*m**3 + 390*m**2 - 8324*m + 27783)*hyper((-m, -m + 1), (-m
+ 2,), -6*x - 3)/(54*m*(-m + 1))

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Mathematica [C]  time = 1.66145, size = 395, normalized size = 2.78 \[ \frac{7}{4} \left (\frac{483 (5-4 x)^2 (8 x+4)^{-m} (12 x+8)^m F_1\left (2;-m,m;3;\frac{3}{23} (5-4 x),\frac{1}{7} (5-4 x)\right )}{483 F_1\left (2;-m,m;3;\frac{3}{23} (5-4 x),\frac{1}{7} (5-4 x)\right )+m (4 x-5) \left (21 F_1\left (3;1-m,m;4;\frac{3}{23} (5-4 x),\frac{1}{7} (5-4 x)\right )-23 F_1\left (3;-m,m+1;4;\frac{3}{23} (5-4 x),\frac{1}{7} (5-4 x)\right )\right )}-\frac{23\ 2^{m+3} (4 x-5)^3 (3 x+2)^m (4 x+2)^{-m} F_1\left (3;-m,m;4;-\frac{3}{23} (4 x-5),\frac{1}{7} (5-4 x)\right )}{3 \left (644 F_1\left (3;-m,m;4;\frac{3}{23} (5-4 x),\frac{1}{7} (5-4 x)\right )+m (4 x-5) \left (21 F_1\left (4;1-m,m;5;\frac{3}{23} (5-4 x),\frac{1}{7} (5-4 x)\right )-23 F_1\left (4;-m,m+1;5;\frac{3}{23} (5-4 x),\frac{1}{7} (5-4 x)\right )\right )\right )}+\frac{7\ 2^{2-m} (2 x+1)^{1-m} \, _2F_1(1-m,-m;2-m;-6 x-3)}{m-1}-\frac{196 (-6 x-3)^m (3 x+2)^{m+1} (2 x+1)^{-m} \, _2F_1(m+1,m+1;m+2;6 x+4)}{m+1}\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

(7*((483*(5 - 4*x)^2*(8 + 12*x)^m*AppellF1[2, -m, m, 3, (3*(5 - 4*x))/23, (5 - 4
*x)/7])/((4 + 8*x)^m*(483*AppellF1[2, -m, m, 3, (3*(5 - 4*x))/23, (5 - 4*x)/7] +
m*(-5 + 4*x)*(21*AppellF1[3, 1 - m, m, 4, (3*(5 - 4*x))/23, (5 - 4*x)/7] - 23*A
ppellF1[3, -m, 1 + m, 4, (3*(5 - 4*x))/23, (5 - 4*x)/7]))) - (23*2^(3 + m)*(2 +
3*x)^m*(-5 + 4*x)^3*AppellF1[3, -m, m, 4, (-3*(-5 + 4*x))/23, (5 - 4*x)/7])/(3*(
2 + 4*x)^m*(644*AppellF1[3, -m, m, 4, (3*(5 - 4*x))/23, (5 - 4*x)/7] + m*(-5 + 4
*x)*(21*AppellF1[4, 1 - m, m, 5, (3*(5 - 4*x))/23, (5 - 4*x)/7] - 23*AppellF1[4,
-m, 1 + m, 5, (3*(5 - 4*x))/23, (5 - 4*x)/7]))) + (7*2^(2 - m)*(1 + 2*x)^(1 - m
)*Hypergeometric2F1[1 - m, -m, 2 - m, -3 - 6*x])/(-1 + m) - (196*(-3 - 6*x)^m*(2
+ 3*x)^(1 + m)*Hypergeometric2F1[1 + m, 1 + m, 2 + m, 4 + 6*x])/((1 + m)*(1 + 2
*x)^m)))/4

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-{\left (64 \, x^{3} - 240 \, x^{2} + 300 \, x - 125\right )}{\left (3 \, x + 2\right )}^{m}{\left (2 \, x + 1\right )}^{-m - 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(64*x^3 - 240*x^2 + 300*x - 125)*(3*x + 2)^m*(2*x + 1)^(-m - 1), 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)**3*(1+2*x)**(-1-m)*(2+3*x)**m,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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