3.3055 \(\int (5-4 x)^2 (1+2 x)^{-1-m} (2+3 x)^m \, dx\)

Optimal. Leaf size=121 \[ \frac{2^{-m-1} \left (2 m^2-86 m+441\right ) (2 x+1)^{1-m} \, _2F_1(1-m,-m;2-m;-3 (2 x+1))}{3 (1-m) m}-\frac{1}{3} (5-4 x) (3 x+2)^{m+1} (2 x+1)^{-m}-\frac{7 (21-m) (3 x+2)^{m+1} (2 x+1)^{-m}}{3 m} \]

[Out]

(-7*(21 - m)*(2 + 3*x)^(1 + m))/(3*m*(1 + 2*x)^m) - ((5 - 4*x)*(2 + 3*x)^(1 + m)
)/(3*(1 + 2*x)^m) + (2^(-1 - m)*(441 - 86*m + 2*m^2)*(1 + 2*x)^(1 - m)*Hypergeom
etric2F1[1 - m, -m, 2 - m, -3*(1 + 2*x)])/(3*(1 - m)*m)

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Rubi [A]  time = 0.267255, antiderivative size = 121, 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 \[ \frac{2^{-m-1} \left (2 m^2-86 m+441\right ) (2 x+1)^{1-m} \, _2F_1(1-m,-m;2-m;-3 (2 x+1))}{3 (1-m) m}-\frac{1}{3} (5-4 x) (3 x+2)^{m+1} (2 x+1)^{-m}-\frac{7 (21-m) (3 x+2)^{m+1} (2 x+1)^{-m}}{3 m} \]

Antiderivative was successfully verified.

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

[Out]

(-7*(21 - m)*(2 + 3*x)^(1 + m))/(3*m*(1 + 2*x)^m) - ((5 - 4*x)*(2 + 3*x)^(1 + m)
)/(3*(1 + 2*x)^m) + (2^(-1 - m)*(441 - 86*m + 2*m^2)*(1 + 2*x)^(1 - m)*Hypergeom
etric2F1[1 - m, -m, 2 - m, -3*(1 + 2*x)])/(3*(1 - m)*m)

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Rubi in Sympy [A]  time = 17.5722, size = 87, normalized size = 0.72 \[ - \frac{\left (- 16 x + 20\right ) \left (2 x + 1\right )^{- m} \left (3 x + 2\right )^{m + 1}}{12} - \frac{7 \left (- m + 21\right ) \left (2 x + 1\right )^{- m} \left (3 x + 2\right )^{m + 1}}{3 m} + \frac{2^{- m} \left (2 x + 1\right )^{- m + 1} \left (2 m^{2} - 86 m + 441\right ){{}_{2}F_{1}\left (\begin{matrix} - m, - m + 1 \\ - m + 2 \end{matrix}\middle |{- 6 x - 3} \right )}}{6 m \left (- m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-16*x + 20)*(2*x + 1)**(-m)*(3*x + 2)**(m + 1)/12 - 7*(-m + 21)*(2*x + 1)**(-m
)*(3*x + 2)**(m + 1)/(3*m) + 2**(-m)*(2*x + 1)**(-m + 1)*(2*m**2 - 86*m + 441)*h
yper((-m, -m + 1), (-m + 2,), -6*x - 3)/(6*m*(-m + 1))

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Mathematica [C]  time = 0.406355, size = 241, normalized size = 1.99 \[ \frac{7}{4} \left (\frac{69 (5-4 x)^2 (8 x+4)^{-m} (12 x+8)^m F_1\left (2;-m,m;3;\frac{3}{23} (5-4 x),\frac{1}{7} (5-4 x)\right )}{483 F_1\left (2;-m,m;3;\frac{3}{23} (5-4 x),\frac{1}{7} (5-4 x)\right )+m (4 x-5) \left (21 F_1\left (3;1-m,m;4;\frac{3}{23} (5-4 x),\frac{1}{7} (5-4 x)\right )-23 F_1\left (3;-m,m+1;4;\frac{3}{23} (5-4 x),\frac{1}{7} (5-4 x)\right )\right )}+\frac{2^{2-m} (2 x+1)^{1-m} \, _2F_1(1-m,-m;2-m;-6 x-3)}{m-1}-\frac{28 (-6 x-3)^m (3 x+2)^{m+1} (2 x+1)^{-m} \, _2F_1(m+1,m+1;m+2;6 x+4)}{m+1}\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

(7*((69*(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*Ap
pellF1[3, -m, 1 + m, 4, (3*(5 - 4*x))/23, (5 - 4*x)/7]))) + (2^(2 - m)*(1 + 2*x)
^(1 - m)*Hypergeometric2F1[1 - m, -m, 2 - m, -3 - 6*x])/(-1 + m) - (28*(-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 = 0.082, size = 0, normalized size = 0. \[ \int \left ( 5-4\,x \right ) ^{2} \left ( 1+2\,x \right ) ^{-1-m} \left ( 2+3\,x \right ) ^{m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (16 \, x^{2} - 40 \, x + 25\right )}{\left (3 \, x + 2\right )}^{m}{\left (2 \, x + 1\right )}^{-m - 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((16*x^2 - 40*x + 25)*(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)**2*(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. \[ \int{\left (3 \, x + 2\right )}^{m}{\left (2 \, x + 1\right )}^{-m - 1}{\left (4 \, x - 5\right )}^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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