3.3063 \(\int (5-4 x)^3 (1+2 x)^{-2-m} (2+3 x)^m \, dx\)

Optimal. Leaf size=132 \[ \frac{2^{-m} \left (2 m^2-128 m+1323\right ) (2 x+1)^{-m} \, _2F_1(-m,-m;1-m;-3 (2 x+1))}{9 m}-\frac{(3 x+2)^{m+1} \left (4 m^2-8 (43-m) (m+1) x-315 m+2768\right ) (2 x+1)^{-m-1}}{9 (m+1)}-\frac{1}{3} (5-4 x)^2 (3 x+2)^{m+1} (2 x+1)^{-m-1} \]

[Out]

-((5 - 4*x)^2*(1 + 2*x)^(-1 - m)*(2 + 3*x)^(1 + m))/3 - ((1 + 2*x)^(-1 - m)*(2 +
3*x)^(1 + m)*(2768 - 315*m + 4*m^2 - 8*(43 - m)*(1 + m)*x))/(9*(1 + m)) + ((132
3 - 128*m + 2*m^2)*Hypergeometric2F1[-m, -m, 1 - m, -3*(1 + 2*x)])/(9*2^m*m*(1 +
2*x)^m)

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Rubi [A]  time = 0.342759, antiderivative size = 132, 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 \[ \frac{2^{-m} \left (2 m^2-128 m+1323\right ) (2 x+1)^{-m} \, _2F_1(-m,-m;1-m;-3 (2 x+1))}{9 m}-\frac{(3 x+2)^{m+1} \left (4 m^2-8 (43-m) (m+1) x-315 m+2768\right ) (2 x+1)^{-m-1}}{9 (m+1)}-\frac{1}{3} (5-4 x)^2 (3 x+2)^{m+1} (2 x+1)^{-m-1} \]

Antiderivative was successfully verified.

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

[Out]

-((5 - 4*x)^2*(1 + 2*x)^(-1 - m)*(2 + 3*x)^(1 + m))/3 - ((1 + 2*x)^(-1 - m)*(2 +
3*x)^(1 + m)*(2768 - 315*m + 4*m^2 - 8*(43 - m)*(1 + m)*x))/(9*(1 + m)) + ((132
3 - 128*m + 2*m^2)*Hypergeometric2F1[-m, -m, 1 - m, -3*(1 + 2*x)])/(9*2^m*m*(1 +
2*x)^m)

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Rubi in Sympy [A]  time = 20.1208, size = 107, normalized size = 0.81 \[ - \frac{\left (- 4 x + 5\right )^{2} \left (2 x + 1\right )^{- m - 1} \left (3 x + 2\right )^{m + 1}}{3} - \frac{\left (2 x + 1\right )^{- m - 1} \left (3 x + 2\right )^{m + 1} \left (64 m^{2} - 5040 m - 128 x \left (- m + 43\right ) \left (m + 1\right ) + 44288\right )}{144 \left (m + 1\right )} + \frac{2^{- m} \left (2 x + 1\right )^{- m} \left (2 m^{2} - 128 m + 1323\right ){{}_{2}F_{1}\left (\begin{matrix} - m, - m \\ - m + 1 \end{matrix}\middle |{- 6 x - 3} \right )}}{9 m} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-4*x + 5)**2*(2*x + 1)**(-m - 1)*(3*x + 2)**(m + 1)/3 - (2*x + 1)**(-m - 1)*(3
*x + 2)**(m + 1)*(64*m**2 - 5040*m - 128*x*(-m + 43)*(m + 1) + 44288)/(144*(m +
1)) + 2**(-m)*(2*x + 1)**(-m)*(2*m**2 - 128*m + 1323)*hyper((-m, -m), (-m + 1,),
-6*x - 3)/(9*m)

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Mathematica [C]  time = 0.796228, size = 273, normalized size = 2.07 \[ \frac{7}{2} \left (-\frac{69 (5-4 x)^2 (4 x+2)^{-m} (6 x+4)^m F_1\left (2;-m,m;3;-\frac{3}{23} (4 x-5),\frac{1}{7} (5-4 x)\right )}{483 F_1\left (2;-m,m;3;\frac{3}{23} (5-4 x),\frac{1}{7} (5-4 x)\right )+m (4 x-5) \left (21 F_1\left (3;1-m,m;4;\frac{3}{23} (5-4 x),\frac{1}{7} (5-4 x)\right )-23 F_1\left (3;-m,m+1;4;\frac{3}{23} (5-4 x),\frac{1}{7} (5-4 x)\right )\right )}+\frac{2^{3-m} (2 x+1)^{1-m} \, _2F_1(1-m,-m;2-m;-6 x-3)}{1-m}+\frac{84 (3 x+2) (-2 x-1)^m (9 x+6)^m (2 x+1)^{-m} \, _2F_1(m+1,m+1;m+2;6 x+4)}{m+1}-\frac{98 (3 x+2)^{m+1} (2 x+1)^{-m-1}}{m+1}\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

(7*((-98*(1 + 2*x)^(-1 - m)*(2 + 3*x)^(1 + m))/(1 + m) - (69*(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*A
ppellF1[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*AppellF1[3, -m, 1 + m, 4, (3
*(5 - 4*x))/23, (5 - 4*x)/7]))) + (2^(3 - m)*(1 + 2*x)^(1 - m)*Hypergeometric2F1
[1 - m, -m, 2 - m, -3 - 6*x])/(1 - m) + (84*(-1 - 2*x)^m*(2 + 3*x)*(6 + 9*x)^m*H
ypergeometric2F1[1 + m, 1 + m, 2 + m, 4 + 6*x])/((1 + m)*(1 + 2*x)^m)))/2

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Maple [F]  time = 0.085, size = 0, normalized size = 0. \[ \int \left ( 5-4\,x \right ) ^{3} \left ( 1+2\,x \right ) ^{-2-m} \left ( 2+3\,x \right ) ^{m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x + 2)^m*(2*x + 1)^(-m - 2)*(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 - 2), 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)**(-2-m)*(2+3*x)**m,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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