3.3072 \(\int (5-4 x)^3 (1+2 x)^{-3-m} (2+3 x)^m \, dx\)

Optimal. Leaf size=139 \[ -\frac{2^{1-m} (63-2 m) (2 x+1)^{-m} \, _2F_1(-m,-m;1-m;-3 (2 x+1))}{3 m}+\frac{7 (3 x+2)^{m+1} \left (2 \left (-8 m^2+102 m+677\right ) x+3 \left (2 m^2-m+186\right )\right ) (2 x+1)^{-m-2}}{3 \left (m^2+3 m+2\right )}-\frac{2}{3} (5-4 x)^2 (3 x+2)^{m+1} (2 x+1)^{-m-2} \]

[Out]

(-2*(5 - 4*x)^2*(1 + 2*x)^(-2 - m)*(2 + 3*x)^(1 + m))/3 + (7*(1 + 2*x)^(-2 - m)*
(2 + 3*x)^(1 + m)*(3*(186 - m + 2*m^2) + 2*(677 + 102*m - 8*m^2)*x))/(3*(2 + 3*m
+ m^2)) - (2^(1 - m)*(63 - 2*m)*Hypergeometric2F1[-m, -m, 1 - m, -3*(1 + 2*x)])
/(3*m*(1 + 2*x)^m)

_______________________________________________________________________________________

Rubi [A]  time = 0.284684, antiderivative size = 139, 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 \[ -\frac{2^{1-m} (63-2 m) (2 x+1)^{-m} \, _2F_1(-m,-m;1-m;-3 (2 x+1))}{3 m}+\frac{7 (3 x+2)^{m+1} \left (2 \left (-8 m^2+102 m+677\right ) x+3 \left (2 m^2-m+186\right )\right ) (2 x+1)^{-m-2}}{3 \left (m^2+3 m+2\right )}-\frac{2}{3} (5-4 x)^2 (3 x+2)^{m+1} (2 x+1)^{-m-2} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(5 - 4*x)^2*(1 + 2*x)^(-2 - m)*(2 + 3*x)^(1 + m))/3 + (7*(1 + 2*x)^(-2 - m)*
(2 + 3*x)^(1 + m)*(3*(186 - m + 2*m^2) + 2*(677 + 102*m - 8*m^2)*x))/(3*(2 + 3*m
+ m^2)) - (2^(1 - m)*(63 - 2*m)*Hypergeometric2F1[-m, -m, 1 - m, -3*(1 + 2*x)])
/(3*m*(1 + 2*x)^m)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 21.505, size = 121, normalized size = 0.87 \[ - \frac{2 \left (- 4 x + 5\right )^{2} \left (2 x + 1\right )^{- m - 2} \left (3 x + 2\right )^{m + 1}}{3} + \frac{\left (2 x + 1\right )^{- m - 2} \left (3 x + 2\right )^{m + 1} \left (644 m^{2} + 2990 m + x \left (- 2208 m^{2} + 31464 m + 179676\right ) + 75348\right )}{54 \left (m + 1\right ) \left (m + 2\right )} - \frac{2 \cdot 2^{- m} \left (- 2 m + 63\right ) \left (2 x + 1\right )^{- m - 2}{{}_{2}F_{1}\left (\begin{matrix} - m - 2, - m - 2 \\ - m - 1 \end{matrix}\middle |{- 6 x - 3} \right )}}{27 \left (m + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(-4*x + 5)**2*(2*x + 1)**(-m - 2)*(3*x + 2)**(m + 1)/3 + (2*x + 1)**(-m - 2)*
(3*x + 2)**(m + 1)*(644*m**2 + 2990*m + x*(-2208*m**2 + 31464*m + 179676) + 7534
8)/(54*(m + 1)*(m + 2)) - 2*2**(-m)*(-2*m + 63)*(2*x + 1)**(-m - 2)*hyper((-m -
2, -m - 2), (-m - 1,), -6*x - 3)/(27*(m + 2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.486791, size = 131, normalized size = 0.94 \[ (2 x+1)^{-m} \left (\frac{2^{2-m} (2 x+1) \, _2F_1(1-m,-m;2-m;-6 x-3)}{m-1}-\frac{21 (3 x+2)^{m+1} \left (4 (2 x+1) (-6 x-3)^m \, _2F_1(m+1,m+1;m+2;6 x+4)+7 \left (-7 (-6 x-3)^{m+1} \, _2F_1(m+1,m+3;m+2;6 x+4)-2\right )\right )}{(m+1) (2 x+1)}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((2^(2 - m)*(1 + 2*x)*Hypergeometric2F1[1 - m, -m, 2 - m, -3 - 6*x])/(-1 + m) -
(21*(2 + 3*x)^(1 + m)*(4*(-3 - 6*x)^m*(1 + 2*x)*Hypergeometric2F1[1 + m, 1 + m,
2 + m, 4 + 6*x] + 7*(-2 - 7*(-3 - 6*x)^(1 + m)*Hypergeometric2F1[1 + m, 3 + m, 2
+ m, 4 + 6*x])))/((1 + m)*(1 + 2*x)))/(1 + 2*x)^m

_______________________________________________________________________________________

Maple [F]  time = 0.079, size = 0, normalized size = 0. \[ \int \left ( 5-4\,x \right ) ^{3} \left ( 1+2\,x \right ) ^{-3-m} \left ( 2+3\,x \right ) ^{m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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