3.3087 \(\int (5-4 x)^5 (1+2 x)^{-5-m} (2+3 x)^m \, dx\)

Optimal. Leaf size=546 \[ -\frac{2^{3-m} (105-2 m) (2 x+1)^{-m-3} \, _2F_1(-m-3,-m-3;-m-2;-3 (2 x+1))}{81 (m+3)}-\frac{322 (13-2 m) \left (2 m^2+52 m+579\right ) (3 x+2)^{m+1} (2 x+1)^{-m-2}}{3 (m+2) (m+3) (m+4)}-\frac{48668 (105-2 m) (3 x+2)^{m+1} (2 x+1)^{-m-2}}{27 \left (m^2+5 m+6\right )}+\frac{4232 (105-2 m) (3 x+2)^{m+2} (2 x+1)^{-m-2}}{9 \left (m^2+5 m+6\right )}+\frac{322 (13-2 m) \left (2 m^2+52 m+579\right ) (3 x+2)^{m+1} (2 x+1)^{-m-1}}{(m+3) (m+4) \left (m^2+3 m+2\right )}+\frac{48668 (105-2 m) (3 x+2)^{m+1} (2 x+1)^{-m-1}}{9 \left (m^3+6 m^2+11 m+6\right )}-\frac{2}{3} (5-4 x)^4 (3 x+2)^{m+1} (2 x+1)^{-m-4}-\frac{7 (13-2 m) (5-4 x)^3 (3 x+2)^{m+1} (2 x+1)^{-m-4}}{3 (m+4)}+\frac{1127 (13-2 m) (2 m+27) (3 x+2)^{m+1} (2 x+1)^{-m-3}}{3 (m+3) (m+4)}-\frac{322 (13-2 m) (5-4 x) (3 x+2)^{m+1} (2 x+1)^{-m-3}}{3 (m+4)}+\frac{24334 (105-2 m) (3 x+2)^{m+1} (2 x+1)^{-m-3}}{81 (m+3)}-\frac{4232 (105-2 m) (3 x+2)^{m+2} (2 x+1)^{-m-3}}{27 (m+3)}+\frac{736 (105-2 m) (3 x+2)^{m+3} (2 x+1)^{-m-3}}{27 (m+3)} \]

[Out]

(-7*(13 - 2*m)*(5 - 4*x)^3*(1 + 2*x)^(-4 - m)*(2 + 3*x)^(1 + m))/(3*(4 + m)) - (
2*(5 - 4*x)^4*(1 + 2*x)^(-4 - m)*(2 + 3*x)^(1 + m))/3 + (24334*(105 - 2*m)*(1 +
2*x)^(-3 - m)*(2 + 3*x)^(1 + m))/(81*(3 + m)) + (1127*(13 - 2*m)*(27 + 2*m)*(1 +
2*x)^(-3 - m)*(2 + 3*x)^(1 + m))/(3*(3 + m)*(4 + m)) - (322*(13 - 2*m)*(5 - 4*x
)*(1 + 2*x)^(-3 - m)*(2 + 3*x)^(1 + m))/(3*(4 + m)) - (48668*(105 - 2*m)*(1 + 2*
x)^(-2 - m)*(2 + 3*x)^(1 + m))/(27*(6 + 5*m + m^2)) - (322*(13 - 2*m)*(579 + 52*
m + 2*m^2)*(1 + 2*x)^(-2 - m)*(2 + 3*x)^(1 + m))/(3*(2 + m)*(3 + m)*(4 + m)) + (
322*(13 - 2*m)*(579 + 52*m + 2*m^2)*(1 + 2*x)^(-1 - m)*(2 + 3*x)^(1 + m))/((3 +
m)*(4 + m)*(2 + 3*m + m^2)) + (48668*(105 - 2*m)*(1 + 2*x)^(-1 - m)*(2 + 3*x)^(1
+ m))/(9*(6 + 11*m + 6*m^2 + m^3)) - (4232*(105 - 2*m)*(1 + 2*x)^(-3 - m)*(2 +
3*x)^(2 + m))/(27*(3 + m)) + (4232*(105 - 2*m)*(1 + 2*x)^(-2 - m)*(2 + 3*x)^(2 +
m))/(9*(6 + 5*m + m^2)) + (736*(105 - 2*m)*(1 + 2*x)^(-3 - m)*(2 + 3*x)^(3 + m)
)/(27*(3 + m)) - (2^(3 - m)*(105 - 2*m)*(1 + 2*x)^(-3 - m)*Hypergeometric2F1[-3
- m, -3 - m, -2 - m, -3*(1 + 2*x)])/(81*(3 + m))

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Rubi [A]  time = 1.15352, antiderivative size = 546, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346 \[ -\frac{2^{3-m} (105-2 m) (2 x+1)^{-m-3} \, _2F_1(-m-3,-m-3;-m-2;-3 (2 x+1))}{81 (m+3)}-\frac{322 (13-2 m) \left (2 m^2+52 m+579\right ) (3 x+2)^{m+1} (2 x+1)^{-m-2}}{3 (m+2) (m+3) (m+4)}-\frac{48668 (105-2 m) (3 x+2)^{m+1} (2 x+1)^{-m-2}}{27 \left (m^2+5 m+6\right )}+\frac{4232 (105-2 m) (3 x+2)^{m+2} (2 x+1)^{-m-2}}{9 \left (m^2+5 m+6\right )}+\frac{322 (13-2 m) \left (2 m^2+52 m+579\right ) (3 x+2)^{m+1} (2 x+1)^{-m-1}}{(m+3) (m+4) \left (m^2+3 m+2\right )}+\frac{48668 (105-2 m) (3 x+2)^{m+1} (2 x+1)^{-m-1}}{9 \left (m^3+6 m^2+11 m+6\right )}-\frac{2}{3} (5-4 x)^4 (3 x+2)^{m+1} (2 x+1)^{-m-4}-\frac{7 (13-2 m) (5-4 x)^3 (3 x+2)^{m+1} (2 x+1)^{-m-4}}{3 (m+4)}+\frac{1127 (13-2 m) (2 m+27) (3 x+2)^{m+1} (2 x+1)^{-m-3}}{3 (m+3) (m+4)}-\frac{322 (13-2 m) (5-4 x) (3 x+2)^{m+1} (2 x+1)^{-m-3}}{3 (m+4)}+\frac{24334 (105-2 m) (3 x+2)^{m+1} (2 x+1)^{-m-3}}{81 (m+3)}-\frac{4232 (105-2 m) (3 x+2)^{m+2} (2 x+1)^{-m-3}}{27 (m+3)}+\frac{736 (105-2 m) (3 x+2)^{m+3} (2 x+1)^{-m-3}}{27 (m+3)} \]

Antiderivative was successfully verified.

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

[Out]

(-7*(13 - 2*m)*(5 - 4*x)^3*(1 + 2*x)^(-4 - m)*(2 + 3*x)^(1 + m))/(3*(4 + m)) - (
2*(5 - 4*x)^4*(1 + 2*x)^(-4 - m)*(2 + 3*x)^(1 + m))/3 + (24334*(105 - 2*m)*(1 +
2*x)^(-3 - m)*(2 + 3*x)^(1 + m))/(81*(3 + m)) + (1127*(13 - 2*m)*(27 + 2*m)*(1 +
2*x)^(-3 - m)*(2 + 3*x)^(1 + m))/(3*(3 + m)*(4 + m)) - (322*(13 - 2*m)*(5 - 4*x
)*(1 + 2*x)^(-3 - m)*(2 + 3*x)^(1 + m))/(3*(4 + m)) - (48668*(105 - 2*m)*(1 + 2*
x)^(-2 - m)*(2 + 3*x)^(1 + m))/(27*(6 + 5*m + m^2)) - (322*(13 - 2*m)*(579 + 52*
m + 2*m^2)*(1 + 2*x)^(-2 - m)*(2 + 3*x)^(1 + m))/(3*(2 + m)*(3 + m)*(4 + m)) + (
322*(13 - 2*m)*(579 + 52*m + 2*m^2)*(1 + 2*x)^(-1 - m)*(2 + 3*x)^(1 + m))/((3 +
m)*(4 + m)*(2 + 3*m + m^2)) + (48668*(105 - 2*m)*(1 + 2*x)^(-1 - m)*(2 + 3*x)^(1
+ m))/(9*(6 + 11*m + 6*m^2 + m^3)) - (4232*(105 - 2*m)*(1 + 2*x)^(-3 - m)*(2 +
3*x)^(2 + m))/(27*(3 + m)) + (4232*(105 - 2*m)*(1 + 2*x)^(-2 - m)*(2 + 3*x)^(2 +
m))/(9*(6 + 5*m + m^2)) + (736*(105 - 2*m)*(1 + 2*x)^(-3 - m)*(2 + 3*x)^(3 + m)
)/(27*(3 + m)) - (2^(3 - m)*(105 - 2*m)*(1 + 2*x)^(-3 - m)*Hypergeometric2F1[-3
- m, -3 - m, -2 - m, -3*(1 + 2*x)])/(81*(3 + m))

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Rubi in Sympy [A]  time = 139.688, size = 479, normalized size = 0.88 \[ - \frac{161 \left (- 2 m + 13\right ) \left (- 16 x + 20\right ) \left (2 x + 1\right )^{- m - 3} \left (3 x + 2\right )^{m + 1}}{6 \left (m + 4\right )} - \frac{7 \left (- 2 m + 13\right ) \left (- 4 x + 5\right )^{3} \left (2 x + 1\right )^{- m - 4} \left (3 x + 2\right )^{m + 1}}{3 \left (m + 4\right )} - \frac{322 \left (- 2 m + 13\right ) \left (2 x + 1\right )^{- m - 2} \left (3 x + 2\right )^{m + 1} \left (2 m^{2} + 52 m + 579\right )}{3 \left (m + 4\right ) \left (m^{2} + 5 m + 6\right )} + \frac{1127 \left (- 2 m + 13\right ) \left (2 m + 27\right ) \left (2 x + 1\right )^{- m - 3} \left (3 x + 2\right )^{m + 1}}{3 \left (m + 3\right ) \left (m + 4\right )} + \frac{322 \left (- 2 m + 13\right ) \left (2 x + 1\right )^{- m - 1} \left (3 x + 2\right )^{m + 1} \left (2 m^{2} + 52 m + 579\right )}{\left (m + 1\right ) \left (m + 2\right ) \left (m + 3\right ) \left (m + 4\right )} + \frac{24334 \left (- 2 m + 105\right ) \left (2 x + 1\right )^{- m - 3} \left (3 x + 2\right )^{m + 1}}{81 \left (m + 3\right )} - \frac{4232 \left (- 2 m + 105\right ) \left (2 x + 1\right )^{- m - 3} \left (3 x + 2\right )^{m + 2}}{27 \left (m + 3\right )} + \frac{736 \left (- 2 m + 105\right ) \left (2 x + 1\right )^{- m - 3} \left (3 x + 2\right )^{m + 3}}{27 \left (m + 3\right )} - \frac{48668 \left (- 2 m + 105\right ) \left (2 x + 1\right )^{- m - 2} \left (3 x + 2\right )^{m + 1}}{27 \left (m + 2\right ) \left (m + 3\right )} + \frac{4232 \left (- 2 m + 105\right ) \left (2 x + 1\right )^{- m - 2} \left (3 x + 2\right )^{m + 2}}{9 \left (m + 2\right ) \left (m + 3\right )} + \frac{48668 \left (- 2 m + 105\right ) \left (2 x + 1\right )^{- m - 1} \left (3 x + 2\right )^{m + 1}}{9 \left (m + 1\right ) \left (m + 2\right ) \left (m + 3\right )} - \frac{2 \left (- 4 x + 5\right )^{4} \left (2 x + 1\right )^{- m - 4} \left (3 x + 2\right )^{m + 1}}{3} - \frac{8 \cdot 2^{- m} \left (- 2 m + 105\right ) \left (2 x + 1\right )^{- m - 3}{{}_{2}F_{1}\left (\begin{matrix} - m - 3, - m - 3 \\ - m - 2 \end{matrix}\middle |{- 6 x - 3} \right )}}{81 \left (m + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-161*(-2*m + 13)*(-16*x + 20)*(2*x + 1)**(-m - 3)*(3*x + 2)**(m + 1)/(6*(m + 4))
- 7*(-2*m + 13)*(-4*x + 5)**3*(2*x + 1)**(-m - 4)*(3*x + 2)**(m + 1)/(3*(m + 4)
) - 322*(-2*m + 13)*(2*x + 1)**(-m - 2)*(3*x + 2)**(m + 1)*(2*m**2 + 52*m + 579)
/(3*(m + 4)*(m**2 + 5*m + 6)) + 1127*(-2*m + 13)*(2*m + 27)*(2*x + 1)**(-m - 3)*
(3*x + 2)**(m + 1)/(3*(m + 3)*(m + 4)) + 322*(-2*m + 13)*(2*x + 1)**(-m - 1)*(3*
x + 2)**(m + 1)*(2*m**2 + 52*m + 579)/((m + 1)*(m + 2)*(m + 3)*(m + 4)) + 24334*
(-2*m + 105)*(2*x + 1)**(-m - 3)*(3*x + 2)**(m + 1)/(81*(m + 3)) - 4232*(-2*m +
105)*(2*x + 1)**(-m - 3)*(3*x + 2)**(m + 2)/(27*(m + 3)) + 736*(-2*m + 105)*(2*x
+ 1)**(-m - 3)*(3*x + 2)**(m + 3)/(27*(m + 3)) - 48668*(-2*m + 105)*(2*x + 1)**
(-m - 2)*(3*x + 2)**(m + 1)/(27*(m + 2)*(m + 3)) + 4232*(-2*m + 105)*(2*x + 1)**
(-m - 2)*(3*x + 2)**(m + 2)/(9*(m + 2)*(m + 3)) + 48668*(-2*m + 105)*(2*x + 1)**
(-m - 1)*(3*x + 2)**(m + 1)/(9*(m + 1)*(m + 2)*(m + 3)) - 2*(-4*x + 5)**4*(2*x +
1)**(-m - 4)*(3*x + 2)**(m + 1)/3 - 8*2**(-m)*(-2*m + 105)*(2*x + 1)**(-m - 3)*
hyper((-m - 3, -m - 3), (-m - 2,), -6*x - 3)/(81*(m + 3))

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Mathematica [A]  time = 0.869329, size = 274, normalized size = 0.5 \[ \frac{2^{4-m} (2 x+1)^{1-m} \, _2F_1(1-m,-m;2-m;-6 x-3)}{m-1}-\frac{560 (-6 x-3)^m (3 x+2)^{m+1} (2 x+1)^{-m} \, _2F_1(m+1,m+1;m+2;6 x+4)}{m+1}-\frac{13720\ 3^{m+2} (-2 x-1)^m (3 x+2)^{m+1} (2 x+1)^{-m} \, _2F_1(m+1,m+3;m+2;6 x+4)}{m+1}-\frac{16807\ 3^{m+4} (-2 x-1)^m (3 x+2)^{m+1} (2 x+1)^{-m} \, _2F_1(m+1,m+5;m+2;6 x+4)}{m+1}-\frac{12005\ 3^{m+3} (-2 x-1)^m (4 x+2)^{-m} (6 x+4)^{m+1} \, _2F_1(m+1,m+4;m+2;6 x+4)}{m+1}+\frac{3920 (3 x+2)^{m+1} (2 x+1)^{-m-1}}{m+1} \]

Warning: Unable to verify antiderivative.

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

[Out]

(3920*(1 + 2*x)^(-1 - m)*(2 + 3*x)^(1 + m))/(1 + m) + (2^(4 - m)*(1 + 2*x)^(1 -
m)*Hypergeometric2F1[1 - m, -m, 2 - m, -3 - 6*x])/(-1 + m) - (560*(-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) - (13720*3^(2 + m)*(-1 - 2*x)^m*(2 + 3*x)^(1 + m)*Hypergeometric2F1[1 +
m, 3 + m, 2 + m, 4 + 6*x])/((1 + m)*(1 + 2*x)^m) - (12005*3^(3 + m)*(-1 - 2*x)^m
*(4 + 6*x)^(1 + m)*Hypergeometric2F1[1 + m, 4 + m, 2 + m, 4 + 6*x])/((1 + m)*(2
+ 4*x)^m) - (16807*3^(4 + m)*(-1 - 2*x)^m*(2 + 3*x)^(1 + m)*Hypergeometric2F1[1
+ m, 5 + m, 2 + m, 4 + 6*x])/((1 + m)*(1 + 2*x)^m)

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Maple [F]  time = 0.083, size = 0, normalized size = 0. \[ \int \left ( 5-4\,x \right ) ^{5} \left ( 1+2\,x \right ) ^{-5-m} \left ( 2+3\,x \right ) ^{m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-{\left (1024 \, x^{5} - 6400 \, x^{4} + 16000 \, x^{3} - 20000 \, x^{2} + 12500 \, x - 3125\right )}{\left (3 \, x + 2\right )}^{m}{\left (2 \, x + 1\right )}^{-m - 5}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(1024*x^5 - 6400*x^4 + 16000*x^3 - 20000*x^2 + 12500*x - 3125)*(3*x +
2)^m*(2*x + 1)^(-m - 5), 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)**5*(1+2*x)**(-5-m)*(2+3*x)**m,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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