3.3079 \(\int (5-4 x)^4 (1+2 x)^{-4-m} (2+3 x)^m \, dx\)

Optimal. Leaf size=333 \[ \frac{2^{3-m} (42-m) (2 x+1)^{-m} \, _2F_1(-m,-m;1-m;-3 (2 x+1))}{3 m}+\frac{14 (15-2 m) \left (2 m^2+52 m+579\right ) (3 x+2)^{m+1} (2 x+1)^{-m-2}}{9 (m+2) (m+3)}-\frac{14 (15-2 m) \left (2 m^2+52 m+579\right ) (3 x+2)^{m+1} (2 x+1)^{-m-1}}{3 (m+3) \left (m^2+3 m+2\right )}-\frac{2}{3} (5-4 x)^3 (3 x+2)^{m+1} (2 x+1)^{-m-3}-\frac{49 (15-2 m) (2 m+27) (3 x+2)^{m+1} (2 x+1)^{-m-3}}{9 (m+3)}+\frac{14}{9} (15-2 m) (5-4 x) (3 x+2)^{m+1} (2 x+1)^{-m-3}+\frac{196 (42-m) (3 x+2)^{m+1} (2 x+1)^{-m-2}}{3 (m+2)}-\frac{28 (42-m) (4 m+29) (3 x+2)^{m+1} (2 x+1)^{-m-1}}{3 (m+1) (m+2)} \]

[Out]

(-49*(15 - 2*m)*(27 + 2*m)*(1 + 2*x)^(-3 - m)*(2 + 3*x)^(1 + m))/(9*(3 + m)) + (
14*(15 - 2*m)*(5 - 4*x)*(1 + 2*x)^(-3 - m)*(2 + 3*x)^(1 + m))/9 - (2*(5 - 4*x)^3
*(1 + 2*x)^(-3 - m)*(2 + 3*x)^(1 + m))/3 + (196*(42 - m)*(1 + 2*x)^(-2 - m)*(2 +
3*x)^(1 + m))/(3*(2 + m)) + (14*(15 - 2*m)*(579 + 52*m + 2*m^2)*(1 + 2*x)^(-2 -
m)*(2 + 3*x)^(1 + m))/(9*(2 + m)*(3 + m)) - (28*(42 - m)*(29 + 4*m)*(1 + 2*x)^(
-1 - m)*(2 + 3*x)^(1 + m))/(3*(1 + m)*(2 + m)) - (14*(15 - 2*m)*(579 + 52*m + 2*
m^2)*(1 + 2*x)^(-1 - m)*(2 + 3*x)^(1 + m))/(3*(3 + m)*(2 + 3*m + m^2)) + (2^(3 -
m)*(42 - m)*Hypergeometric2F1[-m, -m, 1 - m, -3*(1 + 2*x)])/(3*m*(1 + 2*x)^m)

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Rubi [A]  time = 0.816337, antiderivative size = 333, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{2^{3-m} (42-m) (2 x+1)^{-m} \, _2F_1(-m,-m;1-m;-3 (2 x+1))}{3 m}+\frac{14 (15-2 m) \left (2 m^2+52 m+579\right ) (3 x+2)^{m+1} (2 x+1)^{-m-2}}{9 (m+2) (m+3)}-\frac{14 (15-2 m) \left (2 m^2+52 m+579\right ) (3 x+2)^{m+1} (2 x+1)^{-m-1}}{3 (m+3) \left (m^2+3 m+2\right )}-\frac{2}{3} (5-4 x)^3 (3 x+2)^{m+1} (2 x+1)^{-m-3}-\frac{49 (15-2 m) (2 m+27) (3 x+2)^{m+1} (2 x+1)^{-m-3}}{9 (m+3)}+\frac{14}{9} (15-2 m) (5-4 x) (3 x+2)^{m+1} (2 x+1)^{-m-3}+\frac{196 (42-m) (3 x+2)^{m+1} (2 x+1)^{-m-2}}{3 (m+2)}-\frac{28 (42-m) (4 m+29) (3 x+2)^{m+1} (2 x+1)^{-m-1}}{3 (m+1) (m+2)} \]

Antiderivative was successfully verified.

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

[Out]

(-49*(15 - 2*m)*(27 + 2*m)*(1 + 2*x)^(-3 - m)*(2 + 3*x)^(1 + m))/(9*(3 + m)) + (
14*(15 - 2*m)*(5 - 4*x)*(1 + 2*x)^(-3 - m)*(2 + 3*x)^(1 + m))/9 - (2*(5 - 4*x)^3
*(1 + 2*x)^(-3 - m)*(2 + 3*x)^(1 + m))/3 + (196*(42 - m)*(1 + 2*x)^(-2 - m)*(2 +
3*x)^(1 + m))/(3*(2 + m)) + (14*(15 - 2*m)*(579 + 52*m + 2*m^2)*(1 + 2*x)^(-2 -
m)*(2 + 3*x)^(1 + m))/(9*(2 + m)*(3 + m)) - (28*(42 - m)*(29 + 4*m)*(1 + 2*x)^(
-1 - m)*(2 + 3*x)^(1 + m))/(3*(1 + m)*(2 + m)) - (14*(15 - 2*m)*(579 + 52*m + 2*
m^2)*(1 + 2*x)^(-1 - m)*(2 + 3*x)^(1 + m))/(3*(3 + m)*(2 + 3*m + m^2)) + (2^(3 -
m)*(42 - m)*Hypergeometric2F1[-m, -m, 1 - m, -3*(1 + 2*x)])/(3*m*(1 + 2*x)^m)

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Rubi in Sympy [A]  time = 79.323, size = 286, normalized size = 0.86 \[ - \frac{49 \left (- 2 m + 15\right ) \left (2 m + 27\right ) \left (2 x + 1\right )^{- m - 3} \left (3 x + 2\right )^{m + 1}}{9 \left (m + 3\right )} + \frac{14 \left (- 2 m + 15\right ) \left (2 x + 1\right )^{- m - 2} \left (3 x + 2\right )^{m + 1} \left (2 m^{2} + 52 m + 579\right )}{9 \left (m + 2\right ) \left (m + 3\right )} - \frac{14 \left (- 2 m + 15\right ) \left (2 x + 1\right )^{- m - 1} \left (3 x + 2\right )^{m + 1} \left (2 m^{2} + 52 m + 579\right )}{3 \left (m + 1\right ) \left (m + 2\right ) \left (m + 3\right )} + \frac{196 \left (- m + 42\right ) \left (2 x + 1\right )^{- m - 2} \left (3 x + 2\right )^{m + 1}}{3 \left (m + 2\right )} - \frac{28 \left (- m + 42\right ) \left (4 m + 29\right ) \left (2 x + 1\right )^{- m - 1} \left (3 x + 2\right )^{m + 1}}{3 \left (m + 1\right ) \left (m + 2\right )} + \left (- \frac{7 m}{9} + \frac{35}{6}\right ) \left (- 16 x + 20\right ) \left (2 x + 1\right )^{- m - 3} \left (3 x + 2\right )^{m + 1} - \frac{2 \left (- 4 x + 5\right )^{3} \left (2 x + 1\right )^{- m - 3} \left (3 x + 2\right )^{m + 1}}{3} + \frac{8 \cdot 2^{- m} \left (- m + 42\right ) \left (2 x + 1\right )^{- m}{{}_{2}F_{1}\left (\begin{matrix} - m, - m \\ - m + 1 \end{matrix}\middle |{- 6 x - 3} \right )}}{3 m} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-49*(-2*m + 15)*(2*m + 27)*(2*x + 1)**(-m - 3)*(3*x + 2)**(m + 1)/(9*(m + 3)) +
14*(-2*m + 15)*(2*x + 1)**(-m - 2)*(3*x + 2)**(m + 1)*(2*m**2 + 52*m + 579)/(9*(
m + 2)*(m + 3)) - 14*(-2*m + 15)*(2*x + 1)**(-m - 1)*(3*x + 2)**(m + 1)*(2*m**2
+ 52*m + 579)/(3*(m + 1)*(m + 2)*(m + 3)) + 196*(-m + 42)*(2*x + 1)**(-m - 2)*(3
*x + 2)**(m + 1)/(3*(m + 2)) - 28*(-m + 42)*(4*m + 29)*(2*x + 1)**(-m - 1)*(3*x
+ 2)**(m + 1)/(3*(m + 1)*(m + 2)) + (-7*m/9 + 35/6)*(-16*x + 20)*(2*x + 1)**(-m
- 3)*(3*x + 2)**(m + 1) - 2*(-4*x + 5)**3*(2*x + 1)**(-m - 3)*(3*x + 2)**(m + 1)
/3 + 8*2**(-m)*(-m + 42)*(2*x + 1)**(-m)*hyper((-m, -m), (-m + 1,), -6*x - 3)/(3
*m)

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Mathematica [A]  time = 0.742909, size = 166, normalized size = 0.5 \[ (2 x+1)^{-m} \left (\frac{7 (3 x+2)^{m+1} \left (32 (2 x+1) (-6 x-3)^m \, _2F_1(m+1,m+1;m+2;6 x+4)+21 \left (168 (2 x+1) (-6 x-3)^m \, _2F_1(m+1,m+3;m+2;6 x+4)-49\ 3^{m+2} (-2 x-1)^{m+1} \, _2F_1(m+1,m+4;m+2;6 x+4)-8\right )\right )}{(m+1) (2 x+1)}-\frac{2^{3-m} (2 x+1) \, _2F_1(1-m,-m;2-m;-6 x-3)}{m-1}\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

(-((2^(3 - m)*(1 + 2*x)*Hypergeometric2F1[1 - m, -m, 2 - m, -3 - 6*x])/(-1 + m))
+ (7*(2 + 3*x)^(1 + m)*(32*(-3 - 6*x)^m*(1 + 2*x)*Hypergeometric2F1[1 + m, 1 +
m, 2 + m, 4 + 6*x] + 21*(-8 + 168*(-3 - 6*x)^m*(1 + 2*x)*Hypergeometric2F1[1 + m
, 3 + m, 2 + m, 4 + 6*x] - 49*3^(2 + m)*(-1 - 2*x)^(1 + m)*Hypergeometric2F1[1 +
m, 4 + m, 2 + m, 4 + 6*x])))/((1 + m)*(1 + 2*x)))/(1 + 2*x)^m

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Maple [F]  time = 0.094, size = 0, normalized size = 0. \[ \int \left ( 5-4\,x \right ) ^{4} \left ( 1+2\,x \right ) ^{-4-m} \left ( 2+3\,x \right ) ^{m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (256 \, x^{4} - 1280 \, x^{3} + 2400 \, x^{2} - 2000 \, x + 625\right )}{\left (3 \, x + 2\right )}^{m}{\left (2 \, x + 1\right )}^{-m - 4}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((256*x^4 - 1280*x^3 + 2400*x^2 - 2000*x + 625)*(3*x + 2)^m*(2*x + 1)^(-
m - 4), 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)**4*(1+2*x)**(-4-m)*(2+3*x)**m,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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