3.3043 \(\int (5-4 x)^4 (1+2 x)^{-m} (2+3 x)^m \, dx\)

Optimal. Leaf size=188 \[ \frac{2^{-m-1} \left (2 m^4-440 m^3+29050 m^2-639760 m+3528363\right ) (2 x+1)^{1-m} \, _2F_1(1-m,-m;2-m;-3 (2 x+1))}{1215 (1-m)}-\frac{(3 x+2)^{m+1} \left (-2 m^3-24 \left (m^2-154 m+4359\right ) x+426 m^2-25441 m+386850\right ) (2 x+1)^{1-m}}{1215}-\frac{2}{15} (5-4 x)^3 (3 x+2)^{m+1} (2 x+1)^{1-m}-\frac{1}{45} (88-m) (5-4 x)^2 (3 x+2)^{m+1} (2 x+1)^{1-m} \]

[Out]

-((88 - m)*(5 - 4*x)^2*(1 + 2*x)^(1 - m)*(2 + 3*x)^(1 + m))/45 - (2*(5 - 4*x)^3*
(1 + 2*x)^(1 - m)*(2 + 3*x)^(1 + m))/15 - ((1 + 2*x)^(1 - m)*(2 + 3*x)^(1 + m)*(
386850 - 25441*m + 426*m^2 - 2*m^3 - 24*(4359 - 154*m + m^2)*x))/1215 + (2^(-1 -
m)*(3528363 - 639760*m + 29050*m^2 - 440*m^3 + 2*m^4)*(1 + 2*x)^(1 - m)*Hyperge
ometric2F1[1 - m, -m, 2 - m, -3*(1 + 2*x)])/(1215*(1 - m))

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Rubi [A]  time = 0.540879, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{2^{-m-1} \left (2 m^4-440 m^3+29050 m^2-639760 m+3528363\right ) (2 x+1)^{1-m} \, _2F_1(1-m,-m;2-m;-3 (2 x+1))}{1215 (1-m)}-\frac{(3 x+2)^{m+1} \left (-2 m^3-24 \left (m^2-154 m+4359\right ) x+426 m^2-25441 m+386850\right ) (2 x+1)^{1-m}}{1215}-\frac{2}{15} (5-4 x)^3 (3 x+2)^{m+1} (2 x+1)^{1-m}-\frac{1}{45} (88-m) (5-4 x)^2 (3 x+2)^{m+1} (2 x+1)^{1-m} \]

Antiderivative was successfully verified.

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

[Out]

-((88 - m)*(5 - 4*x)^2*(1 + 2*x)^(1 - m)*(2 + 3*x)^(1 + m))/45 - (2*(5 - 4*x)^3*
(1 + 2*x)^(1 - m)*(2 + 3*x)^(1 + m))/15 - ((1 + 2*x)^(1 - m)*(2 + 3*x)^(1 + m)*(
386850 - 25441*m + 426*m^2 - 2*m^3 - 24*(4359 - 154*m + m^2)*x))/1215 + (2^(-1 -
m)*(3528363 - 639760*m + 29050*m^2 - 440*m^3 + 2*m^4)*(1 + 2*x)^(1 - m)*Hyperge
ometric2F1[1 - m, -m, 2 - m, -3*(1 + 2*x)])/(1215*(1 - m))

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Rubi in Sympy [A]  time = 48.6082, size = 153, normalized size = 0.81 \[ - \left (- \frac{m}{45} + \frac{88}{45}\right ) \left (- 4 x + 5\right )^{2} \left (2 x + 1\right )^{- m + 1} \left (3 x + 2\right )^{m + 1} - \frac{2 \left (- 4 x + 5\right )^{3} \left (2 x + 1\right )^{- m + 1} \left (3 x + 2\right )^{m + 1}}{15} - \frac{\left (2 x + 1\right )^{- m + 1} \left (3 x + 2\right )^{m + 1} \left (- 256 m^{3} + 54528 m^{2} - 3256448 m - x \left (3072 m^{2} - 473088 m + 13390848\right ) + 49516800\right )}{155520} + \frac{2^{- m} \left (2 x + 1\right )^{- m + 1} \left (2 m^{4} - 440 m^{3} + 29050 m^{2} - 639760 m + 3528363\right ){{}_{2}F_{1}\left (\begin{matrix} - m, - m + 1 \\ - m + 2 \end{matrix}\middle |{- 6 x - 3} \right )}}{2430 \left (- m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-m/45 + 88/45)*(-4*x + 5)**2*(2*x + 1)**(-m + 1)*(3*x + 2)**(m + 1) - 2*(-4*x
+ 5)**3*(2*x + 1)**(-m + 1)*(3*x + 2)**(m + 1)/15 - (2*x + 1)**(-m + 1)*(3*x + 2
)**(m + 1)*(-256*m**3 + 54528*m**2 - 3256448*m - x*(3072*m**2 - 473088*m + 13390
848) + 49516800)/155520 + 2**(-m)*(2*x + 1)**(-m + 1)*(2*m**4 - 440*m**3 + 29050
*m**2 - 639760*m + 3528363)*hyper((-m, -m + 1), (-m + 2,), -6*x - 3)/(2430*(-m +
1))

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Mathematica [C]  time = 0.252948, size = 155, normalized size = 0.82 \[ \frac{483\ 2^{-m-1} (4 x-5)^5 (4 x+2)^{-m} (12 x+8)^m F_1\left (5;-m,m;6;\frac{3}{23} (5-4 x),\frac{1}{7} (5-4 x)\right )}{5 \left (966 F_1\left (5;-m,m;6;\frac{3}{23} (5-4 x),\frac{1}{7} (5-4 x)\right )+m (4 x-5) \left (21 F_1\left (6;1-m,m;7;\frac{3}{23} (5-4 x),\frac{1}{7} (5-4 x)\right )-23 F_1\left (6;-m,m+1;7;\frac{3}{23} (5-4 x),\frac{1}{7} (5-4 x)\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(483*2^(-1 - m)*(-5 + 4*x)^5*(8 + 12*x)^m*AppellF1[5, -m, m, 6, (3*(5 - 4*x))/23
, (5 - 4*x)/7])/(5*(2 + 4*x)^m*(966*AppellF1[5, -m, m, 6, (3*(5 - 4*x))/23, (5 -
4*x)/7] + m*(-5 + 4*x)*(21*AppellF1[6, 1 - m, m, 7, (3*(5 - 4*x))/23, (5 - 4*x)
/7] - 23*AppellF1[6, -m, 1 + m, 7, (3*(5 - 4*x))/23, (5 - 4*x)/7])))

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Maple [F]  time = 0.109, size = 0, normalized size = 0. \[ \int{\frac{ \left ( 5-4\,x \right ) ^{4} \left ( 2+3\,x \right ) ^{m}}{ \left ( 1+2\,x \right ) ^{m}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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