3.3052 \(\int \frac{(1+2 x)^{-m} (2+3 x)^m}{(5-4 x)^5} \, dx\)

Optimal. Leaf size=179 \[ \frac{\left (m^3+132 m^2+4358 m+32010\right ) (3 x+2)^{m-1} (2 x+1)^{1-m} \, _2F_1\left (2,1-m;2-m;\frac{23 (2 x+1)}{14 (3 x+2)}\right )}{2453889228 (1-m)}+\frac{\left (2 m^2+220 m+4359\right ) (3 x+2)^{m+1} (2 x+1)^{1-m}}{25039686 (5-4 x)^2}+\frac{(m+66) (3 x+2)^{m+1} (2 x+1)^{1-m}}{77763 (5-4 x)^3}+\frac{(3 x+2)^{m+1} (2 x+1)^{1-m}}{322 (5-4 x)^4} \]

[Out]

((1 + 2*x)^(1 - m)*(2 + 3*x)^(1 + m))/(322*(5 - 4*x)^4) + ((66 + m)*(1 + 2*x)^(1
- m)*(2 + 3*x)^(1 + m))/(77763*(5 - 4*x)^3) + ((4359 + 220*m + 2*m^2)*(1 + 2*x)
^(1 - m)*(2 + 3*x)^(1 + m))/(25039686*(5 - 4*x)^2) + ((32010 + 4358*m + 132*m^2
+ m^3)*(1 + 2*x)^(1 - m)*(2 + 3*x)^(-1 + m)*Hypergeometric2F1[2, 1 - m, 2 - m, (
23*(1 + 2*x))/(14*(2 + 3*x))])/(2453889228*(1 - m))

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Rubi [A]  time = 0.326122, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{\left (m^3+132 m^2+4358 m+32010\right ) (3 x+2)^{m-1} (2 x+1)^{1-m} \, _2F_1\left (2,1-m;2-m;\frac{23 (2 x+1)}{14 (3 x+2)}\right )}{2453889228 (1-m)}+\frac{\left (2 m^2+220 m+4359\right ) (3 x+2)^{m+1} (2 x+1)^{1-m}}{25039686 (5-4 x)^2}+\frac{(m+66) (3 x+2)^{m+1} (2 x+1)^{1-m}}{77763 (5-4 x)^3}+\frac{(3 x+2)^{m+1} (2 x+1)^{1-m}}{322 (5-4 x)^4} \]

Antiderivative was successfully verified.

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

[Out]

((1 + 2*x)^(1 - m)*(2 + 3*x)^(1 + m))/(322*(5 - 4*x)^4) + ((66 + m)*(1 + 2*x)^(1
- m)*(2 + 3*x)^(1 + m))/(77763*(5 - 4*x)^3) + ((4359 + 220*m + 2*m^2)*(1 + 2*x)
^(1 - m)*(2 + 3*x)^(1 + m))/(25039686*(5 - 4*x)^2) + ((32010 + 4358*m + 132*m^2
+ m^3)*(1 + 2*x)^(1 - m)*(2 + 3*x)^(-1 + m)*Hypergeometric2F1[2, 1 - m, 2 - m, (
23*(1 + 2*x))/(14*(2 + 3*x))])/(2453889228*(1 - m))

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Rubi in Sympy [A]  time = 43.0198, size = 143, normalized size = 0.8 \[ \frac{\left (\frac{m}{77763} + \frac{22}{25921}\right ) \left (2 x + 1\right )^{- m + 1} \left (3 x + 2\right )^{m + 1}}{\left (- 4 x + 5\right )^{3}} + \frac{\left (2 x + 1\right )^{- m + 1} \left (3 x + 2\right )^{m + 1} \left (\frac{m^{2}}{12519843} + \frac{110 m}{12519843} + \frac{1453}{8346562}\right )}{\left (- 4 x + 5\right )^{2}} + \frac{\left (2 x + 1\right )^{- m + 1} \left (3 x + 2\right )^{m + 1}}{322 \left (- 4 x + 5\right )^{4}} + \frac{\left (2 x + 1\right )^{- m + 1} \left (3 x + 2\right )^{m - 1} \left (m^{3} + 132 m^{2} + 4358 m + 32010\right ){{}_{2}F_{1}\left (\begin{matrix} - m + 1, 2 \\ - m + 2 \end{matrix}\middle |{\frac{46 x + 23}{42 x + 28}} \right )}}{2453889228 \left (- m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(m/77763 + 22/25921)*(2*x + 1)**(-m + 1)*(3*x + 2)**(m + 1)/(-4*x + 5)**3 + (2*x
+ 1)**(-m + 1)*(3*x + 2)**(m + 1)*(m**2/12519843 + 110*m/12519843 + 1453/834656
2)/(-4*x + 5)**2 + (2*x + 1)**(-m + 1)*(3*x + 2)**(m + 1)/(322*(-4*x + 5)**4) +
(2*x + 1)**(-m + 1)*(3*x + 2)**(m - 1)*(m**3 + 132*m**2 + 4358*m + 32010)*hyper(
(-m + 1, 2), (-m + 2,), (46*x + 23)/(42*x + 28))/(2453889228*(-m + 1))

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Mathematica [C]  time = 0.417887, size = 153, normalized size = 0.85 \[ \frac{15\ 2^{-m-4} (4 x+2)^{-m} (12 x+8)^m F_1\left (4;-m,m;5;\frac{23}{15-12 x},\frac{7}{5-4 x}\right )}{(4 x-5)^3 \left (15 (4 x-5) F_1\left (4;-m,m;5;\frac{23}{15-12 x},\frac{7}{5-4 x}\right )+m \left (23 F_1\left (5;1-m,m;6;\frac{23}{15-12 x},\frac{7}{5-4 x}\right )-21 F_1\left (5;-m,m+1;6;\frac{23}{15-12 x},\frac{7}{5-4 x}\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(15*2^(-4 - m)*(8 + 12*x)^m*AppellF1[4, -m, m, 5, 23/(15 - 12*x), 7/(5 - 4*x)])/
((-5 + 4*x)^3*(2 + 4*x)^m*(15*(-5 + 4*x)*AppellF1[4, -m, m, 5, 23/(15 - 12*x), 7
/(5 - 4*x)] + m*(23*AppellF1[5, 1 - m, m, 6, 23/(15 - 12*x), 7/(5 - 4*x)] - 21*A
ppellF1[5, -m, 1 + m, 6, 23/(15 - 12*x), 7/(5 - 4*x)])))

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Maple [F]  time = 0.108, size = 0, normalized size = 0. \[ \int{\frac{ \left ( 2+3\,x \right ) ^{m}}{ \left ( 5-4\,x \right ) ^{5} \left ( 1+2\,x \right ) ^{m}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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