3.286 \(\int \frac{1}{\left (5+4 x-3 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=47 \[ -\frac{2 (2-3 x)}{361 \sqrt{-3 x^2+4 x+5}}-\frac{2-3 x}{57 \left (-3 x^2+4 x+5\right )^{3/2}} \]

[Out]

-(2 - 3*x)/(57*(5 + 4*x - 3*x^2)^(3/2)) - (2*(2 - 3*x))/(361*Sqrt[5 + 4*x - 3*x^
2])

_______________________________________________________________________________________

Rubi [A]  time = 0.0199695, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{2 (2-3 x)}{361 \sqrt{-3 x^2+4 x+5}}-\frac{2-3 x}{57 \left (-3 x^2+4 x+5\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(5 + 4*x - 3*x^2)^(-5/2),x]

[Out]

-(2 - 3*x)/(57*(5 + 4*x - 3*x^2)^(3/2)) - (2*(2 - 3*x))/(361*Sqrt[5 + 4*x - 3*x^
2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 0.847117, size = 41, normalized size = 0.87 \[ - \frac{- 12 x + 8}{722 \sqrt{- 3 x^{2} + 4 x + 5}} - \frac{- 6 x + 4}{114 \left (- 3 x^{2} + 4 x + 5\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-12*x + 8)/(722*sqrt(-3*x**2 + 4*x + 5)) - (-6*x + 4)/(114*(-3*x**2 + 4*x + 5)
**(3/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0276974, size = 33, normalized size = 0.7 \[ -\frac{(3 x-2) \left (18 x^2-24 x-49\right )}{1083 \left (-3 x^2+4 x+5\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-((-2 + 3*x)*(-49 - 24*x + 18*x^2))/(1083*(5 + 4*x - 3*x^2)^(3/2))

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 30, normalized size = 0.6 \[ -{\frac{54\,{x}^{3}-108\,{x}^{2}-99\,x+98}{1083} \left ( -3\,{x}^{2}+4\,x+5 \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-3*x^2+4*x+5)^(5/2),x)

[Out]

-1/1083*(54*x^3-108*x^2-99*x+98)/(-3*x^2+4*x+5)^(3/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.43435, size = 80, normalized size = 1.7 \[ \frac{6 \, x}{361 \, \sqrt{-3 \, x^{2} + 4 \, x + 5}} - \frac{4}{361 \, \sqrt{-3 \, x^{2} + 4 \, x + 5}} + \frac{x}{19 \,{\left (-3 \, x^{2} + 4 \, x + 5\right )}^{\frac{3}{2}}} - \frac{2}{57 \,{\left (-3 \, x^{2} + 4 \, x + 5\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6/361*x/sqrt(-3*x^2 + 4*x + 5) - 4/361/sqrt(-3*x^2 + 4*x + 5) + 1/19*x/(-3*x^2 +
4*x + 5)^(3/2) - 2/57/(-3*x^2 + 4*x + 5)^(3/2)

_______________________________________________________________________________________

Fricas [A]  time = 0.211885, size = 69, normalized size = 1.47 \[ -\frac{{\left (54 \, x^{3} - 108 \, x^{2} - 99 \, x + 98\right )} \sqrt{-3 \, x^{2} + 4 \, x + 5}}{1083 \,{\left (9 \, x^{4} - 24 \, x^{3} - 14 \, x^{2} + 40 \, x + 25\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1083*(54*x^3 - 108*x^2 - 99*x + 98)*sqrt(-3*x^2 + 4*x + 5)/(9*x^4 - 24*x^3 -
14*x^2 + 40*x + 25)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((-3*x**2 + 4*x + 5)**(-5/2), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.215196, size = 53, normalized size = 1.13 \[ -\frac{{\left (9 \,{\left (6 \,{\left (x - 2\right )} x - 11\right )} x + 98\right )} \sqrt{-3 \, x^{2} + 4 \, x + 5}}{1083 \,{\left (3 \, x^{2} - 4 \, x - 5\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1083*(9*(6*(x - 2)*x - 11)*x + 98)*sqrt(-3*x^2 + 4*x + 5)/(3*x^2 - 4*x - 5)^2