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

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 = 0.0076841, 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, Rules used = {614, 613} \[ -\frac{2 (2-3 x)}{361 \sqrt{-3 x^2+4 x+5}}-\frac{2-3 x}{57 \left (-3 x^2+4 x+5\right )^{3/2}} \]

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])

Rule 614

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^(p + 1))/((p +
1)*(b^2 - 4*a*c)), x] - Dist[(2*c*(2*p + 3))/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

Rule 613

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[(-2*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[a + b*x
+ c*x^2]), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rubi steps

\begin{align*} \int \frac{1}{\left (5+4 x-3 x^2\right )^{5/2}} \, dx &=-\frac{2-3 x}{57 \left (5+4 x-3 x^2\right )^{3/2}}+\frac{2}{19} \int \frac{1}{\left (5+4 x-3 x^2\right )^{3/2}} \, dx\\ &=-\frac{2-3 x}{57 \left (5+4 x-3 x^2\right )^{3/2}}-\frac{2 (2-3 x)}{361 \sqrt{5+4 x-3 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.011454, 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 = 0.003, size = 30, normalized size = 0.6 \begin{align*} -{\frac{54\,{x}^{3}-108\,{x}^{2}-99\,x+98}{1083} \left ( -3\,{x}^{2}+4\,x+5 \right ) ^{-{\frac{3}{2}}}} \end{align*}

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 = 0.95672, size = 80, normalized size = 1.7 \begin{align*} \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}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-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 = 1.84248, size = 136, normalized size = 2.89 \begin{align*} -\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-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. \begin{align*} \int \frac{1}{\left (- 3 x^{2} + 4 x + 5\right )^{\frac{5}{2}}}\, dx \end{align*}

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 [A]  time = 1.0955, size = 53, normalized size = 1.13 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-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