3.238 \(\int \frac{x^2}{(a+b x^3+c x^6)^{3/2}} \, dx\)

Optimal. Leaf size=38 \[ -\frac{2 \left (b+2 c x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt{a+b x^3+c x^6}} \]

[Out]

(-2*(b + 2*c*x^3))/(3*(b^2 - 4*a*c)*Sqrt[a + b*x^3 + c*x^6])

________________________________________________________________________________________

Rubi [A]  time = 0.0247371, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {1352, 613} \[ -\frac{2 \left (b+2 c x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt{a+b x^3+c x^6}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*x^3 + c*x^6)^(3/2),x]

[Out]

(-2*(b + 2*c*x^3))/(3*(b^2 - 4*a*c)*Sqrt[a + b*x^3 + c*x^6])

Rule 1352

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[(a + b*x +
c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[Simplify[m - n + 1], 0]

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{x^2}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{\left (a+b x+c x^2\right )^{3/2}} \, dx,x,x^3\right )\\ &=-\frac{2 \left (b+2 c x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt{a+b x^3+c x^6}}\\ \end{align*}

Mathematica [A]  time = 0.0236791, size = 38, normalized size = 1. \[ -\frac{2 \left (b+2 c x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt{a+b x^3+c x^6}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b*x^3 + c*x^6)^(3/2),x]

[Out]

(-2*(b + 2*c*x^3))/(3*(b^2 - 4*a*c)*Sqrt[a + b*x^3 + c*x^6])

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 37, normalized size = 1. \begin{align*}{\frac{4\,c{x}^{3}+2\,b}{12\,ac-3\,{b}^{2}}{\frac{1}{\sqrt{c{x}^{6}+b{x}^{3}+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(c*x^6+b*x^3+a)^(3/2),x)

[Out]

2/3/(c*x^6+b*x^3+a)^(1/2)*(2*c*x^3+b)/(4*a*c-b^2)

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(c*x^6+b*x^3+a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.68983, size = 146, normalized size = 3.84 \begin{align*} -\frac{2 \, \sqrt{c x^{6} + b x^{3} + a}{\left (2 \, c x^{3} + b\right )}}{3 \,{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{6} +{\left (b^{3} - 4 \, a b c\right )} x^{3} + a b^{2} - 4 \, a^{2} c\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(c*x^6+b*x^3+a)^(3/2),x, algorithm="fricas")

[Out]

-2/3*sqrt(c*x^6 + b*x^3 + a)*(2*c*x^3 + b)/((b^2*c - 4*a*c^2)*x^6 + (b^3 - 4*a*b*c)*x^3 + a*b^2 - 4*a^2*c)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\left (a + b x^{3} + c x^{6}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(c*x**6+b*x**3+a)**(3/2),x)

[Out]

Integral(x**2/(a + b*x**3 + c*x**6)**(3/2), x)

________________________________________________________________________________________

Giac [A]  time = 1.48891, size = 61, normalized size = 1.61 \begin{align*} -\frac{2 \,{\left (\frac{2 \, c x^{3}}{b^{2} - 4 \, a c} + \frac{b}{b^{2} - 4 \, a c}\right )}}{3 \, \sqrt{c x^{6} + b x^{3} + a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(c*x^6+b*x^3+a)^(3/2),x, algorithm="giac")

[Out]

-2/3*(2*c*x^3/(b^2 - 4*a*c) + b/(b^2 - 4*a*c))/sqrt(c*x^6 + b*x^3 + a)