3.237 \(\int \frac{x^5}{(a+b x^3+c x^6)^{3/2}} \, dx\)

Optimal. Leaf size=39 \[ \frac{2 \left (2 a+b x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt{a+b x^3+c x^6}} \]

[Out]

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

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Rubi [A]  time = 0.0293345, antiderivative size = 39, 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 = {1357, 636} \[ \frac{2 \left (2 a+b x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt{a+b x^3+c x^6}} \]

Antiderivative was successfully verified.

[In]

Int[x^5/(a + b*x^3 + c*x^6)^(3/2),x]

[Out]

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

Rule 1357

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[
b^2 - 4*a*c, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 636

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

Rubi steps

\begin{align*} \int \frac{x^5}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x}{\left (a+b x+c x^2\right )^{3/2}} \, dx,x,x^3\right )\\ &=\frac{2 \left (2 a+b x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt{a+b x^3+c x^6}}\\ \end{align*}

Mathematica [A]  time = 0.0989696, size = 41, normalized size = 1.05 \[ -\frac{2 \left (2 a+b x^3\right )}{3 \left (4 a c-b^2\right ) \sqrt{a+b x^3+c x^6}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(a + b*x^3 + c*x^6)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.005, size = 38, normalized size = 1. \begin{align*} -{\frac{2\,b{x}^{3}+4\,a}{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^5/(c*x^6+b*x^3+a)^(3/2),x)

[Out]

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

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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^5/(c*x^6+b*x^3+a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.68782, size = 144, normalized size = 3.69 \begin{align*} \frac{2 \, \sqrt{c x^{6} + b x^{3} + a}{\left (b x^{3} + 2 \, a\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^5/(c*x^6+b*x^3+a)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{\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**5/(c*x**6+b*x**3+a)**(3/2),x)

[Out]

Integral(x**5/(a + b*x**3 + c*x**6)**(3/2), x)

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Giac [A]  time = 1.45755, size = 61, normalized size = 1.56 \begin{align*} \frac{2 \,{\left (\frac{b x^{3}}{b^{2} - 4 \, a c} + \frac{2 \, a}{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^5/(c*x^6+b*x^3+a)^(3/2),x, algorithm="giac")

[Out]

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