3.222 \(\int \frac{x^5}{\sqrt{a+b x^3+c x^6}} \, dx\)

Optimal. Leaf size=68 \[ \frac{\sqrt{a+b x^3+c x^6}}{3 c}-\frac{b \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{6 c^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0565937, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1357, 640, 621, 206} \[ \frac{\sqrt{a+b x^3+c x^6}}{3 c}-\frac{b \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{6 c^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[x^5/Sqrt[a + b*x^3 + c*x^6],x]

[Out]

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

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 640

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

Rule 621

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0161302, size = 68, normalized size = 1. \[ \frac{\sqrt{a+b x^3+c x^6}}{3 c}-\frac{b \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{6 c^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/Sqrt[a + b*x^3 + c*x^6],x]

[Out]

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

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Maple [F]  time = 0.017, size = 0, normalized size = 0. \begin{align*} \int{{x}^{5}{\frac{1}{\sqrt{c{x}^{6}+b{x}^{3}+a}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(c*x^6+b*x^3+a)^(1/2),x)

[Out]

int(x^5/(c*x^6+b*x^3+a)^(1/2),x)

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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)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.57334, size = 385, normalized size = 5.66 \begin{align*} \left [\frac{b \sqrt{c} \log \left (-8 \, c^{2} x^{6} - 8 \, b c x^{3} - b^{2} + 4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (2 \, c x^{3} + b\right )} \sqrt{c} - 4 \, a c\right ) + 4 \, \sqrt{c x^{6} + b x^{3} + a} c}{12 \, c^{2}}, \frac{b \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{6} + b x^{3} + a}{\left (2 \, c x^{3} + b\right )} \sqrt{-c}}{2 \,{\left (c^{2} x^{6} + b c x^{3} + a c\right )}}\right ) + 2 \, \sqrt{c x^{6} + b x^{3} + a} c}{6 \, c^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(c*x^6+b*x^3+a)^(1/2),x, algorithm="fricas")

[Out]

[1/12*(b*sqrt(c)*log(-8*c^2*x^6 - 8*b*c*x^3 - b^2 + 4*sqrt(c*x^6 + b*x^3 + a)*(2*c*x^3 + b)*sqrt(c) - 4*a*c) +
4*sqrt(c*x^6 + b*x^3 + a)*c)/c^2, 1/6*(b*sqrt(-c)*arctan(1/2*sqrt(c*x^6 + b*x^3 + a)*(2*c*x^3 + b)*sqrt(-c)/(
c^2*x^6 + b*c*x^3 + a*c)) + 2*sqrt(c*x^6 + b*x^3 + a)*c)/c^2]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(c*x**6+b*x**3+a)**(1/2),x)

[Out]

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

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Giac [A]  time = 1.15987, size = 82, normalized size = 1.21 \begin{align*} \frac{b \log \left ({\left | -2 \,{\left (\sqrt{c} x^{3} - \sqrt{c x^{6} + b x^{3} + a}\right )} \sqrt{c} - b \right |}\right )}{6 \, c^{\frac{3}{2}}} + \frac{\sqrt{c x^{6} + b x^{3} + a}}{3 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(c*x^6+b*x^3+a)^(1/2),x, algorithm="giac")

[Out]

1/6*b*log(abs(-2*(sqrt(c)*x^3 - sqrt(c*x^6 + b*x^3 + a))*sqrt(c) - b))/c^(3/2) + 1/3*sqrt(c*x^6 + b*x^3 + a)/c