3.1011 \(\int x^4 \sqrt [6]{a+b x^2} \, dx\)

Optimal. Leaf size=321 \[ \frac{27\ 3^{3/4} \sqrt{2-\sqrt{3}} a^3 \sqrt [6]{a+b x^2} \left (1-\sqrt [3]{\frac{a}{a+b x^2}}\right ) \sqrt{\frac{\left (\frac{a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac{a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{a}{a+b x^2}}+\sqrt{3}+1}{-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1}\right ),4 \sqrt{3}-7\right )}{640 b^3 x \sqrt [3]{\frac{a}{a+b x^2}} \sqrt{-\frac{1-\sqrt [3]{\frac{a}{a+b x^2}}}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}}}-\frac{27 a^2 x \sqrt [6]{a+b x^2}}{640 b^2}+\frac{3}{16} x^5 \sqrt [6]{a+b x^2}+\frac{3 a x^3 \sqrt [6]{a+b x^2}}{160 b} \]

[Out]

(-27*a^2*x*(a + b*x^2)^(1/6))/(640*b^2) + (3*a*x^3*(a + b*x^2)^(1/6))/(160*b) + (3*x^5*(a + b*x^2)^(1/6))/16 +
(27*3^(3/4)*Sqrt[2 - Sqrt[3]]*a^3*(a + b*x^2)^(1/6)*(1 - (a/(a + b*x^2))^(1/3))*Sqrt[(1 + (a/(a + b*x^2))^(1/
3) + (a/(a + b*x^2))^(2/3))/(1 - Sqrt[3] - (a/(a + b*x^2))^(1/3))^2]*EllipticF[ArcSin[(1 + Sqrt[3] - (a/(a + b
*x^2))^(1/3))/(1 - Sqrt[3] - (a/(a + b*x^2))^(1/3))], -7 + 4*Sqrt[3]])/(640*b^3*x*(a/(a + b*x^2))^(1/3)*Sqrt[-
((1 - (a/(a + b*x^2))^(1/3))/(1 - Sqrt[3] - (a/(a + b*x^2))^(1/3))^2)])

________________________________________________________________________________________

Rubi [A]  time = 0.291074, antiderivative size = 321, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {279, 321, 241, 236, 219} \[ -\frac{27 a^2 x \sqrt [6]{a+b x^2}}{640 b^2}+\frac{27\ 3^{3/4} \sqrt{2-\sqrt{3}} a^3 \sqrt [6]{a+b x^2} \left (1-\sqrt [3]{\frac{a}{a+b x^2}}\right ) \sqrt{\frac{\left (\frac{a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac{a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt [3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{640 b^3 x \sqrt [3]{\frac{a}{a+b x^2}} \sqrt{-\frac{1-\sqrt [3]{\frac{a}{a+b x^2}}}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}}}+\frac{3}{16} x^5 \sqrt [6]{a+b x^2}+\frac{3 a x^3 \sqrt [6]{a+b x^2}}{160 b} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(a + b*x^2)^(1/6),x]

[Out]

(-27*a^2*x*(a + b*x^2)^(1/6))/(640*b^2) + (3*a*x^3*(a + b*x^2)^(1/6))/(160*b) + (3*x^5*(a + b*x^2)^(1/6))/16 +
(27*3^(3/4)*Sqrt[2 - Sqrt[3]]*a^3*(a + b*x^2)^(1/6)*(1 - (a/(a + b*x^2))^(1/3))*Sqrt[(1 + (a/(a + b*x^2))^(1/
3) + (a/(a + b*x^2))^(2/3))/(1 - Sqrt[3] - (a/(a + b*x^2))^(1/3))^2]*EllipticF[ArcSin[(1 + Sqrt[3] - (a/(a + b
*x^2))^(1/3))/(1 - Sqrt[3] - (a/(a + b*x^2))^(1/3))], -7 + 4*Sqrt[3]])/(640*b^3*x*(a/(a + b*x^2))^(1/3)*Sqrt[-
((1 - (a/(a + b*x^2))^(1/3))/(1 - Sqrt[3] - (a/(a + b*x^2))^(1/3))^2)])

Rule 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
n*p + 1)), x] + Dist[(a*n*p)/(m + n*p + 1), Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x]
&& IGtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 241

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a/(a + b*x^n))^(p + 1/n)*(a + b*x^n)^(p + 1/n), Subst[In
t[1/(1 - b*x^n)^(p + 1/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p,
0] && NeQ[p, -2^(-1)] && LtQ[Denominator[p + 1/n], Denominator[p]]

Rule 236

Int[((a_) + (b_.)*(x_)^2)^(-2/3), x_Symbol] :> Dist[(3*Sqrt[b*x^2])/(2*b*x), Subst[Int[1/Sqrt[-a + x^3], x], x
, (a + b*x^2)^(1/3)], x] /; FreeQ[{a, b}, x]

Rule 219

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(2*Sqr
t[2 - Sqrt[3]]*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]*EllipticF[ArcSin[((1 + Sqrt[3
])*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]])/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[-((s*(s + r*x))/((1 - S
qrt[3])*s + r*x)^2)]), x]] /; FreeQ[{a, b}, x] && NegQ[a]

Rubi steps

\begin{align*} \int x^4 \sqrt [6]{a+b x^2} \, dx &=\frac{3}{16} x^5 \sqrt [6]{a+b x^2}+\frac{1}{16} a \int \frac{x^4}{\left (a+b x^2\right )^{5/6}} \, dx\\ &=\frac{3 a x^3 \sqrt [6]{a+b x^2}}{160 b}+\frac{3}{16} x^5 \sqrt [6]{a+b x^2}-\frac{\left (9 a^2\right ) \int \frac{x^2}{\left (a+b x^2\right )^{5/6}} \, dx}{160 b}\\ &=-\frac{27 a^2 x \sqrt [6]{a+b x^2}}{640 b^2}+\frac{3 a x^3 \sqrt [6]{a+b x^2}}{160 b}+\frac{3}{16} x^5 \sqrt [6]{a+b x^2}+\frac{\left (27 a^3\right ) \int \frac{1}{\left (a+b x^2\right )^{5/6}} \, dx}{640 b^2}\\ &=-\frac{27 a^2 x \sqrt [6]{a+b x^2}}{640 b^2}+\frac{3 a x^3 \sqrt [6]{a+b x^2}}{160 b}+\frac{3}{16} x^5 \sqrt [6]{a+b x^2}+\frac{\left (27 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-b x^2\right )^{2/3}} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{640 b^2 \sqrt [3]{\frac{a}{a+b x^2}} \sqrt [3]{a+b x^2}}\\ &=-\frac{27 a^2 x \sqrt [6]{a+b x^2}}{640 b^2}+\frac{3 a x^3 \sqrt [6]{a+b x^2}}{160 b}+\frac{3}{16} x^5 \sqrt [6]{a+b x^2}-\frac{\left (81 a^3 \sqrt{-\frac{b x^2}{a+b x^2}} \sqrt [6]{a+b x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x^3}} \, dx,x,\sqrt [3]{\frac{a}{a+b x^2}}\right )}{1280 b^3 x \sqrt [3]{\frac{a}{a+b x^2}}}\\ &=-\frac{27 a^2 x \sqrt [6]{a+b x^2}}{640 b^2}+\frac{3 a x^3 \sqrt [6]{a+b x^2}}{160 b}+\frac{3}{16} x^5 \sqrt [6]{a+b x^2}+\frac{27\ 3^{3/4} \sqrt{2-\sqrt{3}} a^3 \sqrt{-\frac{b x^2}{a+b x^2}} \sqrt [6]{a+b x^2} \left (1-\sqrt [3]{\frac{a}{a+b x^2}}\right ) \sqrt{\frac{1+\sqrt [3]{\frac{a}{a+b x^2}}+\left (\frac{a}{a+b x^2}\right )^{2/3}}{\left (1-\sqrt{3}-\sqrt [3]{\frac{a}{a+b x^2}}\right )^2}} F\left (\sin ^{-1}\left (\frac{1+\sqrt{3}-\sqrt [3]{\frac{a}{a+b x^2}}}{1-\sqrt{3}-\sqrt [3]{\frac{a}{a+b x^2}}}\right )|-7+4 \sqrt{3}\right )}{640 b^3 x \sqrt [3]{\frac{a}{a+b x^2}} \sqrt{-\frac{1-\sqrt [3]{\frac{a}{a+b x^2}}}{\left (1-\sqrt{3}-\sqrt [3]{\frac{a}{a+b x^2}}\right )^2}} \sqrt{-1+\frac{a}{a+b x^2}}}\\ \end{align*}

Mathematica [C]  time = 0.0460341, size = 93, normalized size = 0.29 \[ \frac{3 x \sqrt [6]{a+b x^2} \left (\sqrt [6]{\frac{b x^2}{a}+1} \left (-9 a^2+a b x^2+10 b^2 x^4\right )+9 a^2 \, _2F_1\left (-\frac{1}{6},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right )\right )}{160 b^2 \sqrt [6]{\frac{b x^2}{a}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(a + b*x^2)^(1/6),x]

[Out]

(3*x*(a + b*x^2)^(1/6)*((1 + (b*x^2)/a)^(1/6)*(-9*a^2 + a*b*x^2 + 10*b^2*x^4) + 9*a^2*Hypergeometric2F1[-1/6,
1/2, 3/2, -((b*x^2)/a)]))/(160*b^2*(1 + (b*x^2)/a)^(1/6))

________________________________________________________________________________________

Maple [F]  time = 0.025, size = 0, normalized size = 0. \begin{align*} \int{x}^{4}\sqrt [6]{b{x}^{2}+a}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(b*x^2+a)^(1/6),x)

[Out]

int(x^4*(b*x^2+a)^(1/6),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{2} + a\right )}^{\frac{1}{6}} x^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x^2+a)^(1/6),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^(1/6)*x^4, x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b x^{2} + a\right )}^{\frac{1}{6}} x^{4}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x^2+a)^(1/6),x, algorithm="fricas")

[Out]

integral((b*x^2 + a)^(1/6)*x^4, x)

________________________________________________________________________________________

Sympy [A]  time = 1.23883, size = 29, normalized size = 0.09 \begin{align*} \frac{\sqrt [6]{a} x^{5}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{6}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(b*x**2+a)**(1/6),x)

[Out]

a**(1/6)*x**5*hyper((-1/6, 5/2), (7/2,), b*x**2*exp_polar(I*pi)/a)/5

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{2} + a\right )}^{\frac{1}{6}} x^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x^2+a)^(1/6),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^(1/6)*x^4, x)