3.1017 \(\int \frac{\sqrt [6]{a+b x^2}}{x^8} \, dx\)

Optimal. Leaf size=347 \[ -\frac{32 \sqrt{2-\sqrt{3}} b^3 \sqrt [6]{a+b x^2} \left (1-\sqrt [3]{\frac{a}{a+b x^2}}\right ) \sqrt{\frac{\left (\frac{a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac{a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{a}{a+b x^2}}+\sqrt{3}+1}{-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1}\right ),4 \sqrt{3}-7\right )}{405 \sqrt [4]{3} a^3 x \sqrt [3]{\frac{a}{a+b x^2}} \sqrt{-\frac{1-\sqrt [3]{\frac{a}{a+b x^2}}}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}}}-\frac{16 b^3 \sqrt [6]{a+b x^2}}{405 a^3 x}+\frac{2 b^2 \sqrt [6]{a+b x^2}}{135 a^2 x^3}-\frac{b \sqrt [6]{a+b x^2}}{105 a x^5}-\frac{\sqrt [6]{a+b x^2}}{7 x^7} \]

[Out]

-(a + b*x^2)^(1/6)/(7*x^7) - (b*(a + b*x^2)^(1/6))/(105*a*x^5) + (2*b^2*(a + b*x^2)^(1/6))/(135*a^2*x^3) - (16
*b^3*(a + b*x^2)^(1/6))/(405*a^3*x) - (32*Sqrt[2 - Sqrt[3]]*b^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[Ar
cSin[(1 + Sqrt[3] - (a/(a + b*x^2))^(1/3))/(1 - Sqrt[3] - (a/(a + b*x^2))^(1/3))], -7 + 4*Sqrt[3]])/(405*3^(1/
4)*a^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)])

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Rubi [A]  time = 0.317764, antiderivative size = 347, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {277, 325, 241, 236, 219} \[ -\frac{16 b^3 \sqrt [6]{a+b x^2}}{405 a^3 x}+\frac{2 b^2 \sqrt [6]{a+b x^2}}{135 a^2 x^3}-\frac{32 \sqrt{2-\sqrt{3}} b^3 \sqrt [6]{a+b x^2} \left (1-\sqrt [3]{\frac{a}{a+b x^2}}\right ) \sqrt{\frac{\left (\frac{a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac{a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt [3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{405 \sqrt [4]{3} a^3 x \sqrt [3]{\frac{a}{a+b x^2}} \sqrt{-\frac{1-\sqrt [3]{\frac{a}{a+b x^2}}}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}}}-\frac{b \sqrt [6]{a+b x^2}}{105 a x^5}-\frac{\sqrt [6]{a+b x^2}}{7 x^7} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(1/6)/x^8,x]

[Out]

-(a + b*x^2)^(1/6)/(7*x^7) - (b*(a + b*x^2)^(1/6))/(105*a*x^5) + (2*b^2*(a + b*x^2)^(1/6))/(135*a^2*x^3) - (16
*b^3*(a + b*x^2)^(1/6))/(405*a^3*x) - (32*Sqrt[2 - Sqrt[3]]*b^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[Ar
cSin[(1 + Sqrt[3] - (a/(a + b*x^2))^(1/3))/(1 - Sqrt[3] - (a/(a + b*x^2))^(1/3))], -7 + 4*Sqrt[3]])/(405*3^(1/
4)*a^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 277

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

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && 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 \frac{\sqrt [6]{a+b x^2}}{x^8} \, dx &=-\frac{\sqrt [6]{a+b x^2}}{7 x^7}+\frac{1}{21} b \int \frac{1}{x^6 \left (a+b x^2\right )^{5/6}} \, dx\\ &=-\frac{\sqrt [6]{a+b x^2}}{7 x^7}-\frac{b \sqrt [6]{a+b x^2}}{105 a x^5}-\frac{\left (2 b^2\right ) \int \frac{1}{x^4 \left (a+b x^2\right )^{5/6}} \, dx}{45 a}\\ &=-\frac{\sqrt [6]{a+b x^2}}{7 x^7}-\frac{b \sqrt [6]{a+b x^2}}{105 a x^5}+\frac{2 b^2 \sqrt [6]{a+b x^2}}{135 a^2 x^3}+\frac{\left (16 b^3\right ) \int \frac{1}{x^2 \left (a+b x^2\right )^{5/6}} \, dx}{405 a^2}\\ &=-\frac{\sqrt [6]{a+b x^2}}{7 x^7}-\frac{b \sqrt [6]{a+b x^2}}{105 a x^5}+\frac{2 b^2 \sqrt [6]{a+b x^2}}{135 a^2 x^3}-\frac{16 b^3 \sqrt [6]{a+b x^2}}{405 a^3 x}-\frac{\left (32 b^4\right ) \int \frac{1}{\left (a+b x^2\right )^{5/6}} \, dx}{1215 a^3}\\ &=-\frac{\sqrt [6]{a+b x^2}}{7 x^7}-\frac{b \sqrt [6]{a+b x^2}}{105 a x^5}+\frac{2 b^2 \sqrt [6]{a+b x^2}}{135 a^2 x^3}-\frac{16 b^3 \sqrt [6]{a+b x^2}}{405 a^3 x}-\frac{\left (32 b^4\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 )}{1215 a^3 \sqrt [3]{\frac{a}{a+b x^2}} \sqrt [3]{a+b x^2}}\\ &=-\frac{\sqrt [6]{a+b x^2}}{7 x^7}-\frac{b \sqrt [6]{a+b x^2}}{105 a x^5}+\frac{2 b^2 \sqrt [6]{a+b x^2}}{135 a^2 x^3}-\frac{16 b^3 \sqrt [6]{a+b x^2}}{405 a^3 x}+\frac{\left (16 b^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 )}{405 a^3 x \sqrt [3]{\frac{a}{a+b x^2}}}\\ &=-\frac{\sqrt [6]{a+b x^2}}{7 x^7}-\frac{b \sqrt [6]{a+b x^2}}{105 a x^5}+\frac{2 b^2 \sqrt [6]{a+b x^2}}{135 a^2 x^3}-\frac{16 b^3 \sqrt [6]{a+b x^2}}{405 a^3 x}-\frac{32 \sqrt{2-\sqrt{3}} b^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 )}{405 \sqrt [4]{3} a^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 = 0.0091563, size = 51, normalized size = 0.15 \[ -\frac{\sqrt [6]{a+b x^2} \, _2F_1\left (-\frac{7}{2},-\frac{1}{6};-\frac{5}{2};-\frac{b x^2}{a}\right )}{7 x^7 \sqrt [6]{\frac{b x^2}{a}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(1/6)/x^8,x]

[Out]

-((a + b*x^2)^(1/6)*Hypergeometric2F1[-7/2, -1/6, -5/2, -((b*x^2)/a)])/(7*x^7*(1 + (b*x^2)/a)^(1/6))

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Maple [F]  time = 0.031, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{8}}\sqrt [6]{b{x}^{2}+a}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(1/6)/x^8,x)

[Out]

int((b*x^2+a)^(1/6)/x^8,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{2} + a\right )}^{\frac{1}{6}}}{x^{8}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(1/6)/x^8,x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^(1/6)/x^8, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x^{2} + a\right )}^{\frac{1}{6}}}{x^{8}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(1/6)/x^8,x, algorithm="fricas")

[Out]

integral((b*x^2 + a)^(1/6)/x^8, x)

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Sympy [A]  time = 2.03402, size = 34, normalized size = 0.1 \begin{align*} - \frac{\sqrt [6]{a}{{}_{2}F_{1}\left (\begin{matrix} - \frac{7}{2}, - \frac{1}{6} \\ - \frac{5}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{7 x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(1/6)/x**8,x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{2} + a\right )}^{\frac{1}{6}}}{x^{8}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(1/6)/x^8,x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^(1/6)/x^8, x)