Integrand size = 11, antiderivative size = 268 \[ \int \sqrt [6]{a+b x^2} \, dx=\frac {3}{4} x \sqrt [6]{a+b x^2}+\frac {3^{3/4} a^{2/3} \sqrt [6]{a+b x^2} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a}-\left (1-\sqrt {3}\right ) \sqrt [3]{a+b x^2}}{\sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+b x^2}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{8 b x \sqrt {-\frac {\sqrt [3]{a+b x^2} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+b x^2}\right )^2}}} \] Output:
3/4*x*(b*x^2+a)^(1/6)+1/8*3^(3/4)*a^(2/3)*(b*x^2+a)^(1/6)*(a^(1/3)-(b*x^2+ a)^(1/3))*((a^(2/3)+a^(1/3)*(b*x^2+a)^(1/3)+(b*x^2+a)^(2/3))/(a^(1/3)-(1+3 ^(1/2))*(b*x^2+a)^(1/3))^2)^(1/2)*InverseJacobiAM(arccos((a^(1/3)-(1-3^(1/ 2))*(b*x^2+a)^(1/3))/(a^(1/3)-(1+3^(1/2))*(b*x^2+a)^(1/3))),1/4*6^(1/2)+1/ 4*2^(1/2))/b/x/(-(b*x^2+a)^(1/3)*(a^(1/3)-(b*x^2+a)^(1/3))/(a^(1/3)-(1+3^( 1/2))*(b*x^2+a)^(1/3))^2)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.00 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.17 \[ \int \sqrt [6]{a+b x^2} \, dx=\frac {x \sqrt [6]{a+b x^2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {1}{2},\frac {3}{2},-\frac {b x^2}{a}\right )}{\sqrt [6]{1+\frac {b x^2}{a}}} \] Input:
Integrate[(a + b*x^2)^(1/6),x]
Output:
(x*(a + b*x^2)^(1/6)*Hypergeometric2F1[-1/6, 1/2, 3/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^(1/6)
Time = 0.25 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.32, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {211, 236, 234, 760}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sqrt [6]{a+b x^2} \, dx\) |
\(\Big \downarrow \) 211 |
\(\displaystyle \frac {1}{4} a \int \frac {1}{\left (b x^2+a\right )^{5/6}}dx+\frac {3}{4} x \sqrt [6]{a+b x^2}\) |
\(\Big \downarrow \) 236 |
\(\displaystyle \frac {a \int \frac {1}{\left (1-\frac {b x^2}{b x^2+a}\right )^{2/3}}d\frac {x}{\sqrt {b x^2+a}}}{4 \sqrt [3]{\frac {a}{a+b x^2}} \sqrt [3]{a+b x^2}}+\frac {3}{4} x \sqrt [6]{a+b x^2}\) |
\(\Big \downarrow \) 234 |
\(\displaystyle \frac {3}{4} x \sqrt [6]{a+b x^2}-\frac {3 a \sqrt {-\frac {b x^2}{a+b x^2}} \sqrt [6]{a+b x^2} \int \frac {1}{\sqrt {\frac {x^3}{\left (b x^2+a\right )^{3/2}}-1}}d\sqrt [3]{1-\frac {b x^2}{b x^2+a}}}{8 b x \sqrt [3]{\frac {a}{a+b x^2}}}\) |
\(\Big \downarrow \) 760 |
\(\displaystyle \frac {3^{3/4} \sqrt {2-\sqrt {3}} a \sqrt {-\frac {b x^2}{a+b x^2}} \sqrt [6]{a+b x^2} \left (1-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}\right ) \sqrt {\frac {\frac {x^2}{a+b x^2}+\sqrt [3]{1-\frac {b x^2}{a+b x^2}}+1}{\left (-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{4 b x \sqrt [3]{\frac {a}{a+b x^2}} \sqrt {\frac {x^3}{\left (a+b x^2\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}}{\left (-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}-\sqrt {3}+1\right )^2}}}+\frac {3}{4} x \sqrt [6]{a+b x^2}\) |
Input:
Int[(a + b*x^2)^(1/6),x]
Output:
(3*x*(a + b*x^2)^(1/6))/4 + (3^(3/4)*Sqrt[2 - Sqrt[3]]*a*Sqrt[-((b*x^2)/(a + b*x^2))]*(a + b*x^2)^(1/6)*(1 - (1 - (b*x^2)/(a + b*x^2))^(1/3))*Sqrt[( 1 + x^2/(a + b*x^2) + (1 - (b*x^2)/(a + b*x^2))^(1/3))/(1 - Sqrt[3] - (1 - (b*x^2)/(a + b*x^2))^(1/3))^2]*EllipticF[ArcSin[(1 + Sqrt[3] - (1 - (b*x^ 2)/(a + b*x^2))^(1/3))/(1 - Sqrt[3] - (1 - (b*x^2)/(a + b*x^2))^(1/3))], - 7 + 4*Sqrt[3]])/(4*b*x*(a/(a + b*x^2))^(1/3)*Sqrt[-1 + x^3/(a + b*x^2)^(3/ 2)]*Sqrt[-((1 - (1 - (b*x^2)/(a + b*x^2))^(1/3))/(1 - Sqrt[3] - (1 - (b*x^ 2)/(a + b*x^2))^(1/3))^2)])
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
Int[((a_) + (b_.)*(x_)^2)^(-2/3), x_Symbol] :> Simp[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]
Int[((a_) + (b_.)*(x_)^2)^(-5/6), x_Symbol] :> Simp[1/((a/(a + b*x^2))^(1/3 )*(a + b*x^2)^(1/3)) Subst[Int[1/(1 - b*x^2)^(2/3), x], x, x/Sqrt[a + b*x ^2]], x] /; FreeQ[{a, b}, x]
Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 - Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s *x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(- s)*((s + r*x)/((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]], x]] /; FreeQ[{a, b}, x ] && NegQ[a]
\[\int \left (b \,x^{2}+a \right )^{\frac {1}{6}}d x\]
Input:
int((b*x^2+a)^(1/6),x)
Output:
int((b*x^2+a)^(1/6),x)
\[ \int \sqrt [6]{a+b x^2} \, dx=\int { {\left (b x^{2} + a\right )}^{\frac {1}{6}} \,d x } \] Input:
integrate((b*x^2+a)^(1/6),x, algorithm="fricas")
Output:
integral((b*x^2 + a)^(1/6), x)
Result contains complex when optimal does not.
Time = 0.48 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.10 \[ \int \sqrt [6]{a+b x^2} \, dx=\sqrt [6]{a} x {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{6}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )} \] Input:
integrate((b*x**2+a)**(1/6),x)
Output:
a**(1/6)*x*hyper((-1/6, 1/2), (3/2,), b*x**2*exp_polar(I*pi)/a)
\[ \int \sqrt [6]{a+b x^2} \, dx=\int { {\left (b x^{2} + a\right )}^{\frac {1}{6}} \,d x } \] Input:
integrate((b*x^2+a)^(1/6),x, algorithm="maxima")
Output:
integrate((b*x^2 + a)^(1/6), x)
\[ \int \sqrt [6]{a+b x^2} \, dx=\int { {\left (b x^{2} + a\right )}^{\frac {1}{6}} \,d x } \] Input:
integrate((b*x^2+a)^(1/6),x, algorithm="giac")
Output:
integrate((b*x^2 + a)^(1/6), x)
Time = 0.24 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.14 \[ \int \sqrt [6]{a+b x^2} \, dx=\frac {x\,{\left (b\,x^2+a\right )}^{1/6}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{6},\frac {1}{2};\ \frac {3}{2};\ -\frac {b\,x^2}{a}\right )}{{\left (\frac {b\,x^2}{a}+1\right )}^{1/6}} \] Input:
int((a + b*x^2)^(1/6),x)
Output:
(x*(a + b*x^2)^(1/6)*hypergeom([-1/6, 1/2], 3/2, -(b*x^2)/a))/((b*x^2)/a + 1)^(1/6)
\[ \int \sqrt [6]{a+b x^2} \, dx=\int \left (b \,x^{2}+a \right )^{\frac {1}{6}}d x \] Input:
int((b*x^2+a)^(1/6),x)
Output:
int((a + b*x**2)**(1/6),x)