Integrand size = 15, antiderivative size = 276 \[ \int \frac {x^2}{\left (a+b x^2\right )^{5/6}} \, dx=\frac {3 x \sqrt [6]{a+b x^2}}{4 b}-\frac {3\ 3^{3/4} \sqrt {2-\sqrt {3}} a \sqrt [6]{a+b x^2} \left (1-\sqrt [3]{\frac {a}{a+b x^2}}\right ) \sqrt {\frac {1+\sqrt [3]{\frac {a}{a+b x^2}}+\left (\frac {a}{a+b x^2}\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}}{1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}}\right ),-7+4 \sqrt {3}\right )}{4 b^2 x \sqrt [3]{\frac {a}{a+b x^2}} \sqrt {-\frac {1-\sqrt [3]{\frac {a}{a+b x^2}}}{\left (1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}\right )^2}}} \]
3/4*x*(b*x^2+a)^(1/6)/b-3/4*3^(3/4)*a*(b*x^2+a)^(1/6)*(1-(a/(b*x^2+a))^(1/ 3))*EllipticF((1-(a/(b*x^2+a))^(1/3)+3^(1/2))/(1-(a/(b*x^2+a))^(1/3)-3^(1/ 2)),2*I-I*3^(1/2))*(1/2*6^(1/2)-1/2*2^(1/2))*((1+(a/(b*x^2+a))^(1/3)+(a/(b *x^2+a))^(2/3))/(1-(a/(b*x^2+a))^(1/3)-3^(1/2))^2)^(1/2)/b^2/x/(a/(b*x^2+a ))^(1/3)/((-1+(a/(b*x^2+a))^(1/3))/(1-(a/(b*x^2+a))^(1/3)-3^(1/2))^2)^(1/2 )
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 7.25 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.22 \[ \int \frac {x^2}{\left (a+b x^2\right )^{5/6}} \, dx=\frac {3 x \left (a+b x^2-a \left (1+\frac {b x^2}{a}\right )^{5/6} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{6},\frac {3}{2},-\frac {b x^2}{a}\right )\right )}{4 b \left (a+b x^2\right )^{5/6}} \]
(3*x*(a + b*x^2 - a*(1 + (b*x^2)/a)^(5/6)*Hypergeometric2F1[1/2, 5/6, 3/2, -((b*x^2)/a)]))/(4*b*(a + b*x^2)^(5/6))
Time = 0.26 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.29, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {262, 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.
\(\displaystyle \int \frac {x^2}{\left (a+b x^2\right )^{5/6}} \, dx\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {3 x \sqrt [6]{a+b x^2}}{4 b}-\frac {3 a \int \frac {1}{\left (b x^2+a\right )^{5/6}}dx}{4 b}\) |
\(\Big \downarrow \) 236 |
\(\displaystyle \frac {3 x \sqrt [6]{a+b x^2}}{4 b}-\frac {3 a \int \frac {1}{\left (1-\frac {b x^2}{b x^2+a}\right )^{2/3}}d\frac {x}{\sqrt {b x^2+a}}}{4 b \sqrt [3]{\frac {a}{a+b x^2}} \sqrt [3]{a+b x^2}}\) |
\(\Big \downarrow \) 234 |
\(\displaystyle \frac {9 a \sqrt {-\frac {b x^2}{a+b x^2}} \sqrt [6]{a+b x^2} \int \frac {1}{\sqrt {\frac {x^3}{\left (b x^2+a\right )^{3/2}}-1}}d\sqrt [3]{1-\frac {b x^2}{b x^2+a}}}{8 b^2 x \sqrt [3]{\frac {a}{a+b x^2}}}+\frac {3 x \sqrt [6]{a+b x^2}}{4 b}\) |
\(\Big \downarrow \) 760 |
\(\displaystyle \frac {3 x \sqrt [6]{a+b x^2}}{4 b}-\frac {3\ 3^{3/4} \sqrt {2-\sqrt {3}} a \sqrt {-\frac {b x^2}{a+b x^2}} \sqrt [6]{a+b x^2} \left (1-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}\right ) \sqrt {\frac {\frac {x^2}{a+b x^2}+\sqrt [3]{1-\frac {b x^2}{a+b x^2}}+1}{\left (-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{4 b^2 x \sqrt [3]{\frac {a}{a+b x^2}} \sqrt {\frac {x^3}{\left (a+b x^2\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}}{\left (-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}-\sqrt {3}+1\right )^2}}}\) |
(3*x*(a + b*x^2)^(1/6))/(4*b) - (3*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^2*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)])
3.11.27.3.1 Defintions of rubi rules used
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[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, 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]
\[\int \frac {x^{2}}{\left (b \,x^{2}+a \right )^{\frac {5}{6}}}d x\]
\[ \int \frac {x^2}{\left (a+b x^2\right )^{5/6}} \, dx=\int { \frac {x^{2}}{{\left (b x^{2} + a\right )}^{\frac {5}{6}}} \,d x } \]
Time = 0.50 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.10 \[ \int \frac {x^2}{\left (a+b x^2\right )^{5/6}} \, dx=\frac {x^{3} {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{6}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{3 a^{\frac {5}{6}}} \]
\[ \int \frac {x^2}{\left (a+b x^2\right )^{5/6}} \, dx=\int { \frac {x^{2}}{{\left (b x^{2} + a\right )}^{\frac {5}{6}}} \,d x } \]
\[ \int \frac {x^2}{\left (a+b x^2\right )^{5/6}} \, dx=\int { \frac {x^{2}}{{\left (b x^{2} + a\right )}^{\frac {5}{6}}} \,d x } \]
Timed out. \[ \int \frac {x^2}{\left (a+b x^2\right )^{5/6}} \, dx=\int \frac {x^2}{{\left (b\,x^2+a\right )}^{5/6}} \,d x \]