Integrand size = 15, antiderivative size = 577 \[ \int \frac {x^4}{\left (a+b x^2\right )^{4/3}} \, dx=-\frac {3 x^3}{2 b \sqrt [3]{a+b x^2}}+\frac {27 x \left (a+b x^2\right )^{2/3}}{14 b^2}+\frac {81 a x}{14 b^2 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}-\frac {81 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^{4/3} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right )|-7+4 \sqrt {3}\right )}{28 b^3 x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}+\frac {27\ 3^{3/4} a^{4/3} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right ),-7+4 \sqrt {3}\right )}{7 \sqrt {2} b^3 x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}} \]
-3/2*x^3/b/(b*x^2+a)^(1/3)+27/14*x*(b*x^2+a)^(2/3)/b^2+81/14*a*x/b^2/(-(b* x^2+a)^(1/3)+a^(1/3)*(1-3^(1/2)))+27/14*3^(3/4)*a^(4/3)*(a^(1/3)-(b*x^2+a) ^(1/3))*EllipticF((-(b*x^2+a)^(1/3)+a^(1/3)*(1+3^(1/2)))/(-(b*x^2+a)^(1/3) +a^(1/3)*(1-3^(1/2))),2*I-I*3^(1/2))*((a^(2/3)+a^(1/3)*(b*x^2+a)^(1/3)+(b* x^2+a)^(2/3))/(-(b*x^2+a)^(1/3)+a^(1/3)*(1-3^(1/2)))^2)^(1/2)/b^3/x*2^(1/2 )/(-a^(1/3)*(a^(1/3)-(b*x^2+a)^(1/3))/(-(b*x^2+a)^(1/3)+a^(1/3)*(1-3^(1/2) ))^2)^(1/2)-81/28*3^(1/4)*a^(4/3)*(a^(1/3)-(b*x^2+a)^(1/3))*EllipticE((-(b *x^2+a)^(1/3)+a^(1/3)*(1+3^(1/2)))/(-(b*x^2+a)^(1/3)+a^(1/3)*(1-3^(1/2))), 2*I-I*3^(1/2))*((a^(2/3)+a^(1/3)*(b*x^2+a)^(1/3)+(b*x^2+a)^(2/3))/(-(b*x^2 +a)^(1/3)+a^(1/3)*(1-3^(1/2)))^2)^(1/2)*(1/2*6^(1/2)+1/2*2^(1/2))/b^3/x/(- a^(1/3)*(a^(1/3)-(b*x^2+a)^(1/3))/(-(b*x^2+a)^(1/3)+a^(1/3)*(1-3^(1/2)))^2 )^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 5.30 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.11 \[ \int \frac {x^4}{\left (a+b x^2\right )^{4/3}} \, dx=\frac {3 x \left (9 a+2 b x^2-9 a \sqrt [3]{1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},-\frac {b x^2}{a}\right )\right )}{14 b^2 \sqrt [3]{a+b x^2}} \]
(3*x*(9*a + 2*b*x^2 - 9*a*(1 + (b*x^2)/a)^(1/3)*Hypergeometric2F1[1/3, 1/2 , 3/2, -((b*x^2)/a)]))/(14*b^2*(a + b*x^2)^(1/3))
Time = 0.48 (sec) , antiderivative size = 624, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {252, 262, 233, 833, 760, 2418}
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.
\(\displaystyle \int \frac {x^4}{\left (a+b x^2\right )^{4/3}} \, dx\) |
\(\Big \downarrow \) 252 |
\(\displaystyle \frac {9 \int \frac {x^2}{\sqrt [3]{b x^2+a}}dx}{2 b}-\frac {3 x^3}{2 b \sqrt [3]{a+b x^2}}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {9 \left (\frac {3 x \left (a+b x^2\right )^{2/3}}{7 b}-\frac {3 a \int \frac {1}{\sqrt [3]{b x^2+a}}dx}{7 b}\right )}{2 b}-\frac {3 x^3}{2 b \sqrt [3]{a+b x^2}}\) |
\(\Big \downarrow \) 233 |
\(\displaystyle \frac {9 \left (\frac {3 x \left (a+b x^2\right )^{2/3}}{7 b}-\frac {9 a \sqrt {b x^2} \int \frac {\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}}{14 b^2 x}\right )}{2 b}-\frac {3 x^3}{2 b \sqrt [3]{a+b x^2}}\) |
\(\Big \downarrow \) 833 |
\(\displaystyle \frac {9 \left (\frac {3 x \left (a+b x^2\right )^{2/3}}{7 b}-\frac {9 a \sqrt {b x^2} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} \int \frac {1}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}\right )}{14 b^2 x}\right )}{2 b}-\frac {3 x^3}{2 b \sqrt [3]{a+b x^2}}\) |
\(\Big \downarrow \) 760 |
\(\displaystyle \frac {9 \left (\frac {3 x \left (a+b x^2\right )^{2/3}}{7 b}-\frac {9 a \sqrt {b x^2} \left (-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}\right )}{14 b^2 x}\right )}{2 b}-\frac {3 x^3}{2 b \sqrt [3]{a+b x^2}}\) |
\(\Big \downarrow \) 2418 |
\(\displaystyle \frac {9 \left (\frac {3 x \left (a+b x^2\right )^{2/3}}{7 b}-\frac {9 a \sqrt {b x^2} \left (-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right )|-7+4 \sqrt {3}\right )}{\sqrt {b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}-\frac {2 \sqrt {b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right )}{14 b^2 x}\right )}{2 b}-\frac {3 x^3}{2 b \sqrt [3]{a+b x^2}}\) |
(-3*x^3)/(2*b*(a + b*x^2)^(1/3)) + (9*((3*x*(a + b*x^2)^(2/3))/(7*b) - (9* a*Sqrt[b*x^2]*((-2*Sqrt[b*x^2])/((1 - Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3) ) + (3^(1/4)*Sqrt[2 + Sqrt[3]]*a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3))*Sqrt[ (a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - Sqrt[3])*a ^(1/3) - (a + b*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*a^(1/3) - ( a + b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))], -7 + 4*Sqr t[3]])/(Sqrt[b*x^2]*Sqrt[-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - S qrt[3])*a^(1/3) - (a + b*x^2)^(1/3))^2)]) - (2*Sqrt[2 - Sqrt[3]]*(1 + Sqrt [3])*a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a + b* x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3) )^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))/((1 - Sq rt[3])*a^(1/3) - (a + b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(3^(1/4)*Sqrt[b*x^2 ]*Sqrt[-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))^2)])))/(14*b^2*x)))/(2*b)
3.8.33.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1/3), x_Symbol] :> Simp[3*(Sqrt[b*x^2]/(2*b*x)) Subst[Int[x/Sqrt[-a + x^3], x], x, (a + b*x^2)^(1/3)], 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)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, 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[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3] ], s = Denom[Rt[b/a, 3]]}, Simp[(-(1 + Sqrt[3]))*(s/r) Int[1/Sqrt[a + b*x ^3], x], x] + Simp[1/r Int[((1 + Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x ]] /; FreeQ[{a, b}, x] && NegQ[a]
Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N umer[Simplify[(1 + Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 + Sqrt[3])*(d/c) ]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 - Sqrt[3])*s + r*x))), x] + S imp[3^(1/4)*Sqrt[2 + Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( (1 - Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[(-s)*((s + r*x)/((1 - S qrt[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 + Sqrt[3])*s + r*x)/((1 - Sqrt[ 3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && EqQ[b*c^3 - 2*(5 + 3*Sqrt[3])*a*d^3, 0]
\[\int \frac {x^{4}}{\left (b \,x^{2}+a \right )^{\frac {4}{3}}}d x\]
\[ \int \frac {x^4}{\left (a+b x^2\right )^{4/3}} \, dx=\int { \frac {x^{4}}{{\left (b x^{2} + a\right )}^{\frac {4}{3}}} \,d x } \]
Time = 0.52 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.05 \[ \int \frac {x^4}{\left (a+b x^2\right )^{4/3}} \, dx=\frac {x^{5} {{}_{2}F_{1}\left (\begin {matrix} \frac {4}{3}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{5 a^{\frac {4}{3}}} \]
\[ \int \frac {x^4}{\left (a+b x^2\right )^{4/3}} \, dx=\int { \frac {x^{4}}{{\left (b x^{2} + a\right )}^{\frac {4}{3}}} \,d x } \]
\[ \int \frac {x^4}{\left (a+b x^2\right )^{4/3}} \, dx=\int { \frac {x^{4}}{{\left (b x^{2} + a\right )}^{\frac {4}{3}}} \,d x } \]
Timed out. \[ \int \frac {x^4}{\left (a+b x^2\right )^{4/3}} \, dx=\int \frac {x^4}{{\left (b\,x^2+a\right )}^{4/3}} \,d x \]