Integrand size = 15, antiderivative size = 578 \[ \int \frac {1}{x^4 \sqrt [3]{a+b x^2}} \, dx=-\frac {\left (a+b x^2\right )^{2/3}}{3 a x^3}+\frac {5 b \left (a+b x^2\right )^{2/3}}{9 a^2 x}+\frac {5 b^2 x}{9 a^2 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}-\frac {5 \sqrt {2+\sqrt {3}} b \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right )|-7+4 \sqrt {3}\right )}{6\ 3^{3/4} a^{5/3} x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}+\frac {5 \sqrt {2} b \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right ),-7+4 \sqrt {3}\right )}{9 \sqrt [4]{3} a^{5/3} x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}} \] Output:
-1/3*(b*x^2+a)^(2/3)/a/x^3+5/9*b*(b*x^2+a)^(2/3)/a^2/x+5/9*b^2*x/a^2/((1-3 ^(1/2))*a^(1/3)-(b*x^2+a)^(1/3))-5/18*(1/2*6^(1/2)+1/2*2^(1/2))*b*(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))/((1-3 ^(1/2))*a^(1/3)-(b*x^2+a)^(1/3))^2)^(1/2)*EllipticE(((1+3^(1/2))*a^(1/3)-( b*x^2+a)^(1/3))/((1-3^(1/2))*a^(1/3)-(b*x^2+a)^(1/3)),2*I-I*3^(1/2))*3^(1/ 4)/a^(5/3)/x/(-a^(1/3)*(a^(1/3)-(b*x^2+a)^(1/3))/((1-3^(1/2))*a^(1/3)-(b*x ^2+a)^(1/3))^2)^(1/2)+5/27*2^(1/2)*b*(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))/((1-3^(1/2))*a^(1/3)-(b*x^2+a)^(1/ 3))^2)^(1/2)*EllipticF(((1+3^(1/2))*a^(1/3)-(b*x^2+a)^(1/3))/((1-3^(1/2))* a^(1/3)-(b*x^2+a)^(1/3)),2*I-I*3^(1/2))*3^(3/4)/a^(5/3)/x/(-a^(1/3)*(a^(1/ 3)-(b*x^2+a)^(1/3))/((1-3^(1/2))*a^(1/3)-(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 = 10.01 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.09 \[ \int \frac {1}{x^4 \sqrt [3]{a+b x^2}} \, dx=-\frac {\sqrt [3]{1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{3},-\frac {1}{2},-\frac {b x^2}{a}\right )}{3 x^3 \sqrt [3]{a+b x^2}} \] Input:
Integrate[1/(x^4*(a + b*x^2)^(1/3)),x]
Output:
-1/3*((1 + (b*x^2)/a)^(1/3)*Hypergeometric2F1[-3/2, 1/3, -1/2, -((b*x^2)/a )])/(x^3*(a + b*x^2)^(1/3))
Time = 0.50 (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 = {264, 264, 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.
\(\displaystyle \int \frac {1}{x^4 \sqrt [3]{a+b x^2}} \, dx\) |
\(\Big \downarrow \) 264 |
\(\displaystyle -\frac {5 b \int \frac {1}{x^2 \sqrt [3]{b x^2+a}}dx}{9 a}-\frac {\left (a+b x^2\right )^{2/3}}{3 a x^3}\) |
\(\Big \downarrow \) 264 |
\(\displaystyle -\frac {5 b \left (\frac {b \int \frac {1}{\sqrt [3]{b x^2+a}}dx}{3 a}-\frac {\left (a+b x^2\right )^{2/3}}{a x}\right )}{9 a}-\frac {\left (a+b x^2\right )^{2/3}}{3 a x^3}\) |
\(\Big \downarrow \) 233 |
\(\displaystyle -\frac {5 b \left (\frac {\sqrt {b x^2} \int \frac {\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}}{2 a x}-\frac {\left (a+b x^2\right )^{2/3}}{a x}\right )}{9 a}-\frac {\left (a+b x^2\right )^{2/3}}{3 a x^3}\) |
\(\Big \downarrow \) 833 |
\(\displaystyle -\frac {5 b \left (\frac {\sqrt {b x^2} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} \int \frac {1}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}\right )}{2 a x}-\frac {\left (a+b x^2\right )^{2/3}}{a x}\right )}{9 a}-\frac {\left (a+b x^2\right )^{2/3}}{3 a x^3}\) |
\(\Big \downarrow \) 760 |
\(\displaystyle -\frac {5 b \left (\frac {\sqrt {b x^2} \left (-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}\right )}{2 a x}-\frac {\left (a+b x^2\right )^{2/3}}{a x}\right )}{9 a}-\frac {\left (a+b x^2\right )^{2/3}}{3 a x^3}\) |
\(\Big \downarrow \) 2418 |
\(\displaystyle -\frac {5 b \left (\frac {\sqrt {b x^2} \left (-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right )|-7+4 \sqrt {3}\right )}{\sqrt {b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}-\frac {2 \sqrt {b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right )}{2 a x}-\frac {\left (a+b x^2\right )^{2/3}}{a x}\right )}{9 a}-\frac {\left (a+b x^2\right )^{2/3}}{3 a x^3}\) |
Input:
Int[1/(x^4*(a + b*x^2)^(1/3)),x]
Output:
-1/3*(a + b*x^2)^(2/3)/(a*x^3) - (5*b*(-((a + b*x^2)^(2/3)/(a*x)) + (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*Sqrt[3]])/ (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)]) - (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]*El lipticF[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*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)])))/(2*a*x)))/(9*a)
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*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && 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]
\[\int \frac {1}{x^{4} \left (b \,x^{2}+a \right )^{\frac {1}{3}}}d x\]
Input:
int(1/x^4/(b*x^2+a)^(1/3),x)
Output:
int(1/x^4/(b*x^2+a)^(1/3),x)
\[ \int \frac {1}{x^4 \sqrt [3]{a+b x^2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {1}{3}} x^{4}} \,d x } \] Input:
integrate(1/x^4/(b*x^2+a)^(1/3),x, algorithm="fricas")
Output:
integral((b*x^2 + a)^(2/3)/(b*x^6 + a*x^4), x)
Time = 0.54 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.06 \[ \int \frac {1}{x^4 \sqrt [3]{a+b x^2}} \, dx=- \frac {{{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {1}{3} \\ - \frac {1}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{3 \sqrt [3]{a} x^{3}} \] Input:
integrate(1/x**4/(b*x**2+a)**(1/3),x)
Output:
-hyper((-3/2, 1/3), (-1/2,), b*x**2*exp_polar(I*pi)/a)/(3*a**(1/3)*x**3)
\[ \int \frac {1}{x^4 \sqrt [3]{a+b x^2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {1}{3}} x^{4}} \,d x } \] Input:
integrate(1/x^4/(b*x^2+a)^(1/3),x, algorithm="maxima")
Output:
integrate(1/((b*x^2 + a)^(1/3)*x^4), x)
\[ \int \frac {1}{x^4 \sqrt [3]{a+b x^2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {1}{3}} x^{4}} \,d x } \] Input:
integrate(1/x^4/(b*x^2+a)^(1/3),x, algorithm="giac")
Output:
integrate(1/((b*x^2 + a)^(1/3)*x^4), x)
Timed out. \[ \int \frac {1}{x^4 \sqrt [3]{a+b x^2}} \, dx=\int \frac {1}{x^4\,{\left (b\,x^2+a\right )}^{1/3}} \,d x \] Input:
int(1/(x^4*(a + b*x^2)^(1/3)),x)
Output:
int(1/(x^4*(a + b*x^2)^(1/3)), x)
\[ \int \frac {1}{x^4 \sqrt [3]{a+b x^2}} \, dx=\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {1}{3}} x^{4}}d x \] Input:
int(1/x^4/(b*x^2+a)^(1/3),x)
Output:
int(1/((a + b*x**2)**(1/3)*x**4),x)