\(\int \frac {1}{x^4 (a+b x^2)^{4/3}} \, dx\) [785]

Optimal result

Integrand size = 15, antiderivative size = 599 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^{4/3}} \, dx=\frac {3}{2 a x^3 \sqrt [3]{a+b x^2}}-\frac {11 \left (a+b x^2\right )^{2/3}}{6 a^2 x^3}+\frac {55 b \left (a+b x^2\right )^{2/3}}{18 a^3 x}+\frac {55 b^2 x}{18 a^3 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}-\frac {55 \sqrt {2+\sqrt {3}} b \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right )|-7+4 \sqrt {3}\right )}{12\ 3^{3/4} a^{8/3} x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}+\frac {55 b \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right ),-7+4 \sqrt {3}\right )}{9 \sqrt {2} \sqrt [4]{3} a^{8/3} x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}} \] Output:

3/2/a/x^3/(b*x^2+a)^(1/3)-11/6*(b*x^2+a)^(2/3)/a^2/x^3+55/18*b*(b*x^2+a)^( 
2/3)/a^3/x+55/18*b^2*x/a^3/((1-3^(1/2))*a^(1/3)-(b*x^2+a)^(1/3))-55/36*(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^(8/3)/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)+55/54*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))*2^(1/2)* 
3^(3/4)/a^(8/3)/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)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.09 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^{4/3}} \, dx=-\frac {\sqrt [3]{1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {4}{3},-\frac {1}{2},-\frac {b x^2}{a}\right )}{3 a x^3 \sqrt [3]{a+b x^2}} \] Input:

Integrate[1/(x^4*(a + b*x^2)^(4/3)),x]
 

Output:

-1/3*((1 + (b*x^2)/a)^(1/3)*Hypergeometric2F1[-3/2, 4/3, -1/2, -((b*x^2)/a 
)])/(a*x^3*(a + b*x^2)^(1/3))
 

Rubi [A] (warning: unable to verify)

Time = 0.52 (sec) , antiderivative size = 653, normalized size of antiderivative = 1.09, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {253, 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.

\(\displaystyle \int \frac {1}{x^4 \left (a+b x^2\right )^{4/3}} \, dx\)

\(\Big \downarrow \) 253

\(\displaystyle \frac {11 \int \frac {1}{x^4 \sqrt [3]{b x^2+a}}dx}{2 a}+\frac {3}{2 a x^3 \sqrt [3]{a+b x^2}}\)

\(\Big \downarrow \) 264

\(\displaystyle \frac {11 \left (-\frac {5 b \int \frac {1}{x^2 \sqrt [3]{b x^2+a}}dx}{9 a}-\frac {\left (a+b x^2\right )^{2/3}}{3 a x^3}\right )}{2 a}+\frac {3}{2 a x^3 \sqrt [3]{a+b x^2}}\)

\(\Big \downarrow \) 264

\(\displaystyle \frac {11 \left (-\frac {5 b \left (\frac {b \int \frac {1}{\sqrt [3]{b x^2+a}}dx}{3 a}-\frac {\left (a+b x^2\right )^{2/3}}{a x}\right )}{9 a}-\frac {\left (a+b x^2\right )^{2/3}}{3 a x^3}\right )}{2 a}+\frac {3}{2 a x^3 \sqrt [3]{a+b x^2}}\)

\(\Big \downarrow \) 233

\(\displaystyle \frac {11 \left (-\frac {5 b \left (\frac {\sqrt {b x^2} \int \frac {\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}}{2 a x}-\frac {\left (a+b x^2\right )^{2/3}}{a x}\right )}{9 a}-\frac {\left (a+b x^2\right )^{2/3}}{3 a x^3}\right )}{2 a}+\frac {3}{2 a x^3 \sqrt [3]{a+b x^2}}\)

\(\Big \downarrow \) 833

\(\displaystyle \frac {11 \left (-\frac {5 b \left (\frac {\sqrt {b x^2} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} \int \frac {1}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}\right )}{2 a x}-\frac {\left (a+b x^2\right )^{2/3}}{a x}\right )}{9 a}-\frac {\left (a+b x^2\right )^{2/3}}{3 a x^3}\right )}{2 a}+\frac {3}{2 a x^3 \sqrt [3]{a+b x^2}}\)

\(\Big \downarrow \) 760

\(\displaystyle \frac {11 \left (-\frac {5 b \left (\frac {\sqrt {b x^2} \left (-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}\right )}{2 a x}-\frac {\left (a+b x^2\right )^{2/3}}{a x}\right )}{9 a}-\frac {\left (a+b x^2\right )^{2/3}}{3 a x^3}\right )}{2 a}+\frac {3}{2 a x^3 \sqrt [3]{a+b x^2}}\)

\(\Big \downarrow \) 2418

\(\displaystyle \frac {11 \left (-\frac {5 b \left (\frac {\sqrt {b x^2} \left (-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right )|-7+4 \sqrt {3}\right )}{\sqrt {b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}-\frac {2 \sqrt {b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right )}{2 a x}-\frac {\left (a+b x^2\right )^{2/3}}{a x}\right )}{9 a}-\frac {\left (a+b x^2\right )^{2/3}}{3 a x^3}\right )}{2 a}+\frac {3}{2 a x^3 \sqrt [3]{a+b x^2}}\)

Input:

Int[1/(x^4*(a + b*x^2)^(4/3)),x]
 

Output:

3/(2*a*x^3*(a + b*x^2)^(1/3)) + (11*(-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[Arc 
Sin[((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))*S 
qrt[(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 - 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)))/ 
(2*a)
 

Defintions of rubi rules used

rule 233

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]
 

rule 253

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x 
)^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 
2*a*(p + 1))   Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m 
}, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
 

rule 264

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]
 

rule 760

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]
 

rule 833

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]
 

rule 2418

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]
 

Maple [F]

\[\int \frac {1}{x^{4} \left (b \,x^{2}+a \right )^{\frac {4}{3}}}d x\]

Input:

int(1/x^4/(b*x^2+a)^(4/3),x)
 

Output:

int(1/x^4/(b*x^2+a)^(4/3),x)
 

Fricas [F]

\[ \int \frac {1}{x^4 \left (a+b x^2\right )^{4/3}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {4}{3}} x^{4}} \,d x } \] Input:

integrate(1/x^4/(b*x^2+a)^(4/3),x, algorithm="fricas")
 

Output:

integral((b*x^2 + a)^(2/3)/(b^2*x^8 + 2*a*b*x^6 + a^2*x^4), x)
 

Sympy [A] (verification not implemented)

Time = 0.62 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.05 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^{4/3}} \, dx=- \frac {{{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {4}{3} \\ - \frac {1}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{3 a^{\frac {4}{3}} x^{3}} \] Input:

integrate(1/x**4/(b*x**2+a)**(4/3),x)
 

Output:

-hyper((-3/2, 4/3), (-1/2,), b*x**2*exp_polar(I*pi)/a)/(3*a**(4/3)*x**3)
 

Maxima [F]

\[ \int \frac {1}{x^4 \left (a+b x^2\right )^{4/3}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {4}{3}} x^{4}} \,d x } \] Input:

integrate(1/x^4/(b*x^2+a)^(4/3),x, algorithm="maxima")
 

Output:

integrate(1/((b*x^2 + a)^(4/3)*x^4), x)
 

Giac [F]

\[ \int \frac {1}{x^4 \left (a+b x^2\right )^{4/3}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {4}{3}} x^{4}} \,d x } \] Input:

integrate(1/x^4/(b*x^2+a)^(4/3),x, algorithm="giac")
 

Output:

integrate(1/((b*x^2 + a)^(4/3)*x^4), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^4 \left (a+b x^2\right )^{4/3}} \, dx=\int \frac {1}{x^4\,{\left (b\,x^2+a\right )}^{4/3}} \,d x \] Input:

int(1/(x^4*(a + b*x^2)^(4/3)),x)
 

Output:

int(1/(x^4*(a + b*x^2)^(4/3)), x)
 

Reduce [F]

\[ \int \frac {1}{x^4 \left (a+b x^2\right )^{4/3}} \, dx=\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {1}{3}} a \,x^{4}+\left (b \,x^{2}+a \right )^{\frac {1}{3}} b \,x^{6}}d x \] Input:

int(1/x^4/(b*x^2+a)^(4/3),x)
                                                                                    
                                                                                    
 

Output:

int(1/((a + b*x**2)**(1/3)*a*x**4 + (a + b*x**2)**(1/3)*b*x**6),x)