\(\int \frac {1}{x^6 (a+b x^2)^{5/6}} \, dx\) [1111]

Optimal result

Integrand size = 15, antiderivative size = 319 \[ \int \frac {1}{x^6 \left (a+b x^2\right )^{5/6}} \, dx=-\frac {\sqrt [6]{a+b x^2}}{5 a x^5}+\frac {14 b \sqrt [6]{a+b x^2}}{45 a^2 x^3}-\frac {112 b^2 \sqrt [6]{a+b x^2}}{135 a^3 x}-\frac {112 b^2 \sqrt [6]{a+b x^2} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a}-\left (1-\sqrt {3}\right ) \sqrt [3]{a+b x^2}}{\sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+b x^2}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{135 \sqrt [4]{3} a^{10/3} x \sqrt {-\frac {\sqrt [3]{a+b x^2} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+b x^2}\right )^2}}} \] Output:

-1/5*(b*x^2+a)^(1/6)/a/x^5+14/45*b*(b*x^2+a)^(1/6)/a^2/x^3-112/135*b^2*(b* 
x^2+a)^(1/6)/a^3/x-112/405*b^2*(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))* 
3^(3/4)/a^(10/3)/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)
 

Mathematica [C] (verified)

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 = 0.16 \[ \int \frac {1}{x^6 \left (a+b x^2\right )^{5/6}} \, dx=-\frac {\left (1+\frac {b x^2}{a}\right )^{5/6} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {5}{6},-\frac {3}{2},-\frac {b x^2}{a}\right )}{5 x^5 \left (a+b x^2\right )^{5/6}} \] Input:

Integrate[1/(x^6*(a + b*x^2)^(5/6)),x]
 

Output:

-1/5*((1 + (b*x^2)/a)^(5/6)*Hypergeometric2F1[-5/2, 5/6, -3/2, -((b*x^2)/a 
)])/(x^5*(a + b*x^2)^(5/6))
 

Rubi [A] (warning: unable to verify)

Time = 0.31 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.29, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {264, 264, 264, 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.

\(\displaystyle \int \frac {1}{x^6 \left (a+b x^2\right )^{5/6}} \, dx\)

\(\Big \downarrow \) 264

\(\displaystyle -\frac {14 b \int \frac {1}{x^4 \left (b x^2+a\right )^{5/6}}dx}{15 a}-\frac {\sqrt [6]{a+b x^2}}{5 a x^5}\)

\(\Big \downarrow \) 264

\(\displaystyle -\frac {14 b \left (-\frac {8 b \int \frac {1}{x^2 \left (b x^2+a\right )^{5/6}}dx}{9 a}-\frac {\sqrt [6]{a+b x^2}}{3 a x^3}\right )}{15 a}-\frac {\sqrt [6]{a+b x^2}}{5 a x^5}\)

\(\Big \downarrow \) 264

\(\displaystyle -\frac {14 b \left (-\frac {8 b \left (-\frac {2 b \int \frac {1}{\left (b x^2+a\right )^{5/6}}dx}{3 a}-\frac {\sqrt [6]{a+b x^2}}{a x}\right )}{9 a}-\frac {\sqrt [6]{a+b x^2}}{3 a x^3}\right )}{15 a}-\frac {\sqrt [6]{a+b x^2}}{5 a x^5}\)

\(\Big \downarrow \) 236

\(\displaystyle -\frac {14 b \left (-\frac {8 b \left (-\frac {2 b \int \frac {1}{\left (1-\frac {b x^2}{b x^2+a}\right )^{2/3}}d\frac {x}{\sqrt {b x^2+a}}}{3 a \sqrt [3]{\frac {a}{a+b x^2}} \sqrt [3]{a+b x^2}}-\frac {\sqrt [6]{a+b x^2}}{a x}\right )}{9 a}-\frac {\sqrt [6]{a+b x^2}}{3 a x^3}\right )}{15 a}-\frac {\sqrt [6]{a+b x^2}}{5 a x^5}\)

\(\Big \downarrow \) 234

\(\displaystyle -\frac {14 b \left (-\frac {8 b \left (\frac {\sqrt {-\frac {b x^2}{a+b x^2}} \sqrt [6]{a+b x^2} \int \frac {1}{\sqrt {\frac {x^3}{\left (b x^2+a\right )^{3/2}}-1}}d\sqrt [3]{1-\frac {b x^2}{b x^2+a}}}{a x \sqrt [3]{\frac {a}{a+b x^2}}}-\frac {\sqrt [6]{a+b x^2}}{a x}\right )}{9 a}-\frac {\sqrt [6]{a+b x^2}}{3 a x^3}\right )}{15 a}-\frac {\sqrt [6]{a+b x^2}}{5 a x^5}\)

\(\Big \downarrow \) 760

\(\displaystyle -\frac {14 b \left (-\frac {8 b \left (-\frac {2 \sqrt {2-\sqrt {3}} \sqrt {-\frac {b x^2}{a+b x^2}} \sqrt [6]{a+b x^2} \left (1-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}\right ) \sqrt {\frac {\frac {x^2}{a+b x^2}+\sqrt [3]{1-\frac {b x^2}{a+b x^2}}+1}{\left (-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} a x \sqrt [3]{\frac {a}{a+b x^2}} \sqrt {\frac {x^3}{\left (a+b x^2\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}}{\left (-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}-\sqrt {3}+1\right )^2}}}-\frac {\sqrt [6]{a+b x^2}}{a x}\right )}{9 a}-\frac {\sqrt [6]{a+b x^2}}{3 a x^3}\right )}{15 a}-\frac {\sqrt [6]{a+b x^2}}{5 a x^5}\)

Input:

Int[1/(x^6*(a + b*x^2)^(5/6)),x]
 

Output:

-1/5*(a + b*x^2)^(1/6)/(a*x^5) - (14*b*(-1/3*(a + b*x^2)^(1/6)/(a*x^3) - ( 
8*b*(-((a + b*x^2)^(1/6)/(a*x)) - (2*Sqrt[2 - Sqrt[3]]*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]])/(3^(1/4)*a*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)])))/(9*a)))/(15*a)
 

Defintions of rubi rules used

rule 234

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]
 

rule 236

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]
 

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]
 

Maple [F]

\[\int \frac {1}{x^{6} \left (b \,x^{2}+a \right )^{\frac {5}{6}}}d x\]

Input:

int(1/x^6/(b*x^2+a)^(5/6),x)
 

Output:

int(1/x^6/(b*x^2+a)^(5/6),x)
 

Fricas [F]

\[ \int \frac {1}{x^6 \left (a+b x^2\right )^{5/6}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {5}{6}} x^{6}} \,d x } \] Input:

integrate(1/x^6/(b*x^2+a)^(5/6),x, algorithm="fricas")
 

Output:

integral((b*x^2 + a)^(1/6)/(b*x^8 + a*x^6), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.71 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.10 \[ \int \frac {1}{x^6 \left (a+b x^2\right )^{5/6}} \, dx=- \frac {{{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{2}, \frac {5}{6} \\ - \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{5 a^{\frac {5}{6}} x^{5}} \] Input:

integrate(1/x**6/(b*x**2+a)**(5/6),x)
 

Output:

-hyper((-5/2, 5/6), (-3/2,), b*x**2*exp_polar(I*pi)/a)/(5*a**(5/6)*x**5)
 

Maxima [F]

\[ \int \frac {1}{x^6 \left (a+b x^2\right )^{5/6}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {5}{6}} x^{6}} \,d x } \] Input:

integrate(1/x^6/(b*x^2+a)^(5/6),x, algorithm="maxima")
 

Output:

integrate(1/((b*x^2 + a)^(5/6)*x^6), x)
 

Giac [F]

\[ \int \frac {1}{x^6 \left (a+b x^2\right )^{5/6}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {5}{6}} x^{6}} \,d x } \] Input:

integrate(1/x^6/(b*x^2+a)^(5/6),x, algorithm="giac")
 

Output:

integrate(1/((b*x^2 + a)^(5/6)*x^6), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^6 \left (a+b x^2\right )^{5/6}} \, dx=\int \frac {1}{x^6\,{\left (b\,x^2+a\right )}^{5/6}} \,d x \] Input:

int(1/(x^6*(a + b*x^2)^(5/6)),x)
 

Output:

int(1/(x^6*(a + b*x^2)^(5/6)), x)
 

Reduce [F]

\[ \int \frac {1}{x^6 \left (a+b x^2\right )^{5/6}} \, dx=\frac {-\left (b \,x^{2}+a \right )^{\frac {5}{6}}-6 \left (b \,x^{2}+a \right )^{\frac {2}{3}} \left (\int \frac {\left (b \,x^{2}+a \right )^{\frac {7}{6}}}{b^{2} x^{8}+2 a b \,x^{6}+a^{2} x^{4}}d x \right ) b \,x^{5}}{5 \left (b \,x^{2}+a \right )^{\frac {2}{3}} a \,x^{5}} \] Input:

int(1/x^6/(b*x^2+a)^(5/6),x)
 

Output:

( - (a + b*x**2)**(5/6) - 6*(a + b*x**2)**(2/3)*int((a + b*x**2)**(7/6)/(a 
**2*x**4 + 2*a*b*x**6 + b**2*x**8),x)*b*x**5)/(5*(a + b*x**2)**(2/3)*a*x** 
5)