\(\int \frac {\sqrt [8]{-a+b x^2}}{x^8} \, dx\) [1200]

Optimal result

Integrand size = 17, antiderivative size = 543 \[ \int \frac {\sqrt [8]{-a+b x^2}}{x^8} \, dx=-\frac {\sqrt [8]{-a+b x^2}}{7 x^7}+\frac {b \sqrt [8]{-a+b x^2}}{140 a x^5}+\frac {19 b^2 \sqrt [8]{-a+b x^2}}{1680 a^2 x^3}+\frac {209 b^3 \sqrt [8]{-a+b x^2}}{6720 a^3 x}+\frac {209 b^3 \sqrt {-\frac {b x^2}{\sqrt {a} \sqrt {-a+b x^2}}} \left (-a+b x^2\right )^{3/8} \sqrt {\frac {\left (\sqrt [4]{a}+\sqrt [4]{-a+b x^2}\right )^2}{\sqrt [4]{a} \sqrt [4]{-a+b x^2}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \sqrt {-\frac {\sqrt [4]{a} \left (\sqrt {2}-\frac {2 \sqrt [4]{-a+b x^2}}{\sqrt [4]{a}}+\frac {\sqrt {2} \sqrt {-a+b x^2}}{\sqrt {a}}\right )}{\sqrt [4]{-a+b x^2}}}\right ),-2 \left (1-\sqrt {2}\right )\right )}{4480 \sqrt {2+\sqrt {2}} a^3 x \left (\sqrt [4]{a}+\sqrt [4]{-a+b x^2}\right )}-\frac {209 b^3 \sqrt {-\frac {b x^2}{\sqrt {a} \sqrt {-a+b x^2}}} \left (-a+b x^2\right )^{3/8} \sqrt {-\frac {\left (\sqrt [4]{a}-\sqrt [4]{-a+b x^2}\right )^2}{\sqrt [4]{a} \sqrt [4]{-a+b x^2}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \sqrt {\frac {\sqrt [4]{a} \left (\sqrt {2}+\frac {2 \sqrt [4]{-a+b x^2}}{\sqrt [4]{a}}+\frac {\sqrt {2} \sqrt {-a+b x^2}}{\sqrt {a}}\right )}{\sqrt [4]{-a+b x^2}}}\right ),-2 \left (1-\sqrt {2}\right )\right )}{4480 \sqrt {2+\sqrt {2}} a^3 x \left (\sqrt [4]{a}-\sqrt [4]{-a+b x^2}\right )} \] Output:

-1/7*(b*x^2-a)^(1/8)/x^7+1/140*b*(b*x^2-a)^(1/8)/a/x^5+19/1680*b^2*(b*x^2- 
a)^(1/8)/a^2/x^3+209/6720*b^3*(b*x^2-a)^(1/8)/a^3/x+209/4480*b^3*(-b*x^2/a 
^(1/2)/(b*x^2-a)^(1/2))^(1/2)*(b*x^2-a)^(3/8)*((a^(1/4)+(b*x^2-a)^(1/4))^2 
/a^(1/4)/(b*x^2-a)^(1/4))^(1/2)*EllipticF(1/2*(-a^(1/4)*(2^(1/2)-2*(b*x^2- 
a)^(1/4)/a^(1/4)+2^(1/2)*(b*x^2-a)^(1/2)/a^(1/2))/(b*x^2-a)^(1/4))^(1/2),( 
-2+2*2^(1/2))^(1/2))/(2+2^(1/2))^(1/2)/a^3/x/(a^(1/4)+(b*x^2-a)^(1/4))-209 
/4480*b^3*(-b*x^2/a^(1/2)/(b*x^2-a)^(1/2))^(1/2)*(b*x^2-a)^(3/8)*(-(a^(1/4 
)-(b*x^2-a)^(1/4))^2/a^(1/4)/(b*x^2-a)^(1/4))^(1/2)*EllipticF(1/2*(a^(1/4) 
*(2^(1/2)+2*(b*x^2-a)^(1/4)/a^(1/4)+2^(1/2)*(b*x^2-a)^(1/2)/a^(1/2))/(b*x^ 
2-a)^(1/4))^(1/2),(-2+2*2^(1/2))^(1/2))/(2+2^(1/2))^(1/2)/a^3/x/(a^(1/4)-( 
b*x^2-a)^(1/4))
 

Mathematica [C] (verified)

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

Time = 10.01 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.10 \[ \int \frac {\sqrt [8]{-a+b x^2}}{x^8} \, dx=-\frac {\sqrt [8]{-a+b x^2} \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},-\frac {1}{8},-\frac {5}{2},\frac {b x^2}{a}\right )}{7 x^7 \sqrt [8]{1-\frac {b x^2}{a}}} \] Input:

Integrate[(-a + b*x^2)^(1/8)/x^8,x]
 

Output:

-1/7*((-a + b*x^2)^(1/8)*Hypergeometric2F1[-7/2, -1/8, -5/2, (b*x^2)/a])/( 
x^7*(1 - (b*x^2)/a)^(1/8))
 

Rubi [C] (verified)

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

Time = 0.16 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.10, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {279, 278}

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.

\(\displaystyle \int \frac {\sqrt [8]{b x^2-a}}{x^8} \, dx\)

\(\Big \downarrow \) 279

\(\displaystyle \frac {\sqrt [8]{b x^2-a} \int \frac {\sqrt [8]{1-\frac {b x^2}{a}}}{x^8}dx}{\sqrt [8]{1-\frac {b x^2}{a}}}\)

\(\Big \downarrow \) 278

\(\displaystyle -\frac {\sqrt [8]{b x^2-a} \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},-\frac {1}{8},-\frac {5}{2},\frac {b x^2}{a}\right )}{7 x^7 \sqrt [8]{1-\frac {b x^2}{a}}}\)

Input:

Int[(-a + b*x^2)^(1/8)/x^8,x]
 

Output:

-1/7*((-a + b*x^2)^(1/8)*Hypergeometric2F1[-7/2, -1/8, -5/2, (b*x^2)/a])/( 
x^7*(1 - (b*x^2)/a)^(1/8))
 

Defintions of rubi rules used

rule 278

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( 
c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( 
-b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && (ILtQ[p, 0 
] || GtQ[a, 0])
 

rule 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntP 
art[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p])   Int[(c*x)^m* 
(1 + b*(x^2/a))^p, x], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && 
!(ILtQ[p, 0] || GtQ[a, 0])
 

Maple [F]

\[\int \frac {\left (b \,x^{2}-a \right )^{\frac {1}{8}}}{x^{8}}d x\]

Input:

int((b*x^2-a)^(1/8)/x^8,x)
 

Output:

int((b*x^2-a)^(1/8)/x^8,x)
 

Fricas [F]

\[ \int \frac {\sqrt [8]{-a+b x^2}}{x^8} \, dx=\int { \frac {{\left (b x^{2} - a\right )}^{\frac {1}{8}}}{x^{8}} \,d x } \] Input:

integrate((b*x^2-a)^(1/8)/x^8,x, algorithm="fricas")
 

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.84 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.07 \[ \int \frac {\sqrt [8]{-a+b x^2}}{x^8} \, dx=\frac {\sqrt [8]{a} e^{- \frac {7 i \pi }{8}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{2}, - \frac {1}{8} \\ - \frac {5}{2} \end {matrix}\middle | {\frac {b x^{2}}{a}} \right )}}{7 x^{7}} \] Input:

integrate((b*x**2-a)**(1/8)/x**8,x)
 

Output:

a**(1/8)*exp(-7*I*pi/8)*hyper((-7/2, -1/8), (-5/2,), b*x**2/a)/(7*x**7)
 

Maxima [F]

\[ \int \frac {\sqrt [8]{-a+b x^2}}{x^8} \, dx=\int { \frac {{\left (b x^{2} - a\right )}^{\frac {1}{8}}}{x^{8}} \,d x } \] Input:

integrate((b*x^2-a)^(1/8)/x^8,x, algorithm="maxima")
 

Output:

integrate((b*x^2 - a)^(1/8)/x^8, x)
 

Giac [F]

\[ \int \frac {\sqrt [8]{-a+b x^2}}{x^8} \, dx=\int { \frac {{\left (b x^{2} - a\right )}^{\frac {1}{8}}}{x^{8}} \,d x } \] Input:

integrate((b*x^2-a)^(1/8)/x^8,x, algorithm="giac")
 

Output:

integrate((b*x^2 - a)^(1/8)/x^8, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [8]{-a+b x^2}}{x^8} \, dx=\int \frac {{\left (b\,x^2-a\right )}^{1/8}}{x^8} \,d x \] Input:

int((b*x^2 - a)^(1/8)/x^8,x)
 

Output:

int((b*x^2 - a)^(1/8)/x^8, x)
 

Reduce [F]

\[ \int \frac {\sqrt [8]{-a+b x^2}}{x^8} \, dx=\frac {-1584 \left (b \,x^{2}-a \right )^{\frac {7}{8}} a^{3}-324 \left (b \,x^{2}-a \right )^{\frac {7}{8}} a^{2} b \,x^{2}+944 \left (b \,x^{2}-a \right )^{\frac {7}{8}} a \,b^{2} x^{4}-380 \left (b \,x^{2}-a \right )^{\frac {7}{8}} b^{3} x^{6}+4371 \left (b \,x^{2}-a \right )^{\frac {3}{4}} \left (\int \frac {\left (b \,x^{2}-a \right )^{\frac {3}{4}}}{\left (b \,x^{2}-a \right )^{\frac {5}{8}} a \,x^{4}-\left (b \,x^{2}-a \right )^{\frac {5}{8}} b \,x^{6}}d x \right ) a^{2} b^{2} x^{7}-2976 \left (b \,x^{2}-a \right )^{\frac {3}{4}} \left (\int \frac {\left (b \,x^{2}-a \right )^{\frac {3}{4}}}{\left (b \,x^{2}-a \right )^{\frac {5}{8}} a \,x^{2}-\left (b \,x^{2}-a \right )^{\frac {5}{8}} b \,x^{4}}d x \right ) a \,b^{3} x^{7}+285 \left (b \,x^{2}-a \right )^{\frac {3}{4}} \left (\int \frac {\left (b \,x^{2}-a \right )^{\frac {3}{4}}}{\left (b \,x^{2}-a \right )^{\frac {5}{8}} a -\left (b \,x^{2}-a \right )^{\frac {5}{8}} b \,x^{2}}d x \right ) b^{4} x^{7}}{11088 \left (b \,x^{2}-a \right )^{\frac {3}{4}} a^{3} x^{7}} \] Input:

int((b*x^2-a)^(1/8)/x^8,x)
 

Output:

( - 1584*( - a + b*x**2)**(7/8)*a**3 - 324*( - a + b*x**2)**(7/8)*a**2*b*x 
**2 + 944*( - a + b*x**2)**(7/8)*a*b**2*x**4 - 380*( - a + b*x**2)**(7/8)* 
b**3*x**6 + 4371*( - a + b*x**2)**(3/4)*int(( - a + b*x**2)**(3/4)/(( - a 
+ b*x**2)**(5/8)*a*x**4 - ( - a + b*x**2)**(5/8)*b*x**6),x)*a**2*b**2*x**7 
 - 2976*( - a + b*x**2)**(3/4)*int(( - a + b*x**2)**(3/4)/(( - a + b*x**2) 
**(5/8)*a*x**2 - ( - a + b*x**2)**(5/8)*b*x**4),x)*a*b**3*x**7 + 285*( - a 
 + b*x**2)**(3/4)*int(( - a + b*x**2)**(3/4)/(( - a + b*x**2)**(5/8)*a - ( 
 - a + b*x**2)**(5/8)*b*x**2),x)*b**4*x**7)/(11088*( - a + b*x**2)**(3/4)* 
a**3*x**7)