Integrand size = 17, antiderivative size = 997 \[ \int \frac {\left (-a+b x^2\right )^{5/8}}{x^8} \, dx =\text {Too large to display} \] Output:
-1/7*(b*x^2-a)^(5/8)/x^7+1/28*b*(b*x^2-a)^(5/8)/a/x^5+5/112*b^2*(b*x^2-a)^ (5/8)/a^2/x^3+5/64*b^3*(b*x^2-a)^(5/8)/a^3/x+5/128*(2+2^(1/2))^(1/2)*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)*EllipticE(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))/a^(5/2)/x/(a^(1/4)+(b*x^2-a)^(1/4))-5/128*(2+ 2^(1/2))^(1/2)*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)*EllipticE(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))/a^(5/2)/x/(a^(1/4)-(b*x^2 -a)^(1/4))-5/128*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^(5/ 2)/x/(a^(1/4)+(b*x^2-a)^(1/4))+5/128*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^(5/2)/x/(a^(1/4)-(b*x^2-a)^(1/4))
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 = 0.05 \[ \int \frac {\left (-a+b x^2\right )^{5/8}}{x^8} \, dx=-\frac {\left (-a+b x^2\right )^{5/8} \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},-\frac {5}{8},-\frac {5}{2},\frac {b x^2}{a}\right )}{7 x^7 \left (1-\frac {b x^2}{a}\right )^{5/8}} \] Input:
Integrate[(-a + b*x^2)^(5/8)/x^8,x]
Output:
-1/7*((-a + b*x^2)^(5/8)*Hypergeometric2F1[-7/2, -5/8, -5/2, (b*x^2)/a])/( x^7*(1 - (b*x^2)/a)^(5/8))
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 = 0.05, 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.
\(\displaystyle \int \frac {\left (b x^2-a\right )^{5/8}}{x^8} \, dx\) |
\(\Big \downarrow \) 279 |
\(\displaystyle \frac {\left (b x^2-a\right )^{5/8} \int \frac {\left (1-\frac {b x^2}{a}\right )^{5/8}}{x^8}dx}{\left (1-\frac {b x^2}{a}\right )^{5/8}}\) |
\(\Big \downarrow \) 278 |
\(\displaystyle -\frac {\left (b x^2-a\right )^{5/8} \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},-\frac {5}{8},-\frac {5}{2},\frac {b x^2}{a}\right )}{7 x^7 \left (1-\frac {b x^2}{a}\right )^{5/8}}\) |
Input:
Int[(-a + b*x^2)^(5/8)/x^8,x]
Output:
-1/7*((-a + b*x^2)^(5/8)*Hypergeometric2F1[-7/2, -5/8, -5/2, (b*x^2)/a])/( x^7*(1 - (b*x^2)/a)^(5/8))
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])
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])
\[\int \frac {\left (b \,x^{2}-a \right )^{\frac {5}{8}}}{x^{8}}d x\]
Input:
int((b*x^2-a)^(5/8)/x^8,x)
Output:
int((b*x^2-a)^(5/8)/x^8,x)
\[ \int \frac {\left (-a+b x^2\right )^{5/8}}{x^8} \, dx=\int { \frac {{\left (b x^{2} - a\right )}^{\frac {5}{8}}}{x^{8}} \,d x } \] Input:
integrate((b*x^2-a)^(5/8)/x^8,x, algorithm="fricas")
Output:
integral((b*x^2 - a)^(5/8)/x^8, x)
Result contains complex when optimal does not.
Time = 1.06 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.04 \[ \int \frac {\left (-a+b x^2\right )^{5/8}}{x^8} \, dx=\frac {a^{\frac {5}{8}} e^{- \frac {3 i \pi }{8}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{2}, - \frac {5}{8} \\ - \frac {5}{2} \end {matrix}\middle | {\frac {b x^{2}}{a}} \right )}}{7 x^{7}} \] Input:
integrate((b*x**2-a)**(5/8)/x**8,x)
Output:
a**(5/8)*exp(-3*I*pi/8)*hyper((-7/2, -5/8), (-5/2,), b*x**2/a)/(7*x**7)
\[ \int \frac {\left (-a+b x^2\right )^{5/8}}{x^8} \, dx=\int { \frac {{\left (b x^{2} - a\right )}^{\frac {5}{8}}}{x^{8}} \,d x } \] Input:
integrate((b*x^2-a)^(5/8)/x^8,x, algorithm="maxima")
Output:
integrate((b*x^2 - a)^(5/8)/x^8, x)
\[ \int \frac {\left (-a+b x^2\right )^{5/8}}{x^8} \, dx=\int { \frac {{\left (b x^{2} - a\right )}^{\frac {5}{8}}}{x^{8}} \,d x } \] Input:
integrate((b*x^2-a)^(5/8)/x^8,x, algorithm="giac")
Output:
integrate((b*x^2 - a)^(5/8)/x^8, x)
Timed out. \[ \int \frac {\left (-a+b x^2\right )^{5/8}}{x^8} \, dx=\int \frac {{\left (b\,x^2-a\right )}^{5/8}}{x^8} \,d x \] Input:
int((b*x^2 - a)^(5/8)/x^8,x)
Output:
int((b*x^2 - a)^(5/8)/x^8, x)
\[ \int \frac {\left (-a+b x^2\right )^{5/8}}{x^8} \, dx=\frac {16 \left (b \,x^{2}-a \right )^{\frac {3}{8}} a^{2}-20 \left (b \,x^{2}-a \right )^{\frac {3}{8}} a b \,x^{2}+4 \left (b \,x^{2}-a \right )^{\frac {3}{8}} b^{2} x^{4}-20 \left (b \,x^{2}-a \right )^{\frac {3}{4}} \left (\int \frac {1}{\left (b \,x^{2}-a \right )^{\frac {3}{8}} x^{8}}d x \right ) a^{2} x^{7}-20 \left (b \,x^{2}-a \right )^{\frac {3}{4}} \left (\int \frac {1}{\left (b \,x^{2}-a \right )^{\frac {3}{8}} x^{6}}d x \right ) a b \,x^{7}+15 \left (b \,x^{2}-a \right )^{\frac {3}{4}} \left (\int \frac {1}{\left (b \,x^{2}-a \right )^{\frac {3}{8}} x^{4}}d x \right ) b^{2} x^{7}}{132 \left (b \,x^{2}-a \right )^{\frac {3}{4}} a \,x^{7}} \] Input:
int((b*x^2-a)^(5/8)/x^8,x)
Output:
(16*( - a + b*x**2)**(3/8)*a**2 - 20*( - a + b*x**2)**(3/8)*a*b*x**2 + 4*( - a + b*x**2)**(3/8)*b**2*x**4 - 20*( - a + b*x**2)**(3/4)*int(( - a + b* x**2)**(1/4)/(( - a + b*x**2)**(5/8)*x**8),x)*a**2*x**7 - 20*( - a + b*x** 2)**(3/4)*int(( - a + b*x**2)**(1/4)/(( - a + b*x**2)**(5/8)*x**6),x)*a*b* x**7 + 15*( - a + b*x**2)**(3/4)*int(( - a + b*x**2)**(1/4)/(( - a + b*x** 2)**(5/8)*x**4),x)*b**2*x**7)/(132*( - a + b*x**2)**(3/4)*a*x**7)