Integrand size = 17, antiderivative size = 969 \[ \int \frac {x^6}{\left (-a+b x^2\right )^{11/8}} \, dx =\text {Too large to display} \] Output:
-4/3*x^5/b/(b*x^2-a)^(3/8)+320/153*a*x*(b*x^2-a)^(5/8)/b^3+80/51*x^3*(b*x^ 2-a)^(5/8)/b^2-640/153*(2+2^(1/2))^(1/2)*a^(5/2)*(-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))/b^4/x/(a^(1/4)+(b*x^2-a)^(1/4))+640/153*(2+2^(1/2))^(1/2)*a^(5/2)*(-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))/b^4/x/(a^(1/4)-(b*x^2-a)^(1/4))+640/153*a^(5/2) *(-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)/b^4/x/(a^(1/4)+(b*x^2-a)^ (1/4))-640/153*a^(5/2)*(-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)/b^4 /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 = 9.67 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.06 \[ \int \frac {x^6}{\left (-a+b x^2\right )^{11/8}} \, dx=-\frac {x^7 \left (1-\frac {b x^2}{a}\right )^{3/8} \operatorname {Hypergeometric2F1}\left (\frac {11}{8},\frac {7}{2},\frac {9}{2},\frac {b x^2}{a}\right )}{7 a \left (-a+b x^2\right )^{3/8}} \] Input:
Integrate[x^6/(-a + b*x^2)^(11/8),x]
Output:
-1/7*(x^7*(1 - (b*x^2)/a)^(3/8)*Hypergeometric2F1[11/8, 7/2, 9/2, (b*x^2)/ a])/(a*(-a + b*x^2)^(3/8))
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.16 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.06, 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.
\(\displaystyle \int \frac {x^6}{\left (b x^2-a\right )^{11/8}} \, dx\) |
\(\Big \downarrow \) 279 |
\(\displaystyle -\frac {\left (1-\frac {b x^2}{a}\right )^{3/8} \int \frac {x^6}{\left (1-\frac {b x^2}{a}\right )^{11/8}}dx}{a \left (b x^2-a\right )^{3/8}}\) |
\(\Big \downarrow \) 278 |
\(\displaystyle -\frac {x^7 \left (1-\frac {b x^2}{a}\right )^{3/8} \operatorname {Hypergeometric2F1}\left (\frac {11}{8},\frac {7}{2},\frac {9}{2},\frac {b x^2}{a}\right )}{7 a \left (b x^2-a\right )^{3/8}}\) |
Input:
Int[x^6/(-a + b*x^2)^(11/8),x]
Output:
-1/7*(x^7*(1 - (b*x^2)/a)^(3/8)*Hypergeometric2F1[11/8, 7/2, 9/2, (b*x^2)/ a])/(a*(-a + b*x^2)^(3/8))
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])
\[\int \frac {x^{6}}{\left (b \,x^{2}-a \right )^{\frac {11}{8}}}d x\]
Input:
int(x^6/(b*x^2-a)^(11/8),x)
Output:
int(x^6/(b*x^2-a)^(11/8),x)
\[ \int \frac {x^6}{\left (-a+b x^2\right )^{11/8}} \, dx=\int { \frac {x^{6}}{{\left (b x^{2} - a\right )}^{\frac {11}{8}}} \,d x } \] Input:
integrate(x^6/(b*x^2-a)^(11/8),x, algorithm="fricas")
Output:
integral((b*x^2 - a)^(5/8)*x^6/(b^2*x^4 - 2*a*b*x^2 + a^2), x)
Result contains complex when optimal does not.
Time = 0.69 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.03 \[ \int \frac {x^6}{\left (-a+b x^2\right )^{11/8}} \, dx=\frac {x^{7} e^{\frac {5 i \pi }{8}} {{}_{2}F_{1}\left (\begin {matrix} \frac {11}{8}, \frac {7}{2} \\ \frac {9}{2} \end {matrix}\middle | {\frac {b x^{2}}{a}} \right )}}{7 a^{\frac {11}{8}}} \] Input:
integrate(x**6/(b*x**2-a)**(11/8),x)
Output:
x**7*exp(5*I*pi/8)*hyper((11/8, 7/2), (9/2,), b*x**2/a)/(7*a**(11/8))
\[ \int \frac {x^6}{\left (-a+b x^2\right )^{11/8}} \, dx=\int { \frac {x^{6}}{{\left (b x^{2} - a\right )}^{\frac {11}{8}}} \,d x } \] Input:
integrate(x^6/(b*x^2-a)^(11/8),x, algorithm="maxima")
Output:
integrate(x^6/(b*x^2 - a)^(11/8), x)
\[ \int \frac {x^6}{\left (-a+b x^2\right )^{11/8}} \, dx=\int { \frac {x^{6}}{{\left (b x^{2} - a\right )}^{\frac {11}{8}}} \,d x } \] Input:
integrate(x^6/(b*x^2-a)^(11/8),x, algorithm="giac")
Output:
integrate(x^6/(b*x^2 - a)^(11/8), x)
Timed out. \[ \int \frac {x^6}{\left (-a+b x^2\right )^{11/8}} \, dx=\int \frac {x^6}{{\left (b\,x^2-a\right )}^{11/8}} \,d x \] Input:
int(x^6/(b*x^2 - a)^(11/8),x)
Output:
int(x^6/(b*x^2 - a)^(11/8), x)
\[ \int \frac {x^6}{\left (-a+b x^2\right )^{11/8}} \, dx=\frac {76 \left (b \,x^{2}-a \right )^{\frac {3}{8}} a^{2} x -80 \left (b \,x^{2}-a \right )^{\frac {3}{8}} a b \,x^{3}+4 \left (b \,x^{2}-a \right )^{\frac {3}{8}} b^{2} x^{5}-76 \left (b \,x^{2}-a \right )^{\frac {3}{4}} \left (\int \frac {\left (b \,x^{2}-a \right )^{\frac {1}{4}}}{\left (b \,x^{2}-a \right )^{\frac {5}{8}} a -\left (b \,x^{2}-a \right )^{\frac {5}{8}} b \,x^{2}}d x \right ) a^{3}+69 \left (b \,x^{2}-a \right )^{\frac {3}{4}} \left (\int \frac {\left (b \,x^{2}-a \right )^{\frac {1}{4}} x^{2}}{\left (b \,x^{2}-a \right )^{\frac {5}{8}} a -\left (b \,x^{2}-a \right )^{\frac {5}{8}} b \,x^{2}}d x \right ) a^{2} b}{7 \left (b \,x^{2}-a \right )^{\frac {3}{4}} b^{3}} \] Input:
int(x^6/(b*x^2-a)^(11/8),x)
Output:
(76*( - a + b*x**2)**(3/8)*a**2*x - 80*( - a + b*x**2)**(3/8)*a*b*x**3 + 4 *( - a + b*x**2)**(3/8)*b**2*x**5 - 76*( - a + b*x**2)**(3/4)*int(( - a + b*x**2)**(1/4)/(( - a + b*x**2)**(5/8)*a - ( - a + b*x**2)**(5/8)*b*x**2), x)*a**3 + 69*( - a + b*x**2)**(3/4)*int((( - a + b*x**2)**(1/4)*x**2)/(( - a + b*x**2)**(5/8)*a - ( - a + b*x**2)**(5/8)*b*x**2),x)*a**2*b)/(7*( - a + b*x**2)**(3/4)*b**3)