Integrand size = 35, antiderivative size = 155 \[ \int \frac {\sqrt [4]{-b+a x^4} \left (-8 b+a x^8\right )}{x^{10} \left (b+a x^4\right )} \, dx=\frac {\sqrt [4]{-b+a x^4} \left (8 b^2-16 a b x^4+80 a^2 x^8-9 a b x^8\right )}{9 b^2 x^9}+\frac {\left (8 a^{9/4}-a^{5/4} b\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2^{3/4} b^2}-\frac {\left (8 a^{9/4}-a^{5/4} b\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2^{3/4} b^2} \] Output:
1/9*(a*x^4-b)^(1/4)*(80*a^2*x^8-9*a*b*x^8-16*a*b*x^4+8*b^2)/b^2/x^9+1/2*(8 *a^(9/4)-a^(5/4)*b)*arctan(2^(1/4)*a^(1/4)*x/(a*x^4-b)^(1/4))*2^(1/4)/b^2- 1/2*(8*a^(9/4)-a^(5/4)*b)*arctanh(2^(1/4)*a^(1/4)*x/(a*x^4-b)^(1/4))*2^(1/ 4)/b^2
Time = 0.71 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt [4]{-b+a x^4} \left (-8 b+a x^8\right )}{x^{10} \left (b+a x^4\right )} \, dx=\frac {\sqrt [4]{-b+a x^4} \left (8 b^2-16 a b x^4+80 a^2 x^8-9 a b x^8\right )}{9 b^2 x^9}+\frac {a^{5/4} (8 a-b) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2^{3/4} b^2}-\frac {a^{5/4} (8 a-b) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2^{3/4} b^2} \] Input:
Integrate[((-b + a*x^4)^(1/4)*(-8*b + a*x^8))/(x^10*(b + a*x^4)),x]
Output:
((-b + a*x^4)^(1/4)*(8*b^2 - 16*a*b*x^4 + 80*a^2*x^8 - 9*a*b*x^8))/(9*b^2* x^9) + (a^(5/4)*(8*a - b)*ArcTan[(2^(1/4)*a^(1/4)*x)/(-b + a*x^4)^(1/4)])/ (2^(3/4)*b^2) - (a^(5/4)*(8*a - b)*ArcTanh[(2^(1/4)*a^(1/4)*x)/(-b + a*x^4 )^(1/4)])/(2^(3/4)*b^2)
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 0.85 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.50, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {7276, 2009}
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.
\(\displaystyle \int \frac {\sqrt [4]{a x^4-b} \left (a x^8-8 b\right )}{x^{10} \left (a x^4+b\right )} \, dx\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \int \left (\frac {a^2 x^2 (8 a-b) \sqrt [4]{a x^4-b}}{b^2 \left (a x^4+b\right )}-\frac {a (8 a-b) \sqrt [4]{a x^4-b}}{b^2 x^2}-\frac {8 \sqrt [4]{a x^4-b}}{x^{10}}+\frac {8 a \sqrt [4]{a x^4-b}}{b x^6}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^{5/4} (8 a-b) \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 b^2}-\frac {a^{5/4} (8 a-b) \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 b^2}+\frac {a^2 x^3 (8 a-b) \sqrt [4]{a x^4-b} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-\frac {a x^4}{b},\frac {a x^4}{b}\right )}{3 b^3 \sqrt [4]{1-\frac {a x^4}{b}}}+\frac {a (8 a-b) \sqrt [4]{a x^4-b}}{b^2 x}+\frac {8 a \left (a x^4-b\right )^{5/4}}{9 b^2 x^5}-\frac {8 \left (a x^4-b\right )^{5/4}}{9 b x^9}\) |
Input:
Int[((-b + a*x^4)^(1/4)*(-8*b + a*x^8))/(x^10*(b + a*x^4)),x]
Output:
(a*(8*a - b)*(-b + a*x^4)^(1/4))/(b^2*x) - (8*(-b + a*x^4)^(5/4))/(9*b*x^9 ) + (8*a*(-b + a*x^4)^(5/4))/(9*b^2*x^5) + (a^2*(8*a - b)*x^3*(-b + a*x^4) ^(1/4)*AppellF1[3/4, 1, -1/4, 7/4, -((a*x^4)/b), (a*x^4)/b])/(3*b^3*(1 - ( a*x^4)/b)^(1/4)) + (a^(5/4)*(8*a - b)*ArcTan[(a^(1/4)*x)/(-b + a*x^4)^(1/4 )])/(2*b^2) - (a^(5/4)*(8*a - b)*ArcTanh[(a^(1/4)*x)/(-b + a*x^4)^(1/4)])/ (2*b^2)
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 0.51 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.01
method | result | size |
pseudoelliptic | \(\frac {9 x^{9} 2^{\frac {1}{4}} \left (a^{\frac {5}{4}} b -8 a^{\frac {9}{4}}\right ) \ln \left (\frac {-2^{\frac {1}{4}} a^{\frac {1}{4}} x -\left (a \,x^{4}-b \right )^{\frac {1}{4}}}{2^{\frac {1}{4}} a^{\frac {1}{4}} x -\left (a \,x^{4}-b \right )^{\frac {1}{4}}}\right )+18 x^{9} 2^{\frac {1}{4}} \left (a^{\frac {5}{4}} b -8 a^{\frac {9}{4}}\right ) \arctan \left (\frac {2^{\frac {3}{4}} \left (a \,x^{4}-b \right )^{\frac {1}{4}}}{2 a^{\frac {1}{4}} x}\right )+320 \left (a \left (a -\frac {9 b}{80}\right ) x^{8}-\frac {a b \,x^{4}}{5}+\frac {b^{2}}{10}\right ) \left (a \,x^{4}-b \right )^{\frac {1}{4}}}{36 b^{2} x^{9}}\) | \(156\) |
Input:
int((a*x^4-b)^(1/4)*(a*x^8-8*b)/x^10/(a*x^4+b),x,method=_RETURNVERBOSE)
Output:
1/36*(9*x^9*2^(1/4)*(a^(5/4)*b-8*a^(9/4))*ln((-2^(1/4)*a^(1/4)*x-(a*x^4-b) ^(1/4))/(2^(1/4)*a^(1/4)*x-(a*x^4-b)^(1/4)))+18*x^9*2^(1/4)*(a^(5/4)*b-8*a ^(9/4))*arctan(1/2*2^(3/4)/a^(1/4)/x*(a*x^4-b)^(1/4))+320*(a*(a-9/80*b)*x^ 8-1/5*a*b*x^4+1/10*b^2)*(a*x^4-b)^(1/4))/b^2/x^9
Timed out. \[ \int \frac {\sqrt [4]{-b+a x^4} \left (-8 b+a x^8\right )}{x^{10} \left (b+a x^4\right )} \, dx=\text {Timed out} \] Input:
integrate((a*x^4-b)^(1/4)*(a*x^8-8*b)/x^10/(a*x^4+b),x, algorithm="fricas" )
Output:
Timed out
\[ \int \frac {\sqrt [4]{-b+a x^4} \left (-8 b+a x^8\right )}{x^{10} \left (b+a x^4\right )} \, dx=\int \frac {\sqrt [4]{a x^{4} - b} \left (a x^{8} - 8 b\right )}{x^{10} \left (a x^{4} + b\right )}\, dx \] Input:
integrate((a*x**4-b)**(1/4)*(a*x**8-8*b)/x**10/(a*x**4+b),x)
Output:
Integral((a*x**4 - b)**(1/4)*(a*x**8 - 8*b)/(x**10*(a*x**4 + b)), x)
\[ \int \frac {\sqrt [4]{-b+a x^4} \left (-8 b+a x^8\right )}{x^{10} \left (b+a x^4\right )} \, dx=\int { \frac {{\left (a x^{8} - 8 \, b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}}{{\left (a x^{4} + b\right )} x^{10}} \,d x } \] Input:
integrate((a*x^4-b)^(1/4)*(a*x^8-8*b)/x^10/(a*x^4+b),x, algorithm="maxima" )
Output:
integrate((a*x^8 - 8*b)*(a*x^4 - b)^(1/4)/((a*x^4 + b)*x^10), x)
Exception generated. \[ \int \frac {\sqrt [4]{-b+a x^4} \left (-8 b+a x^8\right )}{x^{10} \left (b+a x^4\right )} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a*x^4-b)^(1/4)*(a*x^8-8*b)/x^10/(a*x^4+b),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{-8,[0,1,8,2,0]%%%}+%%%{8,[0,1,4,1,1]%%%}+%%%{1,[0,1,0,1,1] %%%}+%%%{
Timed out. \[ \int \frac {\sqrt [4]{-b+a x^4} \left (-8 b+a x^8\right )}{x^{10} \left (b+a x^4\right )} \, dx=\int -\frac {{\left (a\,x^4-b\right )}^{1/4}\,\left (8\,b-a\,x^8\right )}{x^{10}\,\left (a\,x^4+b\right )} \,d x \] Input:
int(-((a*x^4 - b)^(1/4)*(8*b - a*x^8))/(x^10*(b + a*x^4)),x)
Output:
int(-((a*x^4 - b)^(1/4)*(8*b - a*x^8))/(x^10*(b + a*x^4)), x)
\[ \int \frac {\sqrt [4]{-b+a x^4} \left (-8 b+a x^8\right )}{x^{10} \left (b+a x^4\right )} \, dx=\frac {-64 \left (a \,x^{4}-b \right )^{\frac {1}{4}} a^{2} x^{8}+9 \left (a \,x^{4}-b \right )^{\frac {1}{4}} a b \,x^{8}-16 \left (a \,x^{4}-b \right )^{\frac {1}{4}} a b \,x^{4}+8 \left (a \,x^{4}-b \right )^{\frac {1}{4}} b^{2}+144 \left (\int \frac {\left (a \,x^{4}-b \right )^{\frac {1}{4}}}{a^{2} x^{10}-b^{2} x^{2}}d x \right ) a^{2} b^{2} x^{9}-18 \left (\int \frac {\left (a \,x^{4}-b \right )^{\frac {1}{4}}}{a^{2} x^{10}-b^{2} x^{2}}d x \right ) a \,b^{3} x^{9}}{9 b^{2} x^{9}} \] Input:
int((a*x^4-b)^(1/4)*(a*x^8-8*b)/x^10/(a*x^4+b),x)
Output:
( - 64*(a*x**4 - b)**(1/4)*a**2*x**8 + 9*(a*x**4 - b)**(1/4)*a*b*x**8 - 16 *(a*x**4 - b)**(1/4)*a*b*x**4 + 8*(a*x**4 - b)**(1/4)*b**2 + 144*int((a*x* *4 - b)**(1/4)/(a**2*x**10 - b**2*x**2),x)*a**2*b**2*x**9 - 18*int((a*x**4 - b)**(1/4)/(a**2*x**10 - b**2*x**2),x)*a*b**3*x**9)/(9*b**2*x**9)