Integrand size = 37, antiderivative size = 218 \[ \int \frac {\left (-4 b+a x^4\right ) \sqrt [4]{-b+a x^4}}{x^6 \left (-8 b+a x^8\right )} \, dx=\frac {\sqrt [4]{-b+a x^4} \left (-4 b+9 a x^4\right )}{40 b x^5}+\frac {a \text {RootSum}\left [8 a^2-a b-16 a \text {$\#$1}^4+8 \text {$\#$1}^8\&,\frac {-8 a^2 \log (x)+a b \log (x)+8 a^2 \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )-a b \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )+8 a \log (x) \text {$\#$1}^4+4 b \log (x) \text {$\#$1}^4-8 a \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4-4 b \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}^3-\text {$\#$1}^7}\&\right ]}{512 b} \] Output:
Unintegrable
Time = 0.59 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.98 \[ \int \frac {\left (-4 b+a x^4\right ) \sqrt [4]{-b+a x^4}}{x^6 \left (-8 b+a x^8\right )} \, dx=\frac {\frac {64 \sqrt [4]{-b+a x^4} \left (-4 b+9 a x^4\right )}{x^5}+5 a \text {RootSum}\left [8 a^2-a b-16 a \text {$\#$1}^4+8 \text {$\#$1}^8\&,\frac {8 a^2 \log (x)-a b \log (x)-8 a^2 \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )+a b \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )-8 a \log (x) \text {$\#$1}^4-4 b \log (x) \text {$\#$1}^4+8 a \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4+4 b \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}^3+\text {$\#$1}^7}\&\right ]}{2560 b} \] Input:
Integrate[((-4*b + a*x^4)*(-b + a*x^4)^(1/4))/(x^6*(-8*b + a*x^8)),x]
Output:
((64*(-b + a*x^4)^(1/4)*(-4*b + 9*a*x^4))/x^5 + 5*a*RootSum[8*a^2 - a*b - 16*a*#1^4 + 8*#1^8 & , (8*a^2*Log[x] - a*b*Log[x] - 8*a^2*Log[(-b + a*x^4) ^(1/4) - x*#1] + a*b*Log[(-b + a*x^4)^(1/4) - x*#1] - 8*a*Log[x]*#1^4 - 4* b*Log[x]*#1^4 + 8*a*Log[(-b + a*x^4)^(1/4) - x*#1]*#1^4 + 4*b*Log[(-b + a* x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1^3) + #1^7) & ])/(2560*b)
Leaf count is larger than twice the leaf count of optimal. \(928\) vs. \(2(218)=436\).
Time = 4.17 (sec) , antiderivative size = 928, normalized size of antiderivative = 4.26, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, 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.
| \(\displaystyle \int \frac {\left (a x^4-4 b\right ) \sqrt [4]{a x^4-b}}{x^6 \left (a x^8-8 b\right )} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {\sqrt [4]{a x^4-b}}{2 x^6}-\frac {a \sqrt [4]{a x^4-b}}{8 b x^2}-\frac {a x^2 \left (a x^4-4 b\right ) \sqrt [4]{a x^4-b}}{8 b \left (8 b-a x^8\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a \sqrt [4]{a x^4-b} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-\frac {\sqrt {a} x^4}{2 \sqrt {2} \sqrt {b}},\frac {a x^4}{b}\right ) x^3}{96 b \sqrt [4]{1-\frac {a x^4}{b}}}+\frac {a \sqrt [4]{a x^4-b} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {\sqrt {a} x^4}{2 \sqrt {2} \sqrt {b}},\frac {a x^4}{b}\right ) x^3}{96 b \sqrt [4]{1-\frac {a x^4}{b}}}-\frac {a^{9/8} \arctan \left (\frac {\sqrt [8]{a} \sqrt [4]{2 \sqrt {2} \sqrt {a}-\sqrt {b}} x}{2^{3/8} \sqrt [4]{a x^4-b}}\right )}{32\ 2^{3/8} \left (2 \sqrt {2} \sqrt {a}-\sqrt {b}\right )^{3/4} \sqrt {b}}+\frac {a^{13/8} \arctan \left (\frac {\sqrt [8]{a} \sqrt [4]{2 \sqrt {2} \sqrt {a}-\sqrt {b}} x}{2^{3/8} \sqrt [4]{a x^4-b}}\right )}{8\ 2^{7/8} \left (2 \sqrt {2} \sqrt {a}-\sqrt {b}\right )^{3/4} b}+\frac {a^{9/8} \arctan \left (\frac {\sqrt [8]{a} \sqrt [4]{2 \sqrt {2} \sqrt {a}+\sqrt {b}} x}{2^{3/8} \sqrt [4]{a x^4-b}}\right )}{32\ 2^{3/8} \left (2 \sqrt {2} \sqrt {a}+\sqrt {b}\right )^{3/4} \sqrt {b}}+\frac {a^{13/8} \arctan \left (\frac {\sqrt [8]{a} \sqrt [4]{2 \sqrt {2} \sqrt {a}+\sqrt {b}} x}{2^{3/8} \sqrt [4]{a x^4-b}}\right )}{8\ 2^{7/8} \left (2 \sqrt {2} \sqrt {a}+\sqrt {b}\right )^{3/4} b}+\frac {a^{9/8} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [4]{2 \sqrt {2} \sqrt {a}-\sqrt {b}} x}{2^{3/8} \sqrt [4]{a x^4-b}}\right )}{32\ 2^{3/8} \left (2 \sqrt {2} \sqrt {a}-\sqrt {b}\right )^{3/4} \sqrt {b}}-\frac {a^{13/8} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [4]{2 \sqrt {2} \sqrt {a}-\sqrt {b}} x}{2^{3/8} \sqrt [4]{a x^4-b}}\right )}{8\ 2^{7/8} \left (2 \sqrt {2} \sqrt {a}-\sqrt {b}\right )^{3/4} b}-\frac {a^{9/8} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [4]{2 \sqrt {2} \sqrt {a}+\sqrt {b}} x}{2^{3/8} \sqrt [4]{a x^4-b}}\right )}{32\ 2^{3/8} \left (2 \sqrt {2} \sqrt {a}+\sqrt {b}\right )^{3/4} \sqrt {b}}-\frac {a^{13/8} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [4]{2 \sqrt {2} \sqrt {a}+\sqrt {b}} x}{2^{3/8} \sqrt [4]{a x^4-b}}\right )}{8\ 2^{7/8} \left (2 \sqrt {2} \sqrt {a}+\sqrt {b}\right )^{3/4} b}+\frac {a \sqrt [4]{a x^4-b}}{8 b x}+\frac {\left (a x^4-b\right )^{5/4}}{10 b x^5}\) |
Input:
Int[((-4*b + a*x^4)*(-b + a*x^4)^(1/4))/(x^6*(-8*b + a*x^8)),x]
Output:
(a*(-b + a*x^4)^(1/4))/(8*b*x) + (-b + a*x^4)^(5/4)/(10*b*x^5) + (a*x^3*(- b + a*x^4)^(1/4)*AppellF1[3/4, 1, -1/4, 7/4, -1/2*(Sqrt[a]*x^4)/(Sqrt[2]*S qrt[b]), (a*x^4)/b])/(96*b*(1 - (a*x^4)/b)^(1/4)) + (a*x^3*(-b + a*x^4)^(1 /4)*AppellF1[3/4, 1, -1/4, 7/4, (Sqrt[a]*x^4)/(2*Sqrt[2]*Sqrt[b]), (a*x^4) /b])/(96*b*(1 - (a*x^4)/b)^(1/4)) + (a^(13/8)*ArcTan[(a^(1/8)*(2*Sqrt[2]*S qrt[a] - Sqrt[b])^(1/4)*x)/(2^(3/8)*(-b + a*x^4)^(1/4))])/(8*2^(7/8)*(2*Sq rt[2]*Sqrt[a] - Sqrt[b])^(3/4)*b) - (a^(9/8)*ArcTan[(a^(1/8)*(2*Sqrt[2]*Sq rt[a] - Sqrt[b])^(1/4)*x)/(2^(3/8)*(-b + a*x^4)^(1/4))])/(32*2^(3/8)*(2*Sq rt[2]*Sqrt[a] - Sqrt[b])^(3/4)*Sqrt[b]) + (a^(13/8)*ArcTan[(a^(1/8)*(2*Sqr t[2]*Sqrt[a] + Sqrt[b])^(1/4)*x)/(2^(3/8)*(-b + a*x^4)^(1/4))])/(8*2^(7/8) *(2*Sqrt[2]*Sqrt[a] + Sqrt[b])^(3/4)*b) + (a^(9/8)*ArcTan[(a^(1/8)*(2*Sqrt [2]*Sqrt[a] + Sqrt[b])^(1/4)*x)/(2^(3/8)*(-b + a*x^4)^(1/4))])/(32*2^(3/8) *(2*Sqrt[2]*Sqrt[a] + Sqrt[b])^(3/4)*Sqrt[b]) - (a^(13/8)*ArcTanh[(a^(1/8) *(2*Sqrt[2]*Sqrt[a] - Sqrt[b])^(1/4)*x)/(2^(3/8)*(-b + a*x^4)^(1/4))])/(8* 2^(7/8)*(2*Sqrt[2]*Sqrt[a] - Sqrt[b])^(3/4)*b) + (a^(9/8)*ArcTanh[(a^(1/8) *(2*Sqrt[2]*Sqrt[a] - Sqrt[b])^(1/4)*x)/(2^(3/8)*(-b + a*x^4)^(1/4))])/(32 *2^(3/8)*(2*Sqrt[2]*Sqrt[a] - Sqrt[b])^(3/4)*Sqrt[b]) - (a^(13/8)*ArcTanh[ (a^(1/8)*(2*Sqrt[2]*Sqrt[a] + Sqrt[b])^(1/4)*x)/(2^(3/8)*(-b + a*x^4)^(1/4 ))])/(8*2^(7/8)*(2*Sqrt[2]*Sqrt[a] + Sqrt[b])^(3/4)*b) - (a^(9/8)*ArcTanh[ (a^(1/8)*(2*Sqrt[2]*Sqrt[a] + Sqrt[b])^(1/4)*x)/(2^(3/8)*(-b + a*x^4)^(...
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 = 0.66 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.58
| method | result | size |
| pseudoelliptic | \(\frac {5 a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (8 \textit {\_Z}^{8}-16 a \,\textit {\_Z}^{4}+8 a^{2}-a b \right )}{\sum }\frac {\left (8 \textit {\_R}^{4} a +4 \textit {\_R}^{4} b -8 a^{2}+a b \right ) \ln \left (\frac {-\textit {\_R} x +\left (a \,x^{4}-b \right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{3} \left (\textit {\_R}^{4}-a \right )}\right ) x^{5}+576 a \,x^{4} \left (a \,x^{4}-b \right )^{\frac {1}{4}}-256 b \left (a \,x^{4}-b \right )^{\frac {1}{4}}}{2560 b \,x^{5}}\) | \(127\) |
Input:
int((a*x^4-4*b)*(a*x^4-b)^(1/4)/x^6/(a*x^8-8*b),x,method=_RETURNVERBOSE)
Output:
1/2560*(5*a*sum((8*_R^4*a+4*_R^4*b-8*a^2+a*b)*ln((-_R*x+(a*x^4-b)^(1/4))/x )/_R^3/(_R^4-a),_R=RootOf(8*_Z^8-16*_Z^4*a+8*a^2-a*b))*x^5+576*a*x^4*(a*x^ 4-b)^(1/4)-256*b*(a*x^4-b)^(1/4))/b/x^5
Timed out. \[ \int \frac {\left (-4 b+a x^4\right ) \sqrt [4]{-b+a x^4}}{x^6 \left (-8 b+a x^8\right )} \, dx=\text {Timed out} \] Input:
integrate((a*x^4-4*b)*(a*x^4-b)^(1/4)/x^6/(a*x^8-8*b),x, algorithm="fricas ")
Output:
Timed out
Not integrable
Time = 76.94 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.14 \[ \int \frac {\left (-4 b+a x^4\right ) \sqrt [4]{-b+a x^4}}{x^6 \left (-8 b+a x^8\right )} \, dx=\int \frac {\left (a x^{4} - 4 b\right ) \sqrt [4]{a x^{4} - b}}{x^{6} \left (a x^{8} - 8 b\right )}\, dx \] Input:
integrate((a*x**4-4*b)*(a*x**4-b)**(1/4)/x**6/(a*x**8-8*b),x)
Output:
Integral((a*x**4 - 4*b)*(a*x**4 - b)**(1/4)/(x**6*(a*x**8 - 8*b)), x)
Not integrable
Time = 0.09 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.17 \[ \int \frac {\left (-4 b+a x^4\right ) \sqrt [4]{-b+a x^4}}{x^6 \left (-8 b+a x^8\right )} \, dx=\int { \frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}} {\left (a x^{4} - 4 \, b\right )}}{{\left (a x^{8} - 8 \, b\right )} x^{6}} \,d x } \] Input:
integrate((a*x^4-4*b)*(a*x^4-b)^(1/4)/x^6/(a*x^8-8*b),x, algorithm="maxima ")
Output:
integrate((a*x^4 - b)^(1/4)*(a*x^4 - 4*b)/((a*x^8 - 8*b)*x^6), x)
Exception generated. \[ \int \frac {\left (-4 b+a x^4\right ) \sqrt [4]{-b+a x^4}}{x^6 \left (-8 b+a x^8\right )} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a*x^4-4*b)*(a*x^4-b)^(1/4)/x^6/(a*x^8-8*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%%%{4,[0,1,4,1,0]%%%}+%%%{-1,[0,1,0,0,1]%%%} / %%%{8,[0,0,0,1, 0]%%%} Er
Not integrable
Time = 9.76 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.18 \[ \int \frac {\left (-4 b+a x^4\right ) \sqrt [4]{-b+a x^4}}{x^6 \left (-8 b+a x^8\right )} \, dx=\int \frac {{\left (a\,x^4-b\right )}^{1/4}\,\left (4\,b-a\,x^4\right )}{x^6\,\left (8\,b-a\,x^8\right )} \,d x \] Input:
int(((a*x^4 - b)^(1/4)*(4*b - a*x^4))/(x^6*(8*b - a*x^8)),x)
Output:
int(((a*x^4 - b)^(1/4)*(4*b - a*x^4))/(x^6*(8*b - a*x^8)), x)
Not integrable
Time = 0.19 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.94 \[ \int \frac {\left (-4 b+a x^4\right ) \sqrt [4]{-b+a x^4}}{x^6 \left (-8 b+a x^8\right )} \, dx=\frac {-4 \left (a \,x^{4}-b \right )^{\frac {1}{4}} a \,x^{4}-\left (a \,x^{4}-b \right )^{\frac {1}{4}} b -50 \left (\int \frac {\left (a \,x^{4}-b \right )^{\frac {1}{4}}}{a^{2} x^{14}-a b \,x^{10}-8 a b \,x^{6}+8 b^{2} x^{2}}d x \right ) a \,b^{2} x^{5}+10 \left (\int \frac {\left (a \,x^{4}-b \right )^{\frac {1}{4}} x^{2}}{a^{2} x^{12}-a b \,x^{8}-8 a b \,x^{4}+8 b^{2}}d x \right ) a^{2} b \,x^{5}+5 \left (\int \frac {\left (a \,x^{4}-b \right )^{\frac {1}{4}} x^{2}}{a^{2} x^{12}-a b \,x^{8}-8 a b \,x^{4}+8 b^{2}}d x \right ) a \,b^{2} x^{5}}{10 b \,x^{5}} \] Input:
int((a*x^4-4*b)*(a*x^4-b)^(1/4)/x^6/(a*x^8-8*b),x)
Output:
( - 4*(a*x**4 - b)**(1/4)*a*x**4 - (a*x**4 - b)**(1/4)*b - 50*int((a*x**4 - b)**(1/4)/(a**2*x**14 - a*b*x**10 - 8*a*b*x**6 + 8*b**2*x**2),x)*a*b**2* x**5 + 10*int(((a*x**4 - b)**(1/4)*x**2)/(a**2*x**12 - a*b*x**8 - 8*a*b*x* *4 + 8*b**2),x)*a**2*b*x**5 + 5*int(((a*x**4 - b)**(1/4)*x**2)/(a**2*x**12 - a*b*x**8 - 8*a*b*x**4 + 8*b**2),x)*a*b**2*x**5)/(10*b*x**5)