Integrand size = 38, antiderivative size = 114 \[ \int \frac {-b+a x^4+x^8}{x^8 \left (-b+a x^4\right ) \sqrt [4]{b+a x^4}} \, dx=\frac {\left (b+a x^4\right )^{3/4} \left (-3 b+4 a x^4\right )}{21 b^2 x^7}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b} \]
1/21*(a*x^4+b)^(3/4)*(4*a*x^4-3*b)/b^2/x^7-1/4*arctan(2^(1/4)*a^(1/4)*x/(a *x^4+b)^(1/4))*2^(3/4)/a^(1/4)/b-1/4*arctanh(2^(1/4)*a^(1/4)*x/(a*x^4+b)^( 1/4))*2^(3/4)/a^(1/4)/b
Time = 0.86 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00 \[ \int \frac {-b+a x^4+x^8}{x^8 \left (-b+a x^4\right ) \sqrt [4]{b+a x^4}} \, dx=\frac {\left (b+a x^4\right )^{3/4} \left (-3 b+4 a x^4\right )}{21 b^2 x^7}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b} \]
((b + a*x^4)^(3/4)*(-3*b + 4*a*x^4))/(21*b^2*x^7) - ArcTan[(2^(1/4)*a^(1/4 )*x)/(b + a*x^4)^(1/4)]/(2*2^(1/4)*a^(1/4)*b) - ArcTanh[(2^(1/4)*a^(1/4)*x )/(b + a*x^4)^(1/4)]/(2*2^(1/4)*a^(1/4)*b)
Time = 0.77 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.11, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, 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.
| \(\displaystyle \int \frac {a x^4-b+x^8}{x^8 \left (a x^4-b\right ) \sqrt [4]{a x^4+b}} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {1}{\left (a x^4-b\right ) \sqrt [4]{a x^4+b}}+\frac {1}{x^8 \sqrt [4]{a x^4+b}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b}+\frac {4 a \left (a x^4+b\right )^{3/4}}{21 b^2 x^3}-\frac {\left (a x^4+b\right )^{3/4}}{7 b x^7}\) |
-1/7*(b + a*x^4)^(3/4)/(b*x^7) + (4*a*(b + a*x^4)^(3/4))/(21*b^2*x^3) - Ar cTan[(2^(1/4)*a^(1/4)*x)/(b + a*x^4)^(1/4)]/(2*2^(1/4)*a^(1/4)*b) - ArcTan h[(2^(1/4)*a^(1/4)*x)/(b + a*x^4)^(1/4)]/(2*2^(1/4)*a^(1/4)*b)
3.18.3.3.1 Defintions of rubi rules used
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 = 0.36 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.16
| method | result | size |
| pseudoelliptic | \(\frac {\left (42 \arctan \left (\frac {2^{\frac {3}{4}} \left (a \,x^{4}+b \right )^{\frac {1}{4}}}{2 a^{\frac {1}{4}} x}\right ) b \,x^{7}-21 \ln \left (\frac {-x 2^{\frac {1}{4}} a^{\frac {1}{4}}-\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{x 2^{\frac {1}{4}} a^{\frac {1}{4}}-\left (a \,x^{4}+b \right )^{\frac {1}{4}}}\right ) b \,x^{7}+16 a^{\frac {5}{4}} \left (a \,x^{4}+b \right )^{\frac {3}{4}} x^{4} 2^{\frac {1}{4}}-12 b 2^{\frac {1}{4}} a^{\frac {1}{4}} \left (a \,x^{4}+b \right )^{\frac {3}{4}}\right ) 2^{\frac {3}{4}}}{168 x^{7} a^{\frac {1}{4}} b^{2}}\) | \(132\) |
1/168*(42*arctan(1/2*2^(3/4)/a^(1/4)/x*(a*x^4+b)^(1/4))*b*x^7-21*ln((-x*2^ (1/4)*a^(1/4)-(a*x^4+b)^(1/4))/(x*2^(1/4)*a^(1/4)-(a*x^4+b)^(1/4)))*b*x^7+ 16*a^(5/4)*(a*x^4+b)^(3/4)*x^4*2^(1/4)-12*b*2^(1/4)*a^(1/4)*(a*x^4+b)^(3/4 ))/x^7*2^(3/4)/a^(1/4)/b^2
Result contains complex when optimal does not.
Time = 100.28 (sec) , antiderivative size = 558, normalized size of antiderivative = 4.89 \[ \int \frac {-b+a x^4+x^8}{x^8 \left (-b+a x^4\right ) \sqrt [4]{b+a x^4}} \, dx=-\frac {21 \, \left (\frac {1}{2}\right )^{\frac {1}{4}} b^{2} x^{7} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (\frac {4 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {a x^{4} + b} a b^{3} x^{2} \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}} + 4 \, \sqrt {\frac {1}{2}} {\left (a x^{4} + b\right )}^{\frac {1}{4}} a b^{2} x^{3} \sqrt {\frac {1}{a b^{4}}} + 2 \, {\left (a x^{4} + b\right )}^{\frac {3}{4}} x + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (3 \, a b x^{4} + b^{2}\right )} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}}}{2 \, {\left (a x^{4} - b\right )}}\right ) - 21 \, \left (\frac {1}{2}\right )^{\frac {1}{4}} b^{2} x^{7} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {4 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {a x^{4} + b} a b^{3} x^{2} \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}} - 4 \, \sqrt {\frac {1}{2}} {\left (a x^{4} + b\right )}^{\frac {1}{4}} a b^{2} x^{3} \sqrt {\frac {1}{a b^{4}}} - 2 \, {\left (a x^{4} + b\right )}^{\frac {3}{4}} x + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (3 \, a b x^{4} + b^{2}\right )} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}}}{2 \, {\left (a x^{4} - b\right )}}\right ) + 21 i \, \left (\frac {1}{2}\right )^{\frac {1}{4}} b^{2} x^{7} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {4 i \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {a x^{4} + b} a b^{3} x^{2} \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}} + 4 \, \sqrt {\frac {1}{2}} {\left (a x^{4} + b\right )}^{\frac {1}{4}} a b^{2} x^{3} \sqrt {\frac {1}{a b^{4}}} - 2 \, {\left (a x^{4} + b\right )}^{\frac {3}{4}} x + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (-3 i \, a b x^{4} - i \, b^{2}\right )} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}}}{2 \, {\left (a x^{4} - b\right )}}\right ) - 21 i \, \left (\frac {1}{2}\right )^{\frac {1}{4}} b^{2} x^{7} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {-4 i \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {a x^{4} + b} a b^{3} x^{2} \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}} + 4 \, \sqrt {\frac {1}{2}} {\left (a x^{4} + b\right )}^{\frac {1}{4}} a b^{2} x^{3} \sqrt {\frac {1}{a b^{4}}} - 2 \, {\left (a x^{4} + b\right )}^{\frac {3}{4}} x + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (3 i \, a b x^{4} + i \, b^{2}\right )} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}}}{2 \, {\left (a x^{4} - b\right )}}\right ) - 8 \, {\left (4 \, a x^{4} - 3 \, b\right )} {\left (a x^{4} + b\right )}^{\frac {3}{4}}}{168 \, b^{2} x^{7}} \]
-1/168*(21*(1/2)^(1/4)*b^2*x^7*(1/(a*b^4))^(1/4)*log(1/2*(4*(1/2)^(3/4)*sq rt(a*x^4 + b)*a*b^3*x^2*(1/(a*b^4))^(3/4) + 4*sqrt(1/2)*(a*x^4 + b)^(1/4)* a*b^2*x^3*sqrt(1/(a*b^4)) + 2*(a*x^4 + b)^(3/4)*x + (1/2)^(1/4)*(3*a*b*x^4 + b^2)*(1/(a*b^4))^(1/4))/(a*x^4 - b)) - 21*(1/2)^(1/4)*b^2*x^7*(1/(a*b^4 ))^(1/4)*log(-1/2*(4*(1/2)^(3/4)*sqrt(a*x^4 + b)*a*b^3*x^2*(1/(a*b^4))^(3/ 4) - 4*sqrt(1/2)*(a*x^4 + b)^(1/4)*a*b^2*x^3*sqrt(1/(a*b^4)) - 2*(a*x^4 + b)^(3/4)*x + (1/2)^(1/4)*(3*a*b*x^4 + b^2)*(1/(a*b^4))^(1/4))/(a*x^4 - b)) + 21*I*(1/2)^(1/4)*b^2*x^7*(1/(a*b^4))^(1/4)*log(-1/2*(4*I*(1/2)^(3/4)*sq rt(a*x^4 + b)*a*b^3*x^2*(1/(a*b^4))^(3/4) + 4*sqrt(1/2)*(a*x^4 + b)^(1/4)* a*b^2*x^3*sqrt(1/(a*b^4)) - 2*(a*x^4 + b)^(3/4)*x + (1/2)^(1/4)*(-3*I*a*b* x^4 - I*b^2)*(1/(a*b^4))^(1/4))/(a*x^4 - b)) - 21*I*(1/2)^(1/4)*b^2*x^7*(1 /(a*b^4))^(1/4)*log(-1/2*(-4*I*(1/2)^(3/4)*sqrt(a*x^4 + b)*a*b^3*x^2*(1/(a *b^4))^(3/4) + 4*sqrt(1/2)*(a*x^4 + b)^(1/4)*a*b^2*x^3*sqrt(1/(a*b^4)) - 2 *(a*x^4 + b)^(3/4)*x + (1/2)^(1/4)*(3*I*a*b*x^4 + I*b^2)*(1/(a*b^4))^(1/4) )/(a*x^4 - b)) - 8*(4*a*x^4 - 3*b)*(a*x^4 + b)^(3/4))/(b^2*x^7)
\[ \int \frac {-b+a x^4+x^8}{x^8 \left (-b+a x^4\right ) \sqrt [4]{b+a x^4}} \, dx=\int \frac {a x^{4} - b + x^{8}}{x^{8} \left (a x^{4} - b\right ) \sqrt [4]{a x^{4} + b}}\, dx \]
\[ \int \frac {-b+a x^4+x^8}{x^8 \left (-b+a x^4\right ) \sqrt [4]{b+a x^4}} \, dx=\int { \frac {x^{8} + a x^{4} - b}{{\left (a x^{4} + b\right )}^{\frac {1}{4}} {\left (a x^{4} - b\right )} x^{8}} \,d x } \]
\[ \int \frac {-b+a x^4+x^8}{x^8 \left (-b+a x^4\right ) \sqrt [4]{b+a x^4}} \, dx=\int { \frac {x^{8} + a x^{4} - b}{{\left (a x^{4} + b\right )}^{\frac {1}{4}} {\left (a x^{4} - b\right )} x^{8}} \,d x } \]
Timed out. \[ \int \frac {-b+a x^4+x^8}{x^8 \left (-b+a x^4\right ) \sqrt [4]{b+a x^4}} \, dx=-\int \frac {x^8+a\,x^4-b}{x^8\,{\left (a\,x^4+b\right )}^{1/4}\,\left (b-a\,x^4\right )} \,d x \]