Integrand size = 36, antiderivative size = 59 \[ \int \frac {b+a x^8}{\sqrt [4]{b-a x^8} \left (-b+c x^4+a x^8\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{b-a x^8}}\right )}{2 \sqrt [4]{c}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{b-a x^8}}\right )}{2 \sqrt [4]{c}} \] Output:
-1/2*arctan(c^(1/4)*x/(-a*x^8+b)^(1/4))/c^(1/4)-1/2*arctanh(c^(1/4)*x/(-a* x^8+b)^(1/4))/c^(1/4)
Time = 9.37 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.85 \[ \int \frac {b+a x^8}{\sqrt [4]{b-a x^8} \left (-b+c x^4+a x^8\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{b-a x^8}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{b-a x^8}}\right )}{2 \sqrt [4]{c}} \] Input:
Integrate[(b + a*x^8)/((b - a*x^8)^(1/4)*(-b + c*x^4 + a*x^8)),x]
Output:
-1/2*(ArcTan[(c^(1/4)*x)/(b - a*x^8)^(1/4)] + ArcTanh[(c^(1/4)*x)/(b - a*x ^8)^(1/4)])/c^(1/4)
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 1.49 (sec) , antiderivative size = 460, normalized size of antiderivative = 7.80, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {7279, 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.
\(\displaystyle \int \frac {a x^8+b}{\sqrt [4]{b-a x^8} \left (a x^8-b+c x^4\right )} \, dx\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \int \left (\frac {2 b-c x^4}{\sqrt [4]{b-a x^8} \left (a x^8-b+c x^4\right )}+\frac {1}{\sqrt [4]{b-a x^8}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {x \sqrt [4]{1-\frac {a x^8}{b}} \operatorname {AppellF1}\left (\frac {1}{8},1,\frac {1}{4},\frac {9}{8},\frac {2 a^2 x^8}{c^2-\sqrt {c^2+4 a b} c+2 a b},\frac {a x^8}{b}\right )}{\sqrt [4]{b-a x^8}}-\frac {x \sqrt [4]{1-\frac {a x^8}{b}} \operatorname {AppellF1}\left (\frac {1}{8},1,\frac {1}{4},\frac {9}{8},\frac {2 a^2 x^8}{2 a b+c \left (c+\sqrt {c^2+4 a b}\right )},\frac {a x^8}{b}\right )}{\sqrt [4]{b-a x^8}}+\frac {a x^5 \left (c-\sqrt {4 a b+c^2}\right ) \sqrt [4]{1-\frac {a x^8}{b}} \operatorname {AppellF1}\left (\frac {5}{8},1,\frac {1}{4},\frac {13}{8},\frac {2 a^2 x^8}{c^2-\sqrt {c^2+4 a b} c+2 a b},\frac {a x^8}{b}\right )}{5 \left (c \left (c-\sqrt {4 a b+c^2}\right )+2 a b\right ) \sqrt [4]{b-a x^8}}+\frac {a x^5 \left (\sqrt {4 a b+c^2}+c\right ) \sqrt [4]{1-\frac {a x^8}{b}} \operatorname {AppellF1}\left (\frac {5}{8},1,\frac {1}{4},\frac {13}{8},\frac {2 a^2 x^8}{2 a b+c \left (c+\sqrt {c^2+4 a b}\right )},\frac {a x^8}{b}\right )}{5 \left (c \left (\sqrt {4 a b+c^2}+c\right )+2 a b\right ) \sqrt [4]{b-a x^8}}+\frac {x \sqrt [4]{1-\frac {a x^8}{b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{8},\frac {1}{4},\frac {9}{8},\frac {a x^8}{b}\right )}{\sqrt [4]{b-a x^8}}\) |
Input:
Int[(b + a*x^8)/((b - a*x^8)^(1/4)*(-b + c*x^4 + a*x^8)),x]
Output:
-((x*(1 - (a*x^8)/b)^(1/4)*AppellF1[1/8, 1, 1/4, 9/8, (2*a^2*x^8)/(2*a*b + c^2 - c*Sqrt[4*a*b + c^2]), (a*x^8)/b])/(b - a*x^8)^(1/4)) - (x*(1 - (a*x ^8)/b)^(1/4)*AppellF1[1/8, 1, 1/4, 9/8, (2*a^2*x^8)/(2*a*b + c*(c + Sqrt[4 *a*b + c^2])), (a*x^8)/b])/(b - a*x^8)^(1/4) + (a*(c - Sqrt[4*a*b + c^2])* x^5*(1 - (a*x^8)/b)^(1/4)*AppellF1[5/8, 1, 1/4, 13/8, (2*a^2*x^8)/(2*a*b + c^2 - c*Sqrt[4*a*b + c^2]), (a*x^8)/b])/(5*(2*a*b + c*(c - Sqrt[4*a*b + c ^2]))*(b - a*x^8)^(1/4)) + (a*(c + Sqrt[4*a*b + c^2])*x^5*(1 - (a*x^8)/b)^ (1/4)*AppellF1[5/8, 1, 1/4, 13/8, (2*a^2*x^8)/(2*a*b + c*(c + Sqrt[4*a*b + c^2])), (a*x^8)/b])/(5*(2*a*b + c*(c + Sqrt[4*a*b + c^2]))*(b - a*x^8)^(1 /4)) + (x*(1 - (a*x^8)/b)^(1/4)*Hypergeometric2F1[1/8, 1/4, 9/8, (a*x^8)/b ])/(b - a*x^8)^(1/4)
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 0.56 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.19
method | result | size |
pseudoelliptic | \(\frac {2 \arctan \left (\frac {\left (-a \,x^{8}+b \right )^{\frac {1}{4}}}{c^{\frac {1}{4}} x}\right )-\ln \left (\frac {-c^{\frac {1}{4}} x -\left (-a \,x^{8}+b \right )^{\frac {1}{4}}}{c^{\frac {1}{4}} x -\left (-a \,x^{8}+b \right )^{\frac {1}{4}}}\right )}{4 c^{\frac {1}{4}}}\) | \(70\) |
Input:
int((a*x^8+b)/(-a*x^8+b)^(1/4)/(a*x^8+c*x^4-b),x,method=_RETURNVERBOSE)
Output:
1/4/c^(1/4)*(2*arctan(1/c^(1/4)/x*(-a*x^8+b)^(1/4))-ln((-c^(1/4)*x-(-a*x^8 +b)^(1/4))/(c^(1/4)*x-(-a*x^8+b)^(1/4))))
Timed out. \[ \int \frac {b+a x^8}{\sqrt [4]{b-a x^8} \left (-b+c x^4+a x^8\right )} \, dx=\text {Timed out} \] Input:
integrate((a*x^8+b)/(-a*x^8+b)^(1/4)/(a*x^8+c*x^4-b),x, algorithm="fricas" )
Output:
Timed out
Timed out. \[ \int \frac {b+a x^8}{\sqrt [4]{b-a x^8} \left (-b+c x^4+a x^8\right )} \, dx=\text {Timed out} \] Input:
integrate((a*x**8+b)/(-a*x**8+b)**(1/4)/(a*x**8+c*x**4-b),x)
Output:
Timed out
\[ \int \frac {b+a x^8}{\sqrt [4]{b-a x^8} \left (-b+c x^4+a x^8\right )} \, dx=\int { \frac {a x^{8} + b}{{\left (a x^{8} + c x^{4} - b\right )} {\left (-a x^{8} + b\right )}^{\frac {1}{4}}} \,d x } \] Input:
integrate((a*x^8+b)/(-a*x^8+b)^(1/4)/(a*x^8+c*x^4-b),x, algorithm="maxima" )
Output:
integrate((a*x^8 + b)/((a*x^8 + c*x^4 - b)*(-a*x^8 + b)^(1/4)), x)
\[ \int \frac {b+a x^8}{\sqrt [4]{b-a x^8} \left (-b+c x^4+a x^8\right )} \, dx=\int { \frac {a x^{8} + b}{{\left (a x^{8} + c x^{4} - b\right )} {\left (-a x^{8} + b\right )}^{\frac {1}{4}}} \,d x } \] Input:
integrate((a*x^8+b)/(-a*x^8+b)^(1/4)/(a*x^8+c*x^4-b),x, algorithm="giac")
Output:
integrate((a*x^8 + b)/((a*x^8 + c*x^4 - b)*(-a*x^8 + b)^(1/4)), x)
Timed out. \[ \int \frac {b+a x^8}{\sqrt [4]{b-a x^8} \left (-b+c x^4+a x^8\right )} \, dx=\int \frac {a\,x^8+b}{{\left (b-a\,x^8\right )}^{1/4}\,\left (a\,x^8+c\,x^4-b\right )} \,d x \] Input:
int((b + a*x^8)/((b - a*x^8)^(1/4)*(a*x^8 - b + c*x^4)),x)
Output:
int((b + a*x^8)/((b - a*x^8)^(1/4)*(a*x^8 - b + c*x^4)), x)
\[ \int \frac {b+a x^8}{\sqrt [4]{b-a x^8} \left (-b+c x^4+a x^8\right )} \, dx=\left (\int \frac {x^{8}}{\left (-a \,x^{8}+b \right )^{\frac {1}{4}} a \,x^{8}-\left (-a \,x^{8}+b \right )^{\frac {1}{4}} b +\left (-a \,x^{8}+b \right )^{\frac {1}{4}} c \,x^{4}}d x \right ) a +\left (\int \frac {1}{\left (-a \,x^{8}+b \right )^{\frac {1}{4}} a \,x^{8}-\left (-a \,x^{8}+b \right )^{\frac {1}{4}} b +\left (-a \,x^{8}+b \right )^{\frac {1}{4}} c \,x^{4}}d x \right ) b \] Input:
int((a*x^8+b)/(-a*x^8+b)^(1/4)/(a*x^8+c*x^4-b),x)
Output:
int(x**8/(( - a*x**8 + b)**(1/4)*a*x**8 - ( - a*x**8 + b)**(1/4)*b + ( - a *x**8 + b)**(1/4)*c*x**4),x)*a + int(1/(( - a*x**8 + b)**(1/4)*a*x**8 - ( - a*x**8 + b)**(1/4)*b + ( - a*x**8 + b)**(1/4)*c*x**4),x)*b