\(\int \frac {-b+a x^4+x^8}{x^8 (-b+a x^4) \sqrt [4]{b+a x^4}} \, dx\) [1703]

Optimal result

Integrand size = 38, antiderivative size = 114 \[ \int \frac {-b+a x^4+x^8}{x^8 \left (-b+a x^4\right ) \sqrt [4]{b+a x^4}} \, dx=\frac {\left (b+a x^4\right )^{3/4} \left (-3 b+4 a x^4\right )}{21 b^2 x^7}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.11, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.184, Rules used = {6857, 277, 270, 385, 218, 212, 209} \[ \int \frac {-b+a x^4+x^8}{x^8 \left (-b+a x^4\right ) \sqrt [4]{b+a x^4}} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b}+\frac {4 a \left (a x^4+b\right )^{3/4}}{21 b^2 x^3}-\frac {\left (a x^4+b\right )^{3/4}}{7 b x^7} \]

[In]

Int[(-b + a*x^4 + x^8)/(x^8*(-b + a*x^4)*(b + a*x^4)^(1/4)),x]

[Out]

-1/7*(b + a*x^4)^(3/4)/(b*x^7) + (4*a*(b + a*x^4)^(3/4))/(21*b^2*x^3) - 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)

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 218

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]},
Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !Gt
Q[a/b, 0]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 277

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^n)^(p + 1)/(a*(m + 1))), x]
- Dist[b*((m + n*(p + 1) + 1)/(a*(m + 1))), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{x^8 \sqrt [4]{b+a x^4}}+\frac {1}{\left (-b+a x^4\right ) \sqrt [4]{b+a x^4}}\right ) \, dx \\ & = \int \frac {1}{x^8 \sqrt [4]{b+a x^4}} \, dx+\int \frac {1}{\left (-b+a x^4\right ) \sqrt [4]{b+a x^4}} \, dx \\ & = -\frac {\left (b+a x^4\right )^{3/4}}{7 b x^7}-\frac {(4 a) \int \frac {1}{x^4 \sqrt [4]{b+a x^4}} \, dx}{7 b}+\text {Subst}\left (\int \frac {1}{-b+2 a b x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right ) \\ & = -\frac {\left (b+a x^4\right )^{3/4}}{7 b x^7}+\frac {4 a \left (b+a x^4\right )^{3/4}}{21 b^2 x^3}-\frac {\text {Subst}\left (\int \frac {1}{1-\sqrt {2} \sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{2 b}-\frac {\text {Subst}\left (\int \frac {1}{1+\sqrt {2} \sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{2 b} \\ & = -\frac {\left (b+a x^4\right )^{3/4}}{7 b x^7}+\frac {4 a \left (b+a x^4\right )^{3/4}}{21 b^2 x^3}-\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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.86 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00 \[ \int \frac {-b+a x^4+x^8}{x^8 \left (-b+a x^4\right ) \sqrt [4]{b+a x^4}} \, dx=\frac {\left (b+a x^4\right )^{3/4} \left (-3 b+4 a x^4\right )}{21 b^2 x^7}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b} \]

[In]

Integrate[(-b + a*x^4 + x^8)/(x^8*(-b + a*x^4)*(b + a*x^4)^(1/4)),x]

[Out]

((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)

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.16

method	result	size
pseudoelliptic	\(\frac {\left (42 \arctan \left (\frac {2^{\frac {3}{4}} \left (a \,x^{4}+b \right )^{\frac {1}{4}}}{2 a^{\frac {1}{4}} x}\right ) b \,x^{7}-21 \ln \left (\frac {-x 2^{\frac {1}{4}} a^{\frac {1}{4}}-\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{x 2^{\frac {1}{4}} a^{\frac {1}{4}}-\left (a \,x^{4}+b \right )^{\frac {1}{4}}}\right ) b \,x^{7}+16 a^{\frac {5}{4}} \left (a \,x^{4}+b \right )^{\frac {3}{4}} x^{4} 2^{\frac {1}{4}}-12 b 2^{\frac {1}{4}} a^{\frac {1}{4}} \left (a \,x^{4}+b \right )^{\frac {3}{4}}\right ) 2^{\frac {3}{4}}}{168 x^{7} a^{\frac {1}{4}} b^{2}}\)	\(132\)

[In]

int((x^8+a*x^4-b)/x^8/(a*x^4-b)/(a*x^4+b)^(1/4),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [C] (verification not implemented)

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}} \]

[In]

integrate((x^8+a*x^4-b)/x^8/(a*x^4-b)/(a*x^4+b)^(1/4),x, algorithm="fricas")

[Out]

-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)*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*(1/2)^(1/4)*b^2*x^7*(1/(a*b^4))^(1/4)*log(-1/2*(4*(1/2)^(3/4)*s
qrt(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)*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)) - 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)

Sympy [F]

\[ \int \frac {-b+a x^4+x^8}{x^8 \left (-b+a x^4\right ) \sqrt [4]{b+a x^4}} \, dx=\int \frac {a x^{4} - b + x^{8}}{x^{8} \left (a x^{4} - b\right ) \sqrt [4]{a x^{4} + b}}\, dx \]

[In]

integrate((x**8+a*x**4-b)/x**8/(a*x**4-b)/(a*x**4+b)**(1/4),x)

[Out]

Integral((a*x**4 - b + x**8)/(x**8*(a*x**4 - b)*(a*x**4 + b)**(1/4)), x)

Maxima [F]

\[ \int \frac {-b+a x^4+x^8}{x^8 \left (-b+a x^4\right ) \sqrt [4]{b+a x^4}} \, dx=\int { \frac {x^{8} + a x^{4} - b}{{\left (a x^{4} + b\right )}^{\frac {1}{4}} {\left (a x^{4} - b\right )} x^{8}} \,d x } \]

[In]

integrate((x^8+a*x^4-b)/x^8/(a*x^4-b)/(a*x^4+b)^(1/4),x, algorithm="maxima")

[Out]

integrate((x^8 + a*x^4 - b)/((a*x^4 + b)^(1/4)*(a*x^4 - b)*x^8), x)

Giac [F]

\[ \int \frac {-b+a x^4+x^8}{x^8 \left (-b+a x^4\right ) \sqrt [4]{b+a x^4}} \, dx=\int { \frac {x^{8} + a x^{4} - b}{{\left (a x^{4} + b\right )}^{\frac {1}{4}} {\left (a x^{4} - b\right )} x^{8}} \,d x } \]

[In]

integrate((x^8+a*x^4-b)/x^8/(a*x^4-b)/(a*x^4+b)^(1/4),x, algorithm="giac")

[Out]

integrate((x^8 + a*x^4 - b)/((a*x^4 + b)^(1/4)*(a*x^4 - b)*x^8), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {-b+a x^4+x^8}{x^8 \left (-b+a x^4\right ) \sqrt [4]{b+a x^4}} \, dx=-\int \frac {x^8+a\,x^4-b}{x^8\,{\left (a\,x^4+b\right )}^{1/4}\,\left (b-a\,x^4\right )} \,d x \]

[In]

int(-(a*x^4 - b + x^8)/(x^8*(b + a*x^4)^(1/4)*(b - a*x^4)),x)

[Out]

-int((a*x^4 - b + x^8)/(x^8*(b + a*x^4)^(1/4)*(b - a*x^4)), x)