\(\int \frac {\sqrt [4]{-b+a x^4} (b+c x^4+a x^8)}{x^6 (b+2 a x^8)} \, dx\) [2554]

Optimal result

Integrand size = 39, antiderivative size = 215 \[ \int \frac {\sqrt [4]{-b+a x^4} \left (b+c x^4+a x^8\right )}{x^6 \left (b+2 a x^8\right )} \, dx=\frac {\sqrt [4]{-b+a x^4} \left (-b+a x^4-5 c x^4\right )}{5 b x^5}-\frac {a \text {RootSum}\left [a^2+2 a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-a c \log (x)-2 b c \log (x)+a c \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )+2 b c \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )+b \log (x) \text {$\#$1}^4+c \log (x) \text {$\#$1}^4-b \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4-c \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}^3-\text {$\#$1}^7}\&\right ]}{8 b} \]

[Out]

Unintegrable

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(933\) vs. \(2(215)=430\).

Time = 2.86 (sec) , antiderivative size = 933, normalized size of antiderivative = 4.34, number of steps used = 43, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {6857, 270, 283, 338, 304, 209, 212, 1543, 525, 524, 1533, 508} \[ \int \frac {\sqrt [4]{-b+a x^4} \left (b+c x^4+a x^8\right )}{x^6 \left (b+2 a x^8\right )} \, dx=-\frac {a \sqrt [4]{a x^4-b} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-\frac {\sqrt {2} \sqrt {-a} x^4}{\sqrt {b}},\frac {a x^4}{b}\right ) x^3}{6 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 {2} \sqrt {-a} x^4}{\sqrt {b}},\frac {a x^4}{b}\right ) x^3}{6 b \sqrt [4]{1-\frac {a x^4}{b}}}+\frac {\sqrt {-a} c \arctan \left (\frac {\sqrt [4]{\sqrt {2} a-2 \sqrt {-a} \sqrt {b}} x}{\sqrt [8]{2} \sqrt [4]{a x^4-b}}\right )}{2 \sqrt [8]{2} \left (\sqrt {2} a-2 \sqrt {-a} \sqrt {b}\right )^{3/4} \sqrt {b}}-\frac {a c \arctan \left (\frac {\sqrt [4]{\sqrt {2} a-2 \sqrt {-a} \sqrt {b}} x}{\sqrt [8]{2} \sqrt [4]{a x^4-b}}\right )}{2\ 2^{5/8} \left (\sqrt {2} a-2 \sqrt {-a} \sqrt {b}\right )^{3/4} b}-\frac {\sqrt {-a} c \arctan \left (\frac {\sqrt [4]{\sqrt {2} a+2 \sqrt {-a} \sqrt {b}} x}{\sqrt [8]{2} \sqrt [4]{a x^4-b}}\right )}{2 \sqrt [8]{2} \left (\sqrt {2} a+2 \sqrt {-a} \sqrt {b}\right )^{3/4} \sqrt {b}}-\frac {a c \arctan \left (\frac {\sqrt [4]{\sqrt {2} a+2 \sqrt {-a} \sqrt {b}} x}{\sqrt [8]{2} \sqrt [4]{a x^4-b}}\right )}{2\ 2^{5/8} \left (\sqrt {2} a+2 \sqrt {-a} \sqrt {b}\right )^{3/4} b}-\frac {\sqrt {-a} c \text {arctanh}\left (\frac {\sqrt [4]{\sqrt {2} a-2 \sqrt {-a} \sqrt {b}} x}{\sqrt [8]{2} \sqrt [4]{a x^4-b}}\right )}{2 \sqrt [8]{2} \left (\sqrt {2} a-2 \sqrt {-a} \sqrt {b}\right )^{3/4} \sqrt {b}}+\frac {a c \text {arctanh}\left (\frac {\sqrt [4]{\sqrt {2} a-2 \sqrt {-a} \sqrt {b}} x}{\sqrt [8]{2} \sqrt [4]{a x^4-b}}\right )}{2\ 2^{5/8} \left (\sqrt {2} a-2 \sqrt {-a} \sqrt {b}\right )^{3/4} b}+\frac {\sqrt {-a} c \text {arctanh}\left (\frac {\sqrt [4]{\sqrt {2} a+2 \sqrt {-a} \sqrt {b}} x}{\sqrt [8]{2} \sqrt [4]{a x^4-b}}\right )}{2 \sqrt [8]{2} \left (\sqrt {2} a+2 \sqrt {-a} \sqrt {b}\right )^{3/4} \sqrt {b}}+\frac {a c \text {arctanh}\left (\frac {\sqrt [4]{\sqrt {2} a+2 \sqrt {-a} \sqrt {b}} x}{\sqrt [8]{2} \sqrt [4]{a x^4-b}}\right )}{2\ 2^{5/8} \left (\sqrt {2} a+2 \sqrt {-a} \sqrt {b}\right )^{3/4} b}-\frac {c \sqrt [4]{a x^4-b}}{b x}+\frac {\left (a x^4-b\right )^{5/4}}{5 b x^5} \]

[In]

Int[((-b + a*x^4)^(1/4)*(b + c*x^4 + a*x^8))/(x^6*(b + 2*a*x^8)),x]

[Out]

-((c*(-b + a*x^4)^(1/4))/(b*x)) + (-b + a*x^4)^(5/4)/(5*b*x^5) - (a*x^3*(-b + a*x^4)^(1/4)*AppellF1[3/4, 1, -1
/4, 7/4, -((Sqrt[2]*Sqrt[-a]*x^4)/Sqrt[b]), (a*x^4)/b])/(6*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[2]*Sqrt[-a]*x^4)/Sqrt[b], (a*x^4)/b])/(6*b*(1 - (a*x^4)/b)^(1/4)) - (a*c*A
rcTan[((Sqrt[2]*a - 2*Sqrt[-a]*Sqrt[b])^(1/4)*x)/(2^(1/8)*(-b + a*x^4)^(1/4))])/(2*2^(5/8)*(Sqrt[2]*a - 2*Sqrt
[-a]*Sqrt[b])^(3/4)*b) + (Sqrt[-a]*c*ArcTan[((Sqrt[2]*a - 2*Sqrt[-a]*Sqrt[b])^(1/4)*x)/(2^(1/8)*(-b + a*x^4)^(
1/4))])/(2*2^(1/8)*(Sqrt[2]*a - 2*Sqrt[-a]*Sqrt[b])^(3/4)*Sqrt[b]) - (a*c*ArcTan[((Sqrt[2]*a + 2*Sqrt[-a]*Sqrt
[b])^(1/4)*x)/(2^(1/8)*(-b + a*x^4)^(1/4))])/(2*2^(5/8)*(Sqrt[2]*a + 2*Sqrt[-a]*Sqrt[b])^(3/4)*b) - (Sqrt[-a]*
c*ArcTan[((Sqrt[2]*a + 2*Sqrt[-a]*Sqrt[b])^(1/4)*x)/(2^(1/8)*(-b + a*x^4)^(1/4))])/(2*2^(1/8)*(Sqrt[2]*a + 2*S
qrt[-a]*Sqrt[b])^(3/4)*Sqrt[b]) + (a*c*ArcTanh[((Sqrt[2]*a - 2*Sqrt[-a]*Sqrt[b])^(1/4)*x)/(2^(1/8)*(-b + a*x^4
)^(1/4))])/(2*2^(5/8)*(Sqrt[2]*a - 2*Sqrt[-a]*Sqrt[b])^(3/4)*b) - (Sqrt[-a]*c*ArcTanh[((Sqrt[2]*a - 2*Sqrt[-a]
*Sqrt[b])^(1/4)*x)/(2^(1/8)*(-b + a*x^4)^(1/4))])/(2*2^(1/8)*(Sqrt[2]*a - 2*Sqrt[-a]*Sqrt[b])^(3/4)*Sqrt[b]) +
(a*c*ArcTanh[((Sqrt[2]*a + 2*Sqrt[-a]*Sqrt[b])^(1/4)*x)/(2^(1/8)*(-b + a*x^4)^(1/4))])/(2*2^(5/8)*(Sqrt[2]*a
+ 2*Sqrt[-a]*Sqrt[b])^(3/4)*b) + (Sqrt[-a]*c*ArcTanh[((Sqrt[2]*a + 2*Sqrt[-a]*Sqrt[b])^(1/4)*x)/(2^(1/8)*(-b +
a*x^4)^(1/4))])/(2*2^(1/8)*(Sqrt[2]*a + 2*Sqrt[-a]*Sqrt[b])^(3/4)*Sqrt[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 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 283

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

Rule 304

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

Rule 338

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + (m + 1)/n), Subst[Int[x^m/(1 - b*x^n)^(
p + (m + 1)/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[
p, -2^(-1)] && IntegersQ[m, p + (m + 1)/n]

Rule 508

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{k = Denominato
r[p]}, Dist[k*(a^(p + (m + 1)/n)/n), Subst[Int[x^(k*((m + 1)/n) - 1)*((c - (b*c - a*d)*x^k)^q/(1 - b*x^k)^(p +
q + (m + 1)/n + 1)), x], x, x^(n/k)/(a + b*x^n)^(1/k)], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && Ration
alQ[m, p] && IntegersQ[p + (m + 1)/n, q] && LtQ[-1, p, 0]

Rule 524

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*
((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 525

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPar
t[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]), Int[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x
] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] &&  !(IntegerQ[
p] || GtQ[a, 0])

Rule 1533

Int[(((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_)^(n_))^(q_))/((a_) + (c_.)*(x_)^(n2_.)), x_Symbol] :> Dist[e*(f^n/c
), Int[(f*x)^(m - n)*(d + e*x^n)^(q - 1), x], x] - Dist[f^n/c, Int[(f*x)^(m - n)*(d + e*x^n)^(q - 1)*(Simp[a*e
- c*d*x^n, x]/(a + c*x^(2*n))), x], x] /; FreeQ[{a, c, d, e, f}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] &&  !Intege
rQ[q] && GtQ[q, 0] && GtQ[m, n - 1] && LeQ[m, 2*n - 1]

Rule 1543

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))^(q_))/((a_) + (c_.)*(x_)^(n2_.)), x_Symbol] :> Int[ExpandInte
grand[(d + e*x^n)^q, (f*x)^m/(a + c*x^(2*n)), x], x] /; FreeQ[{a, c, d, e, f, q, n}, x] && EqQ[n2, 2*n] && IGt
Q[n, 0] &&  !IntegerQ[q] && IntegerQ[m]

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 {\sqrt [4]{-b+a x^4}}{x^6}+\frac {c \sqrt [4]{-b+a x^4}}{b x^2}-\frac {a x^2 \sqrt [4]{-b+a x^4} \left (b+2 c x^4\right )}{b \left (b+2 a x^8\right )}\right ) \, dx \\ & = -\frac {a \int \frac {x^2 \sqrt [4]{-b+a x^4} \left (b+2 c x^4\right )}{b+2 a x^8} \, dx}{b}+\frac {c \int \frac {\sqrt [4]{-b+a x^4}}{x^2} \, dx}{b}+\int \frac {\sqrt [4]{-b+a x^4}}{x^6} \, dx \\ & = -\frac {c \sqrt [4]{-b+a x^4}}{b x}+\frac {\left (-b+a x^4\right )^{5/4}}{5 b x^5}-\frac {a \int \left (\frac {b x^2 \sqrt [4]{-b+a x^4}}{b+2 a x^8}+\frac {2 c x^6 \sqrt [4]{-b+a x^4}}{b+2 a x^8}\right ) \, dx}{b}+\frac {(a c) \int \frac {x^2}{\left (-b+a x^4\right )^{3/4}} \, dx}{b} \\ & = -\frac {c \sqrt [4]{-b+a x^4}}{b x}+\frac {\left (-b+a x^4\right )^{5/4}}{5 b x^5}-a \int \frac {x^2 \sqrt [4]{-b+a x^4}}{b+2 a x^8} \, dx+\frac {(a c) \text {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{b}-\frac {(2 a c) \int \frac {x^6 \sqrt [4]{-b+a x^4}}{b+2 a x^8} \, dx}{b} \\ & = -\frac {c \sqrt [4]{-b+a x^4}}{b x}+\frac {\left (-b+a x^4\right )^{5/4}}{5 b x^5}-a \int \left (-\frac {a x^2 \sqrt [4]{-b+a x^4}}{\sqrt {2} \sqrt {-a} \sqrt {b} \left (\sqrt {2} \sqrt {-a} \sqrt {b}-2 a x^4\right )}-\frac {a x^2 \sqrt [4]{-b+a x^4}}{\sqrt {2} \sqrt {-a} \sqrt {b} \left (\sqrt {2} \sqrt {-a} \sqrt {b}+2 a x^4\right )}\right ) \, dx+\frac {c \int \frac {x^2 \left (a b+2 a b x^4\right )}{\left (-b+a x^4\right )^{3/4} \left (b+2 a x^8\right )} \, dx}{b}+\frac {\left (\sqrt {a} c\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 b}-\frac {\left (\sqrt {a} c\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 b}-\frac {(a c) \int \frac {x^2}{\left (-b+a x^4\right )^{3/4}} \, dx}{b} \\ & = -\frac {c \sqrt [4]{-b+a x^4}}{b x}+\frac {\left (-b+a x^4\right )^{5/4}}{5 b x^5}-\frac {\sqrt [4]{a} c \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 b}+\frac {\sqrt [4]{a} c \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 b}+\frac {(-a)^{3/2} \int \frac {x^2 \sqrt [4]{-b+a x^4}}{\sqrt {2} \sqrt {-a} \sqrt {b}-2 a x^4} \, dx}{\sqrt {2} \sqrt {b}}+\frac {(-a)^{3/2} \int \frac {x^2 \sqrt [4]{-b+a x^4}}{\sqrt {2} \sqrt {-a} \sqrt {b}+2 a x^4} \, dx}{\sqrt {2} \sqrt {b}}+\frac {c \int \left (\frac {a b x^2}{\left (-b+a x^4\right )^{3/4} \left (b+2 a x^8\right )}+\frac {2 a b x^6}{\left (-b+a x^4\right )^{3/4} \left (b+2 a x^8\right )}\right ) \, dx}{b}-\frac {(a c) \text {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{b} \\ & = -\frac {c \sqrt [4]{-b+a x^4}}{b x}+\frac {\left (-b+a x^4\right )^{5/4}}{5 b x^5}-\frac {\sqrt [4]{a} c \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 b}+\frac {\sqrt [4]{a} c \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 b}+(a c) \int \frac {x^2}{\left (-b+a x^4\right )^{3/4} \left (b+2 a x^8\right )} \, dx+(2 a c) \int \frac {x^6}{\left (-b+a x^4\right )^{3/4} \left (b+2 a x^8\right )} \, dx-\frac {\left (\sqrt {a} c\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 b}+\frac {\left (\sqrt {a} c\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 b}+\frac {\left ((-a)^{3/2} \sqrt [4]{-b+a x^4}\right ) \int \frac {x^2 \sqrt [4]{1-\frac {a x^4}{b}}}{\sqrt {2} \sqrt {-a} \sqrt {b}-2 a x^4} \, dx}{\sqrt {2} \sqrt {b} \sqrt [4]{1-\frac {a x^4}{b}}}+\frac {\left ((-a)^{3/2} \sqrt [4]{-b+a x^4}\right ) \int \frac {x^2 \sqrt [4]{1-\frac {a x^4}{b}}}{\sqrt {2} \sqrt {-a} \sqrt {b}+2 a x^4} \, dx}{\sqrt {2} \sqrt {b} \sqrt [4]{1-\frac {a x^4}{b}}} \\ & = -\frac {c \sqrt [4]{-b+a x^4}}{b x}+\frac {\left (-b+a x^4\right )^{5/4}}{5 b x^5}-\frac {a x^3 \sqrt [4]{-b+a x^4} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-\frac {\sqrt {2} \sqrt {-a} x^4}{\sqrt {b}},\frac {a x^4}{b}\right )}{6 b \sqrt [4]{1-\frac {a x^4}{b}}}-\frac {a x^3 \sqrt [4]{-b+a x^4} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {\sqrt {2} \sqrt {-a} x^4}{\sqrt {b}},\frac {a x^4}{b}\right )}{6 b \sqrt [4]{1-\frac {a x^4}{b}}}+(a c) \int \left (-\frac {a x^2}{\sqrt {2} \sqrt {-a} \sqrt {b} \left (\sqrt {2} \sqrt {-a} \sqrt {b}-2 a x^4\right ) \left (-b+a x^4\right )^{3/4}}-\frac {a x^2}{\sqrt {2} \sqrt {-a} \sqrt {b} \left (-b+a x^4\right )^{3/4} \left (\sqrt {2} \sqrt {-a} \sqrt {b}+2 a x^4\right )}\right ) \, dx+(2 a c) \int \left (\frac {x^2}{2 \left (-b+a x^4\right )^{3/4} \left (-\sqrt {2} \sqrt {-a} \sqrt {b}+2 a x^4\right )}+\frac {x^2}{2 \left (-b+a x^4\right )^{3/4} \left (\sqrt {2} \sqrt {-a} \sqrt {b}+2 a x^4\right )}\right ) \, dx \\ & = -\frac {c \sqrt [4]{-b+a x^4}}{b x}+\frac {\left (-b+a x^4\right )^{5/4}}{5 b x^5}-\frac {a x^3 \sqrt [4]{-b+a x^4} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-\frac {\sqrt {2} \sqrt {-a} x^4}{\sqrt {b}},\frac {a x^4}{b}\right )}{6 b \sqrt [4]{1-\frac {a x^4}{b}}}-\frac {a x^3 \sqrt [4]{-b+a x^4} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {\sqrt {2} \sqrt {-a} x^4}{\sqrt {b}},\frac {a x^4}{b}\right )}{6 b \sqrt [4]{1-\frac {a x^4}{b}}}+(a c) \int \frac {x^2}{\left (-b+a x^4\right )^{3/4} \left (-\sqrt {2} \sqrt {-a} \sqrt {b}+2 a x^4\right )} \, dx+(a c) \int \frac {x^2}{\left (-b+a x^4\right )^{3/4} \left (\sqrt {2} \sqrt {-a} \sqrt {b}+2 a x^4\right )} \, dx-\frac {\left ((-a)^{3/2} c\right ) \int \frac {x^2}{\left (\sqrt {2} \sqrt {-a} \sqrt {b}-2 a x^4\right ) \left (-b+a x^4\right )^{3/4}} \, dx}{\sqrt {2} \sqrt {b}}-\frac {\left ((-a)^{3/2} c\right ) \int \frac {x^2}{\left (-b+a x^4\right )^{3/4} \left (\sqrt {2} \sqrt {-a} \sqrt {b}+2 a x^4\right )} \, dx}{\sqrt {2} \sqrt {b}} \\ & = -\frac {c \sqrt [4]{-b+a x^4}}{b x}+\frac {\left (-b+a x^4\right )^{5/4}}{5 b x^5}-\frac {a x^3 \sqrt [4]{-b+a x^4} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-\frac {\sqrt {2} \sqrt {-a} x^4}{\sqrt {b}},\frac {a x^4}{b}\right )}{6 b \sqrt [4]{1-\frac {a x^4}{b}}}-\frac {a x^3 \sqrt [4]{-b+a x^4} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {\sqrt {2} \sqrt {-a} x^4}{\sqrt {b}},\frac {a x^4}{b}\right )}{6 b \sqrt [4]{1-\frac {a x^4}{b}}}+(a c) \text {Subst}\left (\int \frac {x^2}{-\sqrt {2} \sqrt {-a} \sqrt {b}-\left (-\sqrt {2} \sqrt {-a} a \sqrt {b}+2 a b\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )+(a c) \text {Subst}\left (\int \frac {x^2}{\sqrt {2} \sqrt {-a} \sqrt {b}-\left (\sqrt {2} \sqrt {-a} a \sqrt {b}+2 a b\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )-\frac {\left ((-a)^{3/2} c\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {2} \sqrt {-a} \sqrt {b}-\left (\sqrt {2} \sqrt {-a} a \sqrt {b}-2 a b\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt {2} \sqrt {b}}-\frac {\left ((-a)^{3/2} c\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {2} \sqrt {-a} \sqrt {b}-\left (\sqrt {2} \sqrt {-a} a \sqrt {b}+2 a b\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt {2} \sqrt {b}} \\ & = -\frac {c \sqrt [4]{-b+a x^4}}{b x}+\frac {\left (-b+a x^4\right )^{5/4}}{5 b x^5}-\frac {a x^3 \sqrt [4]{-b+a x^4} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-\frac {\sqrt {2} \sqrt {-a} x^4}{\sqrt {b}},\frac {a x^4}{b}\right )}{6 b \sqrt [4]{1-\frac {a x^4}{b}}}-\frac {a x^3 \sqrt [4]{-b+a x^4} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {\sqrt {2} \sqrt {-a} x^4}{\sqrt {b}},\frac {a x^4}{b}\right )}{6 b \sqrt [4]{1-\frac {a x^4}{b}}}+\frac {(a c) \text {Subst}\left (\int \frac {1}{\sqrt [4]{2}-\sqrt {\sqrt {2} a-2 \sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt {2} \sqrt {\sqrt {2} a-2 \sqrt {-a} \sqrt {b}} b}-\frac {(a c) \text {Subst}\left (\int \frac {1}{\sqrt [4]{2}+\sqrt {\sqrt {2} a-2 \sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt {2} \sqrt {\sqrt {2} a-2 \sqrt {-a} \sqrt {b}} b}+\frac {(a c) \text {Subst}\left (\int \frac {1}{\sqrt [4]{2}-\sqrt {\sqrt {2} a+2 \sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt {2} \sqrt {\sqrt {2} a+2 \sqrt {-a} \sqrt {b}} b}-\frac {(a c) \text {Subst}\left (\int \frac {1}{\sqrt [4]{2}+\sqrt {\sqrt {2} a+2 \sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt {2} \sqrt {\sqrt {2} a+2 \sqrt {-a} \sqrt {b}} b}-\frac {\left (\sqrt {-a} c\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{2}-\sqrt {\sqrt {2} a-2 \sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt {\sqrt {2} a-2 \sqrt {-a} \sqrt {b}} \sqrt {b}}+\frac {\left (\sqrt {-a} c\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{2}+\sqrt {\sqrt {2} a-2 \sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt {\sqrt {2} a-2 \sqrt {-a} \sqrt {b}} \sqrt {b}}+\frac {\left (\sqrt {-a} c\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{2}-\sqrt {\sqrt {2} a+2 \sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt {\sqrt {2} a+2 \sqrt {-a} \sqrt {b}} \sqrt {b}}-\frac {\left (\sqrt {-a} c\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{2}+\sqrt {\sqrt {2} a+2 \sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt {\sqrt {2} a+2 \sqrt {-a} \sqrt {b}} \sqrt {b}} \\ & = -\frac {c \sqrt [4]{-b+a x^4}}{b x}+\frac {\left (-b+a x^4\right )^{5/4}}{5 b x^5}-\frac {a x^3 \sqrt [4]{-b+a x^4} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-\frac {\sqrt {2} \sqrt {-a} x^4}{\sqrt {b}},\frac {a x^4}{b}\right )}{6 b \sqrt [4]{1-\frac {a x^4}{b}}}-\frac {a x^3 \sqrt [4]{-b+a x^4} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {\sqrt {2} \sqrt {-a} x^4}{\sqrt {b}},\frac {a x^4}{b}\right )}{6 b \sqrt [4]{1-\frac {a x^4}{b}}}-\frac {a c \arctan \left (\frac {\sqrt [4]{\sqrt {2} a-2 \sqrt {-a} \sqrt {b}} x}{\sqrt [8]{2} \sqrt [4]{-b+a x^4}}\right )}{2\ 2^{5/8} \left (\sqrt {2} a-2 \sqrt {-a} \sqrt {b}\right )^{3/4} b}+\frac {\sqrt {-a} c \arctan \left (\frac {\sqrt [4]{\sqrt {2} a-2 \sqrt {-a} \sqrt {b}} x}{\sqrt [8]{2} \sqrt [4]{-b+a x^4}}\right )}{2 \sqrt [8]{2} \left (\sqrt {2} a-2 \sqrt {-a} \sqrt {b}\right )^{3/4} \sqrt {b}}-\frac {a c \arctan \left (\frac {\sqrt [4]{\sqrt {2} a+2 \sqrt {-a} \sqrt {b}} x}{\sqrt [8]{2} \sqrt [4]{-b+a x^4}}\right )}{2\ 2^{5/8} \left (\sqrt {2} a+2 \sqrt {-a} \sqrt {b}\right )^{3/4} b}-\frac {\sqrt {-a} c \arctan \left (\frac {\sqrt [4]{\sqrt {2} a+2 \sqrt {-a} \sqrt {b}} x}{\sqrt [8]{2} \sqrt [4]{-b+a x^4}}\right )}{2 \sqrt [8]{2} \left (\sqrt {2} a+2 \sqrt {-a} \sqrt {b}\right )^{3/4} \sqrt {b}}+\frac {a c \text {arctanh}\left (\frac {\sqrt [4]{\sqrt {2} a-2 \sqrt {-a} \sqrt {b}} x}{\sqrt [8]{2} \sqrt [4]{-b+a x^4}}\right )}{2\ 2^{5/8} \left (\sqrt {2} a-2 \sqrt {-a} \sqrt {b}\right )^{3/4} b}-\frac {\sqrt {-a} c \text {arctanh}\left (\frac {\sqrt [4]{\sqrt {2} a-2 \sqrt {-a} \sqrt {b}} x}{\sqrt [8]{2} \sqrt [4]{-b+a x^4}}\right )}{2 \sqrt [8]{2} \left (\sqrt {2} a-2 \sqrt {-a} \sqrt {b}\right )^{3/4} \sqrt {b}}+\frac {a c \text {arctanh}\left (\frac {\sqrt [4]{\sqrt {2} a+2 \sqrt {-a} \sqrt {b}} x}{\sqrt [8]{2} \sqrt [4]{-b+a x^4}}\right )}{2\ 2^{5/8} \left (\sqrt {2} a+2 \sqrt {-a} \sqrt {b}\right )^{3/4} b}+\frac {\sqrt {-a} c \text {arctanh}\left (\frac {\sqrt [4]{\sqrt {2} a+2 \sqrt {-a} \sqrt {b}} x}{\sqrt [8]{2} \sqrt [4]{-b+a x^4}}\right )}{2 \sqrt [8]{2} \left (\sqrt {2} a+2 \sqrt {-a} \sqrt {b}\right )^{3/4} \sqrt {b}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.64 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.98 \[ \int \frac {\sqrt [4]{-b+a x^4} \left (b+c x^4+a x^8\right )}{x^6 \left (b+2 a x^8\right )} \, dx=-\frac {8 \sqrt [4]{-b+a x^4} \left (b-(a-5 c) x^4\right )+5 a x^5 \text {RootSum}\left [a^2+2 a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {a c \log (x)+2 b c \log (x)-a c \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )-2 b c \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )-b \log (x) \text {$\#$1}^4-c \log (x) \text {$\#$1}^4+b \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4+c \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}^3+\text {$\#$1}^7}\&\right ]}{40 b x^5} \]

[In]

Integrate[((-b + a*x^4)^(1/4)*(b + c*x^4 + a*x^8))/(x^6*(b + 2*a*x^8)),x]

[Out]

-1/40*(8*(-b + a*x^4)^(1/4)*(b - (a - 5*c)*x^4) + 5*a*x^5*RootSum[a^2 + 2*a*b - 2*a*#1^4 + #1^8 & , (a*c*Log[x
] + 2*b*c*Log[x] - a*c*Log[(-b + a*x^4)^(1/4) - x*#1] - 2*b*c*Log[(-b + a*x^4)^(1/4) - x*#1] - b*Log[x]*#1^4 -
c*Log[x]*#1^4 + b*Log[(-b + a*x^4)^(1/4) - x*#1]*#1^4 + c*Log[(-b + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1^3) + #
1^7) & ])/(b*x^5)

Maple [N/A]

Time = 0.42 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.54

method	result	size
pseudoelliptic	\(\frac {-5 a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-2 a \,\textit {\_Z}^{4}+a^{2}+2 a b \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (a \,x^{4}-b \right )^{\frac {1}{4}}}{x}\right ) \left (\left (-b -c \right ) \textit {\_R}^{4}+c \left (a +2 b \right )\right )}{\textit {\_R}^{3} \left (-\textit {\_R}^{4}+a \right )}\right ) x^{5}+8 \left (\left (a -5 c \right ) x^{4}-b \right ) \left (a \,x^{4}-b \right )^{\frac {1}{4}}}{40 b \,x^{5}}\)	\(116\)

[In]

int((a*x^4-b)^(1/4)*(a*x^8+c*x^4+b)/x^6/(2*a*x^8+b),x,method=_RETURNVERBOSE)

[Out]

1/40*(-5*a*sum(ln((-_R*x+(a*x^4-b)^(1/4))/x)*((-b-c)*_R^4+c*(a+2*b))/_R^3/(-_R^4+a),_R=RootOf(_Z^8-2*_Z^4*a+a^
2+2*a*b))*x^5+8*((a-5*c)*x^4-b)*(a*x^4-b)^(1/4))/b/x^5

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt [4]{-b+a x^4} \left (b+c x^4+a x^8\right )}{x^6 \left (b+2 a x^8\right )} \, dx=\text {Timed out} \]

[In]

integrate((a*x^4-b)^(1/4)*(a*x^8+c*x^4+b)/x^6/(2*a*x^8+b),x, algorithm="fricas")

[Out]

Timed out

Sympy [N/A]

Not integrable

Time = 96.93 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.16 \[ \int \frac {\sqrt [4]{-b+a x^4} \left (b+c x^4+a x^8\right )}{x^6 \left (b+2 a x^8\right )} \, dx=\int \frac {\sqrt [4]{a x^{4} - b} \left (a x^{8} + b + c x^{4}\right )}{x^{6} \cdot \left (2 a x^{8} + b\right )}\, dx \]

[In]

integrate((a*x**4-b)**(1/4)*(a*x**8+c*x**4+b)/x**6/(2*a*x**8+b),x)

[Out]

Integral((a*x**4 - b)**(1/4)*(a*x**8 + b + c*x**4)/(x**6*(2*a*x**8 + b)), x)

Maxima [N/A]

Not integrable

Time = 0.22 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.18 \[ \int \frac {\sqrt [4]{-b+a x^4} \left (b+c x^4+a x^8\right )}{x^6 \left (b+2 a x^8\right )} \, dx=\int { \frac {{\left (a x^{8} + c x^{4} + b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}}{{\left (2 \, a x^{8} + b\right )} x^{6}} \,d x } \]

[In]

integrate((a*x^4-b)^(1/4)*(a*x^8+c*x^4+b)/x^6/(2*a*x^8+b),x, algorithm="maxima")

[Out]

integrate((a*x^8 + c*x^4 + b)*(a*x^4 - b)^(1/4)/((2*a*x^8 + b)*x^6), x)

Giac [N/A]

Not integrable

Time = 0.37 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.18 \[ \int \frac {\sqrt [4]{-b+a x^4} \left (b+c x^4+a x^8\right )}{x^6 \left (b+2 a x^8\right )} \, dx=\int { \frac {{\left (a x^{8} + c x^{4} + b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}}{{\left (2 \, a x^{8} + b\right )} x^{6}} \,d x } \]

[In]

integrate((a*x^4-b)^(1/4)*(a*x^8+c*x^4+b)/x^6/(2*a*x^8+b),x, algorithm="giac")

[Out]

integrate((a*x^8 + c*x^4 + b)*(a*x^4 - b)^(1/4)/((2*a*x^8 + b)*x^6), x)

Mupad [N/A]

Not integrable

Time = 8.42 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.18 \[ \int \frac {\sqrt [4]{-b+a x^4} \left (b+c x^4+a x^8\right )}{x^6 \left (b+2 a x^8\right )} \, dx=\int \frac {{\left (a\,x^4-b\right )}^{1/4}\,\left (a\,x^8+c\,x^4+b\right )}{x^6\,\left (2\,a\,x^8+b\right )} \,d x \]

[In]

int(((a*x^4 - b)^(1/4)*(b + a*x^8 + c*x^4))/(x^6*(b + 2*a*x^8)),x)

[Out]

int(((a*x^4 - b)^(1/4)*(b + a*x^8 + c*x^4))/(x^6*(b + 2*a*x^8)), x)