\(\int \frac {\sqrt [4]{b+a x^4} (2 b+3 a x^4)}{x^6 (b+a x^8)} \, dx\) [2482]

Optimal result

Integrand size = 34, antiderivative size = 204 \[ \int \frac {\sqrt [4]{b+a x^4} \left (2 b+3 a x^4\right )}{x^6 \left (b+a x^8\right )} \, dx=\frac {\left (-2 b-17 a x^4\right ) \sqrt [4]{b+a x^4}}{5 b x^5}-\frac {a \text {RootSum}\left [a^2+a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-3 a^2 \log (x)-3 a b \log (x)+3 a^2 \log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right )+3 a b \log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right )+3 a \log (x) \text {$\#$1}^4-2 b \log (x) \text {$\#$1}^4-3 a \log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4+2 b \log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}^3-\text {$\#$1}^7}\&\right ]}{8 b} \]

[Out]

Unintegrable

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(715\) vs. \(2(204)=408\).

Time = 2.09 (sec) , antiderivative size = 715, normalized size of antiderivative = 3.50, number of steps used = 43, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {6857, 270, 283, 338, 304, 209, 212, 1543, 525, 524, 1533, 508} \[ \int \frac {\sqrt [4]{b+a x^4} \left (2 b+3 a x^4\right )}{x^6 \left (b+a x^8\right )} \, dx=-\frac {3 a^2 \arctan \left (\frac {x \sqrt [4]{a-\sqrt {-a} \sqrt {b}}}{\sqrt [4]{a x^4+b}}\right )}{4 b \left (a-\sqrt {-a} \sqrt {b}\right )^{3/4}}-\frac {3 a^2 \arctan \left (\frac {x \sqrt [4]{\sqrt {-a} \sqrt {b}+a}}{\sqrt [4]{a x^4+b}}\right )}{4 b \left (\sqrt {-a} \sqrt {b}+a\right )^{3/4}}+\frac {3 a^2 \text {arctanh}\left (\frac {x \sqrt [4]{a-\sqrt {-a} \sqrt {b}}}{\sqrt [4]{a x^4+b}}\right )}{4 b \left (a-\sqrt {-a} \sqrt {b}\right )^{3/4}}+\frac {3 a^2 \text {arctanh}\left (\frac {x \sqrt [4]{\sqrt {-a} \sqrt {b}+a}}{\sqrt [4]{a x^4+b}}\right )}{4 b \left (\sqrt {-a} \sqrt {b}+a\right )^{3/4}}-\frac {a x^3 \sqrt [4]{a x^4+b} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-\frac {\sqrt {-a} x^4}{\sqrt {b}},-\frac {a x^4}{b}\right )}{3 b \sqrt [4]{\frac {a x^4}{b}+1}}-\frac {a x^3 \sqrt [4]{a x^4+b} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {\sqrt {-a} x^4}{\sqrt {b}},-\frac {a x^4}{b}\right )}{3 b \sqrt [4]{\frac {a x^4}{b}+1}}-\frac {3 (-a)^{3/2} \arctan \left (\frac {x \sqrt [4]{a-\sqrt {-a} \sqrt {b}}}{\sqrt [4]{a x^4+b}}\right )}{4 \sqrt {b} \left (a-\sqrt {-a} \sqrt {b}\right )^{3/4}}+\frac {3 (-a)^{3/2} \arctan \left (\frac {x \sqrt [4]{\sqrt {-a} \sqrt {b}+a}}{\sqrt [4]{a x^4+b}}\right )}{4 \sqrt {b} \left (\sqrt {-a} \sqrt {b}+a\right )^{3/4}}+\frac {3 (-a)^{3/2} \text {arctanh}\left (\frac {x \sqrt [4]{a-\sqrt {-a} \sqrt {b}}}{\sqrt [4]{a x^4+b}}\right )}{4 \sqrt {b} \left (a-\sqrt {-a} \sqrt {b}\right )^{3/4}}-\frac {3 (-a)^{3/2} \text {arctanh}\left (\frac {x \sqrt [4]{\sqrt {-a} \sqrt {b}+a}}{\sqrt [4]{a x^4+b}}\right )}{4 \sqrt {b} \left (\sqrt {-a} \sqrt {b}+a\right )^{3/4}}-\frac {3 a \sqrt [4]{a x^4+b}}{b x}-\frac {2 \left (a x^4+b\right )^{5/4}}{5 b x^5} \]

[In]

Int[((b + a*x^4)^(1/4)*(2*b + 3*a*x^4))/(x^6*(b + a*x^8)),x]

[Out]

(-3*a*(b + a*x^4)^(1/4))/(b*x) - (2*(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[-a]*x^4)/Sqrt[b]), -((a*x^4)/b)])/(3*b*(1 + (a*x^4)/b)^(1/4)) - (a*x^3*(b + a*x^4)^(1/4)*App
ellF1[3/4, 1, -1/4, 7/4, (Sqrt[-a]*x^4)/Sqrt[b], -((a*x^4)/b)])/(3*b*(1 + (a*x^4)/b)^(1/4)) - (3*a^2*ArcTan[((
a - Sqrt[-a]*Sqrt[b])^(1/4)*x)/(b + a*x^4)^(1/4)])/(4*(a - Sqrt[-a]*Sqrt[b])^(3/4)*b) - (3*(-a)^(3/2)*ArcTan[(
(a - Sqrt[-a]*Sqrt[b])^(1/4)*x)/(b + a*x^4)^(1/4)])/(4*(a - Sqrt[-a]*Sqrt[b])^(3/4)*Sqrt[b]) - (3*a^2*ArcTan[(
(a + Sqrt[-a]*Sqrt[b])^(1/4)*x)/(b + a*x^4)^(1/4)])/(4*(a + Sqrt[-a]*Sqrt[b])^(3/4)*b) + (3*(-a)^(3/2)*ArcTan[
((a + Sqrt[-a]*Sqrt[b])^(1/4)*x)/(b + a*x^4)^(1/4)])/(4*(a + Sqrt[-a]*Sqrt[b])^(3/4)*Sqrt[b]) + (3*a^2*ArcTanh
[((a - Sqrt[-a]*Sqrt[b])^(1/4)*x)/(b + a*x^4)^(1/4)])/(4*(a - Sqrt[-a]*Sqrt[b])^(3/4)*b) + (3*(-a)^(3/2)*ArcTa
nh[((a - Sqrt[-a]*Sqrt[b])^(1/4)*x)/(b + a*x^4)^(1/4)])/(4*(a - Sqrt[-a]*Sqrt[b])^(3/4)*Sqrt[b]) + (3*a^2*ArcT
anh[((a + Sqrt[-a]*Sqrt[b])^(1/4)*x)/(b + a*x^4)^(1/4)])/(4*(a + Sqrt[-a]*Sqrt[b])^(3/4)*b) - (3*(-a)^(3/2)*Ar
cTanh[((a + Sqrt[-a]*Sqrt[b])^(1/4)*x)/(b + a*x^4)^(1/4)])/(4*(a + 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 {2 \sqrt [4]{b+a x^4}}{x^6}+\frac {3 a \sqrt [4]{b+a x^4}}{b x^2}-\frac {a x^2 \sqrt [4]{b+a x^4} \left (2 b+3 a x^4\right )}{b \left (b+a x^8\right )}\right ) \, dx \\ & = 2 \int \frac {\sqrt [4]{b+a x^4}}{x^6} \, dx-\frac {a \int \frac {x^2 \sqrt [4]{b+a x^4} \left (2 b+3 a x^4\right )}{b+a x^8} \, dx}{b}+\frac {(3 a) \int \frac {\sqrt [4]{b+a x^4}}{x^2} \, dx}{b} \\ & = -\frac {3 a \sqrt [4]{b+a x^4}}{b x}-\frac {2 \left (b+a x^4\right )^{5/4}}{5 b x^5}-\frac {a \int \left (\frac {2 b x^2 \sqrt [4]{b+a x^4}}{b+a x^8}+\frac {3 a x^6 \sqrt [4]{b+a x^4}}{b+a x^8}\right ) \, dx}{b}+\frac {\left (3 a^2\right ) \int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx}{b} \\ & = -\frac {3 a \sqrt [4]{b+a x^4}}{b x}-\frac {2 \left (b+a x^4\right )^{5/4}}{5 b x^5}-(2 a) \int \frac {x^2 \sqrt [4]{b+a x^4}}{b+a x^8} \, dx-\frac {\left (3 a^2\right ) \int \frac {x^6 \sqrt [4]{b+a x^4}}{b+a x^8} \, dx}{b}+\frac {\left (3 a^2\right ) \text {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{b} \\ & = -\frac {3 a \sqrt [4]{b+a x^4}}{b x}-\frac {2 \left (b+a x^4\right )^{5/4}}{5 b x^5}-(2 a) \int \left (-\frac {a x^2 \sqrt [4]{b+a x^4}}{2 \sqrt {-a} \sqrt {b} \left (\sqrt {-a} \sqrt {b}-a x^4\right )}-\frac {a x^2 \sqrt [4]{b+a x^4}}{2 \sqrt {-a} \sqrt {b} \left (\sqrt {-a} \sqrt {b}+a x^4\right )}\right ) \, dx+\frac {(3 a) \int \frac {x^2 \left (a b-a b x^4\right )}{\left (b+a x^4\right )^{3/4} \left (b+a x^8\right )} \, dx}{b}+\frac {\left (3 a^{3/2}\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 (3 a^{3/2}\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 (3 a^2\right ) \int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx}{b} \\ & = -\frac {3 a \sqrt [4]{b+a x^4}}{b x}-\frac {2 \left (b+a x^4\right )^{5/4}}{5 b x^5}-\frac {3 a^{5/4} \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2 b}+\frac {3 a^{5/4} \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2 b}+\frac {(3 a) \int \left (\frac {a b x^2}{\left (b+a x^4\right )^{3/4} \left (b+a x^8\right )}-\frac {a b x^6}{\left (b+a x^4\right )^{3/4} \left (b+a x^8\right )}\right ) \, dx}{b}-\frac {\left (3 a^2\right ) \text {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{b}+\frac {(-a)^{3/2} \int \frac {x^2 \sqrt [4]{b+a x^4}}{\sqrt {-a} \sqrt {b}-a x^4} \, dx}{\sqrt {b}}+\frac {(-a)^{3/2} \int \frac {x^2 \sqrt [4]{b+a x^4}}{\sqrt {-a} \sqrt {b}+a x^4} \, dx}{\sqrt {b}} \\ & = -\frac {3 a \sqrt [4]{b+a x^4}}{b x}-\frac {2 \left (b+a x^4\right )^{5/4}}{5 b x^5}-\frac {3 a^{5/4} \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2 b}+\frac {3 a^{5/4} \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2 b}+\left (3 a^2\right ) \int \frac {x^2}{\left (b+a x^4\right )^{3/4} \left (b+a x^8\right )} \, dx-\left (3 a^2\right ) \int \frac {x^6}{\left (b+a x^4\right )^{3/4} \left (b+a x^8\right )} \, dx-\frac {\left (3 a^{3/2}\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 (3 a^{3/2}\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 {-a} \sqrt {b}-a x^4} \, dx}{\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 {-a} \sqrt {b}+a x^4} \, dx}{\sqrt {b} \sqrt [4]{1+\frac {a x^4}{b}}} \\ & = -\frac {3 a \sqrt [4]{b+a x^4}}{b x}-\frac {2 \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 {-a} x^4}{\sqrt {b}},-\frac {a x^4}{b}\right )}{3 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 {-a} x^4}{\sqrt {b}},-\frac {a x^4}{b}\right )}{3 b \sqrt [4]{1+\frac {a x^4}{b}}}-\left (3 a^2\right ) \int \left (\frac {x^2}{2 \left (-\sqrt {-a} \sqrt {b}+a x^4\right ) \left (b+a x^4\right )^{3/4}}+\frac {x^2}{2 \left (\sqrt {-a} \sqrt {b}+a x^4\right ) \left (b+a x^4\right )^{3/4}}\right ) \, dx+\left (3 a^2\right ) \int \left (-\frac {a x^2}{2 \sqrt {-a} \sqrt {b} \left (\sqrt {-a} \sqrt {b}-a x^4\right ) \left (b+a x^4\right )^{3/4}}-\frac {a x^2}{2 \sqrt {-a} \sqrt {b} \left (\sqrt {-a} \sqrt {b}+a x^4\right ) \left (b+a x^4\right )^{3/4}}\right ) \, dx \\ & = -\frac {3 a \sqrt [4]{b+a x^4}}{b x}-\frac {2 \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 {-a} x^4}{\sqrt {b}},-\frac {a x^4}{b}\right )}{3 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 {-a} x^4}{\sqrt {b}},-\frac {a x^4}{b}\right )}{3 b \sqrt [4]{1+\frac {a x^4}{b}}}-\frac {1}{2} \left (3 a^2\right ) \int \frac {x^2}{\left (-\sqrt {-a} \sqrt {b}+a x^4\right ) \left (b+a x^4\right )^{3/4}} \, dx-\frac {1}{2} \left (3 a^2\right ) \int \frac {x^2}{\left (\sqrt {-a} \sqrt {b}+a x^4\right ) \left (b+a x^4\right )^{3/4}} \, dx+\frac {\left (3 (-a)^{5/2}\right ) \int \frac {x^2}{\left (\sqrt {-a} \sqrt {b}-a x^4\right ) \left (b+a x^4\right )^{3/4}} \, dx}{2 \sqrt {b}}+\frac {\left (3 (-a)^{5/2}\right ) \int \frac {x^2}{\left (\sqrt {-a} \sqrt {b}+a x^4\right ) \left (b+a x^4\right )^{3/4}} \, dx}{2 \sqrt {b}} \\ & = -\frac {3 a \sqrt [4]{b+a x^4}}{b x}-\frac {2 \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 {-a} x^4}{\sqrt {b}},-\frac {a x^4}{b}\right )}{3 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 {-a} x^4}{\sqrt {b}},-\frac {a x^4}{b}\right )}{3 b \sqrt [4]{1+\frac {a x^4}{b}}}-\frac {1}{2} \left (3 a^2\right ) \text {Subst}\left (\int \frac {x^2}{-\sqrt {-a} \sqrt {b}-\left (-\sqrt {-a} a \sqrt {b}-a b\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )-\frac {1}{2} \left (3 a^2\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {-a} \sqrt {b}-\left (\sqrt {-a} a \sqrt {b}-a b\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )+\frac {\left (3 (-a)^{5/2}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {-a} \sqrt {b}-\left (\sqrt {-a} a \sqrt {b}-a b\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt {b}}+\frac {\left (3 (-a)^{5/2}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {-a} \sqrt {b}-\left (\sqrt {-a} a \sqrt {b}+a b\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt {b}} \\ & = -\frac {3 a \sqrt [4]{b+a x^4}}{b x}-\frac {2 \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 {-a} x^4}{\sqrt {b}},-\frac {a x^4}{b}\right )}{3 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 {-a} x^4}{\sqrt {b}},-\frac {a x^4}{b}\right )}{3 b \sqrt [4]{1+\frac {a x^4}{b}}}+\frac {\left (3 a^2\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {a-\sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a-\sqrt {-a} \sqrt {b}} b}-\frac {\left (3 a^2\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {a-\sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a-\sqrt {-a} \sqrt {b}} b}+\frac {\left (3 a^2\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {a+\sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a+\sqrt {-a} \sqrt {b}} b}-\frac {\left (3 a^2\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {a+\sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a+\sqrt {-a} \sqrt {b}} b}+\frac {\left (3 (-a)^{3/2}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {a-\sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a-\sqrt {-a} \sqrt {b}} \sqrt {b}}-\frac {\left (3 (-a)^{3/2}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {a-\sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a-\sqrt {-a} \sqrt {b}} \sqrt {b}}-\frac {\left (3 (-a)^{3/2}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {a+\sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a+\sqrt {-a} \sqrt {b}} \sqrt {b}}+\frac {\left (3 (-a)^{3/2}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {a+\sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a+\sqrt {-a} \sqrt {b}} \sqrt {b}} \\ & = -\frac {3 a \sqrt [4]{b+a x^4}}{b x}-\frac {2 \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 {-a} x^4}{\sqrt {b}},-\frac {a x^4}{b}\right )}{3 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 {-a} x^4}{\sqrt {b}},-\frac {a x^4}{b}\right )}{3 b \sqrt [4]{1+\frac {a x^4}{b}}}-\frac {3 a^2 \arctan \left (\frac {\sqrt [4]{a-\sqrt {-a} \sqrt {b}} x}{\sqrt [4]{b+a x^4}}\right )}{4 \left (a-\sqrt {-a} \sqrt {b}\right )^{3/4} b}-\frac {3 (-a)^{3/2} \arctan \left (\frac {\sqrt [4]{a-\sqrt {-a} \sqrt {b}} x}{\sqrt [4]{b+a x^4}}\right )}{4 \left (a-\sqrt {-a} \sqrt {b}\right )^{3/4} \sqrt {b}}-\frac {3 a^2 \arctan \left (\frac {\sqrt [4]{a+\sqrt {-a} \sqrt {b}} x}{\sqrt [4]{b+a x^4}}\right )}{4 \left (a+\sqrt {-a} \sqrt {b}\right )^{3/4} b}+\frac {3 (-a)^{3/2} \arctan \left (\frac {\sqrt [4]{a+\sqrt {-a} \sqrt {b}} x}{\sqrt [4]{b+a x^4}}\right )}{4 \left (a+\sqrt {-a} \sqrt {b}\right )^{3/4} \sqrt {b}}+\frac {3 a^2 \text {arctanh}\left (\frac {\sqrt [4]{a-\sqrt {-a} \sqrt {b}} x}{\sqrt [4]{b+a x^4}}\right )}{4 \left (a-\sqrt {-a} \sqrt {b}\right )^{3/4} b}+\frac {3 (-a)^{3/2} \text {arctanh}\left (\frac {\sqrt [4]{a-\sqrt {-a} \sqrt {b}} x}{\sqrt [4]{b+a x^4}}\right )}{4 \left (a-\sqrt {-a} \sqrt {b}\right )^{3/4} \sqrt {b}}+\frac {3 a^2 \text {arctanh}\left (\frac {\sqrt [4]{a+\sqrt {-a} \sqrt {b}} x}{\sqrt [4]{b+a x^4}}\right )}{4 \left (a+\sqrt {-a} \sqrt {b}\right )^{3/4} b}-\frac {3 (-a)^{3/2} \text {arctanh}\left (\frac {\sqrt [4]{a+\sqrt {-a} \sqrt {b}} x}{\sqrt [4]{b+a x^4}}\right )}{4 \left (a+\sqrt {-a} \sqrt {b}\right )^{3/4} \sqrt {b}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.56 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [4]{b+a x^4} \left (2 b+3 a x^4\right )}{x^6 \left (b+a x^8\right )} \, dx=\frac {\left (-2 b-17 a x^4\right ) \sqrt [4]{b+a x^4}}{5 b x^5}-\frac {a \text {RootSum}\left [a^2+a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {3 a^2 \log (x)+3 a b \log (x)-3 a^2 \log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right )-3 a b \log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right )-3 a \log (x) \text {$\#$1}^4+2 b \log (x) \text {$\#$1}^4+3 a \log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4-2 b \log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}^3+\text {$\#$1}^7}\&\right ]}{8 b} \]

[In]

Integrate[((b + a*x^4)^(1/4)*(2*b + 3*a*x^4))/(x^6*(b + a*x^8)),x]

[Out]

((-2*b - 17*a*x^4)*(b + a*x^4)^(1/4))/(5*b*x^5) - (a*RootSum[a^2 + a*b - 2*a*#1^4 + #1^8 & , (3*a^2*Log[x] + 3
*a*b*Log[x] - 3*a^2*Log[(b + a*x^4)^(1/4) - x*#1] - 3*a*b*Log[(b + a*x^4)^(1/4) - x*#1] - 3*a*Log[x]*#1^4 + 2*
b*Log[x]*#1^4 + 3*a*Log[(b + a*x^4)^(1/4) - x*#1]*#1^4 - 2*b*Log[(b + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1^3) +
#1^7) & ])/(8*b)

Maple [N/A]

Time = 0.30 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.55

method	result	size
pseudoelliptic	\(\frac {-5 a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-2 a \,\textit {\_Z}^{4}+a^{2}+a b \right )}{\sum }\frac {3 \left (\left (-a +\frac {2 b}{3}\right ) \textit {\_R}^{4}+a \left (a +b \right )\right ) \ln \left (\frac {-\textit {\_R} x +\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{3} \left (-\textit {\_R}^{4}+a \right )}\right ) x^{5}-136 \left (a \,x^{4}+b \right )^{\frac {1}{4}} a \,x^{4}-16 b \left (a \,x^{4}+b \right )^{\frac {1}{4}}}{40 b \,x^{5}}\)	\(113\)

[In]

int((a*x^4+b)^(1/4)*(3*a*x^4+2*b)/x^6/(a*x^8+b),x,method=_RETURNVERBOSE)

[Out]

1/40*(-5*a*sum(3*((-a+2/3*b)*_R^4+a*(a+b))*ln((-_R*x+(a*x^4+b)^(1/4))/x)/_R^3/(-_R^4+a),_R=RootOf(_Z^8-2*_Z^4*
a+a^2+a*b))*x^5-136*(a*x^4+b)^(1/4)*a*x^4-16*b*(a*x^4+b)^(1/4))/b/x^5

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt [4]{b+a x^4} \left (2 b+3 a x^4\right )}{x^6 \left (b+a x^8\right )} \, dx=\text {Timed out} \]

[In]

integrate((a*x^4+b)^(1/4)*(3*a*x^4+2*b)/x^6/(a*x^8+b),x, algorithm="fricas")

[Out]

Timed out

Sympy [N/A]

Not integrable

Time = 72.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.15 \[ \int \frac {\sqrt [4]{b+a x^4} \left (2 b+3 a x^4\right )}{x^6 \left (b+a x^8\right )} \, dx=\int \frac {\sqrt [4]{a x^{4} + b} \left (3 a x^{4} + 2 b\right )}{x^{6} \left (a x^{8} + b\right )}\, dx \]

[In]

integrate((a*x**4+b)**(1/4)*(3*a*x**4+2*b)/x**6/(a*x**8+b),x)

[Out]

Integral((a*x**4 + b)**(1/4)*(3*a*x**4 + 2*b)/(x**6*(a*x**8 + b)), x)

Maxima [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.17 \[ \int \frac {\sqrt [4]{b+a x^4} \left (2 b+3 a x^4\right )}{x^6 \left (b+a x^8\right )} \, dx=\int { \frac {{\left (3 \, a x^{4} + 2 \, b\right )} {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{{\left (a x^{8} + b\right )} x^{6}} \,d x } \]

[In]

integrate((a*x^4+b)^(1/4)*(3*a*x^4+2*b)/x^6/(a*x^8+b),x, algorithm="maxima")

[Out]

integrate((3*a*x^4 + 2*b)*(a*x^4 + b)^(1/4)/((a*x^8 + b)*x^6), x)

Giac [N/A]

Not integrable

Time = 0.37 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.17 \[ \int \frac {\sqrt [4]{b+a x^4} \left (2 b+3 a x^4\right )}{x^6 \left (b+a x^8\right )} \, dx=\int { \frac {{\left (3 \, a x^{4} + 2 \, b\right )} {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{{\left (a x^{8} + b\right )} x^{6}} \,d x } \]

[In]

integrate((a*x^4+b)^(1/4)*(3*a*x^4+2*b)/x^6/(a*x^8+b),x, algorithm="giac")

[Out]

integrate((3*a*x^4 + 2*b)*(a*x^4 + b)^(1/4)/((a*x^8 + b)*x^6), x)

Mupad [N/A]

Not integrable

Time = 7.76 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.17 \[ \int \frac {\sqrt [4]{b+a x^4} \left (2 b+3 a x^4\right )}{x^6 \left (b+a x^8\right )} \, dx=\int \frac {{\left (a\,x^4+b\right )}^{1/4}\,\left (3\,a\,x^4+2\,b\right )}{x^6\,\left (a\,x^8+b\right )} \,d x \]

[In]

int(((b + a*x^4)^(1/4)*(2*b + 3*a*x^4))/(x^6*(b + a*x^8)),x)

[Out]

int(((b + a*x^4)^(1/4)*(2*b + 3*a*x^4))/(x^6*(b + a*x^8)), x)