3.26.67 \(\int \frac {(-4 b+a x^4) \sqrt [4]{-b+a x^4}}{x^6 (-8 b+a x^8)} \, dx\) [2567]

Optimal. Leaf size=218 \[ \frac {\sqrt [4]{-b+a x^4} \left (-4 b+9 a x^4\right )}{40 b x^5}+\frac {a \text {RootSum}\left [8 a^2-a b-16 a \text {$\#$1}^4+8 \text {$\#$1}^8\& ,\frac {-8 a^2 \log (x)+a b \log (x)+8 a^2 \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )-a b \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )+8 a \log (x) \text {$\#$1}^4+4 b \log (x) \text {$\#$1}^4-8 a \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4-4 b \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}^3-\text {$\#$1}^7}\& \right ]}{512 b} \]

[Out]

Unintegrable

________________________________________________________________________________________

Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(928\) vs. \(2(218)=436\).
time = 2.97, antiderivative size = 928, normalized size of antiderivative = 4.26, number of steps used = 43, number of rules used = 12, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.324, Rules used = {6857, 270, 283, 338, 304, 209, 212, 1543, 525, 524, 1533, 508} \begin {gather*} \frac {a \sqrt [4]{a x^4-b} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-\frac {\sqrt {a} x^4}{2 \sqrt {2} \sqrt {b}},\frac {a x^4}{b}\right ) x^3}{96 b \sqrt [4]{1-\frac {a x^4}{b}}}+\frac {a \sqrt [4]{a x^4-b} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {\sqrt {a} x^4}{2 \sqrt {2} \sqrt {b}},\frac {a x^4}{b}\right ) x^3}{96 b \sqrt [4]{1-\frac {a x^4}{b}}}-\frac {a^{9/8} \text {ArcTan}\left (\frac {\sqrt [8]{a} \sqrt [4]{2 \sqrt {2} \sqrt {a}-\sqrt {b}} x}{2^{3/8} \sqrt [4]{a x^4-b}}\right )}{32\ 2^{3/8} \left (2 \sqrt {2} \sqrt {a}-\sqrt {b}\right )^{3/4} \sqrt {b}}+\frac {a^{13/8} \text {ArcTan}\left (\frac {\sqrt [8]{a} \sqrt [4]{2 \sqrt {2} \sqrt {a}-\sqrt {b}} x}{2^{3/8} \sqrt [4]{a x^4-b}}\right )}{8\ 2^{7/8} \left (2 \sqrt {2} \sqrt {a}-\sqrt {b}\right )^{3/4} b}+\frac {a^{9/8} \text {ArcTan}\left (\frac {\sqrt [8]{a} \sqrt [4]{2 \sqrt {2} \sqrt {a}+\sqrt {b}} x}{2^{3/8} \sqrt [4]{a x^4-b}}\right )}{32\ 2^{3/8} \left (2 \sqrt {2} \sqrt {a}+\sqrt {b}\right )^{3/4} \sqrt {b}}+\frac {a^{13/8} \text {ArcTan}\left (\frac {\sqrt [8]{a} \sqrt [4]{2 \sqrt {2} \sqrt {a}+\sqrt {b}} x}{2^{3/8} \sqrt [4]{a x^4-b}}\right )}{8\ 2^{7/8} \left (2 \sqrt {2} \sqrt {a}+\sqrt {b}\right )^{3/4} b}+\frac {a^{9/8} \tanh ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [4]{2 \sqrt {2} \sqrt {a}-\sqrt {b}} x}{2^{3/8} \sqrt [4]{a x^4-b}}\right )}{32\ 2^{3/8} \left (2 \sqrt {2} \sqrt {a}-\sqrt {b}\right )^{3/4} \sqrt {b}}-\frac {a^{13/8} \tanh ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [4]{2 \sqrt {2} \sqrt {a}-\sqrt {b}} x}{2^{3/8} \sqrt [4]{a x^4-b}}\right )}{8\ 2^{7/8} \left (2 \sqrt {2} \sqrt {a}-\sqrt {b}\right )^{3/4} b}-\frac {a^{9/8} \tanh ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [4]{2 \sqrt {2} \sqrt {a}+\sqrt {b}} x}{2^{3/8} \sqrt [4]{a x^4-b}}\right )}{32\ 2^{3/8} \left (2 \sqrt {2} \sqrt {a}+\sqrt {b}\right )^{3/4} \sqrt {b}}-\frac {a^{13/8} \tanh ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [4]{2 \sqrt {2} \sqrt {a}+\sqrt {b}} x}{2^{3/8} \sqrt [4]{a x^4-b}}\right )}{8\ 2^{7/8} \left (2 \sqrt {2} \sqrt {a}+\sqrt {b}\right )^{3/4} b}+\frac {a \sqrt [4]{a x^4-b}}{8 b x}+\frac {\left (a x^4-b\right )^{5/4}}{10 b x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(-b + a*x^4)^(1/4))/(8*b*x) + (-b + a*x^4)^(5/4)/(10*b*x^5) + (a*x^3*(-b + a*x^4)^(1/4)*AppellF1[3/4, 1, -1
/4, 7/4, -1/2*(Sqrt[a]*x^4)/(Sqrt[2]*Sqrt[b]), (a*x^4)/b])/(96*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[a]*x^4)/(2*Sqrt[2]*Sqrt[b]), (a*x^4)/b])/(96*b*(1 - (a*x^4)/b)^(1/4))
+ (a^(13/8)*ArcTan[(a^(1/8)*(2*Sqrt[2]*Sqrt[a] - Sqrt[b])^(1/4)*x)/(2^(3/8)*(-b + a*x^4)^(1/4))])/(8*2^(7/8)*(
2*Sqrt[2]*Sqrt[a] - Sqrt[b])^(3/4)*b) - (a^(9/8)*ArcTan[(a^(1/8)*(2*Sqrt[2]*Sqrt[a] - Sqrt[b])^(1/4)*x)/(2^(3/
8)*(-b + a*x^4)^(1/4))])/(32*2^(3/8)*(2*Sqrt[2]*Sqrt[a] - Sqrt[b])^(3/4)*Sqrt[b]) + (a^(13/8)*ArcTan[(a^(1/8)*
(2*Sqrt[2]*Sqrt[a] + Sqrt[b])^(1/4)*x)/(2^(3/8)*(-b + a*x^4)^(1/4))])/(8*2^(7/8)*(2*Sqrt[2]*Sqrt[a] + Sqrt[b])
^(3/4)*b) + (a^(9/8)*ArcTan[(a^(1/8)*(2*Sqrt[2]*Sqrt[a] + Sqrt[b])^(1/4)*x)/(2^(3/8)*(-b + a*x^4)^(1/4))])/(32
*2^(3/8)*(2*Sqrt[2]*Sqrt[a] + Sqrt[b])^(3/4)*Sqrt[b]) - (a^(13/8)*ArcTanh[(a^(1/8)*(2*Sqrt[2]*Sqrt[a] - Sqrt[b
])^(1/4)*x)/(2^(3/8)*(-b + a*x^4)^(1/4))])/(8*2^(7/8)*(2*Sqrt[2]*Sqrt[a] - Sqrt[b])^(3/4)*b) + (a^(9/8)*ArcTan
h[(a^(1/8)*(2*Sqrt[2]*Sqrt[a] - Sqrt[b])^(1/4)*x)/(2^(3/8)*(-b + a*x^4)^(1/4))])/(32*2^(3/8)*(2*Sqrt[2]*Sqrt[a
] - Sqrt[b])^(3/4)*Sqrt[b]) - (a^(13/8)*ArcTanh[(a^(1/8)*(2*Sqrt[2]*Sqrt[a] + Sqrt[b])^(1/4)*x)/(2^(3/8)*(-b +
a*x^4)^(1/4))])/(8*2^(7/8)*(2*Sqrt[2]*Sqrt[a] + Sqrt[b])^(3/4)*b) - (a^(9/8)*ArcTanh[(a^(1/8)*(2*Sqrt[2]*Sqrt
[a] + Sqrt[b])^(1/4)*x)/(2^(3/8)*(-b + a*x^4)^(1/4))])/(32*2^(3/8)*(2*Sqrt[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*} \int \frac {\left (-4 b+a x^4\right ) \sqrt [4]{-b+a x^4}}{x^6 \left (-8 b+a x^8\right )} \, dx &=\int \left (\frac {\sqrt [4]{-b+a x^4}}{2 x^6}-\frac {a \sqrt [4]{-b+a x^4}}{8 b x^2}-\frac {a x^2 \left (-4 b+a x^4\right ) \sqrt [4]{-b+a x^4}}{8 b \left (8 b-a x^8\right )}\right ) \, dx\\ &=\frac {1}{2} \int \frac {\sqrt [4]{-b+a x^4}}{x^6} \, dx-\frac {a \int \frac {\sqrt [4]{-b+a x^4}}{x^2} \, dx}{8 b}-\frac {a \int \frac {x^2 \left (-4 b+a x^4\right ) \sqrt [4]{-b+a x^4}}{8 b-a x^8} \, dx}{8 b}\\ &=\frac {a \sqrt [4]{-b+a x^4}}{8 b x}+\frac {\left (-b+a x^4\right )^{5/4}}{10 b x^5}-\frac {a \int \left (-\frac {4 b x^2 \sqrt [4]{-b+a x^4}}{8 b-a x^8}-\frac {a x^6 \sqrt [4]{-b+a x^4}}{-8 b+a x^8}\right ) \, dx}{8 b}-\frac {a^2 \int \frac {x^2}{\left (-b+a x^4\right )^{3/4}} \, dx}{8 b}\\ &=\frac {a \sqrt [4]{-b+a x^4}}{8 b x}+\frac {\left (-b+a x^4\right )^{5/4}}{10 b x^5}+\frac {1}{2} a \int \frac {x^2 \sqrt [4]{-b+a x^4}}{8 b-a x^8} \, dx+\frac {a^2 \int \frac {x^6 \sqrt [4]{-b+a x^4}}{-8 b+a x^8} \, dx}{8 b}-\frac {a^2 \text {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{8 b}\\ &=\frac {a \sqrt [4]{-b+a x^4}}{8 b x}+\frac {\left (-b+a x^4\right )^{5/4}}{10 b x^5}+\frac {1}{2} a \int \left (\frac {\sqrt {a} x^2 \sqrt [4]{-b+a x^4}}{4 \sqrt {2} \sqrt {b} \left (2 \sqrt {2} \sqrt {a} \sqrt {b}-a x^4\right )}+\frac {\sqrt {a} x^2 \sqrt [4]{-b+a x^4}}{4 \sqrt {2} \sqrt {b} \left (2 \sqrt {2} \sqrt {a} \sqrt {b}+a x^4\right )}\right ) \, dx-\frac {a \int \frac {x^2 \left (-8 a b+a b x^4\right )}{\left (-b+a x^4\right )^{3/4} \left (-8 b+a x^8\right )} \, dx}{8 b}-\frac {a^{3/2} \text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{16 b}+\frac {a^{3/2} \text {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{16 b}+\frac {a^2 \int \frac {x^2}{\left (-b+a x^4\right )^{3/4}} \, dx}{8 b}\\ &=\frac {a \sqrt [4]{-b+a x^4}}{8 b x}+\frac {\left (-b+a x^4\right )^{5/4}}{10 b x^5}+\frac {a^{5/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{16 b}-\frac {a^{5/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{16 b}-\frac {a \int \left (-\frac {8 a b x^2}{\left (-b+a x^4\right )^{3/4} \left (-8 b+a x^8\right )}+\frac {a b x^6}{\left (-b+a x^4\right )^{3/4} \left (-8 b+a x^8\right )}\right ) \, dx}{8 b}+\frac {a^2 \text {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{8 b}+\frac {a^{3/2} \int \frac {x^2 \sqrt [4]{-b+a x^4}}{2 \sqrt {2} \sqrt {a} \sqrt {b}-a x^4} \, dx}{8 \sqrt {2} \sqrt {b}}+\frac {a^{3/2} \int \frac {x^2 \sqrt [4]{-b+a x^4}}{2 \sqrt {2} \sqrt {a} \sqrt {b}+a x^4} \, dx}{8 \sqrt {2} \sqrt {b}}\\ &=\frac {a \sqrt [4]{-b+a x^4}}{8 b x}+\frac {\left (-b+a x^4\right )^{5/4}}{10 b x^5}+\frac {a^{5/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{16 b}-\frac {a^{5/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{16 b}-\frac {1}{8} a^2 \int \frac {x^6}{\left (-b+a x^4\right )^{3/4} \left (-8 b+a x^8\right )} \, dx+a^2 \int \frac {x^2}{\left (-b+a x^4\right )^{3/4} \left (-8 b+a x^8\right )} \, dx+\frac {a^{3/2} \text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{16 b}-\frac {a^{3/2} \text {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{16 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}}}{2 \sqrt {2} \sqrt {a} \sqrt {b}-a x^4} \, dx}{8 \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}}}{2 \sqrt {2} \sqrt {a} \sqrt {b}+a x^4} \, dx}{8 \sqrt {2} \sqrt {b} \sqrt [4]{1-\frac {a x^4}{b}}}\\ &=\frac {a \sqrt [4]{-b+a x^4}}{8 b x}+\frac {\left (-b+a x^4\right )^{5/4}}{10 b x^5}+\frac {a x^3 \sqrt [4]{-b+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-\frac {\sqrt {a} x^4}{2 \sqrt {2} \sqrt {b}},\frac {a x^4}{b}\right )}{96 b \sqrt [4]{1-\frac {a x^4}{b}}}+\frac {a x^3 \sqrt [4]{-b+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {\sqrt {a} x^4}{2 \sqrt {2} \sqrt {b}},\frac {a x^4}{b}\right )}{96 b \sqrt [4]{1-\frac {a x^4}{b}}}-\frac {1}{8} a^2 \int \left (\frac {x^2}{2 \left (-2 \sqrt {2} \sqrt {a} \sqrt {b}+a x^4\right ) \left (-b+a x^4\right )^{3/4}}+\frac {x^2}{2 \left (2 \sqrt {2} \sqrt {a} \sqrt {b}+a x^4\right ) \left (-b+a x^4\right )^{3/4}}\right ) \, dx+a^2 \int \left (-\frac {\sqrt {a} x^2}{4 \sqrt {2} \sqrt {b} \left (2 \sqrt {2} \sqrt {a} \sqrt {b}-a x^4\right ) \left (-b+a x^4\right )^{3/4}}-\frac {\sqrt {a} x^2}{4 \sqrt {2} \sqrt {b} \left (2 \sqrt {2} \sqrt {a} \sqrt {b}+a x^4\right ) \left (-b+a x^4\right )^{3/4}}\right ) \, dx\\ &=\frac {a \sqrt [4]{-b+a x^4}}{8 b x}+\frac {\left (-b+a x^4\right )^{5/4}}{10 b x^5}+\frac {a x^3 \sqrt [4]{-b+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-\frac {\sqrt {a} x^4}{2 \sqrt {2} \sqrt {b}},\frac {a x^4}{b}\right )}{96 b \sqrt [4]{1-\frac {a x^4}{b}}}+\frac {a x^3 \sqrt [4]{-b+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {\sqrt {a} x^4}{2 \sqrt {2} \sqrt {b}},\frac {a x^4}{b}\right )}{96 b \sqrt [4]{1-\frac {a x^4}{b}}}-\frac {1}{16} a^2 \int \frac {x^2}{\left (-2 \sqrt {2} \sqrt {a} \sqrt {b}+a x^4\right ) \left (-b+a x^4\right )^{3/4}} \, dx-\frac {1}{16} a^2 \int \frac {x^2}{\left (2 \sqrt {2} \sqrt {a} \sqrt {b}+a x^4\right ) \left (-b+a x^4\right )^{3/4}} \, dx-\frac {a^{5/2} \int \frac {x^2}{\left (2 \sqrt {2} \sqrt {a} \sqrt {b}-a x^4\right ) \left (-b+a x^4\right )^{3/4}} \, dx}{4 \sqrt {2} \sqrt {b}}-\frac {a^{5/2} \int \frac {x^2}{\left (2 \sqrt {2} \sqrt {a} \sqrt {b}+a x^4\right ) \left (-b+a x^4\right )^{3/4}} \, dx}{4 \sqrt {2} \sqrt {b}}\\ &=\frac {a \sqrt [4]{-b+a x^4}}{8 b x}+\frac {\left (-b+a x^4\right )^{5/4}}{10 b x^5}+\frac {a x^3 \sqrt [4]{-b+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-\frac {\sqrt {a} x^4}{2 \sqrt {2} \sqrt {b}},\frac {a x^4}{b}\right )}{96 b \sqrt [4]{1-\frac {a x^4}{b}}}+\frac {a x^3 \sqrt [4]{-b+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {\sqrt {a} x^4}{2 \sqrt {2} \sqrt {b}},\frac {a x^4}{b}\right )}{96 b \sqrt [4]{1-\frac {a x^4}{b}}}-\frac {1}{16} a^2 \text {Subst}\left (\int \frac {x^2}{-2 \sqrt {2} \sqrt {a} \sqrt {b}-\left (-2 \sqrt {2} a^{3/2} \sqrt {b}+a b\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )-\frac {1}{16} a^2 \text {Subst}\left (\int \frac {x^2}{2 \sqrt {2} \sqrt {a} \sqrt {b}-\left (2 \sqrt {2} a^{3/2} \sqrt {b}+a b\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )-\frac {a^{5/2} \text {Subst}\left (\int \frac {x^2}{2 \sqrt {2} \sqrt {a} \sqrt {b}-\left (2 \sqrt {2} a^{3/2} \sqrt {b}-a b\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{4 \sqrt {2} \sqrt {b}}-\frac {a^{5/2} \text {Subst}\left (\int \frac {x^2}{2 \sqrt {2} \sqrt {a} \sqrt {b}-\left (2 \sqrt {2} a^{3/2} \sqrt {b}+a b\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{4 \sqrt {2} \sqrt {b}}\\ &=\frac {a \sqrt [4]{-b+a x^4}}{8 b x}+\frac {\left (-b+a x^4\right )^{5/4}}{10 b x^5}+\frac {a x^3 \sqrt [4]{-b+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-\frac {\sqrt {a} x^4}{2 \sqrt {2} \sqrt {b}},\frac {a x^4}{b}\right )}{96 b \sqrt [4]{1-\frac {a x^4}{b}}}+\frac {a x^3 \sqrt [4]{-b+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {\sqrt {a} x^4}{2 \sqrt {2} \sqrt {b}},\frac {a x^4}{b}\right )}{96 b \sqrt [4]{1-\frac {a x^4}{b}}}-\frac {a^{7/4} \text {Subst}\left (\int \frac {1}{2^{3/4}-\sqrt [4]{a} \sqrt {2 \sqrt {2} \sqrt {a}-\sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{8 \sqrt {2} \sqrt {2 \sqrt {2} \sqrt {a}-\sqrt {b}} b}+\frac {a^{7/4} \text {Subst}\left (\int \frac {1}{2^{3/4}+\sqrt [4]{a} \sqrt {2 \sqrt {2} \sqrt {a}-\sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{8 \sqrt {2} \sqrt {2 \sqrt {2} \sqrt {a}-\sqrt {b}} b}-\frac {a^{7/4} \text {Subst}\left (\int \frac {1}{2^{3/4}-\sqrt [4]{a} \sqrt {2 \sqrt {2} \sqrt {a}+\sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{8 \sqrt {2} \sqrt {2 \sqrt {2} \sqrt {a}+\sqrt {b}} b}+\frac {a^{7/4} \text {Subst}\left (\int \frac {1}{2^{3/4}+\sqrt [4]{a} \sqrt {2 \sqrt {2} \sqrt {a}+\sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{8 \sqrt {2} \sqrt {2 \sqrt {2} \sqrt {a}+\sqrt {b}} b}+\frac {a^{5/4} \text {Subst}\left (\int \frac {1}{2^{3/4}-\sqrt [4]{a} \sqrt {2 \sqrt {2} \sqrt {a}-\sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{32 \sqrt {2 \sqrt {2} \sqrt {a}-\sqrt {b}} \sqrt {b}}-\frac {a^{5/4} \text {Subst}\left (\int \frac {1}{2^{3/4}+\sqrt [4]{a} \sqrt {2 \sqrt {2} \sqrt {a}-\sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{32 \sqrt {2 \sqrt {2} \sqrt {a}-\sqrt {b}} \sqrt {b}}-\frac {a^{5/4} \text {Subst}\left (\int \frac {1}{2^{3/4}-\sqrt [4]{a} \sqrt {2 \sqrt {2} \sqrt {a}+\sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{32 \sqrt {2 \sqrt {2} \sqrt {a}+\sqrt {b}} \sqrt {b}}+\frac {a^{5/4} \text {Subst}\left (\int \frac {1}{2^{3/4}+\sqrt [4]{a} \sqrt {2 \sqrt {2} \sqrt {a}+\sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{32 \sqrt {2 \sqrt {2} \sqrt {a}+\sqrt {b}} \sqrt {b}}\\ &=\frac {a \sqrt [4]{-b+a x^4}}{8 b x}+\frac {\left (-b+a x^4\right )^{5/4}}{10 b x^5}+\frac {a x^3 \sqrt [4]{-b+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-\frac {\sqrt {a} x^4}{2 \sqrt {2} \sqrt {b}},\frac {a x^4}{b}\right )}{96 b \sqrt [4]{1-\frac {a x^4}{b}}}+\frac {a x^3 \sqrt [4]{-b+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {\sqrt {a} x^4}{2 \sqrt {2} \sqrt {b}},\frac {a x^4}{b}\right )}{96 b \sqrt [4]{1-\frac {a x^4}{b}}}+\frac {a^{13/8} \tan ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [4]{2 \sqrt {2} \sqrt {a}-\sqrt {b}} x}{2^{3/8} \sqrt [4]{-b+a x^4}}\right )}{8\ 2^{7/8} \left (2 \sqrt {2} \sqrt {a}-\sqrt {b}\right )^{3/4} b}-\frac {a^{9/8} \tan ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [4]{2 \sqrt {2} \sqrt {a}-\sqrt {b}} x}{2^{3/8} \sqrt [4]{-b+a x^4}}\right )}{32\ 2^{3/8} \left (2 \sqrt {2} \sqrt {a}-\sqrt {b}\right )^{3/4} \sqrt {b}}+\frac {a^{13/8} \tan ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [4]{2 \sqrt {2} \sqrt {a}+\sqrt {b}} x}{2^{3/8} \sqrt [4]{-b+a x^4}}\right )}{8\ 2^{7/8} \left (2 \sqrt {2} \sqrt {a}+\sqrt {b}\right )^{3/4} b}+\frac {a^{9/8} \tan ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [4]{2 \sqrt {2} \sqrt {a}+\sqrt {b}} x}{2^{3/8} \sqrt [4]{-b+a x^4}}\right )}{32\ 2^{3/8} \left (2 \sqrt {2} \sqrt {a}+\sqrt {b}\right )^{3/4} \sqrt {b}}-\frac {a^{13/8} \tanh ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [4]{2 \sqrt {2} \sqrt {a}-\sqrt {b}} x}{2^{3/8} \sqrt [4]{-b+a x^4}}\right )}{8\ 2^{7/8} \left (2 \sqrt {2} \sqrt {a}-\sqrt {b}\right )^{3/4} b}+\frac {a^{9/8} \tanh ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [4]{2 \sqrt {2} \sqrt {a}-\sqrt {b}} x}{2^{3/8} \sqrt [4]{-b+a x^4}}\right )}{32\ 2^{3/8} \left (2 \sqrt {2} \sqrt {a}-\sqrt {b}\right )^{3/4} \sqrt {b}}-\frac {a^{13/8} \tanh ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [4]{2 \sqrt {2} \sqrt {a}+\sqrt {b}} x}{2^{3/8} \sqrt [4]{-b+a x^4}}\right )}{8\ 2^{7/8} \left (2 \sqrt {2} \sqrt {a}+\sqrt {b}\right )^{3/4} b}-\frac {a^{9/8} \tanh ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [4]{2 \sqrt {2} \sqrt {a}+\sqrt {b}} x}{2^{3/8} \sqrt [4]{-b+a x^4}}\right )}{32\ 2^{3/8} \left (2 \sqrt {2} \sqrt {a}+\sqrt {b}\right )^{3/4} \sqrt {b}}\\ \end {align*}

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Mathematica [A]
time = 0.35, size = 214, normalized size = 0.98 \begin {gather*} \frac {\frac {64 \sqrt [4]{-b+a x^4} \left (-4 b+9 a x^4\right )}{x^5}+5 a \text {RootSum}\left [8 a^2-a b-16 a \text {$\#$1}^4+8 \text {$\#$1}^8\&,\frac {8 a^2 \log (x)-a b \log (x)-8 a^2 \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )+a b \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )-8 a \log (x) \text {$\#$1}^4-4 b \log (x) \text {$\#$1}^4+8 a \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4+4 b \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}^3+\text {$\#$1}^7}\&\right ]}{2560 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((64*(-b + a*x^4)^(1/4)*(-4*b + 9*a*x^4))/x^5 + 5*a*RootSum[8*a^2 - a*b - 16*a*#1^4 + 8*#1^8 & , (8*a^2*Log[x]
- a*b*Log[x] - 8*a^2*Log[(-b + a*x^4)^(1/4) - x*#1] + a*b*Log[(-b + a*x^4)^(1/4) - x*#1] - 8*a*Log[x]*#1^4 -
4*b*Log[x]*#1^4 + 8*a*Log[(-b + a*x^4)^(1/4) - x*#1]*#1^4 + 4*b*Log[(-b + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1^3
) + #1^7) & ])/(2560*b)

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{4}-4 b \right ) \left (a \,x^{4}-b \right )^{\frac {1}{4}}}{x^{6} \left (a \,x^{8}-8 b \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a x^{4} - 4 b\right ) \sqrt [4]{a x^{4} - b}}{x^{6} \left (a x^{8} - 8 b\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{4,[0,1,4,1,0]%%%}+%%%{-1,[0,1,0,0,1]%%%} / %%%{8,[0,0,0,
1,0]%%%} Er

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a\,x^4-b\right )}^{1/4}\,\left (4\,b-a\,x^4\right )}{x^6\,\left (8\,b-a\,x^8\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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