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

Optimal. Leaf size=215 \[ \frac {\sqrt [4]{a x^4-b} \left (a x^4-b-5 c x^4\right )}{5 b x^5}-\frac {a \text {RootSum}\left [\text {$\#$1}^8-2 \text {$\#$1}^4 a+a^2+2 a b\& ,\frac {-\text {$\#$1}^4 c \log \left (\sqrt [4]{a x^4-b}-\text {$\#$1} x\right )-\text {$\#$1}^4 b \log \left (\sqrt [4]{a x^4-b}-\text {$\#$1} x\right )+\text {$\#$1}^4 b \log (x)+\text {$\#$1}^4 c \log (x)+a c \log \left (\sqrt [4]{a x^4-b}-\text {$\#$1} x\right )+2 b c \log \left (\sqrt [4]{a x^4-b}-\text {$\#$1} x\right )-a c \log (x)-2 b c \log (x)}{\text {$\#$1}^3 a-\text {$\#$1}^7}\& \right ]}{8 b} \]

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Rubi [B]  time = 4.34, antiderivative size = 933, normalized size of antiderivative = 4.34, number of steps used = 43, number of rules used = 12, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {6725, 264, 277, 331, 298, 203, 206, 1529, 511, 510, 1519, 494} \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 {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} F_1\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 \tan ^{-1}\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 \tan ^{-1}\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 \tan ^{-1}\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 \tan ^{-1}\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 \tanh ^{-1}\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 \tanh ^{-1}\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 \tanh ^{-1}\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 \tanh ^{-1}\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} \end {gather*}

Warning: Unable to verify antiderivative.

[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 203

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

Rule 206

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

Rule 264

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[((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 298

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 331

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 494

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 510

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)*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, -((b*x^n)/a), -((d*x^n)/c)])/(e*(m + 1)), 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 511

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPa
rt[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 1519

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 1529

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 6725

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 {\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 \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) \operatorname {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 ) \operatorname {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 ) \operatorname {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 \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 b}+\frac {\sqrt [4]{a} c \tanh ^{-1}\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) \operatorname {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 \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 b}+\frac {\sqrt [4]{a} c \tanh ^{-1}\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 ) \operatorname {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 ) \operatorname {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} F_1\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} F_1\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} F_1\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} F_1\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} F_1\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} F_1\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) \operatorname {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) \operatorname {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 ) \operatorname {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 ) \operatorname {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} F_1\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} F_1\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) \operatorname {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) \operatorname {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) \operatorname {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) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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} F_1\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} F_1\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 \tan ^{-1}\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 \tan ^{-1}\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 \tan ^{-1}\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 \tan ^{-1}\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 \tanh ^{-1}\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 \tanh ^{-1}\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 \tanh ^{-1}\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 \tanh ^{-1}\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*}

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Mathematica [F]  time = 0.37, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.00, size = 214, normalized size = 1.00 \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-b + a*x^4)^(1/4)*(-b + a*x^4 - 5*c*x^4))/(5*b*x^5) - (a*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) & ])/(8*b)

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

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [F]  time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{4}-b \right )^{\frac {1}{4}} \left (a \,x^{8}+c \,x^{4}+b \right )}{x^{6} \left (2 a \,x^{8}+b \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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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 (a\,x^8+c\,x^4+b\right )}{x^6\,\left (2\,a\,x^8+b\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

[Out]

Timed out

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