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

Optimal. Leaf size=204 \[ \frac {\left (-17 a x^4-2 b\right ) \sqrt [4]{a x^4+b}}{5 b x^5}-\frac {a \text {RootSum}\left [\text {$\#$1}^8-2 \text {$\#$1}^4 a+a^2+a b\& ,\frac {-3 \text {$\#$1}^4 a \log \left (\sqrt [4]{a x^4+b}-\text {$\#$1} x\right )+2 \text {$\#$1}^4 b \log \left (\sqrt [4]{a x^4+b}-\text {$\#$1} x\right )+3 \text {$\#$1}^4 a \log (x)-2 \text {$\#$1}^4 b \log (x)+3 a^2 \log \left (\sqrt [4]{a x^4+b}-\text {$\#$1} x\right )+3 a b \log \left (\sqrt [4]{a x^4+b}-\text {$\#$1} x\right )-3 a^2 \log (x)-3 a b \log (x)}{\text {$\#$1}^3 a-\text {$\#$1}^7}\& \right ]}{8 b} \]

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Rubi [B]  time = 2.61, antiderivative size = 715, normalized size of antiderivative = 3.50, number of steps used = 43, number of rules used = 12, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {6725, 264, 277, 331, 298, 203, 206, 1529, 511, 510, 1519, 494} \begin {gather*} -\frac {3 a^2 \tan ^{-1}\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 \tan ^{-1}\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 \tanh ^{-1}\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 \tanh ^{-1}\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} F_1\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} F_1\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 \sqrt [4]{a x^4+b}}{b x}-\frac {3 (-a)^{3/2} \tan ^{-1}\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} \tan ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\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 {2 \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)*(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 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 (2 b+3 a x^4\right )}{x^6 \left (b+a x^8\right )} \, dx &=\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 ) \operatorname {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 ) \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 (3 a^{3/2}\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 (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} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2 b}+\frac {3 a^{5/4} \tanh ^{-1}\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 ) \operatorname {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} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2 b}+\frac {3 a^{5/4} \tanh ^{-1}\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 ) \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 (3 a^{3/2}\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 {-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} F_1\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} F_1\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} F_1\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} F_1\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} F_1\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} F_1\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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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} F_1\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} F_1\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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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} F_1\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} F_1\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 \tan ^{-1}\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} \tan ^{-1}\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 \tan ^{-1}\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} \tan ^{-1}\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 \tanh ^{-1}\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} \tanh ^{-1}\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 \tanh ^{-1}\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} \tanh ^{-1}\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*}

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Mathematica [F]  time = 0.26, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.00, size = 203, normalized size = 1.00 \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((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)

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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)*(3*a*x^4+2*b)/x^6/(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 (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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [F]  time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{4}+b \right )^{\frac {1}{4}} \left (3 a \,x^{4}+2 b \right )}{x^{6} \left (a \,x^{8}+b \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a*x^4+b)^(1/4)*(3*a*x^4+2*b)/x^6/(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 (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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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