\(\int \frac {(b+a x^2+x^4) \sqrt [4]{-b x^2+a x^4}}{-b+a x^4} \, dx\) [2933]

Optimal result

Integrand size = 38, antiderivative size = 343 \[ \int \frac {\left (b+a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}}{-b+a x^4} \, dx=\frac {x \sqrt [4]{-b x^2+a x^4}}{2 a}+\frac {\left (-4 a^2+b\right ) \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^2+a x^4}}\right )}{4 a^{7/4}}+\frac {\left (4 a^2-b\right ) \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^2+a x^4}}\right )}{4 a^{7/4}}+\frac {\text {RootSum}\left [a^2-a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-a^3 \log (x)+a^2 b \log (x)+a^3 \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right )-a^2 b \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right )+a^2 \log (x) \text {$\#$1}^4-b \log (x) \text {$\#$1}^4-a b \log (x) \text {$\#$1}^4-a^2 \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4+b \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4+a b \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}^3-\text {$\#$1}^7}\&\right ]}{4 a} \]

[Out]

Unintegrable

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 337, normalized size of antiderivative = 0.98, number of steps used = 17, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.289, Rules used = {2081, 6857, 285, 335, 338, 304, 209, 212, 477, 525, 524} \[ \int \frac {\left (b+a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}}{-b+a x^4} \, dx=-\frac {x \left (-\frac {a^{3/2}}{\sqrt {b}}+a+1\right ) \sqrt [4]{a x^4-b x^2} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-\frac {\sqrt {a} x^2}{\sqrt {b}},\frac {a x^2}{b}\right )}{3 a \sqrt [4]{1-\frac {a x^2}{b}}}-\frac {x \left (\frac {a^{3/2}}{\sqrt {b}}+a+1\right ) \sqrt [4]{a x^4-b x^2} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {\sqrt {a} x^2}{\sqrt {b}},\frac {a x^2}{b}\right )}{3 a \sqrt [4]{1-\frac {a x^2}{b}}}+\frac {b \sqrt [4]{a x^4-b x^2} \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{4 a^{7/4} \sqrt {x} \sqrt [4]{a x^2-b}}-\frac {b \sqrt [4]{a x^4-b x^2} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{4 a^{7/4} \sqrt {x} \sqrt [4]{a x^2-b}}+\frac {x \sqrt [4]{a x^4-b x^2}}{2 a} \]

[In]

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

[Out]

(x*(-(b*x^2) + a*x^4)^(1/4))/(2*a) - ((1 + a - a^(3/2)/Sqrt[b])*x*(-(b*x^2) + a*x^4)^(1/4)*AppellF1[3/4, 1, -1
/4, 7/4, -((Sqrt[a]*x^2)/Sqrt[b]), (a*x^2)/b])/(3*a*(1 - (a*x^2)/b)^(1/4)) - ((1 + a + a^(3/2)/Sqrt[b])*x*(-(b
*x^2) + a*x^4)^(1/4)*AppellF1[3/4, 1, -1/4, 7/4, (Sqrt[a]*x^2)/Sqrt[b], (a*x^2)/b])/(3*a*(1 - (a*x^2)/b)^(1/4)
) + (b*(-(b*x^2) + a*x^4)^(1/4)*ArcTan[(a^(1/4)*Sqrt[x])/(-b + a*x^2)^(1/4)])/(4*a^(7/4)*Sqrt[x]*(-b + a*x^2)^
(1/4)) - (b*(-(b*x^2) + a*x^4)^(1/4)*ArcTanh[(a^(1/4)*Sqrt[x])/(-b + a*x^2)^(1/4)])/(4*a^(7/4)*Sqrt[x]*(-b + a
*x^2)^(1/4))

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 285

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^p/(c*(m + n
*p + 1))), x] + Dist[a*n*(p/(m + n*p + 1)), Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x]
&& IGtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 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 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

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 477

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/e^n))^p*(c + d*(x^(k*n)/e^n))^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

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 2081

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.18 \[ \int \frac {\left (b+a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}}{-b+a x^4} \, dx=\frac {\sqrt [4]{-b x^2+a x^4} \left (4 a^{3/4} x^{3/2} \sqrt [4]{-b+a x^2}-8 a^2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )+2 b \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )+\left (8 a^2-2 b\right ) \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )-a^{3/4} \text {RootSum}\left [a^2-a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-a^3 \log (x)+a^2 b \log (x)+2 a^3 \log \left (\sqrt [4]{-b+a x^2}-\sqrt {x} \text {$\#$1}\right )-2 a^2 b \log \left (\sqrt [4]{-b+a x^2}-\sqrt {x} \text {$\#$1}\right )+a^2 \log (x) \text {$\#$1}^4-b \log (x) \text {$\#$1}^4-a b \log (x) \text {$\#$1}^4-2 a^2 \log \left (\sqrt [4]{-b+a x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4+2 b \log \left (\sqrt [4]{-b+a x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4+2 a b \log \left (\sqrt [4]{-b+a x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}^3+\text {$\#$1}^7}\&\right ]\right )}{8 a^{7/4} \sqrt {x} \sqrt [4]{-b+a x^2}} \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Time = 0.00 (sec) , antiderivative size = 280, normalized size of antiderivative = 0.82

method	result	size
pseudoelliptic	\(\frac {4 x \,a^{\frac {3}{4}} \left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-2 a \,\textit {\_Z}^{4}+a^{2}-a b \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{x}\right ) \left (\left (-a^{2}+a b +b \right ) \textit {\_R}^{4}+a^{2} \left (a -b \right )\right )}{\textit {\_R}^{3} \left (-\textit {\_R}^{4}+a \right )}\right ) a^{\frac {3}{4}}+4 \ln \left (\frac {-a^{\frac {1}{4}} x -\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x -\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}\right ) a^{2}+8 \arctan \left (\frac {\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right ) a^{2}-\ln \left (\frac {-a^{\frac {1}{4}} x -\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x -\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}\right ) b -2 \arctan \left (\frac {\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right ) b}{8 a^{\frac {7}{4}}}\)	\(280\)

[In]

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

[Out]

1/8/a^(7/4)*(4*x*a^(3/4)*(x^2*(a*x^2-b))^(1/4)+2*sum(ln((-_R*x+(x^2*(a*x^2-b))^(1/4))/x)*((-a^2+a*b+b)*_R^4+a^
2*(a-b))/_R^3/(-_R^4+a),_R=RootOf(_Z^8-2*_Z^4*a+a^2-a*b))*a^(3/4)+4*ln((-a^(1/4)*x-(x^2*(a*x^2-b))^(1/4))/(a^(
1/4)*x-(x^2*(a*x^2-b))^(1/4)))*a^2+8*arctan(1/a^(1/4)/x*(x^2*(a*x^2-b))^(1/4))*a^2-ln((-a^(1/4)*x-(x^2*(a*x^2-
b))^(1/4))/(a^(1/4)*x-(x^2*(a*x^2-b))^(1/4)))*b-2*arctan(1/a^(1/4)/x*(x^2*(a*x^2-b))^(1/4))*b)

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [N/A]

Not integrable

Time = 13.45 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.09 \[ \int \frac {\left (b+a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}}{-b+a x^4} \, dx=\int \frac {\sqrt [4]{x^{2} \left (a x^{2} - b\right )} \left (a x^{2} + b + x^{4}\right )}{a x^{4} - b}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.22 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.11 \[ \int \frac {\left (b+a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}}{-b+a x^4} \, dx=\int { \frac {{\left (a x^{4} - b x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} + a x^{2} + b\right )}}{a x^{4} - b} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 1.37 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.11 \[ \int \frac {\left (b+a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}}{-b+a x^4} \, dx=\int { \frac {{\left (a x^{4} - b x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} + a x^{2} + b\right )}}{a x^{4} - b} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.11 \[ \int \frac {\left (b+a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}}{-b+a x^4} \, dx=\int -\frac {{\left (a\,x^4-b\,x^2\right )}^{1/4}\,\left (x^4+a\,x^2+b\right )}{b-a\,x^4} \,d x \]

[In]

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

[Out]

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