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

Optimal. Leaf size=342 \[ -\frac {\text {RootSum}\left [\text {$\#$1}^8-2 \text {$\#$1}^4 a+a^2+a b\& ,\frac {-\text {$\#$1}^4 a^2 \log \left (\sqrt [4]{a x^4-b x^2}-\text {$\#$1} x\right )+\text {$\#$1}^4 a^2 \log (x)-\text {$\#$1}^4 a b \log \left (\sqrt [4]{a x^4-b x^2}-\text {$\#$1} x\right )-\text {$\#$1}^4 b \log \left (\sqrt [4]{a x^4-b x^2}-\text {$\#$1} x\right )+\text {$\#$1}^4 a b \log (x)+\text {$\#$1}^4 b \log (x)+a^3 \log \left (\sqrt [4]{a x^4-b x^2}-\text {$\#$1} x\right )+a^2 b \log \left (\sqrt [4]{a x^4-b x^2}-\text {$\#$1} x\right )-a^3 \log (x)-a^2 b \log (x)}{\text {$\#$1}^3 a-\text {$\#$1}^7}\& \right ]}{4 a}+\frac {\left (4 a^2+b\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b x^2}}\right )}{4 a^{7/4}}+\frac {\left (-4 a^2-b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b x^2}}\right )}{4 a^{7/4}}+\frac {x \sqrt [4]{a x^4-b x^2}}{2 a} \]

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Rubi [A]  time = 1.29, antiderivative size = 345, normalized size of antiderivative = 1.01, number of steps used = 17, number of rules used = 11, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.282, Rules used = {2056, 6725, 279, 329, 331, 298, 203, 206, 466, 511, 510} \begin {gather*} \frac {b \sqrt [4]{a x^4-b x^2} \tan ^{-1}\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} \tanh ^{-1}\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 \left (\frac {\sqrt {-a} a}{\sqrt {b}}+a+1\right ) \sqrt [4]{a x^4-b x^2} F_1\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} F_1\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 \sqrt [4]{a x^4-b x^2}}{2 a} \end {gather*}

Warning: Unable to verify antiderivative.

[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 + (Sqrt[-a]*a)/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 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 279

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 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 329

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 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 466

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 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 2056

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

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Mathematica [B]  time = 2.49, size = 756, normalized size = 2.21 \begin {gather*} \frac {x \sqrt [4]{a x^4-b x^2} \left (8-\frac {\left (\frac {b}{x^2}-a\right )^{3/4} \left (\frac {8 \left (a^2+a b+b\right ) \left (\sqrt [4]{\sqrt {-a}-\sqrt {b}} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {b}{x^2}-a}}{\sqrt [8]{-a} \sqrt [4]{\sqrt {-a}-\sqrt {b}}}\right )-\sqrt [4]{\sqrt {-a}+\sqrt {b}} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {b}{x^2}-a}}{\sqrt [8]{-a} \sqrt [4]{\sqrt {-a}+\sqrt {b}}}\right )+\sqrt [4]{\sqrt {-a}-\sqrt {b}} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {b}{x^2}-a}}{\sqrt [8]{-a} \sqrt [4]{\sqrt {-a}-\sqrt {b}}}\right )-\sqrt [4]{\sqrt {-a}+\sqrt {b}} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {b}{x^2}-a}}{\sqrt [8]{-a} \sqrt [4]{\sqrt {-a}+\sqrt {b}}}\right )\right )}{(-a)^{3/8} \sqrt {b}}+\frac {\sqrt {2} \left (4 a^2+b\right ) \left (\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{\frac {b}{x^2}-a}+\sqrt {\frac {b}{x^2}-a}+\sqrt {a}\right )-\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{\frac {b}{x^2}-a}+\sqrt {\frac {b}{x^2}-a}+\sqrt {a}\right )\right )}{a^{3/4}}+\frac {2 \sqrt {2} \left (4 a^2+b\right ) \left (\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{\frac {b}{x^2}-a}}{\sqrt [4]{a}}\right )-\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{\frac {b}{x^2}-a}}{\sqrt [4]{a}}+1\right )\right )}{a^{3/4}}+\frac {8 (-a)^{9/8} (a+b) \left (\left (\sqrt {-a}+\sqrt {b}\right )^{3/4} \left (\tan ^{-1}\left (\frac {\sqrt [4]{\frac {b}{x^2}-a}}{\sqrt [8]{-a} \sqrt [4]{\sqrt {-a}-\sqrt {b}}}\right )+\tanh ^{-1}\left (\frac {\sqrt [4]{\frac {b}{x^2}-a}}{\sqrt [8]{-a} \sqrt [4]{\sqrt {-a}-\sqrt {b}}}\right )\right )-\left (\sqrt {-a}-\sqrt {b}\right )^{3/4} \left (\tan ^{-1}\left (\frac {\sqrt [4]{\frac {b}{x^2}-a}}{\sqrt [8]{-a} \sqrt [4]{\sqrt {-a}+\sqrt {b}}}\right )+\tanh ^{-1}\left (\frac {\sqrt [4]{\frac {b}{x^2}-a}}{\sqrt [8]{-a} \sqrt [4]{\sqrt {-a}+\sqrt {b}}}\right )\right )\right )}{\sqrt {b} \left (\sqrt {-a}-\sqrt {b}\right )^{3/4} \left (\sqrt {-a}+\sqrt {b}\right )^{3/4}}\right )}{a x^2-b}\right )}{16 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-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)*(8 - ((-a + b/x^2)^(3/4)*((2*Sqrt[2]*(4*a^2 + b)*(ArcTan[1 - (Sqrt[2]*(-a + b/x^2)
^(1/4))/a^(1/4)] - ArcTan[1 + (Sqrt[2]*(-a + b/x^2)^(1/4))/a^(1/4)]))/a^(3/4) + (8*(a^2 + b + a*b)*((Sqrt[-a]
- Sqrt[b])^(1/4)*ArcTan[(-a + b/x^2)^(1/4)/((-a)^(1/8)*(Sqrt[-a] - Sqrt[b])^(1/4))] - (Sqrt[-a] + Sqrt[b])^(1/
4)*ArcTan[(-a + b/x^2)^(1/4)/((-a)^(1/8)*(Sqrt[-a] + Sqrt[b])^(1/4))] + (Sqrt[-a] - Sqrt[b])^(1/4)*ArcTanh[(-a
+ b/x^2)^(1/4)/((-a)^(1/8)*(Sqrt[-a] - Sqrt[b])^(1/4))] - (Sqrt[-a] + Sqrt[b])^(1/4)*ArcTanh[(-a + b/x^2)^(1/
4)/((-a)^(1/8)*(Sqrt[-a] + Sqrt[b])^(1/4))]))/((-a)^(3/8)*Sqrt[b]) + (8*(-a)^(9/8)*(a + b)*((Sqrt[-a] + Sqrt[b
])^(3/4)*(ArcTan[(-a + b/x^2)^(1/4)/((-a)^(1/8)*(Sqrt[-a] - Sqrt[b])^(1/4))] + ArcTanh[(-a + b/x^2)^(1/4)/((-a
)^(1/8)*(Sqrt[-a] - Sqrt[b])^(1/4))]) - (Sqrt[-a] - Sqrt[b])^(3/4)*(ArcTan[(-a + b/x^2)^(1/4)/((-a)^(1/8)*(Sqr
t[-a] + Sqrt[b])^(1/4))] + ArcTanh[(-a + b/x^2)^(1/4)/((-a)^(1/8)*(Sqrt[-a] + Sqrt[b])^(1/4))])))/((Sqrt[-a] -
Sqrt[b])^(3/4)*(Sqrt[-a] + Sqrt[b])^(3/4)*Sqrt[b]) + (Sqrt[2]*(4*a^2 + b)*(Log[Sqrt[a] - Sqrt[2]*a^(1/4)*(-a
+ b/x^2)^(1/4) + Sqrt[-a + b/x^2]] - Log[Sqrt[a] + Sqrt[2]*a^(1/4)*(-a + b/x^2)^(1/4) + Sqrt[-a + b/x^2]]))/a^
(3/4)))/(-b + a*x^2)))/(16*a)

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IntegrateAlgebraic [A]  time = 0.00, size = 341, normalized size = 1.00 \begin {gather*} \frac {x \sqrt [4]{-b x^2+a x^4}}{2 a}+\frac {\left (4 a^2+b\right ) \tan ^{-1}\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 ) \tanh ^{-1}\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

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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((x^4-a*x^2-b)*(a*x^4-b*x^2)^(1/4)/(a*x^4+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^{4} - b x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - a x^{2} - b\right )}}{a x^{4} + b}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int -\frac {{\left (a\,x^4-b\,x^2\right )}^{1/4}\,\left (-x^4+a\,x^2+b\right )}{a\,x^4+b} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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