3.18.42 \(\int \frac {-b-a x^4+2 x^8}{\sqrt [4]{b+a x^4} (-b-a x^4+x^8)} \, dx\) [1742]

Optimal. Leaf size=117 \[ \frac {\text {ArcTan}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {1}{4} \text {RootSum}\left [b+a \text {$\#$1}^4-\text {$\#$1}^8\& ,\frac {-\log (x) \text {$\#$1}^3+\log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-a+2 \text {$\#$1}^4}\& \right ] \]

[Out]

Unintegrable

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(507\) vs. \(2(117)=234\).
time = 0.46, antiderivative size = 507, normalized size of antiderivative = 4.33, number of steps used = 25, number of rules used = 10, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {6860, 246, 218, 212, 209, 1442, 399, 385, 214, 211} \begin {gather*} \frac {\left (-a \sqrt {a^2+4 b}+a^2+2 b\right )^{3/4} \text {ArcTan}\left (\frac {x \sqrt [4]{-a \sqrt {a^2+4 b}+a^2+2 b}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{a x^4+b}}\right )}{2 \sqrt {a^2+4 b} \left (a-\sqrt {a^2+4 b}\right )^{3/4}}-\frac {\left (a \sqrt {a^2+4 b}+a^2+2 b\right )^{3/4} \text {ArcTan}\left (\frac {x \sqrt [4]{a \sqrt {a^2+4 b}+a^2+2 b}}{\sqrt [4]{\sqrt {a^2+4 b}+a} \sqrt [4]{a x^4+b}}\right )}{2 \sqrt {a^2+4 b} \left (\sqrt {a^2+4 b}+a\right )^{3/4}}+\frac {\left (-a \sqrt {a^2+4 b}+a^2+2 b\right )^{3/4} \tanh ^{-1}\left (\frac {x \sqrt [4]{-a \sqrt {a^2+4 b}+a^2+2 b}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{a x^4+b}}\right )}{2 \sqrt {a^2+4 b} \left (a-\sqrt {a^2+4 b}\right )^{3/4}}-\frac {\left (a \sqrt {a^2+4 b}+a^2+2 b\right )^{3/4} \tanh ^{-1}\left (\frac {x \sqrt [4]{a \sqrt {a^2+4 b}+a^2+2 b}}{\sqrt [4]{\sqrt {a^2+4 b}+a} \sqrt [4]{a x^4+b}}\right )}{2 \sqrt {a^2+4 b} \left (\sqrt {a^2+4 b}+a\right )^{3/4}}+\frac {\text {ArcTan}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{\sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{\sqrt [4]{a}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 211

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

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 214

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

Rule 218

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]},
Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !Gt
Q[a/b, 0]

Rule 246

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + 1/n), Subst[Int[1/(1 - b*x^n)^(p + 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)] && IntegerQ[p
+ 1/n]

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 399

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Dist[b/d, Int[(a + b*x^n)^(p - 1), x]
, x] - Dist[(b*c - a*d)/d, Int[(a + b*x^n)^(p - 1)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c
- a*d, 0] && EqQ[n*(p - 1) + 1, 0] && IntegerQ[n]

Rule 1442

Int[((d_) + (e_.)*(x_)^(n_))^(q_)/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{r = Rt[b^2 -
4*a*c, 2]}, Dist[2*(c/r), Int[(d + e*x^n)^q/(b - r + 2*c*x^n), x], x] - Dist[2*(c/r), Int[(d + e*x^n)^q/(b +
r + 2*c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] &&  !IntegerQ[q]

Rule 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {-b-a x^4+2 x^8}{\sqrt [4]{b+a x^4} \left (-b-a x^4+x^8\right )} \, dx &=\int \left (\frac {2}{\sqrt [4]{b+a x^4}}+\frac {\left (b+a x^4\right )^{3/4}}{-b-a x^4+x^8}\right ) \, dx\\ &=2 \int \frac {1}{\sqrt [4]{b+a x^4}} \, dx+\int \frac {\left (b+a x^4\right )^{3/4}}{-b-a x^4+x^8} \, dx\\ &=2 \text {Subst}\left (\int \frac {1}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )+\frac {2 \int \frac {\left (b+a x^4\right )^{3/4}}{-a-\sqrt {a^2+4 b}+2 x^4} \, dx}{\sqrt {a^2+4 b}}-\frac {2 \int \frac {\left (b+a x^4\right )^{3/4}}{-a+\sqrt {a^2+4 b}+2 x^4} \, dx}{\sqrt {a^2+4 b}}\\ &=\frac {\left (a^2+2 b+a \sqrt {a^2+4 b}\right ) \int \frac {1}{\left (-a-\sqrt {a^2+4 b}+2 x^4\right ) \sqrt [4]{b+a x^4}} \, dx}{\sqrt {a^2+4 b}}+\frac {\left (-2 b+a \left (-a+\sqrt {a^2+4 b}\right )\right ) \int \frac {1}{\left (-a+\sqrt {a^2+4 b}+2 x^4\right ) \sqrt [4]{b+a x^4}} \, dx}{\sqrt {a^2+4 b}}+\text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )+\text {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {\left (a^2+2 b+a \sqrt {a^2+4 b}\right ) \text {Subst}\left (\int \frac {1}{-a-\sqrt {a^2+4 b}-\left (-2 b+a \left (-a-\sqrt {a^2+4 b}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt {a^2+4 b}}+\frac {\left (-2 b+a \left (-a+\sqrt {a^2+4 b}\right )\right ) \text {Subst}\left (\int \frac {1}{-a+\sqrt {a^2+4 b}-\left (-2 b+a \left (-a+\sqrt {a^2+4 b}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt {a^2+4 b}}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}-\frac {\left (a^2+2 b+a \sqrt {a^2+4 b}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2+4 b}}-\sqrt {a^2+2 b+a \sqrt {a^2+4 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt {a^2+4 b} \sqrt {a+\sqrt {a^2+4 b}}}-\frac {\left (a^2+2 b+a \sqrt {a^2+4 b}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2+4 b}}+\sqrt {a^2+2 b+a \sqrt {a^2+4 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt {a^2+4 b} \sqrt {a+\sqrt {a^2+4 b}}}-\frac {\left (-2 b+a \left (-a+\sqrt {a^2+4 b}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2+4 b}}-\sqrt {a^2+2 b-a \sqrt {a^2+4 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt {a^2+4 b} \sqrt {a-\sqrt {a^2+4 b}}}-\frac {\left (-2 b+a \left (-a+\sqrt {a^2+4 b}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2+4 b}}+\sqrt {a^2+2 b-a \sqrt {a^2+4 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt {a^2+4 b} \sqrt {a-\sqrt {a^2+4 b}}}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {\left (a^2+2 b-a \sqrt {a^2+4 b}\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a^2+2 b-a \sqrt {a^2+4 b}} x}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{b+a x^4}}\right )}{2 \sqrt {a^2+4 b} \left (a-\sqrt {a^2+4 b}\right )^{3/4}}-\frac {\left (a^2+2 b+a \sqrt {a^2+4 b}\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a^2+2 b+a \sqrt {a^2+4 b}} x}{\sqrt [4]{a+\sqrt {a^2+4 b}} \sqrt [4]{b+a x^4}}\right )}{2 \sqrt {a^2+4 b} \left (a+\sqrt {a^2+4 b}\right )^{3/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {\left (a^2+2 b-a \sqrt {a^2+4 b}\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2+2 b-a \sqrt {a^2+4 b}} x}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{b+a x^4}}\right )}{2 \sqrt {a^2+4 b} \left (a-\sqrt {a^2+4 b}\right )^{3/4}}-\frac {\left (a^2+2 b+a \sqrt {a^2+4 b}\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2+2 b+a \sqrt {a^2+4 b}} x}{\sqrt [4]{a+\sqrt {a^2+4 b}} \sqrt [4]{b+a x^4}}\right )}{2 \sqrt {a^2+4 b} \left (a+\sqrt {a^2+4 b}\right )^{3/4}}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 112, normalized size = 0.96 \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )+\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {1}{4} \text {RootSum}\left [b+a \text {$\#$1}^4-\text {$\#$1}^8\&,\frac {-\log (x) \text {$\#$1}^3+\log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-a+2 \text {$\#$1}^4}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((2*x^8-a*x^4-b)/(a*x^4+b)^(1/4)/(x^8-a*x^4-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((2*x^8-a*x^4-b)/(a*x^4+b)^(1/4)/(x^8-a*x^4-b),x, algorithm="maxima")

[Out]

integrate((2*x^8 - a*x^4 - b)/((x^8 - a*x^4 - b)*(a*x^4 + b)^(1/4)), x)

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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 0.67, size = 3119, normalized size = 26.66 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^8-a*x^4-b)/(a*x^4+b)^(1/4)/(x^8-a*x^4-b),x, algorithm="fricas")

[Out]

-sqrt(sqrt(1/2)*sqrt((a^3 + 3*a*b - (a^4 + 8*a^2*b + 16*b^2)*sqrt((a^4 + 2*a^2*b + b^2)/(a^6 + 12*a^4*b + 48*a
^2*b^2 + 64*b^3)))/(a^4 + 8*a^2*b + 16*b^2)))*arctan(1/2*(sqrt(1/2)*((a^5 + 8*a^3*b + 16*a*b^2)*x*sqrt((a^4 +
2*a^2*b + b^2)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)) + (a^4 + 5*a^2*b + 4*b^2)*x)*sqrt(-(sqrt(1/2)*((a^8*b^3
+ 13*a^6*b^4 + 60*a^4*b^5 + 112*a^2*b^6 + 64*b^7)*x^2*sqrt((a^4 + 2*a^2*b + b^2)/(a^6 + 12*a^4*b + 48*a^2*b^2
+ 64*b^3)) + (a^7*b^3 + 6*a^5*b^4 + 9*a^3*b^5 + 4*a*b^6)*x^2)*sqrt((a^3 + 3*a*b - (a^4 + 8*a^2*b + 16*b^2)*sq
rt((a^4 + 2*a^2*b + b^2)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)))/(a^4 + 8*a^2*b + 16*b^2)) - 2*(a^4*b^4 + 2*a
^2*b^5 + b^6)*sqrt(a*x^4 + b))/x^2) - (a^6*b^2 + 6*a^4*b^3 + 9*a^2*b^4 + 4*b^5 + (a^7*b^2 + 9*a^5*b^3 + 24*a^3
*b^4 + 16*a*b^5)*sqrt((a^4 + 2*a^2*b + b^2)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)))*(a*x^4 + b)^(1/4))*sqrt(s
qrt(1/2)*sqrt((a^3 + 3*a*b - (a^4 + 8*a^2*b + 16*b^2)*sqrt((a^4 + 2*a^2*b + b^2)/(a^6 + 12*a^4*b + 48*a^2*b^2
+ 64*b^3)))/(a^4 + 8*a^2*b + 16*b^2)))/((a^4*b^3 + 2*a^2*b^4 + b^5)*x)) + sqrt(sqrt(1/2)*sqrt((a^3 + 3*a*b + (
a^4 + 8*a^2*b + 16*b^2)*sqrt((a^4 + 2*a^2*b + b^2)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)))/(a^4 + 8*a^2*b + 1
6*b^2)))*arctan(1/2*(sqrt(1/2)*((a^5 + 8*a^3*b + 16*a*b^2)*x*sqrt((a^4 + 2*a^2*b + b^2)/(a^6 + 12*a^4*b + 48*a
^2*b^2 + 64*b^3)) - (a^4 + 5*a^2*b + 4*b^2)*x)*sqrt(sqrt(1/2)*sqrt((a^3 + 3*a*b + (a^4 + 8*a^2*b + 16*b^2)*sqr
t((a^4 + 2*a^2*b + b^2)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)))/(a^4 + 8*a^2*b + 16*b^2)))*sqrt((sqrt(1/2)*((
a^8*b^3 + 13*a^6*b^4 + 60*a^4*b^5 + 112*a^2*b^6 + 64*b^7)*x^2*sqrt((a^4 + 2*a^2*b + b^2)/(a^6 + 12*a^4*b + 48*
a^2*b^2 + 64*b^3)) - (a^7*b^3 + 6*a^5*b^4 + 9*a^3*b^5 + 4*a*b^6)*x^2)*sqrt((a^3 + 3*a*b + (a^4 + 8*a^2*b + 16*
b^2)*sqrt((a^4 + 2*a^2*b + b^2)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)))/(a^4 + 8*a^2*b + 16*b^2)) + 2*(a^4*b^
4 + 2*a^2*b^5 + b^6)*sqrt(a*x^4 + b))/x^2) + (a^6*b^2 + 6*a^4*b^3 + 9*a^2*b^4 + 4*b^5 - (a^7*b^2 + 9*a^5*b^3 +
24*a^3*b^4 + 16*a*b^5)*sqrt((a^4 + 2*a^2*b + b^2)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)))*(a*x^4 + b)^(1/4)*
sqrt(sqrt(1/2)*sqrt((a^3 + 3*a*b + (a^4 + 8*a^2*b + 16*b^2)*sqrt((a^4 + 2*a^2*b + b^2)/(a^6 + 12*a^4*b + 48*a^
2*b^2 + 64*b^3)))/(a^4 + 8*a^2*b + 16*b^2))))/((a^4*b^3 + 2*a^2*b^4 + b^5)*x)) - 1/4*sqrt(sqrt(1/2)*sqrt((a^3
+ 3*a*b + (a^4 + 8*a^2*b + 16*b^2)*sqrt((a^4 + 2*a^2*b + b^2)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)))/(a^4 +
8*a^2*b + 16*b^2)))*log(1/2*(sqrt(1/2)*((a^8 + 14*a^6*b + 72*a^4*b^2 + 160*a^2*b^3 + 128*b^4)*x*sqrt((a^4 + 2*
a^2*b + b^2)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)) - (a^7 + 9*a^5*b + 24*a^3*b^2 + 16*a*b^3)*x)*sqrt(sqrt(1/
2)*sqrt((a^3 + 3*a*b + (a^4 + 8*a^2*b + 16*b^2)*sqrt((a^4 + 2*a^2*b + b^2)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b
^3)))/(a^4 + 8*a^2*b + 16*b^2)))*sqrt((a^3 + 3*a*b + (a^4 + 8*a^2*b + 16*b^2)*sqrt((a^4 + 2*a^2*b + b^2)/(a^6
+ 12*a^4*b + 48*a^2*b^2 + 64*b^3)))/(a^4 + 8*a^2*b + 16*b^2)) + 2*(a*x^4 + b)^(1/4)*(a^2*b^2 + b^3))/x) + 1/4*
sqrt(sqrt(1/2)*sqrt((a^3 + 3*a*b + (a^4 + 8*a^2*b + 16*b^2)*sqrt((a^4 + 2*a^2*b + b^2)/(a^6 + 12*a^4*b + 48*a^
2*b^2 + 64*b^3)))/(a^4 + 8*a^2*b + 16*b^2)))*log(-1/2*(sqrt(1/2)*((a^8 + 14*a^6*b + 72*a^4*b^2 + 160*a^2*b^3 +
128*b^4)*x*sqrt((a^4 + 2*a^2*b + b^2)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)) - (a^7 + 9*a^5*b + 24*a^3*b^2 +
16*a*b^3)*x)*sqrt(sqrt(1/2)*sqrt((a^3 + 3*a*b + (a^4 + 8*a^2*b + 16*b^2)*sqrt((a^4 + 2*a^2*b + b^2)/(a^6 + 12
*a^4*b + 48*a^2*b^2 + 64*b^3)))/(a^4 + 8*a^2*b + 16*b^2)))*sqrt((a^3 + 3*a*b + (a^4 + 8*a^2*b + 16*b^2)*sqrt((
a^4 + 2*a^2*b + b^2)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)))/(a^4 + 8*a^2*b + 16*b^2)) - 2*(a*x^4 + b)^(1/4)*
(a^2*b^2 + b^3))/x) + 1/4*sqrt(sqrt(1/2)*sqrt((a^3 + 3*a*b - (a^4 + 8*a^2*b + 16*b^2)*sqrt((a^4 + 2*a^2*b + b^
2)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)))/(a^4 + 8*a^2*b + 16*b^2)))*log(1/2*(sqrt(1/2)*((a^8 + 14*a^6*b + 7
2*a^4*b^2 + 160*a^2*b^3 + 128*b^4)*x*sqrt((a^4 + 2*a^2*b + b^2)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)) + (a^7
+ 9*a^5*b + 24*a^3*b^2 + 16*a*b^3)*x)*sqrt(sqrt(1/2)*sqrt((a^3 + 3*a*b - (a^4 + 8*a^2*b + 16*b^2)*sqrt((a^4 +
2*a^2*b + b^2)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)))/(a^4 + 8*a^2*b + 16*b^2)))*sqrt((a^3 + 3*a*b - (a^4 +
8*a^2*b + 16*b^2)*sqrt((a^4 + 2*a^2*b + b^2)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)))/(a^4 + 8*a^2*b + 16*b^2
)) + 2*(a*x^4 + b)^(1/4)*(a^2*b^2 + b^3))/x) - 1/4*sqrt(sqrt(1/2)*sqrt((a^3 + 3*a*b - (a^4 + 8*a^2*b + 16*b^2)
*sqrt((a^4 + 2*a^2*b + b^2)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)))/(a^4 + 8*a^2*b + 16*b^2)))*log(-1/2*(sqrt
(1/2)*((a^8 + 14*a^6*b + 72*a^4*b^2 + 160*a^2*b^3 + 128*b^4)*x*sqrt((a^4 + 2*a^2*b + b^2)/(a^6 + 12*a^4*b + 48
*a^2*b^2 + 64*b^3)) + (a^7 + 9*a^5*b + 24*a^3*b^2 + 16*a*b^3)*x)*sqrt(sqrt(1/2)*sqrt((a^3 + 3*a*b - (a^4 + 8*a
^2*b + 16*b^2)*sqrt((a^4 + 2*a^2*b + b^2)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)))/(a^4 + 8*a^2*b + 16*b^2)))*
sqrt((a^3 + 3*a*b - (a^4 + 8*a^2*b + 16*b^2)*sqrt((a^4 + 2*a^2*b + b^2)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)
))/(a^4 + 8*a^2*b + 16*b^2)) - 2*(a*x^4 + b)^(1...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {- a x^{4} - b + 2 x^{8}}{\sqrt [4]{a x^{4} + b} \left (- a x^{4} - b + x^{8}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**8-a*x**4-b)/(a*x**4+b)**(1/4)/(x**8-a*x**4-b),x)

[Out]

Integral((-a*x**4 - b + 2*x**8)/((a*x**4 + b)**(1/4)*(-a*x**4 - b + x**8)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^8-a*x^4-b)/(a*x^4+b)^(1/4)/(x^8-a*x^4-b),x, algorithm="giac")

[Out]

integrate((2*x^8 - a*x^4 - b)/((x^8 - a*x^4 - b)*(a*x^4 + b)^(1/4)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {-2\,x^8+a\,x^4+b}{{\left (a\,x^4+b\right )}^{1/4}\,\left (-x^8+a\,x^4+b\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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