3.18.42 \(\int \frac {-b-a x^4+2 x^8}{\sqrt [4]{b+a x^4} (-b-a x^4+x^8)} \, dx\)
Optimal. Leaf size=117 \[ \frac {1}{4} \text {RootSum}\left [-\text {$\#$1}^8+\text {$\#$1}^4 a+b\& ,\frac {\text {$\#$1}^3 \log \left (\sqrt [4]{a x^4+b}-\text {$\#$1} x\right )-\text {$\#$1}^3 \log (x)}{2 \text {$\#$1}^4-a}\& \right ]+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{\sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{\sqrt [4]{a}} \]
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Rubi [B] time = 0.70, 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
= {6728, 240, 212, 206, 203, 1428, 408, 377, 208, 205} \begin {gather*} \frac {\left (-a \sqrt {a^2+4 b}+a^2+2 b\right )^{3/4} \tan ^{-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} \tan ^{-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 {\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 {\tan ^{-1}\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 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 205
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]
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 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rule 212
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] &&
!GtQ[a/b, 0]
Rule 240
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 377
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 408
Int[((a_) + (b_.)*(x_)^4)^(p_)/((c_) + (d_.)*(x_)^4), x_Symbol] :> Dist[b/d, Int[(a + b*x^4)^(p - 1), x], x] -
Dist[(b*c - a*d)/d, Int[(a + b*x^4)^(p - 1)/(c + d*x^4), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0
] && (EqQ[p, 3/4] || EqQ[p, 5/4])
Rule 1428
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 6728
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 \operatorname {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}}+\operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )+\operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 [F] time = 0.08, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-b-a x^4+2 x^8}{\sqrt [4]{b+a x^4} \left (-b-a x^4+x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Integrate[(-b - a*x^4 + 2*x^8)/((b + a*x^4)^(1/4)*(-b - a*x^4 + x^8)),x]
[Out]
Integrate[(-b - a*x^4 + 2*x^8)/((b + a*x^4)^(1/4)*(-b - a*x^4 + x^8)), x]
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IntegrateAlgebraic [A] time = 0.00, size = 117, normalized size = 1.00 \begin {gather*} \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 {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]
IntegrateAlgebraic[(-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) + 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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fricas [B] time = 0.80, size = 3119, normalized size = 26.66
result too large to display
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)^(1/4)*(a^2*b^2 + b^3))/x) + 2*arctan((x*sqrt((sqrt(a)*x^2 + sqrt(
a*x^4 + b))/x^2)/a^(1/4) - (a*x^4 + b)^(1/4)/a^(1/4))/x)/a^(1/4) + 1/2*log((a^(1/4)*x + (a*x^4 + b)^(1/4))/x)/
a^(1/4) - 1/2*log(-(a^(1/4)*x - (a*x^4 + b)^(1/4))/x)/a^(1/4)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, x^{8} - a x^{4} - b}{{\left (x^{8} - a x^{4} - b\right )} {\left (a x^{4} + b\right )}^{\frac {1}{4}}}\,{d x} \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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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*} \int \frac {2 \, x^{8} - a x^{4} - b}{{\left (x^{8} - a x^{4} - b\right )} {\left (a x^{4} + b\right )}^{\frac {1}{4}}}\,{d x} \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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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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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((2*x**8-a*x**4-b)/(a*x**4+b)**(1/4)/(x**8-a*x**4-b),x)
[Out]
Timed out
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