3.22.28 \(\int \frac {2 b-a x^4+2 x^8}{\sqrt [4]{b+a x^4} (-2 b+a x^4+x^8)} \, dx\)
Optimal. Leaf size=154 \[ -\frac {3}{4} \text {RootSum}\left [2 \text {$\#$1}^8-5 \text {$\#$1}^4 a+3 a^2-b\& ,\frac {2 \text {$\#$1}^4 \log \left (\sqrt [4]{a x^4+b}-\text {$\#$1} x\right )-2 \text {$\#$1}^4 \log (x)-3 a \log \left (\sqrt [4]{a x^4+b}-\text {$\#$1} x\right )+3 a \log (x)}{5 \text {$\#$1} a-4 \text {$\#$1}^5}\& \right ]+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{\sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{\sqrt [4]{a}} \]
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Rubi [B] time = 1.74, antiderivative size = 549, normalized size of antiderivative = 3.56,
number of steps used = 16, number of rules used = 8, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.195, Rules used
= {6728, 240, 212, 206, 203, 377, 208, 205} \begin {gather*} -\frac {3 \left (a-\frac {a^2+4 b}{\sqrt {a^2+8 b}}\right ) \tan ^{-1}\left (\frac {x \sqrt [4]{-a \sqrt {a^2+8 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2+8 b}} \sqrt [4]{a x^4+b}}\right )}{2 \left (a-\sqrt {a^2+8 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+8 b}+a^2-2 b}}-\frac {3 \left (\frac {a^2+4 b}{\sqrt {a^2+8 b}}+a\right ) \tan ^{-1}\left (\frac {x \sqrt [4]{a \sqrt {a^2+8 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2+8 b}+a} \sqrt [4]{a x^4+b}}\right )}{2 \left (\sqrt {a^2+8 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+8 b}+a^2-2 b}}-\frac {3 \left (a-\frac {a^2+4 b}{\sqrt {a^2+8 b}}\right ) \tanh ^{-1}\left (\frac {x \sqrt [4]{-a \sqrt {a^2+8 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2+8 b}} \sqrt [4]{a x^4+b}}\right )}{2 \left (a-\sqrt {a^2+8 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+8 b}+a^2-2 b}}-\frac {3 \left (\frac {a^2+4 b}{\sqrt {a^2+8 b}}+a\right ) \tanh ^{-1}\left (\frac {x \sqrt [4]{a \sqrt {a^2+8 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2+8 b}+a} \sqrt [4]{a x^4+b}}\right )}{2 \left (\sqrt {a^2+8 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+8 b}+a^2-2 b}}+\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[(2*b - a*x^4 + 2*x^8)/((b + a*x^4)^(1/4)*(-2*b + a*x^4 + x^8)),x]
[Out]
ArcTan[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/a^(1/4) - (3*(a - (a^2 + 4*b)/Sqrt[a^2 + 8*b])*ArcTan[((a^2 - 2*b - a*Sq
rt[a^2 + 8*b])^(1/4)*x)/((a - Sqrt[a^2 + 8*b])^(1/4)*(b + a*x^4)^(1/4))])/(2*(a - Sqrt[a^2 + 8*b])^(3/4)*(a^2
- 2*b - a*Sqrt[a^2 + 8*b])^(1/4)) - (3*(a + (a^2 + 4*b)/Sqrt[a^2 + 8*b])*ArcTan[((a^2 - 2*b + a*Sqrt[a^2 + 8*b
])^(1/4)*x)/((a + Sqrt[a^2 + 8*b])^(1/4)*(b + a*x^4)^(1/4))])/(2*(a + Sqrt[a^2 + 8*b])^(3/4)*(a^2 - 2*b + a*Sq
rt[a^2 + 8*b])^(1/4)) + ArcTanh[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/a^(1/4) - (3*(a - (a^2 + 4*b)/Sqrt[a^2 + 8*b])*
ArcTanh[((a^2 - 2*b - a*Sqrt[a^2 + 8*b])^(1/4)*x)/((a - Sqrt[a^2 + 8*b])^(1/4)*(b + a*x^4)^(1/4))])/(2*(a - Sq
rt[a^2 + 8*b])^(3/4)*(a^2 - 2*b - a*Sqrt[a^2 + 8*b])^(1/4)) - (3*(a + (a^2 + 4*b)/Sqrt[a^2 + 8*b])*ArcTanh[((a
^2 - 2*b + a*Sqrt[a^2 + 8*b])^(1/4)*x)/((a + Sqrt[a^2 + 8*b])^(1/4)*(b + a*x^4)^(1/4))])/(2*(a + Sqrt[a^2 + 8*
b])^(3/4)*(a^2 - 2*b + a*Sqrt[a^2 + 8*b])^(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 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 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 {2 b-a x^4+2 x^8}{\sqrt [4]{b+a x^4} \left (-2 b+a x^4+x^8\right )} \, dx &=\int \left (\frac {2}{\sqrt [4]{b+a x^4}}+\frac {3 \left (2 b-a x^4\right )}{\sqrt [4]{b+a x^4} \left (-2 b+a x^4+x^8\right )}\right ) \, dx\\ &=2 \int \frac {1}{\sqrt [4]{b+a x^4}} \, dx+3 \int \frac {2 b-a x^4}{\sqrt [4]{b+a x^4} \left (-2 b+a x^4+x^8\right )} \, dx\\ &=2 \operatorname {Subst}\left (\int \frac {1}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )+3 \int \left (\frac {-a+\frac {a^2+4 b}{\sqrt {a^2+8 b}}}{\left (a-\sqrt {a^2+8 b}+2 x^4\right ) \sqrt [4]{b+a x^4}}+\frac {-a-\frac {a^2+4 b}{\sqrt {a^2+8 b}}}{\left (a+\sqrt {a^2+8 b}+2 x^4\right ) \sqrt [4]{b+a x^4}}\right ) \, dx\\ &=-\left (\left (3 \left (a-\frac {a^2+4 b}{\sqrt {a^2+8 b}}\right )\right ) \int \frac {1}{\left (a-\sqrt {a^2+8 b}+2 x^4\right ) \sqrt [4]{b+a x^4}} \, dx\right )-\left (3 \left (a+\frac {a^2+4 b}{\sqrt {a^2+8 b}}\right )\right ) \int \frac {1}{\left (a+\sqrt {a^2+8 b}+2 x^4\right ) \sqrt [4]{b+a x^4}} \, dx+\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}}-\left (3 \left (a-\frac {a^2+4 b}{\sqrt {a^2+8 b}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a-\sqrt {a^2+8 b}-\left (-2 b+a \left (a-\sqrt {a^2+8 b}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )-\left (3 \left (a+\frac {a^2+4 b}{\sqrt {a^2+8 b}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+\sqrt {a^2+8 b}-\left (-2 b+a \left (a+\sqrt {a^2+8 b}\right )\right ) x^4} \, 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 (3 \left (a-\frac {a^2+4 b}{\sqrt {a^2+8 b}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2+8 b}}-\sqrt {a^2-2 b-a \sqrt {a^2+8 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt {a-\sqrt {a^2+8 b}}}-\frac {\left (3 \left (a-\frac {a^2+4 b}{\sqrt {a^2+8 b}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2+8 b}}+\sqrt {a^2-2 b-a \sqrt {a^2+8 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt {a-\sqrt {a^2+8 b}}}-\frac {\left (3 \left (a+\frac {a^2+4 b}{\sqrt {a^2+8 b}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2+8 b}}-\sqrt {a^2-2 b+a \sqrt {a^2+8 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt {a+\sqrt {a^2+8 b}}}-\frac {\left (3 \left (a+\frac {a^2+4 b}{\sqrt {a^2+8 b}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2+8 b}}+\sqrt {a^2-2 b+a \sqrt {a^2+8 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt {a+\sqrt {a^2+8 b}}}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}-\frac {3 \left (a-\frac {a^2+4 b}{\sqrt {a^2+8 b}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a^2-2 b-a \sqrt {a^2+8 b}} x}{\sqrt [4]{a-\sqrt {a^2+8 b}} \sqrt [4]{b+a x^4}}\right )}{2 \left (a-\sqrt {a^2+8 b}\right )^{3/4} \sqrt [4]{a^2-2 b-a \sqrt {a^2+8 b}}}-\frac {3 \left (a+\frac {a^2+4 b}{\sqrt {a^2+8 b}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a^2-2 b+a \sqrt {a^2+8 b}} x}{\sqrt [4]{a+\sqrt {a^2+8 b}} \sqrt [4]{b+a x^4}}\right )}{2 \left (a+\sqrt {a^2+8 b}\right )^{3/4} \sqrt [4]{a^2-2 b+a \sqrt {a^2+8 b}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}-\frac {3 \left (a-\frac {a^2+4 b}{\sqrt {a^2+8 b}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a^2-2 b-a \sqrt {a^2+8 b}} x}{\sqrt [4]{a-\sqrt {a^2+8 b}} \sqrt [4]{b+a x^4}}\right )}{2 \left (a-\sqrt {a^2+8 b}\right )^{3/4} \sqrt [4]{a^2-2 b-a \sqrt {a^2+8 b}}}-\frac {3 \left (a+\frac {a^2+4 b}{\sqrt {a^2+8 b}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a^2-2 b+a \sqrt {a^2+8 b}} x}{\sqrt [4]{a+\sqrt {a^2+8 b}} \sqrt [4]{b+a x^4}}\right )}{2 \left (a+\sqrt {a^2+8 b}\right )^{3/4} \sqrt [4]{a^2-2 b+a \sqrt {a^2+8 b}}}\\ \end {align*}
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Mathematica [B] time = 1.03, size = 549, normalized size = 3.56 \begin {gather*} -\frac {3 \left (a-\frac {a^2+4 b}{\sqrt {a^2+8 b}}\right ) \tan ^{-1}\left (\frac {x \sqrt [4]{-a \sqrt {a^2+8 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2+8 b}} \sqrt [4]{a x^4+b}}\right )}{2 \left (a-\sqrt {a^2+8 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+8 b}+a^2-2 b}}-\frac {3 \left (\frac {a^2+4 b}{\sqrt {a^2+8 b}}+a\right ) \tan ^{-1}\left (\frac {x \sqrt [4]{a \sqrt {a^2+8 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2+8 b}+a} \sqrt [4]{a x^4+b}}\right )}{2 \left (\sqrt {a^2+8 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+8 b}+a^2-2 b}}-\frac {3 \left (a-\frac {a^2+4 b}{\sqrt {a^2+8 b}}\right ) \tanh ^{-1}\left (\frac {x \sqrt [4]{-a \sqrt {a^2+8 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2+8 b}} \sqrt [4]{a x^4+b}}\right )}{2 \left (a-\sqrt {a^2+8 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+8 b}+a^2-2 b}}-\frac {3 \left (\frac {a^2+4 b}{\sqrt {a^2+8 b}}+a\right ) \tanh ^{-1}\left (\frac {x \sqrt [4]{a \sqrt {a^2+8 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2+8 b}+a} \sqrt [4]{a x^4+b}}\right )}{2 \left (\sqrt {a^2+8 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+8 b}+a^2-2 b}}+\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]
Integrate[(2*b - a*x^4 + 2*x^8)/((b + a*x^4)^(1/4)*(-2*b + a*x^4 + x^8)),x]
[Out]
ArcTan[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/a^(1/4) - (3*(a - (a^2 + 4*b)/Sqrt[a^2 + 8*b])*ArcTan[((a^2 - 2*b - a*Sq
rt[a^2 + 8*b])^(1/4)*x)/((a - Sqrt[a^2 + 8*b])^(1/4)*(b + a*x^4)^(1/4))])/(2*(a - Sqrt[a^2 + 8*b])^(3/4)*(a^2
- 2*b - a*Sqrt[a^2 + 8*b])^(1/4)) - (3*(a + (a^2 + 4*b)/Sqrt[a^2 + 8*b])*ArcTan[((a^2 - 2*b + a*Sqrt[a^2 + 8*b
])^(1/4)*x)/((a + Sqrt[a^2 + 8*b])^(1/4)*(b + a*x^4)^(1/4))])/(2*(a + Sqrt[a^2 + 8*b])^(3/4)*(a^2 - 2*b + a*Sq
rt[a^2 + 8*b])^(1/4)) + ArcTanh[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/a^(1/4) - (3*(a - (a^2 + 4*b)/Sqrt[a^2 + 8*b])*
ArcTanh[((a^2 - 2*b - a*Sqrt[a^2 + 8*b])^(1/4)*x)/((a - Sqrt[a^2 + 8*b])^(1/4)*(b + a*x^4)^(1/4))])/(2*(a - Sq
rt[a^2 + 8*b])^(3/4)*(a^2 - 2*b - a*Sqrt[a^2 + 8*b])^(1/4)) - (3*(a + (a^2 + 4*b)/Sqrt[a^2 + 8*b])*ArcTanh[((a
^2 - 2*b + a*Sqrt[a^2 + 8*b])^(1/4)*x)/((a + Sqrt[a^2 + 8*b])^(1/4)*(b + a*x^4)^(1/4))])/(2*(a + Sqrt[a^2 + 8*
b])^(3/4)*(a^2 - 2*b + a*Sqrt[a^2 + 8*b])^(1/4))
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IntegrateAlgebraic [A] time = 1.24, size = 154, 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 {3}{4} \text {RootSum}\left [3 a^2-b-5 a \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {-3 a \log (x)+3 a \log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right )+2 \log (x) \text {$\#$1}^4-2 \log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-5 a \text {$\#$1}+4 \text {$\#$1}^5}\&\right ] \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(2*b - a*x^4 + 2*x^8)/((b + a*x^4)^(1/4)*(-2*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) - (3*RootSum[3*
a^2 - b - 5*a*#1^4 + 2*#1^8 & , (-3*a*Log[x] + 3*a*Log[(b + a*x^4)^(1/4) - x*#1] + 2*Log[x]*#1^4 - 2*Log[(b +
a*x^4)^(1/4) - x*#1]*#1^4)/(-5*a*#1 + 4*#1^5) & ])/4
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fricas [B] time = 1.62, size = 5259, normalized size = 34.15
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*x^8-a*x^4+2*b)/(a*x^4+b)^(1/4)/(x^8+a*x^4-2*b),x, algorithm="fricas")
[Out]
3*sqrt(sqrt(1/2)*sqrt((3*a^5 + 26*a^3*b + 36*a*b^2 - (3*a^6 + 47*a^4*b + 176*a^2*b^2 - 64*b^3)*sqrt((9*a^8 + 8
4*a^6*b + 220*a^4*b^2 + 112*a^2*b^3 + 16*b^4)/(9*a^10 + 210*a^8*b + 1585*a^6*b^2 + 3480*a^4*b^3 - 2880*a^2*b^4
+ 512*b^5)))/(3*a^6 + 47*a^4*b + 176*a^2*b^2 - 64*b^3)))*arctan(-1/16*(((3*a^7 + 47*a^5*b + 176*a^3*b^2 - 64*
a*b^3)*x*sqrt((9*a^8 + 84*a^6*b + 220*a^4*b^2 + 112*a^2*b^3 + 16*b^4)/(9*a^10 + 210*a^8*b + 1585*a^6*b^2 + 348
0*a^4*b^3 - 2880*a^2*b^4 + 512*b^5)) + (3*a^6 + 38*a^4*b + 116*a^2*b^2 + 32*b^3)*x)*sqrt((sqrt(1/2)*((27*a^14*
b^4 + 783*a^12*b^5 + 8496*a^10*b^6 + 41456*a^8*b^7 + 82552*a^6*b^8 + 33600*a^4*b^9 - 9728*a^2*b^10 - 4096*b^11
)*x^2*sqrt((9*a^8 + 84*a^6*b + 220*a^4*b^2 + 112*a^2*b^3 + 16*b^4)/(9*a^10 + 210*a^8*b + 1585*a^6*b^2 + 3480*a
^4*b^3 - 2880*a^2*b^4 + 512*b^5)) + (27*a^13*b^4 + 594*a^11*b^5 + 4860*a^9*b^6 + 18104*a^7*b^7 + 28944*a^5*b^8
+ 13152*a^3*b^9 + 1792*a*b^10)*x^2)*sqrt((3*a^5 + 26*a^3*b + 36*a*b^2 - (3*a^6 + 47*a^4*b + 176*a^2*b^2 - 64*
b^3)*sqrt((9*a^8 + 84*a^6*b + 220*a^4*b^2 + 112*a^2*b^3 + 16*b^4)/(9*a^10 + 210*a^8*b + 1585*a^6*b^2 + 3480*a^
4*b^3 - 2880*a^2*b^4 + 512*b^5)))/(3*a^6 + 47*a^4*b + 176*a^2*b^2 - 64*b^3)) + 16*(9*a^8*b^6 + 84*a^6*b^7 + 22
0*a^4*b^8 + 112*a^2*b^9 + 16*b^10)*sqrt(a*x^4 + b))/x^2) - 4*(9*a^10*b^3 + 156*a^8*b^4 + 892*a^6*b^5 + 1872*a^
4*b^6 + 912*a^2*b^7 + 128*b^8 + (9*a^11*b^3 + 183*a^9*b^4 + 1198*a^7*b^5 + 2460*a^5*b^6 - 192*a^3*b^7 - 256*a*
b^8)*sqrt((9*a^8 + 84*a^6*b + 220*a^4*b^2 + 112*a^2*b^3 + 16*b^4)/(9*a^10 + 210*a^8*b + 1585*a^6*b^2 + 3480*a^
4*b^3 - 2880*a^2*b^4 + 512*b^5)))*(a*x^4 + b)^(1/4))*sqrt(sqrt(1/2)*sqrt((3*a^5 + 26*a^3*b + 36*a*b^2 - (3*a^6
+ 47*a^4*b + 176*a^2*b^2 - 64*b^3)*sqrt((9*a^8 + 84*a^6*b + 220*a^4*b^2 + 112*a^2*b^3 + 16*b^4)/(9*a^10 + 210
*a^8*b + 1585*a^6*b^2 + 3480*a^4*b^3 - 2880*a^2*b^4 + 512*b^5)))/(3*a^6 + 47*a^4*b + 176*a^2*b^2 - 64*b^3)))/(
(9*a^8*b^4 + 84*a^6*b^5 + 220*a^4*b^6 + 112*a^2*b^7 + 16*b^8)*x)) - 3*sqrt(sqrt(1/2)*sqrt((3*a^5 + 26*a^3*b +
36*a*b^2 + (3*a^6 + 47*a^4*b + 176*a^2*b^2 - 64*b^3)*sqrt((9*a^8 + 84*a^6*b + 220*a^4*b^2 + 112*a^2*b^3 + 16*b
^4)/(9*a^10 + 210*a^8*b + 1585*a^6*b^2 + 3480*a^4*b^3 - 2880*a^2*b^4 + 512*b^5)))/(3*a^6 + 47*a^4*b + 176*a^2*
b^2 - 64*b^3)))*arctan(-1/16*(((3*a^7 + 47*a^5*b + 176*a^3*b^2 - 64*a*b^3)*x*sqrt((9*a^8 + 84*a^6*b + 220*a^4*
b^2 + 112*a^2*b^3 + 16*b^4)/(9*a^10 + 210*a^8*b + 1585*a^6*b^2 + 3480*a^4*b^3 - 2880*a^2*b^4 + 512*b^5)) - (3*
a^6 + 38*a^4*b + 116*a^2*b^2 + 32*b^3)*x)*sqrt(sqrt(1/2)*sqrt((3*a^5 + 26*a^3*b + 36*a*b^2 + (3*a^6 + 47*a^4*b
+ 176*a^2*b^2 - 64*b^3)*sqrt((9*a^8 + 84*a^6*b + 220*a^4*b^2 + 112*a^2*b^3 + 16*b^4)/(9*a^10 + 210*a^8*b + 15
85*a^6*b^2 + 3480*a^4*b^3 - 2880*a^2*b^4 + 512*b^5)))/(3*a^6 + 47*a^4*b + 176*a^2*b^2 - 64*b^3)))*sqrt(-(sqrt(
1/2)*((27*a^14*b^4 + 783*a^12*b^5 + 8496*a^10*b^6 + 41456*a^8*b^7 + 82552*a^6*b^8 + 33600*a^4*b^9 - 9728*a^2*b
^10 - 4096*b^11)*x^2*sqrt((9*a^8 + 84*a^6*b + 220*a^4*b^2 + 112*a^2*b^3 + 16*b^4)/(9*a^10 + 210*a^8*b + 1585*a
^6*b^2 + 3480*a^4*b^3 - 2880*a^2*b^4 + 512*b^5)) - (27*a^13*b^4 + 594*a^11*b^5 + 4860*a^9*b^6 + 18104*a^7*b^7
+ 28944*a^5*b^8 + 13152*a^3*b^9 + 1792*a*b^10)*x^2)*sqrt((3*a^5 + 26*a^3*b + 36*a*b^2 + (3*a^6 + 47*a^4*b + 17
6*a^2*b^2 - 64*b^3)*sqrt((9*a^8 + 84*a^6*b + 220*a^4*b^2 + 112*a^2*b^3 + 16*b^4)/(9*a^10 + 210*a^8*b + 1585*a^
6*b^2 + 3480*a^4*b^3 - 2880*a^2*b^4 + 512*b^5)))/(3*a^6 + 47*a^4*b + 176*a^2*b^2 - 64*b^3)) - 16*(9*a^8*b^6 +
84*a^6*b^7 + 220*a^4*b^8 + 112*a^2*b^9 + 16*b^10)*sqrt(a*x^4 + b))/x^2) + 4*(9*a^10*b^3 + 156*a^8*b^4 + 892*a^
6*b^5 + 1872*a^4*b^6 + 912*a^2*b^7 + 128*b^8 - (9*a^11*b^3 + 183*a^9*b^4 + 1198*a^7*b^5 + 2460*a^5*b^6 - 192*a
^3*b^7 - 256*a*b^8)*sqrt((9*a^8 + 84*a^6*b + 220*a^4*b^2 + 112*a^2*b^3 + 16*b^4)/(9*a^10 + 210*a^8*b + 1585*a^
6*b^2 + 3480*a^4*b^3 - 2880*a^2*b^4 + 512*b^5)))*(a*x^4 + b)^(1/4)*sqrt(sqrt(1/2)*sqrt((3*a^5 + 26*a^3*b + 36*
a*b^2 + (3*a^6 + 47*a^4*b + 176*a^2*b^2 - 64*b^3)*sqrt((9*a^8 + 84*a^6*b + 220*a^4*b^2 + 112*a^2*b^3 + 16*b^4)
/(9*a^10 + 210*a^8*b + 1585*a^6*b^2 + 3480*a^4*b^3 - 2880*a^2*b^4 + 512*b^5)))/(3*a^6 + 47*a^4*b + 176*a^2*b^2
- 64*b^3))))/((9*a^8*b^4 + 84*a^6*b^5 + 220*a^4*b^6 + 112*a^2*b^7 + 16*b^8)*x)) + 3/4*sqrt(sqrt(1/2)*sqrt((3*
a^5 + 26*a^3*b + 36*a*b^2 + (3*a^6 + 47*a^4*b + 176*a^2*b^2 - 64*b^3)*sqrt((9*a^8 + 84*a^6*b + 220*a^4*b^2 + 1
12*a^2*b^3 + 16*b^4)/(9*a^10 + 210*a^8*b + 1585*a^6*b^2 + 3480*a^4*b^3 - 2880*a^2*b^4 + 512*b^5)))/(3*a^6 + 47
*a^4*b + 176*a^2*b^2 - 64*b^3)))*log(2187/16*(sqrt(1/2)*((9*a^12 + 273*a^10*b + 3100*a^8*b^2 + 15640*a^6*b^3 +
29760*a^4*b^4 + 512*a^2*b^5 - 4096*b^6)*x*sqrt((9*a^8 + 84*a^6*b + 220*a^4*b^2 + 112*a^2*b^3 + 16*b^4)/(9*a^1
0 + 210*a^8*b + 1585*a^6*b^2 + 3480*a^4*b^3 - 2880*a^2*b^4 + 512*b^5)) - (9*a^11 + 210*a^9*b + 1756*a^7*b^2 +
6240*a^5*b^3 + 8448*a^3*b^4 + 2048*a*b^5)*x)*sqrt(sqrt(1/2)*sqrt((3*a^5 + 26*a^3*b + 36*a*b^2 + (3*a^6 + 47*a^
4*b + 176*a^2*b^2 - 64*b^3)*sqrt((9*a^8 + 84*a^6*b + 220*a^4*b^2 + 112*a^2*b^3 + 16*b^4)/(9*a^10 + 210*a^8*b +
1585*a^6*b^2 + 3480*a^4*b^3 - 2880*a^2*b^4 + 512*b^5)))/(3*a^6 + 47*a^4*b + 176*a^2*b^2 - 64*b^3)))*sqrt((3*a
^5 + 26*a^3*b + 36*a*b^2 + (3*a^6 + 47*a^4*b + 176*a^2*b^2 - 64*b^3)*sqrt((9*a^8 + 84*a^6*b + 220*a^4*b^2 + 11
2*a^2*b^3 + 16*b^4)/(9*a^10 + 210*a^8*b + 1585*a^6*b^2 + 3480*a^4*b^3 - 2880*a^2*b^4 + 512*b^5)))/(3*a^6 + 47*
a^4*b + 176*a^2*b^2 - 64*b^3)) + 32*(3*a^4*b^3 + 14*a^2*b^4 + 4*b^5)*(a*x^4 + b)^(1/4))/x) - 3/4*sqrt(sqrt(1/2
)*sqrt((3*a^5 + 26*a^3*b + 36*a*b^2 + (3*a^6 + 47*a^4*b + 176*a^2*b^2 - 64*b^3)*sqrt((9*a^8 + 84*a^6*b + 220*a
^4*b^2 + 112*a^2*b^3 + 16*b^4)/(9*a^10 + 210*a^8*b + 1585*a^6*b^2 + 3480*a^4*b^3 - 2880*a^2*b^4 + 512*b^5)))/(
3*a^6 + 47*a^4*b + 176*a^2*b^2 - 64*b^3)))*log(-2187/16*(sqrt(1/2)*((9*a^12 + 273*a^10*b + 3100*a^8*b^2 + 1564
0*a^6*b^3 + 29760*a^4*b^4 + 512*a^2*b^5 - 4096*b^6)*x*sqrt((9*a^8 + 84*a^6*b + 220*a^4*b^2 + 112*a^2*b^3 + 16*
b^4)/(9*a^10 + 210*a^8*b + 1585*a^6*b^2 + 3480*a^4*b^3 - 2880*a^2*b^4 + 512*b^5)) - (9*a^11 + 210*a^9*b + 1756
*a^7*b^2 + 6240*a^5*b^3 + 8448*a^3*b^4 + 2048*a*b^5)*x)*sqrt(sqrt(1/2)*sqrt((3*a^5 + 26*a^3*b + 36*a*b^2 + (3*
a^6 + 47*a^4*b + 176*a^2*b^2 - 64*b^3)*sqrt((9*a^8 + 84*a^6*b + 220*a^4*b^2 + 112*a^2*b^3 + 16*b^4)/(9*a^10 +
210*a^8*b + 1585*a^6*b^2 + 3480*a^4*b^3 - 2880*a^2*b^4 + 512*b^5)))/(3*a^6 + 47*a^4*b + 176*a^2*b^2 - 64*b^3))
)*sqrt((3*a^5 + 26*a^3*b + 36*a*b^2 + (3*a^6 + 47*a^4*b + 176*a^2*b^2 - 64*b^3)*sqrt((9*a^8 + 84*a^6*b + 220*a
^4*b^2 + 112*a^2*b^3 + 16*b^4)/(9*a^10 + 210*a^8*b + 1585*a^6*b^2 + 3480*a^4*b^3 - 2880*a^2*b^4 + 512*b^5)))/(
3*a^6 + 47*a^4*b + 176*a^2*b^2 - 64*b^3)) - 32*(3*a^4*b^3 + 14*a^2*b^4 + 4*b^5)*(a*x^4 + b)^(1/4))/x) - 3/4*sq
rt(sqrt(1/2)*sqrt((3*a^5 + 26*a^3*b + 36*a*b^2 - (3*a^6 + 47*a^4*b + 176*a^2*b^2 - 64*b^3)*sqrt((9*a^8 + 84*a^
6*b + 220*a^4*b^2 + 112*a^2*b^3 + 16*b^4)/(9*a^10 + 210*a^8*b + 1585*a^6*b^2 + 3480*a^4*b^3 - 2880*a^2*b^4 + 5
12*b^5)))/(3*a^6 + 47*a^4*b + 176*a^2*b^2 - 64*b^3)))*log(2187/16*(sqrt(1/2)*((9*a^12 + 273*a^10*b + 3100*a^8*
b^2 + 15640*a^6*b^3 + 29760*a^4*b^4 + 512*a^2*b^5 - 4096*b^6)*x*sqrt((9*a^8 + 84*a^6*b + 220*a^4*b^2 + 112*a^2
*b^3 + 16*b^4)/(9*a^10 + 210*a^8*b + 1585*a^6*b^2 + 3480*a^4*b^3 - 2880*a^2*b^4 + 512*b^5)) + (9*a^11 + 210*a^
9*b + 1756*a^7*b^2 + 6240*a^5*b^3 + 8448*a^3*b^4 + 2048*a*b^5)*x)*sqrt(sqrt(1/2)*sqrt((3*a^5 + 26*a^3*b + 36*a
*b^2 - (3*a^6 + 47*a^4*b + 176*a^2*b^2 - 64*b^3)*sqrt((9*a^8 + 84*a^6*b + 220*a^4*b^2 + 112*a^2*b^3 + 16*b^4)/
(9*a^10 + 210*a^8*b + 1585*a^6*b^2 + 3480*a^4*b^3 - 2880*a^2*b^4 + 512*b^5)))/(3*a^6 + 47*a^4*b + 176*a^2*b^2
- 64*b^3)))*sqrt((3*a^5 + 26*a^3*b + 36*a*b^2 - (3*a^6 + 47*a^4*b + 176*a^2*b^2 - 64*b^3)*sqrt((9*a^8 + 84*a^6
*b + 220*a^4*b^2 + 112*a^2*b^3 + 16*b^4)/(9*a^10 + 210*a^8*b + 1585*a^6*b^2 + 3480*a^4*b^3 - 2880*a^2*b^4 + 51
2*b^5)))/(3*a^6 + 47*a^4*b + 176*a^2*b^2 - 64*b^3)) + 32*(3*a^4*b^3 + 14*a^2*b^4 + 4*b^5)*(a*x^4 + b)^(1/4))/x
) + 3/4*sqrt(sqrt(1/2)*sqrt((3*a^5 + 26*a^3*b + 36*a*b^2 - (3*a^6 + 47*a^4*b + 176*a^2*b^2 - 64*b^3)*sqrt((9*a
^8 + 84*a^6*b + 220*a^4*b^2 + 112*a^2*b^3 + 16*b^4)/(9*a^10 + 210*a^8*b + 1585*a^6*b^2 + 3480*a^4*b^3 - 2880*a
^2*b^4 + 512*b^5)))/(3*a^6 + 47*a^4*b + 176*a^2*b^2 - 64*b^3)))*log(-2187/16*(sqrt(1/2)*((9*a^12 + 273*a^10*b
+ 3100*a^8*b^2 + 15640*a^6*b^3 + 29760*a^4*b^4 + 512*a^2*b^5 - 4096*b^6)*x*sqrt((9*a^8 + 84*a^6*b + 220*a^4*b^
2 + 112*a^2*b^3 + 16*b^4)/(9*a^10 + 210*a^8*b + 1585*a^6*b^2 + 3480*a^4*b^3 - 2880*a^2*b^4 + 512*b^5)) + (9*a^
11 + 210*a^9*b + 1756*a^7*b^2 + 6240*a^5*b^3 + 8448*a^3*b^4 + 2048*a*b^5)*x)*sqrt(sqrt(1/2)*sqrt((3*a^5 + 26*a
^3*b + 36*a*b^2 - (3*a^6 + 47*a^4*b + 176*a^2*b^2 - 64*b^3)*sqrt((9*a^8 + 84*a^6*b + 220*a^4*b^2 + 112*a^2*b^3
+ 16*b^4)/(9*a^10 + 210*a^8*b + 1585*a^6*b^2 + 3480*a^4*b^3 - 2880*a^2*b^4 + 512*b^5)))/(3*a^6 + 47*a^4*b + 1
76*a^2*b^2 - 64*b^3)))*sqrt((3*a^5 + 26*a^3*b + 36*a*b^2 - (3*a^6 + 47*a^4*b + 176*a^2*b^2 - 64*b^3)*sqrt((9*a
^8 + 84*a^6*b + 220*a^4*b^2 + 112*a^2*b^3 + 16*b^4)/(9*a^10 + 210*a^8*b + 1585*a^6*b^2 + 3480*a^4*b^3 - 2880*a
^2*b^4 + 512*b^5)))/(3*a^6 + 47*a^4*b + 176*a^2*b^2 - 64*b^3)) - 32*(3*a^4*b^3 + 14*a^2*b^4 + 4*b^5)*(a*x^4 +
b)^(1/4))/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} + 2 \, b}{{\left (x^{8} + a x^{4} - 2 \, 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+2*b)/(a*x^4+b)^(1/4)/(x^8+a*x^4-2*b),x, algorithm="giac")
[Out]
integrate((2*x^8 - a*x^4 + 2*b)/((x^8 + a*x^4 - 2*b)*(a*x^4 + b)^(1/4)), x)
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {2 x^{8}-a \,x^{4}+2 b}{\left (a \,x^{4}+b \right )^{\frac {1}{4}} \left (x^{8}+a \,x^{4}-2 b \right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((2*x^8-a*x^4+2*b)/(a*x^4+b)^(1/4)/(x^8+a*x^4-2*b),x)
[Out]
int((2*x^8-a*x^4+2*b)/(a*x^4+b)^(1/4)/(x^8+a*x^4-2*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} + 2 \, b}{{\left (x^{8} + a x^{4} - 2 \, 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+2*b)/(a*x^4+b)^(1/4)/(x^8+a*x^4-2*b),x, algorithm="maxima")
[Out]
integrate((2*x^8 - a*x^4 + 2*b)/((x^8 + a*x^4 - 2*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+2\,b}{{\left (a\,x^4+b\right )}^{1/4}\,\left (x^8+a\,x^4-2\,b\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((2*b - a*x^4 + 2*x^8)/((b + a*x^4)^(1/4)*(a*x^4 - 2*b + x^8)),x)
[Out]
int((2*b - a*x^4 + 2*x^8)/((b + a*x^4)^(1/4)*(a*x^4 - 2*b + 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+2*b)/(a*x**4+b)**(1/4)/(x**8+a*x**4-2*b),x)
[Out]
Timed out
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