3.21.81 \(\int \frac {-b+2 a x^4+2 x^8}{\sqrt [4]{b+a x^4} (-b+a x^4+x^8)} \, dx\)
Optimal. Leaf size=150 \[ -\frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8-3 \text {$\#$1}^4 a+2 a^2-b\& ,\frac {\text {$\#$1}^4 \log \left (\sqrt [4]{a x^4+b}-\text {$\#$1} x\right )+\text {$\#$1}^4 (-\log (x))-a \log \left (\sqrt [4]{a x^4+b}-\text {$\#$1} x\right )+a \log (x)}{3 \text {$\#$1} a-2 \text {$\#$1}^5}\& \right ]+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{\sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{\sqrt [4]{a}} \]
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Rubi [B] time = 0.75, antiderivative size = 501, normalized size of antiderivative = 3.34,
number of steps used = 15, number of rules used = 9, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.220, Rules used
= {6728, 240, 212, 206, 203, 1428, 377, 208, 205} \begin {gather*} \frac {b \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 )}{\sqrt {a^2+4 b} \left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+4 b}+a^2-2 b}}-\frac {b \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 )}{\sqrt {a^2+4 b} \left (\sqrt {a^2+4 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+4 b}+a^2-2 b}}+\frac {b \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 )}{\sqrt {a^2+4 b} \left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+4 b}+a^2-2 b}}-\frac {b \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 )}{\sqrt {a^2+4 b} \left (\sqrt {a^2+4 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+4 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[(-b + 2*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) + (b*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))])/(Sqrt[a^2 + 4*b]*(a - Sqrt[a^2 + 4*b])^(3/4)*(a^2 - 2*b - a*Sqrt[a^2 +
4*b])^(1/4)) - (b*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))])/(Sqrt[a^2 + 4*b]*(a + Sqrt[a^2 + 4*b])^(3/4)*(a^2 - 2*b + a*Sqrt[a^2 + 4*b])^(1/4)) + ArcTanh[(a^(1/4)
*x)/(b + a*x^4)^(1/4)]/a^(1/4) + (b*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))])/(Sqrt[a^2 + 4*b]*(a - Sqrt[a^2 + 4*b])^(3/4)*(a^2 - 2*b - a*Sqrt[a^2 + 4*b])^(1/4))
- (b*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))])/(Sqrt
[a^2 + 4*b]*(a + Sqrt[a^2 + 4*b])^(3/4)*(a^2 - 2*b + a*Sqrt[a^2 + 4*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 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+2 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 {b}{\sqrt [4]{b+a x^4} \left (-b+a x^4+x^8\right )}\right ) \, dx\\ &=2 \int \frac {1}{\sqrt [4]{b+a x^4}} \, dx+b \int \frac {1}{\sqrt [4]{b+a x^4} \left (-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 )+\frac {(2 b) \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 {(2 b) \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 {(2 b) \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 {(2 b) \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}}+\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 {b \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 )}{\sqrt {a^2+4 b} \sqrt {a-\sqrt {a^2+4 b}}}+\frac {b \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 )}{\sqrt {a^2+4 b} \sqrt {a-\sqrt {a^2+4 b}}}-\frac {b \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 )}{\sqrt {a^2+4 b} \sqrt {a+\sqrt {a^2+4 b}}}-\frac {b \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 )}{\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 {b \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 )}{\sqrt {a^2+4 b} \left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-2 b-a \sqrt {a^2+4 b}}}-\frac {b \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 )}{\sqrt {a^2+4 b} \left (a+\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-2 b+a \sqrt {a^2+4 b}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {b \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 )}{\sqrt {a^2+4 b} \left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-2 b-a \sqrt {a^2+4 b}}}-\frac {b \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 )}{\sqrt {a^2+4 b} \left (a+\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-2 b+a \sqrt {a^2+4 b}}}\\ \end {align*}
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Mathematica [B] time = 0.19, size = 391, normalized size = 2.61 \begin {gather*} \frac {b \left (\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 )+\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 )\right )}{\sqrt {a^2+4 b} \left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+4 b}+a^2-2 b}}-\frac {b \left (\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 )+\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 )\right )}{\sqrt {a^2+4 b} \left (\sqrt {a^2+4 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+4 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[(-b + 2*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) + (b*(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))] + 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))]))/(Sqrt[a^2 + 4*b]*(a - Sqrt[a
^2 + 4*b])^(3/4)*(a^2 - 2*b - a*Sqrt[a^2 + 4*b])^(1/4)) - (b*(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))] + ArcTanh[((a^2 - 2*b + a*Sqrt[a^2 + 4*b])^(1/4)*x)/((a + Sq
rt[a^2 + 4*b])^(1/4)*(b + a*x^4)^(1/4))]))/(Sqrt[a^2 + 4*b]*(a + Sqrt[a^2 + 4*b])^(3/4)*(a^2 - 2*b + a*Sqrt[a^
2 + 4*b])^(1/4))
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IntegrateAlgebraic [A] time = 0.00, size = 150, 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 [2 a^2-b-3 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-a \log (x)+a \log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-3 a \text {$\#$1}+2 \text {$\#$1}^5}\&\right ] \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(-b + 2*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[2*a^2
- b - 3*a*#1^4 + #1^8 & , (-(a*Log[x]) + a*Log[(b + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(b + a*x^4)^(1/4
) - x*#1]*#1^4)/(-3*a*#1 + 2*#1^5) & ]/4
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fricas [B] time = 1.31, size = 5047, normalized size = 33.65
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*x^8+2*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^5 + 3*a^3*b - a*b^2 - (2*a^6 + 15*a^4*b + 24*a^2*b^2 - 16*b^3)*sqrt((a^8 + 2*a^6*b - a
^4*b^2 - 2*a^2*b^3 + b^4)/(4*a^10 + 44*a^8*b + 145*a^6*b^2 + 76*a^4*b^3 - 208*a^2*b^4 + 64*b^5)))/(2*a^6 + 15*
a^4*b + 24*a^2*b^2 - 16*b^3)))*arctan(-1/2*(sqrt(1/2)*((2*a^7 + 15*a^5*b + 24*a^3*b^2 - 16*a*b^3)*x*sqrt((a^8
+ 2*a^6*b - a^4*b^2 - 2*a^2*b^3 + b^4)/(4*a^10 + 44*a^8*b + 145*a^6*b^2 + 76*a^4*b^3 - 208*a^2*b^4 + 64*b^5))
+ (a^6 + 5*a^4*b + 3*a^2*b^2 - 4*b^3)*x)*sqrt((sqrt(1/2)*((2*a^14*b^4 + 23*a^12*b^5 + 80*a^10*b^6 + 36*a^8*b^7
- 209*a^6*b^8 - 76*a^4*b^9 + 208*a^2*b^10 - 64*b^11)*x^2*sqrt((a^8 + 2*a^6*b - a^4*b^2 - 2*a^2*b^3 + b^4)/(4*
a^10 + 44*a^8*b + 145*a^6*b^2 + 76*a^4*b^3 - 208*a^2*b^4 + 64*b^5)) + (a^13*b^4 + 7*a^11*b^5 + 13*a^9*b^6 + a^
7*b^7 - 13*a^5*b^8 - 3*a^3*b^9 + 4*a*b^10)*x^2)*sqrt((a^5 + 3*a^3*b - a*b^2 - (2*a^6 + 15*a^4*b + 24*a^2*b^2 -
16*b^3)*sqrt((a^8 + 2*a^6*b - a^4*b^2 - 2*a^2*b^3 + b^4)/(4*a^10 + 44*a^8*b + 145*a^6*b^2 + 76*a^4*b^3 - 208*
a^2*b^4 + 64*b^5)))/(2*a^6 + 15*a^4*b + 24*a^2*b^2 - 16*b^3)) + 2*(a^8*b^6 + 2*a^6*b^7 - a^4*b^8 - 2*a^2*b^9 +
b^10)*sqrt(a*x^4 + b))/x^2) + (a^10*b^3 + 6*a^8*b^4 + 7*a^6*b^5 - 6*a^4*b^6 - 7*a^2*b^7 + 4*b^8 + (2*a^11*b^3
+ 17*a^9*b^4 + 37*a^7*b^5 - 7*a^5*b^6 - 40*a^3*b^7 + 16*a*b^8)*sqrt((a^8 + 2*a^6*b - a^4*b^2 - 2*a^2*b^3 + b^
4)/(4*a^10 + 44*a^8*b + 145*a^6*b^2 + 76*a^4*b^3 - 208*a^2*b^4 + 64*b^5)))*(a*x^4 + b)^(1/4))*sqrt(sqrt(1/2)*s
qrt((a^5 + 3*a^3*b - a*b^2 - (2*a^6 + 15*a^4*b + 24*a^2*b^2 - 16*b^3)*sqrt((a^8 + 2*a^6*b - a^4*b^2 - 2*a^2*b^
3 + b^4)/(4*a^10 + 44*a^8*b + 145*a^6*b^2 + 76*a^4*b^3 - 208*a^2*b^4 + 64*b^5)))/(2*a^6 + 15*a^4*b + 24*a^2*b^
2 - 16*b^3)))/((a^8*b^4 + 2*a^6*b^5 - a^4*b^6 - 2*a^2*b^7 + b^8)*x)) + sqrt(sqrt(1/2)*sqrt((a^5 + 3*a^3*b - a*
b^2 + (2*a^6 + 15*a^4*b + 24*a^2*b^2 - 16*b^3)*sqrt((a^8 + 2*a^6*b - a^4*b^2 - 2*a^2*b^3 + b^4)/(4*a^10 + 44*a
^8*b + 145*a^6*b^2 + 76*a^4*b^3 - 208*a^2*b^4 + 64*b^5)))/(2*a^6 + 15*a^4*b + 24*a^2*b^2 - 16*b^3)))*arctan(-1
/2*(sqrt(1/2)*((2*a^7 + 15*a^5*b + 24*a^3*b^2 - 16*a*b^3)*x*sqrt((a^8 + 2*a^6*b - a^4*b^2 - 2*a^2*b^3 + b^4)/(
4*a^10 + 44*a^8*b + 145*a^6*b^2 + 76*a^4*b^3 - 208*a^2*b^4 + 64*b^5)) - (a^6 + 5*a^4*b + 3*a^2*b^2 - 4*b^3)*x)
*sqrt(sqrt(1/2)*sqrt((a^5 + 3*a^3*b - a*b^2 + (2*a^6 + 15*a^4*b + 24*a^2*b^2 - 16*b^3)*sqrt((a^8 + 2*a^6*b - a
^4*b^2 - 2*a^2*b^3 + b^4)/(4*a^10 + 44*a^8*b + 145*a^6*b^2 + 76*a^4*b^3 - 208*a^2*b^4 + 64*b^5)))/(2*a^6 + 15*
a^4*b + 24*a^2*b^2 - 16*b^3)))*sqrt(-(sqrt(1/2)*((2*a^14*b^4 + 23*a^12*b^5 + 80*a^10*b^6 + 36*a^8*b^7 - 209*a^
6*b^8 - 76*a^4*b^9 + 208*a^2*b^10 - 64*b^11)*x^2*sqrt((a^8 + 2*a^6*b - a^4*b^2 - 2*a^2*b^3 + b^4)/(4*a^10 + 44
*a^8*b + 145*a^6*b^2 + 76*a^4*b^3 - 208*a^2*b^4 + 64*b^5)) - (a^13*b^4 + 7*a^11*b^5 + 13*a^9*b^6 + a^7*b^7 - 1
3*a^5*b^8 - 3*a^3*b^9 + 4*a*b^10)*x^2)*sqrt((a^5 + 3*a^3*b - a*b^2 + (2*a^6 + 15*a^4*b + 24*a^2*b^2 - 16*b^3)*
sqrt((a^8 + 2*a^6*b - a^4*b^2 - 2*a^2*b^3 + b^4)/(4*a^10 + 44*a^8*b + 145*a^6*b^2 + 76*a^4*b^3 - 208*a^2*b^4 +
64*b^5)))/(2*a^6 + 15*a^4*b + 24*a^2*b^2 - 16*b^3)) - 2*(a^8*b^6 + 2*a^6*b^7 - a^4*b^8 - 2*a^2*b^9 + b^10)*sq
rt(a*x^4 + b))/x^2) - (a^10*b^3 + 6*a^8*b^4 + 7*a^6*b^5 - 6*a^4*b^6 - 7*a^2*b^7 + 4*b^8 - (2*a^11*b^3 + 17*a^9
*b^4 + 37*a^7*b^5 - 7*a^5*b^6 - 40*a^3*b^7 + 16*a*b^8)*sqrt((a^8 + 2*a^6*b - a^4*b^2 - 2*a^2*b^3 + b^4)/(4*a^1
0 + 44*a^8*b + 145*a^6*b^2 + 76*a^4*b^3 - 208*a^2*b^4 + 64*b^5)))*(a*x^4 + b)^(1/4)*sqrt(sqrt(1/2)*sqrt((a^5 +
3*a^3*b - a*b^2 + (2*a^6 + 15*a^4*b + 24*a^2*b^2 - 16*b^3)*sqrt((a^8 + 2*a^6*b - a^4*b^2 - 2*a^2*b^3 + b^4)/(
4*a^10 + 44*a^8*b + 145*a^6*b^2 + 76*a^4*b^3 - 208*a^2*b^4 + 64*b^5)))/(2*a^6 + 15*a^4*b + 24*a^2*b^2 - 16*b^3
))))/((a^8*b^4 + 2*a^6*b^5 - a^4*b^6 - 2*a^2*b^7 + b^8)*x)) + 1/4*sqrt(sqrt(1/2)*sqrt((a^5 + 3*a^3*b - a*b^2 +
(2*a^6 + 15*a^4*b + 24*a^2*b^2 - 16*b^3)*sqrt((a^8 + 2*a^6*b - a^4*b^2 - 2*a^2*b^3 + b^4)/(4*a^10 + 44*a^8*b
+ 145*a^6*b^2 + 76*a^4*b^3 - 208*a^2*b^4 + 64*b^5)))/(2*a^6 + 15*a^4*b + 24*a^2*b^2 - 16*b^3)))*log(-1/2*(sqrt
(1/2)*((2*a^12 + 27*a^10*b + 126*a^8*b^2 + 202*a^6*b^3 - 72*a^4*b^4 - 288*a^2*b^5 + 128*b^6)*x*sqrt((a^8 + 2*a
^6*b - a^4*b^2 - 2*a^2*b^3 + b^4)/(4*a^10 + 44*a^8*b + 145*a^6*b^2 + 76*a^4*b^3 - 208*a^2*b^4 + 64*b^5)) - (a^
11 + 9*a^9*b + 23*a^7*b^2 + 8*a^5*b^3 - 16*a^3*b^4)*x)*sqrt(sqrt(1/2)*sqrt((a^5 + 3*a^3*b - a*b^2 + (2*a^6 + 1
5*a^4*b + 24*a^2*b^2 - 16*b^3)*sqrt((a^8 + 2*a^6*b - a^4*b^2 - 2*a^2*b^3 + b^4)/(4*a^10 + 44*a^8*b + 145*a^6*b
^2 + 76*a^4*b^3 - 208*a^2*b^4 + 64*b^5)))/(2*a^6 + 15*a^4*b + 24*a^2*b^2 - 16*b^3)))*sqrt((a^5 + 3*a^3*b - a*b
^2 + (2*a^6 + 15*a^4*b + 24*a^2*b^2 - 16*b^3)*sqrt((a^8 + 2*a^6*b - a^4*b^2 - 2*a^2*b^3 + b^4)/(4*a^10 + 44*a^
8*b + 145*a^6*b^2 + 76*a^4*b^3 - 208*a^2*b^4 + 64*b^5)))/(2*a^6 + 15*a^4*b + 24*a^2*b^2 - 16*b^3)) + 2*(a^4*b^
3 + a^2*b^4 - b^5)*(a*x^4 + b)^(1/4))/x) - 1/4*sqrt(sqrt(1/2)*sqrt((a^5 + 3*a^3*b - a*b^2 + (2*a^6 + 15*a^4*b
+ 24*a^2*b^2 - 16*b^3)*sqrt((a^8 + 2*a^6*b - a^4*b^2 - 2*a^2*b^3 + b^4)/(4*a^10 + 44*a^8*b + 145*a^6*b^2 + 76*
a^4*b^3 - 208*a^2*b^4 + 64*b^5)))/(2*a^6 + 15*a^4*b + 24*a^2*b^2 - 16*b^3)))*log(1/2*(sqrt(1/2)*((2*a^12 + 27*
a^10*b + 126*a^8*b^2 + 202*a^6*b^3 - 72*a^4*b^4 - 288*a^2*b^5 + 128*b^6)*x*sqrt((a^8 + 2*a^6*b - a^4*b^2 - 2*a
^2*b^3 + b^4)/(4*a^10 + 44*a^8*b + 145*a^6*b^2 + 76*a^4*b^3 - 208*a^2*b^4 + 64*b^5)) - (a^11 + 9*a^9*b + 23*a^
7*b^2 + 8*a^5*b^3 - 16*a^3*b^4)*x)*sqrt(sqrt(1/2)*sqrt((a^5 + 3*a^3*b - a*b^2 + (2*a^6 + 15*a^4*b + 24*a^2*b^2
- 16*b^3)*sqrt((a^8 + 2*a^6*b - a^4*b^2 - 2*a^2*b^3 + b^4)/(4*a^10 + 44*a^8*b + 145*a^6*b^2 + 76*a^4*b^3 - 20
8*a^2*b^4 + 64*b^5)))/(2*a^6 + 15*a^4*b + 24*a^2*b^2 - 16*b^3)))*sqrt((a^5 + 3*a^3*b - a*b^2 + (2*a^6 + 15*a^4
*b + 24*a^2*b^2 - 16*b^3)*sqrt((a^8 + 2*a^6*b - a^4*b^2 - 2*a^2*b^3 + b^4)/(4*a^10 + 44*a^8*b + 145*a^6*b^2 +
76*a^4*b^3 - 208*a^2*b^4 + 64*b^5)))/(2*a^6 + 15*a^4*b + 24*a^2*b^2 - 16*b^3)) - 2*(a^4*b^3 + a^2*b^4 - b^5)*(
a*x^4 + b)^(1/4))/x) - 1/4*sqrt(sqrt(1/2)*sqrt((a^5 + 3*a^3*b - a*b^2 - (2*a^6 + 15*a^4*b + 24*a^2*b^2 - 16*b^
3)*sqrt((a^8 + 2*a^6*b - a^4*b^2 - 2*a^2*b^3 + b^4)/(4*a^10 + 44*a^8*b + 145*a^6*b^2 + 76*a^4*b^3 - 208*a^2*b^
4 + 64*b^5)))/(2*a^6 + 15*a^4*b + 24*a^2*b^2 - 16*b^3)))*log(-1/2*(sqrt(1/2)*((2*a^12 + 27*a^10*b + 126*a^8*b^
2 + 202*a^6*b^3 - 72*a^4*b^4 - 288*a^2*b^5 + 128*b^6)*x*sqrt((a^8 + 2*a^6*b - a^4*b^2 - 2*a^2*b^3 + b^4)/(4*a^
10 + 44*a^8*b + 145*a^6*b^2 + 76*a^4*b^3 - 208*a^2*b^4 + 64*b^5)) + (a^11 + 9*a^9*b + 23*a^7*b^2 + 8*a^5*b^3 -
16*a^3*b^4)*x)*sqrt(sqrt(1/2)*sqrt((a^5 + 3*a^3*b - a*b^2 - (2*a^6 + 15*a^4*b + 24*a^2*b^2 - 16*b^3)*sqrt((a^
8 + 2*a^6*b - a^4*b^2 - 2*a^2*b^3 + b^4)/(4*a^10 + 44*a^8*b + 145*a^6*b^2 + 76*a^4*b^3 - 208*a^2*b^4 + 64*b^5)
))/(2*a^6 + 15*a^4*b + 24*a^2*b^2 - 16*b^3)))*sqrt((a^5 + 3*a^3*b - a*b^2 - (2*a^6 + 15*a^4*b + 24*a^2*b^2 - 1
6*b^3)*sqrt((a^8 + 2*a^6*b - a^4*b^2 - 2*a^2*b^3 + b^4)/(4*a^10 + 44*a^8*b + 145*a^6*b^2 + 76*a^4*b^3 - 208*a^
2*b^4 + 64*b^5)))/(2*a^6 + 15*a^4*b + 24*a^2*b^2 - 16*b^3)) + 2*(a^4*b^3 + a^2*b^4 - b^5)*(a*x^4 + b)^(1/4))/x
) + 1/4*sqrt(sqrt(1/2)*sqrt((a^5 + 3*a^3*b - a*b^2 - (2*a^6 + 15*a^4*b + 24*a^2*b^2 - 16*b^3)*sqrt((a^8 + 2*a^
6*b - a^4*b^2 - 2*a^2*b^3 + b^4)/(4*a^10 + 44*a^8*b + 145*a^6*b^2 + 76*a^4*b^3 - 208*a^2*b^4 + 64*b^5)))/(2*a^
6 + 15*a^4*b + 24*a^2*b^2 - 16*b^3)))*log(1/2*(sqrt(1/2)*((2*a^12 + 27*a^10*b + 126*a^8*b^2 + 202*a^6*b^3 - 72
*a^4*b^4 - 288*a^2*b^5 + 128*b^6)*x*sqrt((a^8 + 2*a^6*b - a^4*b^2 - 2*a^2*b^3 + b^4)/(4*a^10 + 44*a^8*b + 145*
a^6*b^2 + 76*a^4*b^3 - 208*a^2*b^4 + 64*b^5)) + (a^11 + 9*a^9*b + 23*a^7*b^2 + 8*a^5*b^3 - 16*a^3*b^4)*x)*sqrt
(sqrt(1/2)*sqrt((a^5 + 3*a^3*b - a*b^2 - (2*a^6 + 15*a^4*b + 24*a^2*b^2 - 16*b^3)*sqrt((a^8 + 2*a^6*b - a^4*b^
2 - 2*a^2*b^3 + b^4)/(4*a^10 + 44*a^8*b + 145*a^6*b^2 + 76*a^4*b^3 - 208*a^2*b^4 + 64*b^5)))/(2*a^6 + 15*a^4*b
+ 24*a^2*b^2 - 16*b^3)))*sqrt((a^5 + 3*a^3*b - a*b^2 - (2*a^6 + 15*a^4*b + 24*a^2*b^2 - 16*b^3)*sqrt((a^8 + 2
*a^6*b - a^4*b^2 - 2*a^2*b^3 + b^4)/(4*a^10 + 44*a^8*b + 145*a^6*b^2 + 76*a^4*b^3 - 208*a^2*b^4 + 64*b^5)))/(2
*a^6 + 15*a^4*b + 24*a^2*b^2 - 16*b^3)) - 2*(a^4*b^3 + a^2*b^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} + 2 \, 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+2*a*x^4-b)/(a*x^4+b)^(1/4)/(x^8+a*x^4-b),x, algorithm="giac")
[Out]
integrate((2*x^8 + 2*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 \[\int \frac {2 x^{8}+2 a \,x^{4}-b}{\left (a \,x^{4}+b \right )^{\frac {1}{4}} \left (x^{8}+a \,x^{4}-b \right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((2*x^8+2*a*x^4-b)/(a*x^4+b)^(1/4)/(x^8+a*x^4-b),x)
[Out]
int((2*x^8+2*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} + 2 \, 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+2*a*x^4-b)/(a*x^4+b)^(1/4)/(x^8+a*x^4-b),x, algorithm="maxima")
[Out]
integrate((2*x^8 + 2*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+2\,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((2*a*x^4 - b + 2*x^8)/((b + a*x^4)^(1/4)*(a*x^4 - b + x^8)),x)
[Out]
int((2*a*x^4 - b + 2*x^8)/((b + a*x^4)^(1/4)*(a*x^4 - 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+2*a*x**4-b)/(a*x**4+b)**(1/4)/(x**8+a*x**4-b),x)
[Out]
Timed out
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