3.1461 \(\int \frac{x^8}{a+b x^8} \, dx\)

Optimal. Leaf size=272 \[ \frac{\sqrt [8]{-a} \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} b^{9/8}}-\frac{\sqrt [8]{-a} \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} b^{9/8}}-\frac{\sqrt [8]{-a} \tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 b^{9/8}}+\frac{\sqrt [8]{-a} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt{2} b^{9/8}}-\frac{\sqrt [8]{-a} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{4 \sqrt{2} b^{9/8}}-\frac{\sqrt [8]{-a} \tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 b^{9/8}}+\frac{x}{b} \]

[Out]

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

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Rubi [A]  time = 0.29687, antiderivative size = 272, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.846, Rules used = {321, 214, 212, 208, 205, 211, 1165, 628, 1162, 617, 204} \[ \frac{\sqrt [8]{-a} \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} b^{9/8}}-\frac{\sqrt [8]{-a} \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} b^{9/8}}-\frac{\sqrt [8]{-a} \tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 b^{9/8}}+\frac{\sqrt [8]{-a} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt{2} b^{9/8}}-\frac{\sqrt [8]{-a} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{4 \sqrt{2} b^{9/8}}-\frac{\sqrt [8]{-a} \tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 b^{9/8}}+\frac{x}{b} \]

Antiderivative was successfully verified.

[In]

Int[x^8/(a + b*x^8),x]

[Out]

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

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 214

Int[((a_) + (b_.)*(x_)^(n_))^(-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^(n/2)), x], x] + Dist[r/(2*a), Int[1/(r + s*x^(n/2)), x], x]] /; FreeQ[{a,
b}, x] && IGtQ[n/4, 1] &&  !GtQ[a/b, 0]

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 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 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 211

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{x^8}{a+b x^8} \, dx &=\frac{x}{b}-\frac{a \int \frac{1}{a+b x^8} \, dx}{b}\\ &=\frac{x}{b}-\frac{\sqrt{-a} \int \frac{1}{\sqrt{-a}-\sqrt{b} x^4} \, dx}{2 b}-\frac{\sqrt{-a} \int \frac{1}{\sqrt{-a}+\sqrt{b} x^4} \, dx}{2 b}\\ &=\frac{x}{b}-\frac{\sqrt [4]{-a} \int \frac{1}{\sqrt [4]{-a}-\sqrt [4]{b} x^2} \, dx}{4 b}-\frac{\sqrt [4]{-a} \int \frac{1}{\sqrt [4]{-a}+\sqrt [4]{b} x^2} \, dx}{4 b}-\frac{\sqrt [4]{-a} \int \frac{\sqrt [4]{-a}-\sqrt [4]{b} x^2}{\sqrt{-a}+\sqrt{b} x^4} \, dx}{4 b}-\frac{\sqrt [4]{-a} \int \frac{\sqrt [4]{-a}+\sqrt [4]{b} x^2}{\sqrt{-a}+\sqrt{b} x^4} \, dx}{4 b}\\ &=\frac{x}{b}-\frac{\sqrt [8]{-a} \tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 b^{9/8}}-\frac{\sqrt [8]{-a} \tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 b^{9/8}}-\frac{\sqrt [4]{-a} \int \frac{1}{\frac{\sqrt [4]{-a}}{\sqrt [4]{b}}-\frac{\sqrt{2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+x^2} \, dx}{8 b^{5/4}}-\frac{\sqrt [4]{-a} \int \frac{1}{\frac{\sqrt [4]{-a}}{\sqrt [4]{b}}+\frac{\sqrt{2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+x^2} \, dx}{8 b^{5/4}}+\frac{\sqrt [8]{-a} \int \frac{\frac{\sqrt{2} \sqrt [8]{-a}}{\sqrt [8]{b}}+2 x}{-\frac{\sqrt [4]{-a}}{\sqrt [4]{b}}-\frac{\sqrt{2} \sqrt [8]{-a} x}{\sqrt [8]{b}}-x^2} \, dx}{8 \sqrt{2} b^{9/8}}+\frac{\sqrt [8]{-a} \int \frac{\frac{\sqrt{2} \sqrt [8]{-a}}{\sqrt [8]{b}}-2 x}{-\frac{\sqrt [4]{-a}}{\sqrt [4]{b}}+\frac{\sqrt{2} \sqrt [8]{-a} x}{\sqrt [8]{b}}-x^2} \, dx}{8 \sqrt{2} b^{9/8}}\\ &=\frac{x}{b}-\frac{\sqrt [8]{-a} \tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 b^{9/8}}-\frac{\sqrt [8]{-a} \tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 b^{9/8}}+\frac{\sqrt [8]{-a} \log \left (\sqrt [4]{-a}-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} b^{9/8}}-\frac{\sqrt [8]{-a} \log \left (\sqrt [4]{-a}+\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} b^{9/8}}-\frac{\sqrt [8]{-a} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt{2} b^{9/8}}+\frac{\sqrt [8]{-a} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt{2} b^{9/8}}\\ &=\frac{x}{b}-\frac{\sqrt [8]{-a} \tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 b^{9/8}}+\frac{\sqrt [8]{-a} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt{2} b^{9/8}}-\frac{\sqrt [8]{-a} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt{2} b^{9/8}}-\frac{\sqrt [8]{-a} \tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 b^{9/8}}+\frac{\sqrt [8]{-a} \log \left (\sqrt [4]{-a}-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} b^{9/8}}-\frac{\sqrt [8]{-a} \log \left (\sqrt [4]{-a}+\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} b^{9/8}}\\ \end{align*}

Mathematica [A]  time = 0.114981, size = 367, normalized size = 1.35 \[ \frac{\sqrt [8]{a} \sin \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )-\sqrt [8]{a} \sin \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )+\sqrt [8]{a} \cos \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )-\sqrt [8]{a} \cos \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )-2 \sqrt [8]{a} \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{b} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right )-2 \sqrt [8]{a} \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{b} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right )+2 \sqrt [8]{a} \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{b} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )-2 \sqrt [8]{a} \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{b} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )+8 \sqrt [8]{b} x}{8 b^{9/8}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^8/(a + b*x^8),x]

[Out]

(8*b^(1/8)*x - 2*a^(1/8)*ArcTan[(b^(1/8)*x*Sec[Pi/8])/a^(1/8) - Tan[Pi/8]]*Cos[Pi/8] - 2*a^(1/8)*ArcTan[(b^(1/
8)*x*Sec[Pi/8])/a^(1/8) + Tan[Pi/8]]*Cos[Pi/8] + a^(1/8)*Cos[Pi/8]*Log[a^(1/4) + b^(1/4)*x^2 - 2*a^(1/8)*b^(1/
8)*x*Cos[Pi/8]] - a^(1/8)*Cos[Pi/8]*Log[a^(1/4) + b^(1/4)*x^2 + 2*a^(1/8)*b^(1/8)*x*Cos[Pi/8]] + 2*a^(1/8)*Arc
Tan[Cot[Pi/8] - (b^(1/8)*x*Csc[Pi/8])/a^(1/8)]*Sin[Pi/8] - 2*a^(1/8)*ArcTan[Cot[Pi/8] + (b^(1/8)*x*Csc[Pi/8])/
a^(1/8)]*Sin[Pi/8] + a^(1/8)*Log[a^(1/4) + b^(1/4)*x^2 - 2*a^(1/8)*b^(1/8)*x*Sin[Pi/8]]*Sin[Pi/8] - a^(1/8)*Lo
g[a^(1/4) + b^(1/4)*x^2 + 2*a^(1/8)*b^(1/8)*x*Sin[Pi/8]]*Sin[Pi/8])/(8*b^(9/8))

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Maple [C]  time = 0.016, size = 34, normalized size = 0.1 \begin{align*}{\frac{x}{b}}-{\frac{a}{8\,{b}^{2}}\sum _{{\it \_R}={\it RootOf} \left ( b{{\it \_Z}}^{8}+a \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{7}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8/(b*x^8+a),x)

[Out]

x/b-1/8/b^2*a*sum(1/_R^7*ln(x-_R),_R=RootOf(_Z^8*b+a))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{\frac{1}{16} \, a{\left (\frac{2 \, \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} \arctan \left (\frac{2 \, x + \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{a} + \frac{2 \, \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} \arctan \left (\frac{2 \, x - \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{a} + \frac{2 \, \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} \arctan \left (\frac{2 \, x + \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{a} + \frac{2 \, \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} \arctan \left (\frac{2 \, x - \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{a} + \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} \log \left (x^{2} + x \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} + \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}{a} - \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} \log \left (x^{2} - x \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} + \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}{a} + \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} \log \left (x^{2} + x \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} + \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}{a} - \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} \log \left (x^{2} - x \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} + \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}{a}\right )}}{b} + \frac{x}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(b*x^8+a),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.40089, size = 972, normalized size = 3.57 \begin{align*} -\frac{4 \, \sqrt{2} b \left (-\frac{a}{b^{9}}\right )^{\frac{1}{8}} \arctan \left (-\frac{\sqrt{2} b^{8} x \left (-\frac{a}{b^{9}}\right )^{\frac{7}{8}} - \sqrt{2} \sqrt{\sqrt{2} b x \left (-\frac{a}{b^{9}}\right )^{\frac{1}{8}} + b^{2} \left (-\frac{a}{b^{9}}\right )^{\frac{1}{4}} + x^{2}} b^{8} \left (-\frac{a}{b^{9}}\right )^{\frac{7}{8}} - a}{a}\right ) + 4 \, \sqrt{2} b \left (-\frac{a}{b^{9}}\right )^{\frac{1}{8}} \arctan \left (-\frac{\sqrt{2} b^{8} x \left (-\frac{a}{b^{9}}\right )^{\frac{7}{8}} - \sqrt{2} \sqrt{-\sqrt{2} b x \left (-\frac{a}{b^{9}}\right )^{\frac{1}{8}} + b^{2} \left (-\frac{a}{b^{9}}\right )^{\frac{1}{4}} + x^{2}} b^{8} \left (-\frac{a}{b^{9}}\right )^{\frac{7}{8}} + a}{a}\right ) + \sqrt{2} b \left (-\frac{a}{b^{9}}\right )^{\frac{1}{8}} \log \left (\sqrt{2} b x \left (-\frac{a}{b^{9}}\right )^{\frac{1}{8}} + b^{2} \left (-\frac{a}{b^{9}}\right )^{\frac{1}{4}} + x^{2}\right ) - \sqrt{2} b \left (-\frac{a}{b^{9}}\right )^{\frac{1}{8}} \log \left (-\sqrt{2} b x \left (-\frac{a}{b^{9}}\right )^{\frac{1}{8}} + b^{2} \left (-\frac{a}{b^{9}}\right )^{\frac{1}{4}} + x^{2}\right ) + 8 \, b \left (-\frac{a}{b^{9}}\right )^{\frac{1}{8}} \arctan \left (-\frac{b^{8} x \left (-\frac{a}{b^{9}}\right )^{\frac{7}{8}} - \sqrt{b^{2} \left (-\frac{a}{b^{9}}\right )^{\frac{1}{4}} + x^{2}} b^{8} \left (-\frac{a}{b^{9}}\right )^{\frac{7}{8}}}{a}\right ) + 2 \, b \left (-\frac{a}{b^{9}}\right )^{\frac{1}{8}} \log \left (b \left (-\frac{a}{b^{9}}\right )^{\frac{1}{8}} + x\right ) - 2 \, b \left (-\frac{a}{b^{9}}\right )^{\frac{1}{8}} \log \left (-b \left (-\frac{a}{b^{9}}\right )^{\frac{1}{8}} + x\right ) - 16 \, x}{16 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(4*sqrt(2)*b*(-a/b^9)^(1/8)*arctan(-(sqrt(2)*b^8*x*(-a/b^9)^(7/8) - sqrt(2)*sqrt(sqrt(2)*b*x*(-a/b^9)^(1
/8) + b^2*(-a/b^9)^(1/4) + x^2)*b^8*(-a/b^9)^(7/8) - a)/a) + 4*sqrt(2)*b*(-a/b^9)^(1/8)*arctan(-(sqrt(2)*b^8*x
*(-a/b^9)^(7/8) - sqrt(2)*sqrt(-sqrt(2)*b*x*(-a/b^9)^(1/8) + b^2*(-a/b^9)^(1/4) + x^2)*b^8*(-a/b^9)^(7/8) + a)
/a) + sqrt(2)*b*(-a/b^9)^(1/8)*log(sqrt(2)*b*x*(-a/b^9)^(1/8) + b^2*(-a/b^9)^(1/4) + x^2) - sqrt(2)*b*(-a/b^9)
^(1/8)*log(-sqrt(2)*b*x*(-a/b^9)^(1/8) + b^2*(-a/b^9)^(1/4) + x^2) + 8*b*(-a/b^9)^(1/8)*arctan(-(b^8*x*(-a/b^9
)^(7/8) - sqrt(b^2*(-a/b^9)^(1/4) + x^2)*b^8*(-a/b^9)^(7/8))/a) + 2*b*(-a/b^9)^(1/8)*log(b*(-a/b^9)^(1/8) + x)
- 2*b*(-a/b^9)^(1/8)*log(-b*(-a/b^9)^(1/8) + x) - 16*x)/b

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Sympy [A]  time = 0.345156, size = 22, normalized size = 0.08 \begin{align*} \operatorname{RootSum}{\left (16777216 t^{8} b^{9} + a, \left ( t \mapsto t \log{\left (- 8 t b + x \right )} \right )\right )} + \frac{x}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8/(b*x**8+a),x)

[Out]

RootSum(16777216*_t**8*b**9 + a, Lambda(_t, _t*log(-8*_t*b + x))) + x/b

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Giac [B]  time = 1.19555, size = 586, normalized size = 2.15 \begin{align*} -\frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} \arctan \left (\frac{2 \, x + \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{8 \, b} - \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} \arctan \left (\frac{2 \, x - \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{8 \, b} - \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} \arctan \left (\frac{2 \, x + \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{8 \, b} - \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} \arctan \left (\frac{2 \, x - \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{8 \, b} - \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} \log \left (x^{2} + x \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} + \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}{16 \, b} + \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} \log \left (x^{2} - x \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} + \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}{16 \, b} - \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} \log \left (x^{2} + x \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} + \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}{16 \, b} + \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} \log \left (x^{2} - x \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} + \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}{16 \, b} + \frac{x}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*sqrt(sqrt(2) + 2)*(a/b)^(1/8)*arctan((2*x + sqrt(-sqrt(2) + 2)*(a/b)^(1/8))/(sqrt(sqrt(2) + 2)*(a/b)^(1/8
)))/b - 1/8*sqrt(sqrt(2) + 2)*(a/b)^(1/8)*arctan((2*x - sqrt(-sqrt(2) + 2)*(a/b)^(1/8))/(sqrt(sqrt(2) + 2)*(a/
b)^(1/8)))/b - 1/8*sqrt(-sqrt(2) + 2)*(a/b)^(1/8)*arctan((2*x + sqrt(sqrt(2) + 2)*(a/b)^(1/8))/(sqrt(-sqrt(2)
+ 2)*(a/b)^(1/8)))/b - 1/8*sqrt(-sqrt(2) + 2)*(a/b)^(1/8)*arctan((2*x - sqrt(sqrt(2) + 2)*(a/b)^(1/8))/(sqrt(-
sqrt(2) + 2)*(a/b)^(1/8)))/b - 1/16*sqrt(sqrt(2) + 2)*(a/b)^(1/8)*log(x^2 + x*sqrt(sqrt(2) + 2)*(a/b)^(1/8) +
(a/b)^(1/4))/b + 1/16*sqrt(sqrt(2) + 2)*(a/b)^(1/8)*log(x^2 - x*sqrt(sqrt(2) + 2)*(a/b)^(1/8) + (a/b)^(1/4))/b
- 1/16*sqrt(-sqrt(2) + 2)*(a/b)^(1/8)*log(x^2 + x*sqrt(-sqrt(2) + 2)*(a/b)^(1/8) + (a/b)^(1/4))/b + 1/16*sqrt
(-sqrt(2) + 2)*(a/b)^(1/8)*log(x^2 - x*sqrt(-sqrt(2) + 2)*(a/b)^(1/8) + (a/b)^(1/4))/b + x/b