3.1467 \(\int \frac{1}{x^4 (a+b x^8)} \, dx\)

Optimal. Leaf size=277 \[ -\frac{b^{3/8} \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{11/8}}+\frac{b^{3/8} \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{11/8}}-\frac{b^{3/8} \tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{11/8}}-\frac{b^{3/8} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt{2} (-a)^{11/8}}+\frac{b^{3/8} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{4 \sqrt{2} (-a)^{11/8}}-\frac{b^{3/8} \tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{11/8}}-\frac{1}{3 a x^3} \]

[Out]

-1/(3*a*x^3) - (b^(3/8)*ArcTan[(b^(1/8)*x)/(-a)^(1/8)])/(4*(-a)^(11/8)) - (b^(3/8)*ArcTan[1 - (Sqrt[2]*b^(1/8)
*x)/(-a)^(1/8)])/(4*Sqrt[2]*(-a)^(11/8)) + (b^(3/8)*ArcTan[1 + (Sqrt[2]*b^(1/8)*x)/(-a)^(1/8)])/(4*Sqrt[2]*(-a
)^(11/8)) - (b^(3/8)*ArcTanh[(b^(1/8)*x)/(-a)^(1/8)])/(4*(-a)^(11/8)) - (b^(3/8)*Log[(-a)^(1/4) - Sqrt[2]*(-a)
^(1/8)*b^(1/8)*x + b^(1/4)*x^2])/(8*Sqrt[2]*(-a)^(11/8)) + (b^(3/8)*Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*b^(1/8
)*x + b^(1/4)*x^2])/(8*Sqrt[2]*(-a)^(11/8))

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Rubi [A]  time = 0.226553, antiderivative size = 277, 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 = {325, 301, 211, 1165, 628, 1162, 617, 204, 212, 208, 205} \[ -\frac{b^{3/8} \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{11/8}}+\frac{b^{3/8} \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{11/8}}-\frac{b^{3/8} \tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{11/8}}-\frac{b^{3/8} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt{2} (-a)^{11/8}}+\frac{b^{3/8} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{4 \sqrt{2} (-a)^{11/8}}-\frac{b^{3/8} \tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{11/8}}-\frac{1}{3 a x^3} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(a + b*x^8)),x]

[Out]

-1/(3*a*x^3) - (b^(3/8)*ArcTan[(b^(1/8)*x)/(-a)^(1/8)])/(4*(-a)^(11/8)) - (b^(3/8)*ArcTan[1 - (Sqrt[2]*b^(1/8)
*x)/(-a)^(1/8)])/(4*Sqrt[2]*(-a)^(11/8)) + (b^(3/8)*ArcTan[1 + (Sqrt[2]*b^(1/8)*x)/(-a)^(1/8)])/(4*Sqrt[2]*(-a
)^(11/8)) - (b^(3/8)*ArcTanh[(b^(1/8)*x)/(-a)^(1/8)])/(4*(-a)^(11/8)) - (b^(3/8)*Log[(-a)^(1/4) - Sqrt[2]*(-a)
^(1/8)*b^(1/8)*x + b^(1/4)*x^2])/(8*Sqrt[2]*(-a)^(11/8)) + (b^(3/8)*Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*b^(1/8
)*x + b^(1/4)*x^2])/(8*Sqrt[2]*(-a)^(11/8))

Rule 325

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

Rule 301

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(
a/b), 2]]}, Dist[s/(2*b), Int[x^(m - n/2)/(r + s*x^(n/2)), x], x] - Dist[s/(2*b), Int[x^(m - n/2)/(r - s*x^(n/
2)), x], x]] /; FreeQ[{a, b}, x] && IGtQ[n/4, 0] && IGtQ[m, 0] && LeQ[n/2, m] && LtQ[m, n] &&  !GtQ[a/b, 0]

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])

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]

Rubi steps

\begin{align*} \int \frac{1}{x^4 \left (a+b x^8\right )} \, dx &=-\frac{1}{3 a x^3}-\frac{b \int \frac{x^4}{a+b x^8} \, dx}{a}\\ &=-\frac{1}{3 a x^3}+\frac{\sqrt{b} \int \frac{1}{\sqrt{-a}-\sqrt{b} x^4} \, dx}{2 a}-\frac{\sqrt{b} \int \frac{1}{\sqrt{-a}+\sqrt{b} x^4} \, dx}{2 a}\\ &=-\frac{1}{3 a x^3}-\frac{\sqrt{b} \int \frac{1}{\sqrt [4]{-a}-\sqrt [4]{b} x^2} \, dx}{4 (-a)^{5/4}}-\frac{\sqrt{b} \int \frac{1}{\sqrt [4]{-a}+\sqrt [4]{b} x^2} \, dx}{4 (-a)^{5/4}}+\frac{\sqrt{b} \int \frac{\sqrt [4]{-a}-\sqrt [4]{b} x^2}{\sqrt{-a}+\sqrt{b} x^4} \, dx}{4 (-a)^{5/4}}+\frac{\sqrt{b} \int \frac{\sqrt [4]{-a}+\sqrt [4]{b} x^2}{\sqrt{-a}+\sqrt{b} x^4} \, dx}{4 (-a)^{5/4}}\\ &=-\frac{1}{3 a x^3}-\frac{b^{3/8} \tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{11/8}}-\frac{b^{3/8} \tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{11/8}}+\frac{\sqrt [4]{b} \int \frac{1}{\frac{\sqrt [4]{-a}}{\sqrt [4]{b}}-\frac{\sqrt{2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+x^2} \, dx}{8 (-a)^{5/4}}+\frac{\sqrt [4]{b} \int \frac{1}{\frac{\sqrt [4]{-a}}{\sqrt [4]{b}}+\frac{\sqrt{2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+x^2} \, dx}{8 (-a)^{5/4}}-\frac{b^{3/8} \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} (-a)^{11/8}}-\frac{b^{3/8} \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} (-a)^{11/8}}\\ &=-\frac{1}{3 a x^3}-\frac{b^{3/8} \tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{11/8}}-\frac{b^{3/8} \tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{11/8}}-\frac{b^{3/8} \log \left (\sqrt [4]{-a}-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{11/8}}+\frac{b^{3/8} \log \left (\sqrt [4]{-a}+\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{11/8}}+\frac{b^{3/8} \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} (-a)^{11/8}}-\frac{b^{3/8} \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} (-a)^{11/8}}\\ &=-\frac{1}{3 a x^3}-\frac{b^{3/8} \tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{11/8}}-\frac{b^{3/8} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt{2} (-a)^{11/8}}+\frac{b^{3/8} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt{2} (-a)^{11/8}}-\frac{b^{3/8} \tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{11/8}}-\frac{b^{3/8} \log \left (\sqrt [4]{-a}-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{11/8}}+\frac{b^{3/8} \log \left (\sqrt [4]{-a}+\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{11/8}}\\ \end{align*}

Mathematica [A]  time = 0.108984, size = 395, normalized size = 1.43 \[ \frac{-8 a^{3/8}+3 b^{3/8} x^3 \cos \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )-3 b^{3/8} x^3 \cos \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )-3 b^{3/8} x^3 \sin \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )+3 b^{3/8} x^3 \sin \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )+6 b^{3/8} x^3 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{b} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right )+6 b^{3/8} x^3 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{b} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right )+6 b^{3/8} x^3 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{b} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )-6 b^{3/8} x^3 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{b} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )}{24 a^{11/8} x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(a + b*x^8)),x]

[Out]

(-8*a^(3/8) + 6*b^(3/8)*x^3*ArcTan[Cot[Pi/8] - (b^(1/8)*x*Csc[Pi/8])/a^(1/8)]*Cos[Pi/8] - 6*b^(3/8)*x^3*ArcTan
[Cot[Pi/8] + (b^(1/8)*x*Csc[Pi/8])/a^(1/8)]*Cos[Pi/8] + 3*b^(3/8)*x^3*Cos[Pi/8]*Log[a^(1/4) + b^(1/4)*x^2 - 2*
a^(1/8)*b^(1/8)*x*Sin[Pi/8]] - 3*b^(3/8)*x^3*Cos[Pi/8]*Log[a^(1/4) + b^(1/4)*x^2 + 2*a^(1/8)*b^(1/8)*x*Sin[Pi/
8]] + 6*b^(3/8)*x^3*ArcTan[(b^(1/8)*x*Sec[Pi/8])/a^(1/8) - Tan[Pi/8]]*Sin[Pi/8] + 6*b^(3/8)*x^3*ArcTan[(b^(1/8
)*x*Sec[Pi/8])/a^(1/8) + Tan[Pi/8]]*Sin[Pi/8] - 3*b^(3/8)*x^3*Log[a^(1/4) + b^(1/4)*x^2 - 2*a^(1/8)*b^(1/8)*x*
Cos[Pi/8]]*Sin[Pi/8] + 3*b^(3/8)*x^3*Log[a^(1/4) + b^(1/4)*x^2 + 2*a^(1/8)*b^(1/8)*x*Cos[Pi/8]]*Sin[Pi/8])/(24
*a^(11/8)*x^3)

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Maple [C]  time = 0.005, size = 36, normalized size = 0.1 \begin{align*} -{\frac{1}{8\,a}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}b+a \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{3}}}}-{\frac{1}{3\,a{x}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8/a*sum(1/_R^3*ln(x-_R),_R=RootOf(_Z^8*b+a))-1/3/a/x^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{-\frac{1}{16} \,{\left (\frac{2 \, \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{5}{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{5}{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{5}{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{5}{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{5}{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{5}{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{5}{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{5}{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}{a} - \frac{1}{3 \, a x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b*integrate(x^4/(b*x^8 + a), x)/a - 1/3/(a*x^3)

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Fricas [B]  time = 1.45889, size = 1207, normalized size = 4.36 \begin{align*} -\frac{12 \, \sqrt{2} a x^{3} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{8}} \arctan \left (-\frac{\sqrt{2} a^{4} x \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{3}{8}} - \sqrt{2} a^{4} \sqrt{-\frac{\sqrt{2} a^{7} x \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{5}{8}} + a^{3} b \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{4}} - b^{2} x^{2}}{b^{2}}} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{3}{8}} + b}{b}\right ) + 12 \, \sqrt{2} a x^{3} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{8}} \arctan \left (-\frac{\sqrt{2} a^{4} x \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{3}{8}} - \sqrt{2} a^{4} \sqrt{\frac{\sqrt{2} a^{7} x \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{5}{8}} - a^{3} b \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{4}} + b^{2} x^{2}}{b^{2}}} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{3}{8}} - b}{b}\right ) + 3 \, \sqrt{2} a x^{3} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{8}} \log \left (\sqrt{2} a^{7} x \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{5}{8}} - a^{3} b \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{4}} + b^{2} x^{2}\right ) - 3 \, \sqrt{2} a x^{3} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{8}} \log \left (-\sqrt{2} a^{7} x \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{5}{8}} - a^{3} b \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{4}} + b^{2} x^{2}\right ) - 24 \, a x^{3} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{8}} \arctan \left (-\frac{a^{4} x \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{3}{8}} - a^{4} \sqrt{-\frac{a^{3} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{4}} - b x^{2}}{b}} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{3}{8}}}{b}\right ) - 6 \, a x^{3} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{8}} \log \left (a^{7} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{5}{8}} + b^{2} x\right ) + 6 \, a x^{3} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{8}} \log \left (-a^{7} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{5}{8}} + b^{2} x\right ) + 16}{48 \, a x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/48*(12*sqrt(2)*a*x^3*(-b^3/a^11)^(1/8)*arctan(-(sqrt(2)*a^4*x*(-b^3/a^11)^(3/8) - sqrt(2)*a^4*sqrt(-(sqrt(2
)*a^7*x*(-b^3/a^11)^(5/8) + a^3*b*(-b^3/a^11)^(1/4) - b^2*x^2)/b^2)*(-b^3/a^11)^(3/8) + b)/b) + 12*sqrt(2)*a*x
^3*(-b^3/a^11)^(1/8)*arctan(-(sqrt(2)*a^4*x*(-b^3/a^11)^(3/8) - sqrt(2)*a^4*sqrt((sqrt(2)*a^7*x*(-b^3/a^11)^(5
/8) - a^3*b*(-b^3/a^11)^(1/4) + b^2*x^2)/b^2)*(-b^3/a^11)^(3/8) - b)/b) + 3*sqrt(2)*a*x^3*(-b^3/a^11)^(1/8)*lo
g(sqrt(2)*a^7*x*(-b^3/a^11)^(5/8) - a^3*b*(-b^3/a^11)^(1/4) + b^2*x^2) - 3*sqrt(2)*a*x^3*(-b^3/a^11)^(1/8)*log
(-sqrt(2)*a^7*x*(-b^3/a^11)^(5/8) - a^3*b*(-b^3/a^11)^(1/4) + b^2*x^2) - 24*a*x^3*(-b^3/a^11)^(1/8)*arctan(-(a
^4*x*(-b^3/a^11)^(3/8) - a^4*sqrt(-(a^3*(-b^3/a^11)^(1/4) - b*x^2)/b)*(-b^3/a^11)^(3/8))/b) - 6*a*x^3*(-b^3/a^
11)^(1/8)*log(a^7*(-b^3/a^11)^(5/8) + b^2*x) + 6*a*x^3*(-b^3/a^11)^(1/8)*log(-a^7*(-b^3/a^11)^(5/8) + b^2*x) +
16)/(a*x^3)

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Sympy [A]  time = 0.558572, size = 36, normalized size = 0.13 \begin{align*} \operatorname{RootSum}{\left (16777216 t^{8} a^{11} + b^{3}, \left ( t \mapsto t \log{\left (\frac{32768 t^{5} a^{7}}{b^{2}} + x \right )} \right )\right )} - \frac{1}{3 a x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(16777216*_t**8*a**11 + b**3, Lambda(_t, _t*log(32768*_t**5*a**7/b**2 + x))) - 1/(3*a*x**3)

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Giac [B]  time = 1.27444, size = 601, normalized size = 2.17 \begin{align*} \frac{b \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{5}{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 \, a^{2}} + \frac{b \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{5}{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 \, a^{2}} - \frac{b \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{5}{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 \, a^{2}} - \frac{b \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{5}{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 \, a^{2}} + \frac{b \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{5}{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 \, a^{2}} - \frac{b \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{5}{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 \, a^{2}} - \frac{b \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{5}{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 \, a^{2}} + \frac{b \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{5}{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 \, a^{2}} - \frac{1}{3 \, a x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*b*sqrt(-sqrt(2) + 2)*(a/b)^(5/8)*arctan((2*x + sqrt(-sqrt(2) + 2)*(a/b)^(1/8))/(sqrt(sqrt(2) + 2)*(a/b)^(1
/8)))/a^2 + 1/8*b*sqrt(-sqrt(2) + 2)*(a/b)^(5/8)*arctan((2*x - sqrt(-sqrt(2) + 2)*(a/b)^(1/8))/(sqrt(sqrt(2) +
2)*(a/b)^(1/8)))/a^2 - 1/8*b*sqrt(sqrt(2) + 2)*(a/b)^(5/8)*arctan((2*x + sqrt(sqrt(2) + 2)*(a/b)^(1/8))/(sqrt
(-sqrt(2) + 2)*(a/b)^(1/8)))/a^2 - 1/8*b*sqrt(sqrt(2) + 2)*(a/b)^(5/8)*arctan((2*x - sqrt(sqrt(2) + 2)*(a/b)^(
1/8))/(sqrt(-sqrt(2) + 2)*(a/b)^(1/8)))/a^2 + 1/16*b*sqrt(-sqrt(2) + 2)*(a/b)^(5/8)*log(x^2 + x*sqrt(sqrt(2) +
2)*(a/b)^(1/8) + (a/b)^(1/4))/a^2 - 1/16*b*sqrt(-sqrt(2) + 2)*(a/b)^(5/8)*log(x^2 - x*sqrt(sqrt(2) + 2)*(a/b)
^(1/8) + (a/b)^(1/4))/a^2 - 1/16*b*sqrt(sqrt(2) + 2)*(a/b)^(5/8)*log(x^2 + x*sqrt(-sqrt(2) + 2)*(a/b)^(1/8) +
(a/b)^(1/4))/a^2 + 1/16*b*sqrt(sqrt(2) + 2)*(a/b)^(5/8)*log(x^2 - x*sqrt(-sqrt(2) + 2)*(a/b)^(1/8) + (a/b)^(1/
4))/a^2 - 1/3/(a*x^3)