3.145 \(\int \frac{x^4}{a+b x^3+c x^6} \, dx\)

Optimal. Leaf size=558 \[ -\frac{\left (b-\sqrt{b^2-4 a c}\right )^{2/3} \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{b-\sqrt{b^2-4 a c}}+\left (b-\sqrt{b^2-4 a c}\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} c^{2/3} \sqrt{b^2-4 a c}}+\frac{\left (\sqrt{b^2-4 a c}+b\right )^{2/3} \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{\sqrt{b^2-4 a c}+b}+\left (\sqrt{b^2-4 a c}+b\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} c^{2/3} \sqrt{b^2-4 a c}}+\frac{\left (b-\sqrt{b^2-4 a c}\right )^{2/3} \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt{b^2-4 a c}}-\frac{\left (\sqrt{b^2-4 a c}+b\right )^{2/3} \log \left (\sqrt [3]{\sqrt{b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt{b^2-4 a c}}+\frac{\left (b-\sqrt{b^2-4 a c}\right )^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3} c^{2/3} \sqrt{b^2-4 a c}}-\frac{\left (\sqrt{b^2-4 a c}+b\right )^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3} c^{2/3} \sqrt{b^2-4 a c}} \]

[Out]

((b - Sqrt[b^2 - 4*a*c])^(2/3)*ArcTan[(1 - (2*2^(1/3)*c^(1/3)*x)/(b - Sqrt[b^2 - 4*a*c])^(1/3))/Sqrt[3]])/(2^(
2/3)*Sqrt[3]*c^(2/3)*Sqrt[b^2 - 4*a*c]) - ((b + Sqrt[b^2 - 4*a*c])^(2/3)*ArcTan[(1 - (2*2^(1/3)*c^(1/3)*x)/(b
+ Sqrt[b^2 - 4*a*c])^(1/3))/Sqrt[3]])/(2^(2/3)*Sqrt[3]*c^(2/3)*Sqrt[b^2 - 4*a*c]) + ((b - Sqrt[b^2 - 4*a*c])^(
2/3)*Log[(b - Sqrt[b^2 - 4*a*c])^(1/3) + 2^(1/3)*c^(1/3)*x])/(3*2^(2/3)*c^(2/3)*Sqrt[b^2 - 4*a*c]) - ((b + Sqr
t[b^2 - 4*a*c])^(2/3)*Log[(b + Sqrt[b^2 - 4*a*c])^(1/3) + 2^(1/3)*c^(1/3)*x])/(3*2^(2/3)*c^(2/3)*Sqrt[b^2 - 4*
a*c]) - ((b - Sqrt[b^2 - 4*a*c])^(2/3)*Log[(b - Sqrt[b^2 - 4*a*c])^(2/3) - 2^(1/3)*c^(1/3)*(b - Sqrt[b^2 - 4*a
*c])^(1/3)*x + 2^(2/3)*c^(2/3)*x^2])/(6*2^(2/3)*c^(2/3)*Sqrt[b^2 - 4*a*c]) + ((b + Sqrt[b^2 - 4*a*c])^(2/3)*Lo
g[(b + Sqrt[b^2 - 4*a*c])^(2/3) - 2^(1/3)*c^(1/3)*(b + Sqrt[b^2 - 4*a*c])^(1/3)*x + 2^(2/3)*c^(2/3)*x^2])/(6*2
^(2/3)*c^(2/3)*Sqrt[b^2 - 4*a*c])

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Rubi [A]  time = 0.519725, antiderivative size = 558, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {1374, 292, 31, 634, 617, 204, 628} \[ -\frac{\left (b-\sqrt{b^2-4 a c}\right )^{2/3} \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{b-\sqrt{b^2-4 a c}}+\left (b-\sqrt{b^2-4 a c}\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} c^{2/3} \sqrt{b^2-4 a c}}+\frac{\left (\sqrt{b^2-4 a c}+b\right )^{2/3} \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{\sqrt{b^2-4 a c}+b}+\left (\sqrt{b^2-4 a c}+b\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} c^{2/3} \sqrt{b^2-4 a c}}+\frac{\left (b-\sqrt{b^2-4 a c}\right )^{2/3} \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt{b^2-4 a c}}-\frac{\left (\sqrt{b^2-4 a c}+b\right )^{2/3} \log \left (\sqrt [3]{\sqrt{b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt{b^2-4 a c}}+\frac{\left (b-\sqrt{b^2-4 a c}\right )^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3} c^{2/3} \sqrt{b^2-4 a c}}-\frac{\left (\sqrt{b^2-4 a c}+b\right )^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3} c^{2/3} \sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

[In]

Int[x^4/(a + b*x^3 + c*x^6),x]

[Out]

((b - Sqrt[b^2 - 4*a*c])^(2/3)*ArcTan[(1 - (2*2^(1/3)*c^(1/3)*x)/(b - Sqrt[b^2 - 4*a*c])^(1/3))/Sqrt[3]])/(2^(
2/3)*Sqrt[3]*c^(2/3)*Sqrt[b^2 - 4*a*c]) - ((b + Sqrt[b^2 - 4*a*c])^(2/3)*ArcTan[(1 - (2*2^(1/3)*c^(1/3)*x)/(b
+ Sqrt[b^2 - 4*a*c])^(1/3))/Sqrt[3]])/(2^(2/3)*Sqrt[3]*c^(2/3)*Sqrt[b^2 - 4*a*c]) + ((b - Sqrt[b^2 - 4*a*c])^(
2/3)*Log[(b - Sqrt[b^2 - 4*a*c])^(1/3) + 2^(1/3)*c^(1/3)*x])/(3*2^(2/3)*c^(2/3)*Sqrt[b^2 - 4*a*c]) - ((b + Sqr
t[b^2 - 4*a*c])^(2/3)*Log[(b + Sqrt[b^2 - 4*a*c])^(1/3) + 2^(1/3)*c^(1/3)*x])/(3*2^(2/3)*c^(2/3)*Sqrt[b^2 - 4*
a*c]) - ((b - Sqrt[b^2 - 4*a*c])^(2/3)*Log[(b - Sqrt[b^2 - 4*a*c])^(2/3) - 2^(1/3)*c^(1/3)*(b - Sqrt[b^2 - 4*a
*c])^(1/3)*x + 2^(2/3)*c^(2/3)*x^2])/(6*2^(2/3)*c^(2/3)*Sqrt[b^2 - 4*a*c]) + ((b + Sqrt[b^2 - 4*a*c])^(2/3)*Lo
g[(b + Sqrt[b^2 - 4*a*c])^(2/3) - 2^(1/3)*c^(1/3)*(b + Sqrt[b^2 - 4*a*c])^(1/3)*x + 2^(2/3)*c^(2/3)*x^2])/(6*2
^(2/3)*c^(2/3)*Sqrt[b^2 - 4*a*c])

Rule 1374

Int[((d_.)*(x_))^(m_)/((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}
, Dist[(d^n*(b/q + 1))/2, Int[(d*x)^(m - n)/(b/2 + q/2 + c*x^n), x], x] - Dist[(d^n*(b/q - 1))/2, Int[(d*x)^(m
- n)/(b/2 - q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n,
0] && GeQ[m, n]

Rule 292

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> -Dist[(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3
]^2*x^2), x], x] /; FreeQ[{a, b}, x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

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 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]

Rubi steps

\begin{align*} \int \frac{x^4}{a+b x^3+c x^6} \, dx &=-\left (\frac{1}{2} \left (-1+\frac{b}{\sqrt{b^2-4 a c}}\right ) \int \frac{x}{\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^3} \, dx\right )+\frac{1}{2} \left (1+\frac{b}{\sqrt{b^2-4 a c}}\right ) \int \frac{x}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^3} \, dx\\ &=\frac{\left (b-\sqrt{b^2-4 a c}\right )^{2/3} \int \frac{1}{\frac{\sqrt [3]{b-\sqrt{b^2-4 a c}}}{\sqrt [3]{2}}+\sqrt [3]{c} x} \, dx}{3\ 2^{2/3} \sqrt [3]{c} \sqrt{b^2-4 a c}}-\frac{\left (b-\sqrt{b^2-4 a c}\right )^{2/3} \int \frac{\frac{\sqrt [3]{b-\sqrt{b^2-4 a c}}}{\sqrt [3]{2}}+\sqrt [3]{c} x}{\frac{\left (b-\sqrt{b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac{\sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{3\ 2^{2/3} \sqrt [3]{c} \sqrt{b^2-4 a c}}-\frac{\left (b+\sqrt{b^2-4 a c}\right )^{2/3} \int \frac{1}{\frac{\sqrt [3]{b+\sqrt{b^2-4 a c}}}{\sqrt [3]{2}}+\sqrt [3]{c} x} \, dx}{3\ 2^{2/3} \sqrt [3]{c} \sqrt{b^2-4 a c}}+\frac{\left (b+\sqrt{b^2-4 a c}\right )^{2/3} \int \frac{\frac{\sqrt [3]{b+\sqrt{b^2-4 a c}}}{\sqrt [3]{2}}+\sqrt [3]{c} x}{\frac{\left (b+\sqrt{b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac{\sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{3\ 2^{2/3} \sqrt [3]{c} \sqrt{b^2-4 a c}}\\ &=\frac{\left (b-\sqrt{b^2-4 a c}\right )^{2/3} \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt{b^2-4 a c}}-\frac{\left (b+\sqrt{b^2-4 a c}\right )^{2/3} \log \left (\sqrt [3]{b+\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt{b^2-4 a c}}+\frac{\left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \int \frac{1}{\frac{\left (b-\sqrt{b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac{\sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{4 \sqrt [3]{c}}+\frac{\left (1+\frac{b}{\sqrt{b^2-4 a c}}\right ) \int \frac{1}{\frac{\left (b+\sqrt{b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac{\sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{4 \sqrt [3]{c}}-\frac{\left (b-\sqrt{b^2-4 a c}\right )^{2/3} \int \frac{-\frac{\sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}}}{\sqrt [3]{2}}+2 c^{2/3} x}{\frac{\left (b-\sqrt{b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac{\sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{6\ 2^{2/3} c^{2/3} \sqrt{b^2-4 a c}}+\frac{\left (b+\sqrt{b^2-4 a c}\right )^{2/3} \int \frac{-\frac{\sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}}}{\sqrt [3]{2}}+2 c^{2/3} x}{\frac{\left (b+\sqrt{b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac{\sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{6\ 2^{2/3} c^{2/3} \sqrt{b^2-4 a c}}\\ &=\frac{\left (b-\sqrt{b^2-4 a c}\right )^{2/3} \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt{b^2-4 a c}}-\frac{\left (b+\sqrt{b^2-4 a c}\right )^{2/3} \log \left (\sqrt [3]{b+\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt{b^2-4 a c}}-\frac{\left (b-\sqrt{b^2-4 a c}\right )^{2/3} \log \left (\left (b-\sqrt{b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} c^{2/3} \sqrt{b^2-4 a c}}+\frac{\left (b+\sqrt{b^2-4 a c}\right )^{2/3} \log \left (\left (b+\sqrt{b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} c^{2/3} \sqrt{b^2-4 a c}}-\frac{\left (b-\sqrt{b^2-4 a c}\right )^{2/3} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt{b^2-4 a c}}}\right )}{2^{2/3} c^{2/3} \sqrt{b^2-4 a c}}+\frac{\left (b+\sqrt{b^2-4 a c}\right )^{2/3} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b+\sqrt{b^2-4 a c}}}\right )}{2^{2/3} c^{2/3} \sqrt{b^2-4 a c}}\\ &=\frac{\left (b-\sqrt{b^2-4 a c}\right )^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3} c^{2/3} \sqrt{b^2-4 a c}}-\frac{\left (b+\sqrt{b^2-4 a c}\right )^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b+\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3} c^{2/3} \sqrt{b^2-4 a c}}+\frac{\left (b-\sqrt{b^2-4 a c}\right )^{2/3} \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt{b^2-4 a c}}-\frac{\left (b+\sqrt{b^2-4 a c}\right )^{2/3} \log \left (\sqrt [3]{b+\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt{b^2-4 a c}}-\frac{\left (b-\sqrt{b^2-4 a c}\right )^{2/3} \log \left (\left (b-\sqrt{b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} c^{2/3} \sqrt{b^2-4 a c}}+\frac{\left (b+\sqrt{b^2-4 a c}\right )^{2/3} \log \left (\left (b+\sqrt{b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} c^{2/3} \sqrt{b^2-4 a c}}\\ \end{align*}

Mathematica [C]  time = 0.0171399, size = 44, normalized size = 0.08 \[ \frac{1}{3} \text{RootSum}\left [\text{$\#$1}^3 b+\text{$\#$1}^6 c+a\& ,\frac{\text{$\#$1}^2 \log (x-\text{$\#$1})}{2 \text{$\#$1}^3 c+b}\& \right ] \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/(a + b*x^3 + c*x^6),x]

[Out]

RootSum[a + b*#1^3 + c*#1^6 & , (Log[x - #1]*#1^2)/(b + 2*c*#1^3) & ]/3

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Maple [C]  time = 0.003, size = 43, normalized size = 0.1 \begin{align*}{\frac{1}{3}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}c+{{\it \_Z}}^{3}b+a \right ) }{\frac{{{\it \_R}}^{4}\ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{5}c+{{\it \_R}}^{2}b}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(c*x^6+b*x^3+a),x)

[Out]

1/3*sum(_R^4/(2*_R^5*c+_R^2*b)*ln(x-_R),_R=RootOf(_Z^6*c+_Z^3*b+a))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{c x^{6} + b x^{3} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(c*x^6+b*x^3+a),x, algorithm="maxima")

[Out]

integrate(x^4/(c*x^6 + b*x^3 + a), x)

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Fricas [B]  time = 2.9361, size = 8007, normalized size = 14.35 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(c*x^6+b*x^3+a),x, algorithm="fricas")

[Out]

2/3*sqrt(3)*(1/2)^(1/3)*(-((b^2*c^2 - 4*a*c^3)*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(b^6*c^4 - 12*a*b^4*c^5 + 48
*a^2*b^2*c^6 - 64*a^3*c^7)) + b)/(b^2*c^2 - 4*a*c^3))^(1/3)*arctan(-1/3*((1/2)^(5/6)*(sqrt(3)*(b^5*c^2 - 8*a*b
^3*c^3 + 16*a^2*b*c^4)*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^
7)) - sqrt(3)*(b^4 - 6*a*b^2*c + 8*a^2*c^2))*sqrt((2*(a*b^2 - 2*a^2*c)*x^2 + (1/2)^(2/3)*((b^6*c^2 - 12*a*b^4*
c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*x*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c
^6 - 64*a^3*c^7)) - (b^5 - 6*a*b^3*c + 8*a^2*b*c^2)*x)*(-((b^2*c^2 - 4*a*c^3)*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^
2)/(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)) + b)/(b^2*c^2 - 4*a*c^3))^(2/3) - 2*(1/2)^(1/3)*(a*
b^4*c^2 - 8*a^2*b^2*c^3 + 16*a^3*c^4)*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*
c^6 - 64*a^3*c^7))*(-((b^2*c^2 - 4*a*c^3)*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*
b^2*c^6 - 64*a^3*c^7)) + b)/(b^2*c^2 - 4*a*c^3))^(1/3))/(a*b^2 - 2*a^2*c))*(-((b^2*c^2 - 4*a*c^3)*sqrt((b^4 -
4*a*b^2*c + 4*a^2*c^2)/(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)) + b)/(b^2*c^2 - 4*a*c^3))^(1/3)
- (1/2)^(1/3)*(sqrt(3)*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*x*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(b^6*c^4 -
12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)) - sqrt(3)*(b^4 - 6*a*b^2*c + 8*a^2*c^2)*x)*(-((b^2*c^2 - 4*a*c^3
)*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)) + b)/(b^2*c^2 - 4
*a*c^3))^(1/3) - sqrt(3)*(a*b^2 - 2*a^2*c))/(a*b^2 - 2*a^2*c)) - 2/3*sqrt(3)*(1/2)^(1/3)*(((b^2*c^2 - 4*a*c^3)
*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)) - b)/(b^2*c^2 - 4*
a*c^3))^(1/3)*arctan(-1/3*((1/2)^(5/6)*(sqrt(3)*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*sqrt((b^4 - 4*a*b^2*c +
4*a^2*c^2)/(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)) + sqrt(3)*(b^4 - 6*a*b^2*c + 8*a^2*c^2))*s
qrt((2*(a*b^2 - 2*a^2*c)*x^2 - (1/2)^(2/3)*((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*x*sqrt((b^4
- 4*a*b^2*c + 4*a^2*c^2)/(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)) + (b^5 - 6*a*b^3*c + 8*a^2*b
*c^2)*x)*(((b^2*c^2 - 4*a*c^3)*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 6
4*a^3*c^7)) - b)/(b^2*c^2 - 4*a*c^3))^(2/3) + 2*(1/2)^(1/3)*(a*b^4*c^2 - 8*a^2*b^2*c^3 + 16*a^3*c^4)*sqrt((b^4
- 4*a*b^2*c + 4*a^2*c^2)/(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7))*(((b^2*c^2 - 4*a*c^3)*sqrt((
b^4 - 4*a*b^2*c + 4*a^2*c^2)/(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)) - b)/(b^2*c^2 - 4*a*c^3))
^(1/3))/(a*b^2 - 2*a^2*c))*(((b^2*c^2 - 4*a*c^3)*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(b^6*c^4 - 12*a*b^4*c^5 +
48*a^2*b^2*c^6 - 64*a^3*c^7)) - b)/(b^2*c^2 - 4*a*c^3))^(1/3) - (1/2)^(1/3)*(sqrt(3)*(b^5*c^2 - 8*a*b^3*c^3 +
16*a^2*b*c^4)*x*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)) + s
qrt(3)*(b^4 - 6*a*b^2*c + 8*a^2*c^2)*x)*(((b^2*c^2 - 4*a*c^3)*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(b^6*c^4 - 12
*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)) - b)/(b^2*c^2 - 4*a*c^3))^(1/3) + sqrt(3)*(a*b^2 - 2*a^2*c))/(a*b^2
- 2*a^2*c)) - 1/6*(1/2)^(1/3)*(-((b^2*c^2 - 4*a*c^3)*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(b^6*c^4 - 12*a*b^4*c
^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)) + b)/(b^2*c^2 - 4*a*c^3))^(1/3)*log(-2*(a*b^2 - 2*a^2*c)*x^2 - (1/2)^(2/3)*
((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*x*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(b^6*c^4 - 12*a*b
^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)) - (b^5 - 6*a*b^3*c + 8*a^2*b*c^2)*x)*(-((b^2*c^2 - 4*a*c^3)*sqrt((b^4 -
4*a*b^2*c + 4*a^2*c^2)/(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)) + b)/(b^2*c^2 - 4*a*c^3))^(2/3
) + 2*(1/2)^(1/3)*(a*b^4*c^2 - 8*a^2*b^2*c^3 + 16*a^3*c^4)*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(b^6*c^4 - 12*a*
b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7))*(-((b^2*c^2 - 4*a*c^3)*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(b^6*c^4 - 1
2*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)) + b)/(b^2*c^2 - 4*a*c^3))^(1/3)) - 1/6*(1/2)^(1/3)*(((b^2*c^2 - 4*
a*c^3)*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)) - b)/(b^2*c^
2 - 4*a*c^3))^(1/3)*log(-2*(a*b^2 - 2*a^2*c)*x^2 + (1/2)^(2/3)*((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*
a^3*c^5)*x*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)) + (b^5 -
6*a*b^3*c + 8*a^2*b*c^2)*x)*(((b^2*c^2 - 4*a*c^3)*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(b^6*c^4 - 12*a*b^4*c^5
+ 48*a^2*b^2*c^6 - 64*a^3*c^7)) - b)/(b^2*c^2 - 4*a*c^3))^(2/3) - 2*(1/2)^(1/3)*(a*b^4*c^2 - 8*a^2*b^2*c^3 + 1
6*a^3*c^4)*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7))*(((b^2*c
^2 - 4*a*c^3)*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)) - b)/
(b^2*c^2 - 4*a*c^3))^(1/3)) + 1/3*(1/2)^(1/3)*(-((b^2*c^2 - 4*a*c^3)*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(b^6*c
^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)) + b)/(b^2*c^2 - 4*a*c^3))^(1/3)*log(-(1/2)^(2/3)*(b^5 - 6*a*
b^3*c + 8*a^2*b*c^2 - (b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2
)/(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)))*(-((b^2*c^2 - 4*a*c^3)*sqrt((b^4 - 4*a*b^2*c + 4*a^
2*c^2)/(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)) + b)/(b^2*c^2 - 4*a*c^3))^(2/3) - 2*(a*b^2 - 2*
a^2*c)*x) + 1/3*(1/2)^(1/3)*(((b^2*c^2 - 4*a*c^3)*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(b^6*c^4 - 12*a*b^4*c^5 +
48*a^2*b^2*c^6 - 64*a^3*c^7)) - b)/(b^2*c^2 - 4*a*c^3))^(1/3)*log(-(1/2)^(2/3)*(b^5 - 6*a*b^3*c + 8*a^2*b*c^2
+ (b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(b^6*c^4 - 12*a*b
^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)))*(((b^2*c^2 - 4*a*c^3)*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(b^6*c^4 - 12
*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)) - b)/(b^2*c^2 - 4*a*c^3))^(2/3) - 2*(a*b^2 - 2*a^2*c)*x)

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Sympy [A]  time = 1.75708, size = 175, normalized size = 0.31 \begin{align*} \operatorname{RootSum}{\left (t^{6} \left (46656 a^{3} c^{5} - 34992 a^{2} b^{2} c^{4} + 8748 a b^{4} c^{3} - 729 b^{6} c^{2}\right ) + t^{3} \left (- 432 a^{2} b c^{2} + 216 a b^{3} c - 27 b^{5}\right ) + a^{2}, \left ( t \mapsto t \log{\left (x + \frac{15552 t^{5} a^{3} c^{5} - 11664 t^{5} a^{2} b^{2} c^{4} + 2916 t^{5} a b^{4} c^{3} - 243 t^{5} b^{6} c^{2} - 108 t^{2} a^{2} b c^{2} + 63 t^{2} a b^{3} c - 9 t^{2} b^{5}}{2 a^{2} c - a b^{2}} \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(c*x**6+b*x**3+a),x)

[Out]

RootSum(_t**6*(46656*a**3*c**5 - 34992*a**2*b**2*c**4 + 8748*a*b**4*c**3 - 729*b**6*c**2) + _t**3*(-432*a**2*b
*c**2 + 216*a*b**3*c - 27*b**5) + a**2, Lambda(_t, _t*log(x + (15552*_t**5*a**3*c**5 - 11664*_t**5*a**2*b**2*c
**4 + 2916*_t**5*a*b**4*c**3 - 243*_t**5*b**6*c**2 - 108*_t**2*a**2*b*c**2 + 63*_t**2*a*b**3*c - 9*_t**2*b**5)
/(2*a**2*c - a*b**2))))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{c x^{6} + b x^{3} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(c*x^6+b*x^3+a),x, algorithm="giac")

[Out]

integrate(x^4/(c*x^6 + b*x^3 + a), x)