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

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

[Out]

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

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Rubi [A]  time = 0.573605, 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, 200, 31, 634, 617, 204, 628} \[ \frac{\sqrt [3]{b-\sqrt{b^2-4 a c}} \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{b-\sqrt{b^2-4 a c}}+\left (b-\sqrt{b^2-4 a c}\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{6 \sqrt [3]{2} \sqrt [3]{c} \sqrt{b^2-4 a c}}-\frac{\sqrt [3]{\sqrt{b^2-4 a c}+b} \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{\sqrt{b^2-4 a c}+b}+\left (\sqrt{b^2-4 a c}+b\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{6 \sqrt [3]{2} \sqrt [3]{c} \sqrt{b^2-4 a c}}-\frac{\sqrt [3]{b-\sqrt{b^2-4 a c}} \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt{b^2-4 a c}}+\frac{\sqrt [3]{\sqrt{b^2-4 a c}+b} \log \left (\sqrt [3]{\sqrt{b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt{b^2-4 a c}}+\frac{\sqrt [3]{b-\sqrt{b^2-4 a c}} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3} \sqrt [3]{c} \sqrt{b^2-4 a c}}-\frac{\sqrt [3]{\sqrt{b^2-4 a c}+b} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3} \sqrt [3]{c} \sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*x^3 + c*x^6),x]

[Out]

((b - Sqrt[b^2 - 4*a*c])^(1/3)*ArcTan[(1 - (2*2^(1/3)*c^(1/3)*x)/(b - Sqrt[b^2 - 4*a*c])^(1/3))/Sqrt[3]])/(2^(
1/3)*Sqrt[3]*c^(1/3)*Sqrt[b^2 - 4*a*c]) - ((b + Sqrt[b^2 - 4*a*c])^(1/3)*ArcTan[(1 - (2*2^(1/3)*c^(1/3)*x)/(b
+ Sqrt[b^2 - 4*a*c])^(1/3))/Sqrt[3]])/(2^(1/3)*Sqrt[3]*c^(1/3)*Sqrt[b^2 - 4*a*c]) - ((b - Sqrt[b^2 - 4*a*c])^(
1/3)*Log[(b - Sqrt[b^2 - 4*a*c])^(1/3) + 2^(1/3)*c^(1/3)*x])/(3*2^(1/3)*c^(1/3)*Sqrt[b^2 - 4*a*c]) + ((b + Sqr
t[b^2 - 4*a*c])^(1/3)*Log[(b + Sqrt[b^2 - 4*a*c])^(1/3) + 2^(1/3)*c^(1/3)*x])/(3*2^(1/3)*c^(1/3)*Sqrt[b^2 - 4*
a*c]) + ((b - Sqrt[b^2 - 4*a*c])^(1/3)*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^(1/3)*c^(1/3)*Sqrt[b^2 - 4*a*c]) - ((b + Sqrt[b^2 - 4*a*c])^(1/3)*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
^(1/3)*c^(1/3)*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 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*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^3}{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{1}{\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{1}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^3} \, dx\\ &=-\frac{\sqrt [3]{b-\sqrt{b^2-4 a c}} \int \frac{1}{\frac{\sqrt [3]{b-\sqrt{b^2-4 a c}}}{\sqrt [3]{2}}+\sqrt [3]{c} x} \, dx}{3 \sqrt [3]{2} \sqrt{b^2-4 a c}}-\frac{\sqrt [3]{b-\sqrt{b^2-4 a c}} \int \frac{2^{2/3} \sqrt [3]{b-\sqrt{b^2-4 a c}}-\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 \sqrt [3]{2} \sqrt{b^2-4 a c}}+\frac{\sqrt [3]{b+\sqrt{b^2-4 a c}} \int \frac{1}{\frac{\sqrt [3]{b+\sqrt{b^2-4 a c}}}{\sqrt [3]{2}}+\sqrt [3]{c} x} \, dx}{3 \sqrt [3]{2} \sqrt{b^2-4 a c}}+\frac{\sqrt [3]{b+\sqrt{b^2-4 a c}} \int \frac{2^{2/3} \sqrt [3]{b+\sqrt{b^2-4 a c}}-\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 \sqrt [3]{2} \sqrt{b^2-4 a c}}\\ &=-\frac{\sqrt [3]{b-\sqrt{b^2-4 a c}} \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt{b^2-4 a c}}+\frac{\sqrt [3]{b+\sqrt{b^2-4 a c}} \log \left (\sqrt [3]{b+\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt{b^2-4 a c}}+\frac{\sqrt [3]{b-\sqrt{b^2-4 a c}} \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 \sqrt [3]{2} \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{\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}{2\ 2^{2/3} \sqrt{b^2-4 a c}}-\frac{\sqrt [3]{b+\sqrt{b^2-4 a c}} \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 \sqrt [3]{2} \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{\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}{2\ 2^{2/3} \sqrt{b^2-4 a c}}\\ &=-\frac{\sqrt [3]{b-\sqrt{b^2-4 a c}} \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt{b^2-4 a c}}+\frac{\sqrt [3]{b+\sqrt{b^2-4 a c}} \log \left (\sqrt [3]{b+\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt{b^2-4 a c}}+\frac{\sqrt [3]{b-\sqrt{b^2-4 a c}} \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 \sqrt [3]{2} \sqrt [3]{c} \sqrt{b^2-4 a c}}-\frac{\sqrt [3]{b+\sqrt{b^2-4 a c}} \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 \sqrt [3]{2} \sqrt [3]{c} \sqrt{b^2-4 a c}}-\frac{\sqrt [3]{b-\sqrt{b^2-4 a c}} \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 )}{\sqrt [3]{2} \sqrt [3]{c} \sqrt{b^2-4 a c}}+\frac{\sqrt [3]{b+\sqrt{b^2-4 a c}} \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 )}{\sqrt [3]{2} \sqrt [3]{c} \sqrt{b^2-4 a c}}\\ &=\frac{\sqrt [3]{b-\sqrt{b^2-4 a c}} \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 )}{\sqrt [3]{2} \sqrt{3} \sqrt [3]{c} \sqrt{b^2-4 a c}}-\frac{\sqrt [3]{b+\sqrt{b^2-4 a c}} \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 )}{\sqrt [3]{2} \sqrt{3} \sqrt [3]{c} \sqrt{b^2-4 a c}}-\frac{\sqrt [3]{b-\sqrt{b^2-4 a c}} \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt{b^2-4 a c}}+\frac{\sqrt [3]{b+\sqrt{b^2-4 a c}} \log \left (\sqrt [3]{b+\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt{b^2-4 a c}}+\frac{\sqrt [3]{b-\sqrt{b^2-4 a c}} \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 \sqrt [3]{2} \sqrt [3]{c} \sqrt{b^2-4 a c}}-\frac{\sqrt [3]{b+\sqrt{b^2-4 a c}} \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 \sqrt [3]{2} \sqrt [3]{c} \sqrt{b^2-4 a c}}\\ \end{align*}

Mathematica [C]  time = 0.0160134, size = 42, normalized size = 0.08 \[ \frac{1}{3} \text{RootSum}\left [\text{$\#$1}^3 b+\text{$\#$1}^6 c+a\& ,\frac{\text{$\#$1} \log (x-\text{$\#$1})}{2 \text{$\#$1}^3 c+b}\& \right ] \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b*x^3 + c*x^6),x]

[Out]

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

[Out]

1/3*sum(_R^3/(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^{3}}{c x^{6} + b x^{3} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^3/(c*x^6 + b*x^3 + a), x)

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Fricas [B]  time = 1.99071, size = 5501, normalized size = 9.86 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.8966, size = 122, normalized size = 0.22 \begin{align*} \operatorname{RootSum}{\left (t^{6} \left (46656 a^{3} c^{4} - 34992 a^{2} b^{2} c^{3} + 8748 a b^{4} c^{2} - 729 b^{6} c\right ) + t^{3} \left (432 a^{2} c^{2} - 216 a b^{2} c + 27 b^{4}\right ) + a, \left ( t \mapsto t \log{\left (x + \frac{2592 t^{4} a^{2} c^{3} - 1296 t^{4} a b^{2} c^{2} + 162 t^{4} b^{4} c + 12 t a c - 3 t b^{2}}{b} \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(_t**6*(46656*a**3*c**4 - 34992*a**2*b**2*c**3 + 8748*a*b**4*c**2 - 729*b**6*c) + _t**3*(432*a**2*c**2
- 216*a*b**2*c + 27*b**4) + a, Lambda(_t, _t*log(x + (2592*_t**4*a**2*c**3 - 1296*_t**4*a*b**2*c**2 + 162*_t**
4*b**4*c + 12*_t*a*c - 3*_t*b**2)/b)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^3/(c*x^6 + b*x^3 + a), x)