3.138 \(\int \frac{x^2}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx\)

Optimal. Leaf size=334 \[ \frac{2 (-1)^{2/3} \tan ^{-1}\left (\frac{3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x}{\sqrt{3} \sqrt{a} \sqrt{4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}\right )}{9 \sqrt{3} \left (1+\sqrt [3]{-1}\right )^2 a^{11/6} c^{2/3} \sqrt{4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}+\frac{2 \tan ^{-1}\left (\frac{3 a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt{3} \sqrt{a} \sqrt{4 b-3 \sqrt [3]{a} c^{2/3}}}\right )}{27 \sqrt{3} a^{11/6} c^{2/3} \sqrt{4 b-3 \sqrt [3]{a} c^{2/3}}}+\frac{2 (-1)^{2/3} \tan ^{-1}\left (\frac{3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt{3} \sqrt{a} \sqrt{3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}\right )}{9 \sqrt{3} \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{11/6} c^{2/3} \sqrt{3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}} \]

[Out]

(2*(-1)^(2/3)*ArcTan[(3*(-1)^(1/3)*a^(2/3)*c^(1/3) - 2*b*x)/(Sqrt[3]*Sqrt[a]*Sqrt[4*b - 3*(-1)^(2/3)*a^(1/3)*c
^(2/3)])])/(9*Sqrt[3]*(1 + (-1)^(1/3))^2*a^(11/6)*Sqrt[4*b - 3*(-1)^(2/3)*a^(1/3)*c^(2/3)]*c^(2/3)) + (2*ArcTa
n[(3*a^(2/3)*c^(1/3) + 2*b*x)/(Sqrt[3]*Sqrt[a]*Sqrt[4*b - 3*a^(1/3)*c^(2/3)])])/(27*Sqrt[3]*a^(11/6)*Sqrt[4*b
- 3*a^(1/3)*c^(2/3)]*c^(2/3)) + (2*(-1)^(2/3)*ArcTan[(3*(-1)^(2/3)*a^(2/3)*c^(1/3) + 2*b*x)/(Sqrt[3]*Sqrt[a]*S
qrt[4*b + 3*(-1)^(1/3)*a^(1/3)*c^(2/3)])])/(9*Sqrt[3]*(1 - (-1)^(1/3))*(1 + (-1)^(1/3))^2*a^(11/6)*Sqrt[4*b +
3*(-1)^(1/3)*a^(1/3)*c^(2/3)]*c^(2/3))

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Rubi [A]  time = 0.469659, antiderivative size = 334, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {2097, 618, 204} \[ \frac{2 (-1)^{2/3} \tan ^{-1}\left (\frac{3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x}{\sqrt{3} \sqrt{a} \sqrt{4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}\right )}{9 \sqrt{3} \left (1+\sqrt [3]{-1}\right )^2 a^{11/6} c^{2/3} \sqrt{4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}+\frac{2 \tan ^{-1}\left (\frac{3 a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt{3} \sqrt{a} \sqrt{4 b-3 \sqrt [3]{a} c^{2/3}}}\right )}{27 \sqrt{3} a^{11/6} c^{2/3} \sqrt{4 b-3 \sqrt [3]{a} c^{2/3}}}+\frac{2 (-1)^{2/3} \tan ^{-1}\left (\frac{3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt{3} \sqrt{a} \sqrt{3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}\right )}{9 \sqrt{3} \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{11/6} c^{2/3} \sqrt{3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(27*a^3 + 27*a^2*b*x^2 + 27*a^2*c*x^3 + 9*a*b^2*x^4 + b^3*x^6),x]

[Out]

(2*(-1)^(2/3)*ArcTan[(3*(-1)^(1/3)*a^(2/3)*c^(1/3) - 2*b*x)/(Sqrt[3]*Sqrt[a]*Sqrt[4*b - 3*(-1)^(2/3)*a^(1/3)*c
^(2/3)])])/(9*Sqrt[3]*(1 + (-1)^(1/3))^2*a^(11/6)*Sqrt[4*b - 3*(-1)^(2/3)*a^(1/3)*c^(2/3)]*c^(2/3)) + (2*ArcTa
n[(3*a^(2/3)*c^(1/3) + 2*b*x)/(Sqrt[3]*Sqrt[a]*Sqrt[4*b - 3*a^(1/3)*c^(2/3)])])/(27*Sqrt[3]*a^(11/6)*Sqrt[4*b
- 3*a^(1/3)*c^(2/3)]*c^(2/3)) + (2*(-1)^(2/3)*ArcTan[(3*(-1)^(2/3)*a^(2/3)*c^(1/3) + 2*b*x)/(Sqrt[3]*Sqrt[a]*S
qrt[4*b + 3*(-1)^(1/3)*a^(1/3)*c^(2/3)])])/(9*Sqrt[3]*(1 - (-1)^(1/3))*(1 + (-1)^(1/3))^2*a^(11/6)*Sqrt[4*b +
3*(-1)^(1/3)*a^(1/3)*c^(2/3)]*c^(2/3))

Rule 2097

Int[(Q6_)^(p_)*(u_), x_Symbol] :> With[{a = Coeff[Q6, x, 0], b = Coeff[Q6, x, 2], c = Coeff[Q6, x, 3], d = Coe
ff[Q6, x, 4], e = Coeff[Q6, x, 6]}, Dist[1/(3^(3*p)*a^(2*p)), Int[ExpandIntegrand[u*(3*a + 3*Rt[a, 3]^2*Rt[c,
3]*x + b*x^2)^p*(3*a - 3*(-1)^(1/3)*Rt[a, 3]^2*Rt[c, 3]*x + b*x^2)^p*(3*a + 3*(-1)^(2/3)*Rt[a, 3]^2*Rt[c, 3]*x
+ b*x^2)^p, x], x], x] /; EqQ[b^2 - 3*a*d, 0] && EqQ[b^3 - 27*a^2*e, 0]] /; ILtQ[p, 0] && PolyQ[Q6, x, 6] &&
EqQ[Coeff[Q6, x, 1], 0] && EqQ[Coeff[Q6, x, 5], 0] && RationalFunctionQ[u, x]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; 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^2}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx &=\left (19683 a^6\right ) \int \left (\frac{(-1)^{2/3}}{177147 \left (1+\sqrt [3]{-1}\right )^2 a^{22/3} c^{2/3} \left (-3 a+3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x-b x^2\right )}+\frac{1}{531441 a^{22/3} c^{2/3} \left (3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2\right )}-\frac{(-1)^{2/3}}{177147 \left (-1+\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{22/3} c^{2/3} \left (3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{1}{3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{27 a^{4/3} c^{2/3}}+\frac{(-1)^{2/3} \int \frac{1}{3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{27 a^{4/3} c^{2/3}}+\frac{(-1)^{2/3} \int \frac{1}{-3 a+3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x-b x^2} \, dx}{9 \left (1+\sqrt [3]{-1}\right )^2 a^{4/3} c^{2/3}}\\ &=-\frac{2 \operatorname{Subst}\left (\int \frac{1}{-3 a \left (4 b-3 \sqrt [3]{a} c^{2/3}\right )-x^2} \, dx,x,3 a^{2/3} \sqrt [3]{c}+2 b x\right )}{27 a^{4/3} c^{2/3}}-\frac{\left (2 (-1)^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{-3 a \left (4 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}\right )-x^2} \, dx,x,3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x\right )}{27 a^{4/3} c^{2/3}}-\frac{\left (2 (-1)^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{-3 a \left (4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}\right )-x^2} \, dx,x,3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x\right )}{9 \left (1+\sqrt [3]{-1}\right )^2 a^{4/3} c^{2/3}}\\ &=\frac{2 (-1)^{2/3} \tan ^{-1}\left (\frac{3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x}{\sqrt{3} \sqrt{a} \sqrt{4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}\right )}{9 \sqrt{3} \left (1+\sqrt [3]{-1}\right )^2 a^{11/6} \sqrt{4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}} c^{2/3}}+\frac{2 \tan ^{-1}\left (\frac{3 a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt{3} \sqrt{a} \sqrt{4 b-3 \sqrt [3]{a} c^{2/3}}}\right )}{27 \sqrt{3} a^{11/6} \sqrt{4 b-3 \sqrt [3]{a} c^{2/3}} c^{2/3}}+\frac{2 (-1)^{2/3} \tan ^{-1}\left (\frac{3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt{3} \sqrt{a} \sqrt{4 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}}}\right )}{27 \sqrt{3} a^{11/6} \sqrt{4 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}} c^{2/3}}\\ \end{align*}

Mathematica [C]  time = 0.0413876, size = 97, normalized size = 0.29 \[ \frac{1}{3} \text{RootSum}\left [27 \text{$\#$1}^2 a^2 b+27 \text{$\#$1}^3 a^2 c+9 \text{$\#$1}^4 a b^2+\text{$\#$1}^6 b^3+27 a^3\& ,\frac{\text{$\#$1} \log (x-\text{$\#$1})}{12 \text{$\#$1}^2 a b^2+2 \text{$\#$1}^4 b^3+27 \text{$\#$1} a^2 c+18 a^2 b}\& \right ] \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(27*a^3 + 27*a^2*b*x^2 + 27*a^2*c*x^3 + 9*a*b^2*x^4 + b^3*x^6),x]

[Out]

RootSum[27*a^3 + 27*a^2*b*#1^2 + 27*a^2*c*#1^3 + 9*a*b^2*#1^4 + b^3*#1^6 & , (Log[x - #1]*#1)/(18*a^2*b + 27*a
^2*c*#1 + 12*a*b^2*#1^2 + 2*b^3*#1^4) & ]/3

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Maple [C]  time = 0.003, size = 93, normalized size = 0.3 \begin{align*}{\frac{1}{3}\sum _{{\it \_R}={\it RootOf} \left ({b}^{3}{{\it \_Z}}^{6}+9\,a{b}^{2}{{\it \_Z}}^{4}+27\,{a}^{2}c{{\it \_Z}}^{3}+27\,{a}^{2}b{{\it \_Z}}^{2}+27\,{a}^{3} \right ) }{\frac{{{\it \_R}}^{2}\ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{5}{b}^{3}+12\,{{\it \_R}}^{3}a{b}^{2}+27\,{{\it \_R}}^{2}{a}^{2}c+18\,{\it \_R}\,{a}^{2}b}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(b^3*x^6+9*a*b^2*x^4+27*a^2*c*x^3+27*a^2*b*x^2+27*a^3),x)

[Out]

1/3*sum(_R^2/(2*_R^5*b^3+12*_R^3*a*b^2+27*_R^2*a^2*c+18*_R*a^2*b)*ln(x-_R),_R=RootOf(_Z^6*b^3+9*_Z^4*a*b^2+27*
_Z^3*a^2*c+27*_Z^2*a^2*b+27*a^3))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{b^{3} x^{6} + 9 \, a b^{2} x^{4} + 27 \, a^{2} c x^{3} + 27 \, a^{2} b x^{2} + 27 \, a^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(b^3*x^6+9*a*b^2*x^4+27*a^2*c*x^3+27*a^2*b*x^2+27*a^3),x, algorithm="maxima")

[Out]

integrate(x^2/(b^3*x^6 + 9*a*b^2*x^4 + 27*a^2*c*x^3 + 27*a^2*b*x^2 + 27*a^3), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(b^3*x^6+9*a*b^2*x^4+27*a^2*c*x^3+27*a^2*b*x^2+27*a^3),x, algorithm="fricas")

[Out]

Timed out

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Sympy [A]  time = 28.1814, size = 167, normalized size = 0.5 \begin{align*} \operatorname{RootSum}{\left (t^{6} \left (282429536481 a^{12} c^{6} - 669462604992 a^{11} b^{3} c^{4}\right ) - 129140163 t^{4} a^{8} c^{4} + 19683 t^{2} a^{4} c^{2} - 1, \left ( t \mapsto t \log{\left (x + \frac{62762119218 t^{5} a^{11} c^{6} - 148769467776 t^{5} a^{10} b^{3} c^{4} - 387420489 t^{4} a^{9} c^{5} + 918330048 t^{4} a^{8} b^{3} c^{3} - 23914845 t^{3} a^{7} c^{4} - 11337408 t^{3} a^{6} b^{3} c^{2} + 177147 t^{2} a^{5} c^{3} + 2187 t a^{3} c^{2} - 18 a c}{8 b^{2}} \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(b**3*x**6+9*a*b**2*x**4+27*a**2*c*x**3+27*a**2*b*x**2+27*a**3),x)

[Out]

RootSum(_t**6*(282429536481*a**12*c**6 - 669462604992*a**11*b**3*c**4) - 129140163*_t**4*a**8*c**4 + 19683*_t*
*2*a**4*c**2 - 1, Lambda(_t, _t*log(x + (62762119218*_t**5*a**11*c**6 - 148769467776*_t**5*a**10*b**3*c**4 - 3
87420489*_t**4*a**9*c**5 + 918330048*_t**4*a**8*b**3*c**3 - 23914845*_t**3*a**7*c**4 - 11337408*_t**3*a**6*b**
3*c**2 + 177147*_t**2*a**5*c**3 + 2187*_t*a**3*c**2 - 18*a*c)/(8*b**2))))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{b^{3} x^{6} + 9 \, a b^{2} x^{4} + 27 \, a^{2} c x^{3} + 27 \, a^{2} b x^{2} + 27 \, a^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(b^3*x^6+9*a*b^2*x^4+27*a^2*c*x^3+27*a^2*b*x^2+27*a^3),x, algorithm="giac")

[Out]

integrate(x^2/(b^3*x^6 + 9*a*b^2*x^4 + 27*a^2*c*x^3 + 27*a^2*b*x^2 + 27*a^3), x)