3.11 \(\int \frac{(a+b x^3)^2}{c+d x^3} \, dx\)

Optimal. Leaf size=173 \[ -\frac{(b c-a d)^2 \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{2/3} d^{7/3}}+\frac{(b c-a d)^2 \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} d^{7/3}}-\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{2/3} d^{7/3}}-\frac{b x (b c-2 a d)}{d^2}+\frac{b^2 x^4}{4 d} \]

[Out]

-((b*(b*c - 2*a*d)*x)/d^2) + (b^2*x^4)/(4*d) - ((b*c - a*d)^2*ArcTan[(c^(1/3) - 2*d^(1/3)*x)/(Sqrt[3]*c^(1/3))
])/(Sqrt[3]*c^(2/3)*d^(7/3)) + ((b*c - a*d)^2*Log[c^(1/3) + d^(1/3)*x])/(3*c^(2/3)*d^(7/3)) - ((b*c - a*d)^2*L
og[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/(6*c^(2/3)*d^(7/3))

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Rubi [A]  time = 0.128459, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {390, 200, 31, 634, 617, 204, 628} \[ -\frac{(b c-a d)^2 \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{2/3} d^{7/3}}+\frac{(b c-a d)^2 \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} d^{7/3}}-\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{2/3} d^{7/3}}-\frac{b x (b c-2 a d)}{d^2}+\frac{b^2 x^4}{4 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^2/(c + d*x^3),x]

[Out]

-((b*(b*c - 2*a*d)*x)/d^2) + (b^2*x^4)/(4*d) - ((b*c - a*d)^2*ArcTan[(c^(1/3) - 2*d^(1/3)*x)/(Sqrt[3]*c^(1/3))
])/(Sqrt[3]*c^(2/3)*d^(7/3)) + ((b*c - a*d)^2*Log[c^(1/3) + d^(1/3)*x])/(3*c^(2/3)*d^(7/3)) - ((b*c - a*d)^2*L
og[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/(6*c^(2/3)*d^(7/3))

Rule 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

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{\left (a+b x^3\right )^2}{c+d x^3} \, dx &=\int \left (-\frac{b (b c-2 a d)}{d^2}+\frac{b^2 x^3}{d}+\frac{b^2 c^2-2 a b c d+a^2 d^2}{d^2 \left (c+d x^3\right )}\right ) \, dx\\ &=-\frac{b (b c-2 a d) x}{d^2}+\frac{b^2 x^4}{4 d}+\frac{(b c-a d)^2 \int \frac{1}{c+d x^3} \, dx}{d^2}\\ &=-\frac{b (b c-2 a d) x}{d^2}+\frac{b^2 x^4}{4 d}+\frac{(b c-a d)^2 \int \frac{1}{\sqrt [3]{c}+\sqrt [3]{d} x} \, dx}{3 c^{2/3} d^2}+\frac{(b c-a d)^2 \int \frac{2 \sqrt [3]{c}-\sqrt [3]{d} x}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{3 c^{2/3} d^2}\\ &=-\frac{b (b c-2 a d) x}{d^2}+\frac{b^2 x^4}{4 d}+\frac{(b c-a d)^2 \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} d^{7/3}}-\frac{(b c-a d)^2 \int \frac{-\sqrt [3]{c} \sqrt [3]{d}+2 d^{2/3} x}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{6 c^{2/3} d^{7/3}}+\frac{(b c-a d)^2 \int \frac{1}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{2 \sqrt [3]{c} d^2}\\ &=-\frac{b (b c-2 a d) x}{d^2}+\frac{b^2 x^4}{4 d}+\frac{(b c-a d)^2 \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} d^{7/3}}-\frac{(b c-a d)^2 \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{2/3} d^{7/3}}+\frac{(b c-a d)^2 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}\right )}{c^{2/3} d^{7/3}}\\ &=-\frac{b (b c-2 a d) x}{d^2}+\frac{b^2 x^4}{4 d}-\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{2/3} d^{7/3}}+\frac{(b c-a d)^2 \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} d^{7/3}}-\frac{(b c-a d)^2 \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{2/3} d^{7/3}}\\ \end{align*}

Mathematica [A]  time = 0.0908675, size = 167, normalized size = 0.97 \[ \frac{-2 (b c-a d)^2 \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )-12 b c^{2/3} \sqrt [3]{d} x (b c-2 a d)+4 (b c-a d)^2 \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )+4 \sqrt{3} (b c-a d)^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{d} x-\sqrt [3]{c}}{\sqrt{3} \sqrt [3]{c}}\right )+3 b^2 c^{2/3} d^{4/3} x^4}{12 c^{2/3} d^{7/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^2/(c + d*x^3),x]

[Out]

(-12*b*c^(2/3)*d^(1/3)*(b*c - 2*a*d)*x + 3*b^2*c^(2/3)*d^(4/3)*x^4 + 4*Sqrt[3]*(b*c - a*d)^2*ArcTan[(-c^(1/3)
+ 2*d^(1/3)*x)/(Sqrt[3]*c^(1/3))] + 4*(b*c - a*d)^2*Log[c^(1/3) + d^(1/3)*x] - 2*(b*c - a*d)^2*Log[c^(2/3) - c
^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/(12*c^(2/3)*d^(7/3))

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Maple [B]  time = 0.003, size = 334, normalized size = 1.9 \begin{align*}{\frac{{b}^{2}{x}^{4}}{4\,d}}+2\,{\frac{xab}{d}}-{\frac{{b}^{2}xc}{{d}^{2}}}+{\frac{{a}^{2}}{3\,d}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{2\,abc}{3\,{d}^{2}}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{{b}^{2}{c}^{2}}{3\,{d}^{3}}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{{a}^{2}}{6\,d}\ln \left ({x}^{2}-\sqrt [3]{{\frac{c}{d}}}x+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{abc}{3\,{d}^{2}}\ln \left ({x}^{2}-\sqrt [3]{{\frac{c}{d}}}x+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{{b}^{2}{c}^{2}}{6\,{d}^{3}}\ln \left ({x}^{2}-\sqrt [3]{{\frac{c}{d}}}x+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}{a}^{2}}{3\,d}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{2\,\sqrt{3}cab}{3\,{d}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}{b}^{2}{c}^{2}}{3\,{d}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^2/(d*x^3+c),x)

[Out]

1/4*b^2*x^4/d+2*b/d*a*x-b^2/d^2*x*c+1/3/d/(c/d)^(2/3)*ln(x+(c/d)^(1/3))*a^2-2/3/d^2/(c/d)^(2/3)*ln(x+(c/d)^(1/
3))*c*a*b+1/3/d^3/(c/d)^(2/3)*ln(x+(c/d)^(1/3))*b^2*c^2-1/6/d/(c/d)^(2/3)*ln(x^2-(c/d)^(1/3)*x+(c/d)^(2/3))*a^
2+1/3/d^2/(c/d)^(2/3)*ln(x^2-(c/d)^(1/3)*x+(c/d)^(2/3))*c*a*b-1/6/d^3/(c/d)^(2/3)*ln(x^2-(c/d)^(1/3)*x+(c/d)^(
2/3))*b^2*c^2+1/3/d/(c/d)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(c/d)^(1/3)*x-1))*a^2-2/3/d^2/(c/d)^(2/3)*3^(1/2
)*arctan(1/3*3^(1/2)*(2/(c/d)^(1/3)*x-1))*c*a*b+1/3/d^3/(c/d)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(c/d)^(1/3)*
x-1))*b^2*c^2

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2/(d*x^3+c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.68486, size = 1183, normalized size = 6.84 \begin{align*} \left [\frac{3 \, b^{2} c^{2} d^{2} x^{4} + 6 \, \sqrt{\frac{1}{3}}{\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} \sqrt{-\frac{\left (c^{2} d\right )^{\frac{1}{3}}}{d}} \log \left (\frac{2 \, c d x^{3} - 3 \, \left (c^{2} d\right )^{\frac{1}{3}} c x - c^{2} + 3 \, \sqrt{\frac{1}{3}}{\left (2 \, c d x^{2} + \left (c^{2} d\right )^{\frac{2}{3}} x - \left (c^{2} d\right )^{\frac{1}{3}} c\right )} \sqrt{-\frac{\left (c^{2} d\right )^{\frac{1}{3}}}{d}}}{d x^{3} + c}\right ) - 2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \left (c^{2} d\right )^{\frac{2}{3}} \log \left (c d x^{2} - \left (c^{2} d\right )^{\frac{2}{3}} x + \left (c^{2} d\right )^{\frac{1}{3}} c\right ) + 4 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \left (c^{2} d\right )^{\frac{2}{3}} \log \left (c d x + \left (c^{2} d\right )^{\frac{2}{3}}\right ) - 12 \,{\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2}\right )} x}{12 \, c^{2} d^{3}}, \frac{3 \, b^{2} c^{2} d^{2} x^{4} + 12 \, \sqrt{\frac{1}{3}}{\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} \sqrt{\frac{\left (c^{2} d\right )^{\frac{1}{3}}}{d}} \arctan \left (\frac{\sqrt{\frac{1}{3}}{\left (2 \, \left (c^{2} d\right )^{\frac{2}{3}} x - \left (c^{2} d\right )^{\frac{1}{3}} c\right )} \sqrt{\frac{\left (c^{2} d\right )^{\frac{1}{3}}}{d}}}{c^{2}}\right ) - 2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \left (c^{2} d\right )^{\frac{2}{3}} \log \left (c d x^{2} - \left (c^{2} d\right )^{\frac{2}{3}} x + \left (c^{2} d\right )^{\frac{1}{3}} c\right ) + 4 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \left (c^{2} d\right )^{\frac{2}{3}} \log \left (c d x + \left (c^{2} d\right )^{\frac{2}{3}}\right ) - 12 \,{\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2}\right )} x}{12 \, c^{2} d^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2/(d*x^3+c),x, algorithm="fricas")

[Out]

[1/12*(3*b^2*c^2*d^2*x^4 + 6*sqrt(1/3)*(b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*sqrt(-(c^2*d)^(1/3)/d)*log((2*c
*d*x^3 - 3*(c^2*d)^(1/3)*c*x - c^2 + 3*sqrt(1/3)*(2*c*d*x^2 + (c^2*d)^(2/3)*x - (c^2*d)^(1/3)*c)*sqrt(-(c^2*d)
^(1/3)/d))/(d*x^3 + c)) - 2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*(c^2*d)^(2/3)*log(c*d*x^2 - (c^2*d)^(2/3)*x + (c^2
*d)^(1/3)*c) + 4*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*(c^2*d)^(2/3)*log(c*d*x + (c^2*d)^(2/3)) - 12*(b^2*c^3*d - 2*
a*b*c^2*d^2)*x)/(c^2*d^3), 1/12*(3*b^2*c^2*d^2*x^4 + 12*sqrt(1/3)*(b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*sqrt
((c^2*d)^(1/3)/d)*arctan(sqrt(1/3)*(2*(c^2*d)^(2/3)*x - (c^2*d)^(1/3)*c)*sqrt((c^2*d)^(1/3)/d)/c^2) - 2*(b^2*c
^2 - 2*a*b*c*d + a^2*d^2)*(c^2*d)^(2/3)*log(c*d*x^2 - (c^2*d)^(2/3)*x + (c^2*d)^(1/3)*c) + 4*(b^2*c^2 - 2*a*b*
c*d + a^2*d^2)*(c^2*d)^(2/3)*log(c*d*x + (c^2*d)^(2/3)) - 12*(b^2*c^3*d - 2*a*b*c^2*d^2)*x)/(c^2*d^3)]

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Sympy [A]  time = 1.30891, size = 156, normalized size = 0.9 \begin{align*} \frac{b^{2} x^{4}}{4 d} + \operatorname{RootSum}{\left (27 t^{3} c^{2} d^{7} - a^{6} d^{6} + 6 a^{5} b c d^{5} - 15 a^{4} b^{2} c^{2} d^{4} + 20 a^{3} b^{3} c^{3} d^{3} - 15 a^{2} b^{4} c^{4} d^{2} + 6 a b^{5} c^{5} d - b^{6} c^{6}, \left ( t \mapsto t \log{\left (\frac{3 t c d^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )} \right )\right )} + \frac{x \left (2 a b d - b^{2} c\right )}{d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**2/(d*x**3+c),x)

[Out]

b**2*x**4/(4*d) + RootSum(27*_t**3*c**2*d**7 - a**6*d**6 + 6*a**5*b*c*d**5 - 15*a**4*b**2*c**2*d**4 + 20*a**3*
b**3*c**3*d**3 - 15*a**2*b**4*c**4*d**2 + 6*a*b**5*c**5*d - b**6*c**6, Lambda(_t, _t*log(3*_t*c*d**2/(a**2*d**
2 - 2*a*b*c*d + b**2*c**2) + x))) + x*(2*a*b*d - b**2*c)/d**2

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Giac [A]  time = 1.24444, size = 336, normalized size = 1.94 \begin{align*} \frac{\sqrt{3}{\left (\left (-c d^{2}\right )^{\frac{1}{3}} b^{2} c^{2} - 2 \, \left (-c d^{2}\right )^{\frac{1}{3}} a b c d + \left (-c d^{2}\right )^{\frac{1}{3}} a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{c}{d}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{c}{d}\right )^{\frac{1}{3}}}\right )}{3 \, c d^{3}} + \frac{{\left (\left (-c d^{2}\right )^{\frac{1}{3}} b^{2} c^{2} - 2 \, \left (-c d^{2}\right )^{\frac{1}{3}} a b c d + \left (-c d^{2}\right )^{\frac{1}{3}} a^{2} d^{2}\right )} \log \left (x^{2} + x \left (-\frac{c}{d}\right )^{\frac{1}{3}} + \left (-\frac{c}{d}\right )^{\frac{2}{3}}\right )}{6 \, c d^{3}} - \frac{{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} \left (-\frac{c}{d}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{c}{d}\right )^{\frac{1}{3}} \right |}\right )}{3 \, c d^{4}} + \frac{b^{2} d^{3} x^{4} - 4 \, b^{2} c d^{2} x + 8 \, a b d^{3} x}{4 \, d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2/(d*x^3+c),x, algorithm="giac")

[Out]

1/3*sqrt(3)*((-c*d^2)^(1/3)*b^2*c^2 - 2*(-c*d^2)^(1/3)*a*b*c*d + (-c*d^2)^(1/3)*a^2*d^2)*arctan(1/3*sqrt(3)*(2
*x + (-c/d)^(1/3))/(-c/d)^(1/3))/(c*d^3) + 1/6*((-c*d^2)^(1/3)*b^2*c^2 - 2*(-c*d^2)^(1/3)*a*b*c*d + (-c*d^2)^(
1/3)*a^2*d^2)*log(x^2 + x*(-c/d)^(1/3) + (-c/d)^(2/3))/(c*d^3) - 1/3*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(-c
/d)^(1/3)*log(abs(x - (-c/d)^(1/3)))/(c*d^4) + 1/4*(b^2*d^3*x^4 - 4*b^2*c*d^2*x + 8*a*b*d^3*x)/d^4