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

Optimal. Leaf size=282 \[ -\frac{\left (b \sqrt [3]{c} \left (b^3 c-4 a^3 d\right )-\sqrt [3]{d} \left (4 a b^3 c-a^4 d\right )\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{2/3} d^{5/3}}+\frac{\left (b \sqrt [3]{c} \left (b^3 c-4 a^3 d\right )-\sqrt [3]{d} \left (4 a b^3 c-a^4 d\right )\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} d^{5/3}}+\frac{\left (-4 a^3 b \sqrt [3]{c} d+a^4 \left (-d^{4/3}\right )+4 a b^3 c \sqrt [3]{d}+b^4 c^{4/3}\right ) \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{2/3} d^{5/3}}+\frac{2 a^2 b^2 \log \left (c+d x^3\right )}{d}+\frac{4 a b^3 x}{d}+\frac{b^4 x^2}{2 d} \]

[Out]

(4*a*b^3*x)/d + (b^4*x^2)/(2*d) + ((b^4*c^(4/3) + 4*a*b^3*c*d^(1/3) - 4*a^3*b*c^(1/3)*d - a^4*d^(4/3))*ArcTan[
(c^(1/3) - 2*d^(1/3)*x)/(Sqrt[3]*c^(1/3))])/(Sqrt[3]*c^(2/3)*d^(5/3)) + ((b*c^(1/3)*(b^3*c - 4*a^3*d) - d^(1/3
)*(4*a*b^3*c - a^4*d))*Log[c^(1/3) + d^(1/3)*x])/(3*c^(2/3)*d^(5/3)) - ((b*c^(1/3)*(b^3*c - 4*a^3*d) - d^(1/3)
*(4*a*b^3*c - a^4*d))*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/(6*c^(2/3)*d^(5/3)) + (2*a^2*b^2*Log[c +
d*x^3])/d

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Rubi [A]  time = 0.443201, antiderivative size = 280, normalized size of antiderivative = 0.99, number of steps used = 10, number of rules used = 9, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.529, Rules used = {1887, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ \frac{\left (-\frac{b \sqrt [3]{c} \left (b^3 c-4 a^3 d\right )}{\sqrt [3]{d}}+a^4 (-d)+4 a b^3 c\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{2/3} d^{4/3}}+\frac{\left (b \sqrt [3]{c} \left (b^3 c-4 a^3 d\right )-\sqrt [3]{d} \left (4 a b^3 c-a^4 d\right )\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} d^{5/3}}+\frac{\left (-4 a^3 b \sqrt [3]{c} d+a^4 \left (-d^{4/3}\right )+4 a b^3 c \sqrt [3]{d}+b^4 c^{4/3}\right ) \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{2/3} d^{5/3}}+\frac{2 a^2 b^2 \log \left (c+d x^3\right )}{d}+\frac{4 a b^3 x}{d}+\frac{b^4 x^2}{2 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^4/(c + d*x^3),x]

[Out]

(4*a*b^3*x)/d + (b^4*x^2)/(2*d) + ((b^4*c^(4/3) + 4*a*b^3*c*d^(1/3) - 4*a^3*b*c^(1/3)*d - a^4*d^(4/3))*ArcTan[
(c^(1/3) - 2*d^(1/3)*x)/(Sqrt[3]*c^(1/3))])/(Sqrt[3]*c^(2/3)*d^(5/3)) + ((b*c^(1/3)*(b^3*c - 4*a^3*d) - d^(1/3
)*(4*a*b^3*c - a^4*d))*Log[c^(1/3) + d^(1/3)*x])/(3*c^(2/3)*d^(5/3)) + ((4*a*b^3*c - a^4*d - (b*c^(1/3)*(b^3*c
- 4*a^3*d))/d^(1/3))*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/(6*c^(2/3)*d^(4/3)) + (2*a^2*b^2*Log[c +
d*x^3])/d

Rule 1887

Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[Pq/(a + b*x^n), x], x] /; FreeQ[{a, b}, x
] && PolyQ[Pq, x] && IntegerQ[n]

Rule 1871

Int[(P2_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x,
2]}, Int[(A + B*x)/(a + b*x^3), x] + Dist[C, Int[x^2/(a + b*x^3), x], x] /; EqQ[a*B^3 - b*A^3, 0] ||  !Ration
alQ[a/b]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]

Rule 1860

Int[((A_) + (B_.)*(x_))/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{r = Numerator[Rt[a/b, 3]], s = Denominator[R
t[a/b, 3]]}, -Dist[(r*(B*r - A*s))/(3*a*s), Int[1/(r + s*x), x], x] + Dist[r/(3*a*s), Int[(r*(B*r + 2*A*s) + s
*(B*r - A*s)*x)/(r^2 - r*s*x + s^2*x^2), x], x]] /; FreeQ[{a, b, A, B}, x] && NeQ[a*B^3 - b*A^3, 0] && PosQ[a/
b]

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]

Rule 260

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

Rubi steps

\begin{align*} \int \frac{(a+b x)^4}{c+d x^3} \, dx &=\int \left (\frac{4 a b^3}{d}+\frac{b^4 x}{d}-\frac{4 a b^3 c-a^4 d+b \left (b^3 c-4 a^3 d\right ) x-6 a^2 b^2 d x^2}{d \left (c+d x^3\right )}\right ) \, dx\\ &=\frac{4 a b^3 x}{d}+\frac{b^4 x^2}{2 d}-\frac{\int \frac{4 a b^3 c-a^4 d+b \left (b^3 c-4 a^3 d\right ) x-6 a^2 b^2 d x^2}{c+d x^3} \, dx}{d}\\ &=\frac{4 a b^3 x}{d}+\frac{b^4 x^2}{2 d}+\left (6 a^2 b^2\right ) \int \frac{x^2}{c+d x^3} \, dx-\frac{\int \frac{4 a b^3 c-a^4 d+b \left (b^3 c-4 a^3 d\right ) x}{c+d x^3} \, dx}{d}\\ &=\frac{4 a b^3 x}{d}+\frac{b^4 x^2}{2 d}+\frac{2 a^2 b^2 \log \left (c+d x^3\right )}{d}-\frac{\int \frac{\sqrt [3]{c} \left (b \sqrt [3]{c} \left (b^3 c-4 a^3 d\right )+2 \sqrt [3]{d} \left (4 a b^3 c-a^4 d\right )\right )+\sqrt [3]{d} \left (b \sqrt [3]{c} \left (b^3 c-4 a^3 d\right )-\sqrt [3]{d} \left (4 a b^3 c-a^4 d\right )\right ) x}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{3 c^{2/3} d^{4/3}}-\frac{\left (4 a b^3 c-a^4 d-\frac{b \sqrt [3]{c} \left (b^3 c-4 a^3 d\right )}{\sqrt [3]{d}}\right ) \int \frac{1}{\sqrt [3]{c}+\sqrt [3]{d} x} \, dx}{3 c^{2/3} d}\\ &=\frac{4 a b^3 x}{d}+\frac{b^4 x^2}{2 d}-\frac{\left (4 a b^3 c-a^4 d-\frac{b \sqrt [3]{c} \left (b^3 c-4 a^3 d\right )}{\sqrt [3]{d}}\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} d^{4/3}}+\frac{2 a^2 b^2 \log \left (c+d x^3\right )}{d}-\frac{\left (b^4 c^{4/3}+4 a b^3 c \sqrt [3]{d}-4 a^3 b \sqrt [3]{c} d-a^4 d^{4/3}\right ) \int \frac{1}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{2 \sqrt [3]{c} d^{4/3}}+\frac{\left (4 a b^3 c-a^4 d-\frac{b \sqrt [3]{c} \left (b^3 c-4 a^3 d\right )}{\sqrt [3]{d}}\right ) \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^{4/3}}\\ &=\frac{4 a b^3 x}{d}+\frac{b^4 x^2}{2 d}-\frac{\left (4 a b^3 c-a^4 d-\frac{b \sqrt [3]{c} \left (b^3 c-4 a^3 d\right )}{\sqrt [3]{d}}\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} d^{4/3}}+\frac{\left (4 a b^3 c-a^4 d-\frac{b \sqrt [3]{c} \left (b^3 c-4 a^3 d\right )}{\sqrt [3]{d}}\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{2/3} d^{4/3}}+\frac{2 a^2 b^2 \log \left (c+d x^3\right )}{d}-\frac{\left (b^4 c^{4/3}+4 a b^3 c \sqrt [3]{d}-4 a^3 b \sqrt [3]{c} d-a^4 d^{4/3}\right ) \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^{5/3}}\\ &=\frac{4 a b^3 x}{d}+\frac{b^4 x^2}{2 d}+\frac{\left (b^4 c^{4/3}+4 a b^3 c \sqrt [3]{d}-4 a^3 b \sqrt [3]{c} d-a^4 d^{4/3}\right ) \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{2/3} d^{5/3}}-\frac{\left (4 a b^3 c-a^4 d-\frac{b \sqrt [3]{c} \left (b^3 c-4 a^3 d\right )}{\sqrt [3]{d}}\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} d^{4/3}}+\frac{\left (4 a b^3 c-a^4 d-\frac{b \sqrt [3]{c} \left (b^3 c-4 a^3 d\right )}{\sqrt [3]{d}}\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{2/3} d^{4/3}}+\frac{2 a^2 b^2 \log \left (c+d x^3\right )}{d}\\ \end{align*}

Mathematica [A]  time = 0.244274, size = 277, normalized size = 0.98 \[ \frac{-\frac{\left (-4 a^3 b \sqrt [3]{c} d+a^4 d^{4/3}-4 a b^3 c \sqrt [3]{d}+b^4 c^{4/3}\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{c^{2/3}}+\frac{2 \left (-4 a^3 b \sqrt [3]{c} d+a^4 d^{4/3}-4 a b^3 c \sqrt [3]{d}+b^4 c^{4/3}\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{c^{2/3}}+\frac{2 \sqrt{3} \left (-4 a^3 b \sqrt [3]{c} d+a^4 \left (-d^{4/3}\right )+4 a b^3 c \sqrt [3]{d}+b^4 c^{4/3}\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt{3}}\right )}{c^{2/3}}+12 a^2 b^2 d^{2/3} \log \left (c+d x^3\right )+24 a b^3 d^{2/3} x+3 b^4 d^{2/3} x^2}{6 d^{5/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^4/(c + d*x^3),x]

[Out]

(24*a*b^3*d^(2/3)*x + 3*b^4*d^(2/3)*x^2 + (2*Sqrt[3]*(b^4*c^(4/3) + 4*a*b^3*c*d^(1/3) - 4*a^3*b*c^(1/3)*d - a^
4*d^(4/3))*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt[3]])/c^(2/3) + (2*(b^4*c^(4/3) - 4*a*b^3*c*d^(1/3) - 4*a^3*
b*c^(1/3)*d + a^4*d^(4/3))*Log[c^(1/3) + d^(1/3)*x])/c^(2/3) - ((b^4*c^(4/3) - 4*a*b^3*c*d^(1/3) - 4*a^3*b*c^(
1/3)*d + a^4*d^(4/3))*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/c^(2/3) + 12*a^2*b^2*d^(2/3)*Log[c + d*x
^3])/(6*d^(5/3))

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Maple [A]  time = 0.003, size = 446, normalized size = 1.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [C]  time = 57.8475, size = 18692, normalized size = 66.28 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(6*b^4*x^2 + 48*a*b^3*x + 2*(12*a^2*b^2/d - 2*(1/2)^(2/3)*(36*a^4*b^4/d^2 - (4*a*b^7*c^2 + 19*a^4*b^4*c*d
+ 4*a^7*b*d^2)/(c*d^3))*(-I*sqrt(3) + 1)/(432*a^6*b^6/d^3 - 18*(4*a*b^7*c^2 + 19*a^4*b^4*c*d + 4*a^7*b*d^2)*a
^2*b^2/(c*d^4) - (b^12*c^4 + 52*a^3*b^9*c^3*d - 52*a^9*b^3*c*d^3 - a^12*d^4)/(c^2*d^5) + (b^12*c^4 - 4*a^3*b^9
*c^3*d + 6*a^6*b^6*c^2*d^2 - 4*a^9*b^3*c*d^3 + a^12*d^4)/(c^2*d^5))^(1/3) - (1/2)^(1/3)*(432*a^6*b^6/d^3 - 18*
(4*a*b^7*c^2 + 19*a^4*b^4*c*d + 4*a^7*b*d^2)*a^2*b^2/(c*d^4) - (b^12*c^4 + 52*a^3*b^9*c^3*d - 52*a^9*b^3*c*d^3
- a^12*d^4)/(c^2*d^5) + (b^12*c^4 - 4*a^3*b^9*c^3*d + 6*a^6*b^6*c^2*d^2 - 4*a^9*b^3*c*d^3 + a^12*d^4)/(c^2*d^
5))^(1/3)*(I*sqrt(3) + 1))*d*log(-8*a*b^11*c^4 - 66*a^4*b^8*c^3*d + 48*a^7*b^5*c^2*d^2 + 26*a^10*b^2*c*d^3 - 1
/4*(b^4*c^3*d^3 - 4*a^3*b*c^2*d^4)*(12*a^2*b^2/d - 2*(1/2)^(2/3)*(36*a^4*b^4/d^2 - (4*a*b^7*c^2 + 19*a^4*b^4*c
*d + 4*a^7*b*d^2)/(c*d^3))*(-I*sqrt(3) + 1)/(432*a^6*b^6/d^3 - 18*(4*a*b^7*c^2 + 19*a^4*b^4*c*d + 4*a^7*b*d^2)
*a^2*b^2/(c*d^4) - (b^12*c^4 + 52*a^3*b^9*c^3*d - 52*a^9*b^3*c*d^3 - a^12*d^4)/(c^2*d^5) + (b^12*c^4 - 4*a^3*b
^9*c^3*d + 6*a^6*b^6*c^2*d^2 - 4*a^9*b^3*c*d^3 + a^12*d^4)/(c^2*d^5))^(1/3) - (1/2)^(1/3)*(432*a^6*b^6/d^3 - 1
8*(4*a*b^7*c^2 + 19*a^4*b^4*c*d + 4*a^7*b*d^2)*a^2*b^2/(c*d^4) - (b^12*c^4 + 52*a^3*b^9*c^3*d - 52*a^9*b^3*c*d
^3 - a^12*d^4)/(c^2*d^5) + (b^12*c^4 - 4*a^3*b^9*c^3*d + 6*a^6*b^6*c^2*d^2 - 4*a^9*b^3*c*d^3 + a^12*d^4)/(c^2*
d^5))^(1/3)*(I*sqrt(3) + 1))^2 + 1/2*(28*a^2*b^6*c^3*d^2 - 56*a^5*b^3*c^2*d^3 + a^8*c*d^4)*(12*a^2*b^2/d - 2*(
1/2)^(2/3)*(36*a^4*b^4/d^2 - (4*a*b^7*c^2 + 19*a^4*b^4*c*d + 4*a^7*b*d^2)/(c*d^3))*(-I*sqrt(3) + 1)/(432*a^6*b
^6/d^3 - 18*(4*a*b^7*c^2 + 19*a^4*b^4*c*d + 4*a^7*b*d^2)*a^2*b^2/(c*d^4) - (b^12*c^4 + 52*a^3*b^9*c^3*d - 52*a
^9*b^3*c*d^3 - a^12*d^4)/(c^2*d^5) + (b^12*c^4 - 4*a^3*b^9*c^3*d + 6*a^6*b^6*c^2*d^2 - 4*a^9*b^3*c*d^3 + a^12*
d^4)/(c^2*d^5))^(1/3) - (1/2)^(1/3)*(432*a^6*b^6/d^3 - 18*(4*a*b^7*c^2 + 19*a^4*b^4*c*d + 4*a^7*b*d^2)*a^2*b^2
/(c*d^4) - (b^12*c^4 + 52*a^3*b^9*c^3*d - 52*a^9*b^3*c*d^3 - a^12*d^4)/(c^2*d^5) + (b^12*c^4 - 4*a^3*b^9*c^3*d
+ 6*a^6*b^6*c^2*d^2 - 4*a^9*b^3*c*d^3 + a^12*d^4)/(c^2*d^5))^(1/3)*(I*sqrt(3) + 1)) - (b^12*c^4 + 52*a^3*b^9*
c^3*d - 52*a^9*b^3*c*d^3 - a^12*d^4)*x) + (36*a^2*b^2 - (12*a^2*b^2/d - 2*(1/2)^(2/3)*(36*a^4*b^4/d^2 - (4*a*b
^7*c^2 + 19*a^4*b^4*c*d + 4*a^7*b*d^2)/(c*d^3))*(-I*sqrt(3) + 1)/(432*a^6*b^6/d^3 - 18*(4*a*b^7*c^2 + 19*a^4*b
^4*c*d + 4*a^7*b*d^2)*a^2*b^2/(c*d^4) - (b^12*c^4 + 52*a^3*b^9*c^3*d - 52*a^9*b^3*c*d^3 - a^12*d^4)/(c^2*d^5)
+ (b^12*c^4 - 4*a^3*b^9*c^3*d + 6*a^6*b^6*c^2*d^2 - 4*a^9*b^3*c*d^3 + a^12*d^4)/(c^2*d^5))^(1/3) - (1/2)^(1/3)
*(432*a^6*b^6/d^3 - 18*(4*a*b^7*c^2 + 19*a^4*b^4*c*d + 4*a^7*b*d^2)*a^2*b^2/(c*d^4) - (b^12*c^4 + 52*a^3*b^9*c
^3*d - 52*a^9*b^3*c*d^3 - a^12*d^4)/(c^2*d^5) + (b^12*c^4 - 4*a^3*b^9*c^3*d + 6*a^6*b^6*c^2*d^2 - 4*a^9*b^3*c*
d^3 + a^12*d^4)/(c^2*d^5))^(1/3)*(I*sqrt(3) + 1))*d + 3*sqrt(1/3)*d*sqrt(-(64*a*b^7*c^2 - 128*a^4*b^4*c*d + 64
*a^7*b*d^2 - 24*(12*a^2*b^2/d - 2*(1/2)^(2/3)*(36*a^4*b^4/d^2 - (4*a*b^7*c^2 + 19*a^4*b^4*c*d + 4*a^7*b*d^2)/(
c*d^3))*(-I*sqrt(3) + 1)/(432*a^6*b^6/d^3 - 18*(4*a*b^7*c^2 + 19*a^4*b^4*c*d + 4*a^7*b*d^2)*a^2*b^2/(c*d^4) -
(b^12*c^4 + 52*a^3*b^9*c^3*d - 52*a^9*b^3*c*d^3 - a^12*d^4)/(c^2*d^5) + (b^12*c^4 - 4*a^3*b^9*c^3*d + 6*a^6*b^
6*c^2*d^2 - 4*a^9*b^3*c*d^3 + a^12*d^4)/(c^2*d^5))^(1/3) - (1/2)^(1/3)*(432*a^6*b^6/d^3 - 18*(4*a*b^7*c^2 + 19
*a^4*b^4*c*d + 4*a^7*b*d^2)*a^2*b^2/(c*d^4) - (b^12*c^4 + 52*a^3*b^9*c^3*d - 52*a^9*b^3*c*d^3 - a^12*d^4)/(c^2
*d^5) + (b^12*c^4 - 4*a^3*b^9*c^3*d + 6*a^6*b^6*c^2*d^2 - 4*a^9*b^3*c*d^3 + a^12*d^4)/(c^2*d^5))^(1/3)*(I*sqrt
(3) + 1))*a^2*b^2*c*d^2 + (12*a^2*b^2/d - 2*(1/2)^(2/3)*(36*a^4*b^4/d^2 - (4*a*b^7*c^2 + 19*a^4*b^4*c*d + 4*a^
7*b*d^2)/(c*d^3))*(-I*sqrt(3) + 1)/(432*a^6*b^6/d^3 - 18*(4*a*b^7*c^2 + 19*a^4*b^4*c*d + 4*a^7*b*d^2)*a^2*b^2/
(c*d^4) - (b^12*c^4 + 52*a^3*b^9*c^3*d - 52*a^9*b^3*c*d^3 - a^12*d^4)/(c^2*d^5) + (b^12*c^4 - 4*a^3*b^9*c^3*d
+ 6*a^6*b^6*c^2*d^2 - 4*a^9*b^3*c*d^3 + a^12*d^4)/(c^2*d^5))^(1/3) - (1/2)^(1/3)*(432*a^6*b^6/d^3 - 18*(4*a*b^
7*c^2 + 19*a^4*b^4*c*d + 4*a^7*b*d^2)*a^2*b^2/(c*d^4) - (b^12*c^4 + 52*a^3*b^9*c^3*d - 52*a^9*b^3*c*d^3 - a^12
*d^4)/(c^2*d^5) + (b^12*c^4 - 4*a^3*b^9*c^3*d + 6*a^6*b^6*c^2*d^2 - 4*a^9*b^3*c*d^3 + a^12*d^4)/(c^2*d^5))^(1/
3)*(I*sqrt(3) + 1))^2*c*d^3)/(c*d^3)))*log(8*a*b^11*c^4 + 66*a^4*b^8*c^3*d - 48*a^7*b^5*c^2*d^2 - 26*a^10*b^2*
c*d^3 + 1/4*(b^4*c^3*d^3 - 4*a^3*b*c^2*d^4)*(12*a^2*b^2/d - 2*(1/2)^(2/3)*(36*a^4*b^4/d^2 - (4*a*b^7*c^2 + 19*
a^4*b^4*c*d + 4*a^7*b*d^2)/(c*d^3))*(-I*sqrt(3) + 1)/(432*a^6*b^6/d^3 - 18*(4*a*b^7*c^2 + 19*a^4*b^4*c*d + 4*a
^7*b*d^2)*a^2*b^2/(c*d^4) - (b^12*c^4 + 52*a^3*b^9*c^3*d - 52*a^9*b^3*c*d^3 - a^12*d^4)/(c^2*d^5) + (b^12*c^4
- 4*a^3*b^9*c^3*d + 6*a^6*b^6*c^2*d^2 - 4*a^9*b^3*c*d^3 + a^12*d^4)/(c^2*d^5))^(1/3) - (1/2)^(1/3)*(432*a^6*b^
6/d^3 - 18*(4*a*b^7*c^2 + 19*a^4*b^4*c*d + 4*a^7*b*d^2)*a^2*b^2/(c*d^4) - (b^12*c^4 + 52*a^3*b^9*c^3*d - 52*a^
9*b^3*c*d^3 - a^12*d^4)/(c^2*d^5) + (b^12*c^4 - 4*a^3*b^9*c^3*d + 6*a^6*b^6*c^2*d^2 - 4*a^9*b^3*c*d^3 + a^12*d
^4)/(c^2*d^5))^(1/3)*(I*sqrt(3) + 1))^2 - 1/2*(28*a^2*b^6*c^3*d^2 - 56*a^5*b^3*c^2*d^3 + a^8*c*d^4)*(12*a^2*b^
2/d - 2*(1/2)^(2/3)*(36*a^4*b^4/d^2 - (4*a*b^7*c^2 + 19*a^4*b^4*c*d + 4*a^7*b*d^2)/(c*d^3))*(-I*sqrt(3) + 1)/(
432*a^6*b^6/d^3 - 18*(4*a*b^7*c^2 + 19*a^4*b^4*c*d + 4*a^7*b*d^2)*a^2*b^2/(c*d^4) - (b^12*c^4 + 52*a^3*b^9*c^3
*d - 52*a^9*b^3*c*d^3 - a^12*d^4)/(c^2*d^5) + (b^12*c^4 - 4*a^3*b^9*c^3*d + 6*a^6*b^6*c^2*d^2 - 4*a^9*b^3*c*d^
3 + a^12*d^4)/(c^2*d^5))^(1/3) - (1/2)^(1/3)*(432*a^6*b^6/d^3 - 18*(4*a*b^7*c^2 + 19*a^4*b^4*c*d + 4*a^7*b*d^2
)*a^2*b^2/(c*d^4) - (b^12*c^4 + 52*a^3*b^9*c^3*d - 52*a^9*b^3*c*d^3 - a^12*d^4)/(c^2*d^5) + (b^12*c^4 - 4*a^3*
b^9*c^3*d + 6*a^6*b^6*c^2*d^2 - 4*a^9*b^3*c*d^3 + a^12*d^4)/(c^2*d^5))^(1/3)*(I*sqrt(3) + 1)) - 2*(b^12*c^4 +
52*a^3*b^9*c^3*d - 52*a^9*b^3*c*d^3 - a^12*d^4)*x + 3/4*sqrt(1/3)*(20*a^2*b^6*c^3*d^2 + 32*a^5*b^3*c^2*d^3 + 2
*a^8*c*d^4 + (b^4*c^3*d^3 - 4*a^3*b*c^2*d^4)*(12*a^2*b^2/d - 2*(1/2)^(2/3)*(36*a^4*b^4/d^2 - (4*a*b^7*c^2 + 19
*a^4*b^4*c*d + 4*a^7*b*d^2)/(c*d^3))*(-I*sqrt(3) + 1)/(432*a^6*b^6/d^3 - 18*(4*a*b^7*c^2 + 19*a^4*b^4*c*d + 4*
a^7*b*d^2)*a^2*b^2/(c*d^4) - (b^12*c^4 + 52*a^3*b^9*c^3*d - 52*a^9*b^3*c*d^3 - a^12*d^4)/(c^2*d^5) + (b^12*c^4
- 4*a^3*b^9*c^3*d + 6*a^6*b^6*c^2*d^2 - 4*a^9*b^3*c*d^3 + a^12*d^4)/(c^2*d^5))^(1/3) - (1/2)^(1/3)*(432*a^6*b
^6/d^3 - 18*(4*a*b^7*c^2 + 19*a^4*b^4*c*d + 4*a^7*b*d^2)*a^2*b^2/(c*d^4) - (b^12*c^4 + 52*a^3*b^9*c^3*d - 52*a
^9*b^3*c*d^3 - a^12*d^4)/(c^2*d^5) + (b^12*c^4 - 4*a^3*b^9*c^3*d + 6*a^6*b^6*c^2*d^2 - 4*a^9*b^3*c*d^3 + a^12*
d^4)/(c^2*d^5))^(1/3)*(I*sqrt(3) + 1)))*sqrt(-(64*a*b^7*c^2 - 128*a^4*b^4*c*d + 64*a^7*b*d^2 - 24*(12*a^2*b^2/
d - 2*(1/2)^(2/3)*(36*a^4*b^4/d^2 - (4*a*b^7*c^2 + 19*a^4*b^4*c*d + 4*a^7*b*d^2)/(c*d^3))*(-I*sqrt(3) + 1)/(43
2*a^6*b^6/d^3 - 18*(4*a*b^7*c^2 + 19*a^4*b^4*c*d + 4*a^7*b*d^2)*a^2*b^2/(c*d^4) - (b^12*c^4 + 52*a^3*b^9*c^3*d
- 52*a^9*b^3*c*d^3 - a^12*d^4)/(c^2*d^5) + (b^12*c^4 - 4*a^3*b^9*c^3*d + 6*a^6*b^6*c^2*d^2 - 4*a^9*b^3*c*d^3
+ a^12*d^4)/(c^2*d^5))^(1/3) - (1/2)^(1/3)*(432*a^6*b^6/d^3 - 18*(4*a*b^7*c^2 + 19*a^4*b^4*c*d + 4*a^7*b*d^2)*
a^2*b^2/(c*d^4) - (b^12*c^4 + 52*a^3*b^9*c^3*d - 52*a^9*b^3*c*d^3 - a^12*d^4)/(c^2*d^5) + (b^12*c^4 - 4*a^3*b^
9*c^3*d + 6*a^6*b^6*c^2*d^2 - 4*a^9*b^3*c*d^3 + a^12*d^4)/(c^2*d^5))^(1/3)*(I*sqrt(3) + 1))*a^2*b^2*c*d^2 + (1
2*a^2*b^2/d - 2*(1/2)^(2/3)*(36*a^4*b^4/d^2 - (4*a*b^7*c^2 + 19*a^4*b^4*c*d + 4*a^7*b*d^2)/(c*d^3))*(-I*sqrt(3
) + 1)/(432*a^6*b^6/d^3 - 18*(4*a*b^7*c^2 + 19*a^4*b^4*c*d + 4*a^7*b*d^2)*a^2*b^2/(c*d^4) - (b^12*c^4 + 52*a^3
*b^9*c^3*d - 52*a^9*b^3*c*d^3 - a^12*d^4)/(c^2*d^5) + (b^12*c^4 - 4*a^3*b^9*c^3*d + 6*a^6*b^6*c^2*d^2 - 4*a^9*
b^3*c*d^3 + a^12*d^4)/(c^2*d^5))^(1/3) - (1/2)^(1/3)*(432*a^6*b^6/d^3 - 18*(4*a*b^7*c^2 + 19*a^4*b^4*c*d + 4*a
^7*b*d^2)*a^2*b^2/(c*d^4) - (b^12*c^4 + 52*a^3*b^9*c^3*d - 52*a^9*b^3*c*d^3 - a^12*d^4)/(c^2*d^5) + (b^12*c^4
- 4*a^3*b^9*c^3*d + 6*a^6*b^6*c^2*d^2 - 4*a^9*b^3*c*d^3 + a^12*d^4)/(c^2*d^5))^(1/3)*(I*sqrt(3) + 1))^2*c*d^3)
/(c*d^3))) + (36*a^2*b^2 - (12*a^2*b^2/d - 2*(1/2)^(2/3)*(36*a^4*b^4/d^2 - (4*a*b^7*c^2 + 19*a^4*b^4*c*d + 4*a
^7*b*d^2)/(c*d^3))*(-I*sqrt(3) + 1)/(432*a^6*b^6/d^3 - 18*(4*a*b^7*c^2 + 19*a^4*b^4*c*d + 4*a^7*b*d^2)*a^2*b^2
/(c*d^4) - (b^12*c^4 + 52*a^3*b^9*c^3*d - 52*a^9*b^3*c*d^3 - a^12*d^4)/(c^2*d^5) + (b^12*c^4 - 4*a^3*b^9*c^3*d
+ 6*a^6*b^6*c^2*d^2 - 4*a^9*b^3*c*d^3 + a^12*d^4)/(c^2*d^5))^(1/3) - (1/2)^(1/3)*(432*a^6*b^6/d^3 - 18*(4*a*b
^7*c^2 + 19*a^4*b^4*c*d + 4*a^7*b*d^2)*a^2*b^2/(c*d^4) - (b^12*c^4 + 52*a^3*b^9*c^3*d - 52*a^9*b^3*c*d^3 - a^1
2*d^4)/(c^2*d^5) + (b^12*c^4 - 4*a^3*b^9*c^3*d + 6*a^6*b^6*c^2*d^2 - 4*a^9*b^3*c*d^3 + a^12*d^4)/(c^2*d^5))^(1
/3)*(I*sqrt(3) + 1))*d - 3*sqrt(1/3)*d*sqrt(-(64*a*b^7*c^2 - 128*a^4*b^4*c*d + 64*a^7*b*d^2 - 24*(12*a^2*b^2/d
- 2*(1/2)^(2/3)*(36*a^4*b^4/d^2 - (4*a*b^7*c^2 + 19*a^4*b^4*c*d + 4*a^7*b*d^2)/(c*d^3))*(-I*sqrt(3) + 1)/(432
*a^6*b^6/d^3 - 18*(4*a*b^7*c^2 + 19*a^4*b^4*c*d + 4*a^7*b*d^2)*a^2*b^2/(c*d^4) - (b^12*c^4 + 52*a^3*b^9*c^3*d
- 52*a^9*b^3*c*d^3 - a^12*d^4)/(c^2*d^5) + (b^12*c^4 - 4*a^3*b^9*c^3*d + 6*a^6*b^6*c^2*d^2 - 4*a^9*b^3*c*d^3 +
a^12*d^4)/(c^2*d^5))^(1/3) - (1/2)^(1/3)*(432*a^6*b^6/d^3 - 18*(4*a*b^7*c^2 + 19*a^4*b^4*c*d + 4*a^7*b*d^2)*a
^2*b^2/(c*d^4) - (b^12*c^4 + 52*a^3*b^9*c^3*d - 52*a^9*b^3*c*d^3 - a^12*d^4)/(c^2*d^5) + (b^12*c^4 - 4*a^3*b^9
*c^3*d + 6*a^6*b^6*c^2*d^2 - 4*a^9*b^3*c*d^3 + a^12*d^4)/(c^2*d^5))^(1/3)*(I*sqrt(3) + 1))*a^2*b^2*c*d^2 + (12
*a^2*b^2/d - 2*(1/2)^(2/3)*(36*a^4*b^4/d^2 - (4*a*b^7*c^2 + 19*a^4*b^4*c*d + 4*a^7*b*d^2)/(c*d^3))*(-I*sqrt(3)
+ 1)/(432*a^6*b^6/d^3 - 18*(4*a*b^7*c^2 + 19*a^4*b^4*c*d + 4*a^7*b*d^2)*a^2*b^2/(c*d^4) - (b^12*c^4 + 52*a^3*
b^9*c^3*d - 52*a^9*b^3*c*d^3 - a^12*d^4)/(c^2*d^5) + (b^12*c^4 - 4*a^3*b^9*c^3*d + 6*a^6*b^6*c^2*d^2 - 4*a^9*b
^3*c*d^3 + a^12*d^4)/(c^2*d^5))^(1/3) - (1/2)^(1/3)*(432*a^6*b^6/d^3 - 18*(4*a*b^7*c^2 + 19*a^4*b^4*c*d + 4*a^
7*b*d^2)*a^2*b^2/(c*d^4) - (b^12*c^4 + 52*a^3*b^9*c^3*d - 52*a^9*b^3*c*d^3 - a^12*d^4)/(c^2*d^5) + (b^12*c^4 -
4*a^3*b^9*c^3*d + 6*a^6*b^6*c^2*d^2 - 4*a^9*b^3*c*d^3 + a^12*d^4)/(c^2*d^5))^(1/3)*(I*sqrt(3) + 1))^2*c*d^3)/
(c*d^3)))*log(8*a*b^11*c^4 + 66*a^4*b^8*c^3*d - 48*a^7*b^5*c^2*d^2 - 26*a^10*b^2*c*d^3 + 1/4*(b^4*c^3*d^3 - 4*
a^3*b*c^2*d^4)*(12*a^2*b^2/d - 2*(1/2)^(2/3)*(36*a^4*b^4/d^2 - (4*a*b^7*c^2 + 19*a^4*b^4*c*d + 4*a^7*b*d^2)/(c
*d^3))*(-I*sqrt(3) + 1)/(432*a^6*b^6/d^3 - 18*(4*a*b^7*c^2 + 19*a^4*b^4*c*d + 4*a^7*b*d^2)*a^2*b^2/(c*d^4) - (
b^12*c^4 + 52*a^3*b^9*c^3*d - 52*a^9*b^3*c*d^3 - a^12*d^4)/(c^2*d^5) + (b^12*c^4 - 4*a^3*b^9*c^3*d + 6*a^6*b^6
*c^2*d^2 - 4*a^9*b^3*c*d^3 + a^12*d^4)/(c^2*d^5))^(1/3) - (1/2)^(1/3)*(432*a^6*b^6/d^3 - 18*(4*a*b^7*c^2 + 19*
a^4*b^4*c*d + 4*a^7*b*d^2)*a^2*b^2/(c*d^4) - (b^12*c^4 + 52*a^3*b^9*c^3*d - 52*a^9*b^3*c*d^3 - a^12*d^4)/(c^2*
d^5) + (b^12*c^4 - 4*a^3*b^9*c^3*d + 6*a^6*b^6*c^2*d^2 - 4*a^9*b^3*c*d^3 + a^12*d^4)/(c^2*d^5))^(1/3)*(I*sqrt(
3) + 1))^2 - 1/2*(28*a^2*b^6*c^3*d^2 - 56*a^5*b^3*c^2*d^3 + a^8*c*d^4)*(12*a^2*b^2/d - 2*(1/2)^(2/3)*(36*a^4*b
^4/d^2 - (4*a*b^7*c^2 + 19*a^4*b^4*c*d + 4*a^7*b*d^2)/(c*d^3))*(-I*sqrt(3) + 1)/(432*a^6*b^6/d^3 - 18*(4*a*b^7
*c^2 + 19*a^4*b^4*c*d + 4*a^7*b*d^2)*a^2*b^2/(c*d^4) - (b^12*c^4 + 52*a^3*b^9*c^3*d - 52*a^9*b^3*c*d^3 - a^12*
d^4)/(c^2*d^5) + (b^12*c^4 - 4*a^3*b^9*c^3*d + 6*a^6*b^6*c^2*d^2 - 4*a^9*b^3*c*d^3 + a^12*d^4)/(c^2*d^5))^(1/3
) - (1/2)^(1/3)*(432*a^6*b^6/d^3 - 18*(4*a*b^7*c^2 + 19*a^4*b^4*c*d + 4*a^7*b*d^2)*a^2*b^2/(c*d^4) - (b^12*c^4
+ 52*a^3*b^9*c^3*d - 52*a^9*b^3*c*d^3 - a^12*d^4)/(c^2*d^5) + (b^12*c^4 - 4*a^3*b^9*c^3*d + 6*a^6*b^6*c^2*d^2
- 4*a^9*b^3*c*d^3 + a^12*d^4)/(c^2*d^5))^(1/3)*(I*sqrt(3) + 1)) - 2*(b^12*c^4 + 52*a^3*b^9*c^3*d - 52*a^9*b^3
*c*d^3 - a^12*d^4)*x - 3/4*sqrt(1/3)*(20*a^2*b^6*c^3*d^2 + 32*a^5*b^3*c^2*d^3 + 2*a^8*c*d^4 + (b^4*c^3*d^3 - 4
*a^3*b*c^2*d^4)*(12*a^2*b^2/d - 2*(1/2)^(2/3)*(36*a^4*b^4/d^2 - (4*a*b^7*c^2 + 19*a^4*b^4*c*d + 4*a^7*b*d^2)/(
c*d^3))*(-I*sqrt(3) + 1)/(432*a^6*b^6/d^3 - 18*(4*a*b^7*c^2 + 19*a^4*b^4*c*d + 4*a^7*b*d^2)*a^2*b^2/(c*d^4) -
(b^12*c^4 + 52*a^3*b^9*c^3*d - 52*a^9*b^3*c*d^3 - a^12*d^4)/(c^2*d^5) + (b^12*c^4 - 4*a^3*b^9*c^3*d + 6*a^6*b^
6*c^2*d^2 - 4*a^9*b^3*c*d^3 + a^12*d^4)/(c^2*d^5))^(1/3) - (1/2)^(1/3)*(432*a^6*b^6/d^3 - 18*(4*a*b^7*c^2 + 19
*a^4*b^4*c*d + 4*a^7*b*d^2)*a^2*b^2/(c*d^4) - (b^12*c^4 + 52*a^3*b^9*c^3*d - 52*a^9*b^3*c*d^3 - a^12*d^4)/(c^2
*d^5) + (b^12*c^4 - 4*a^3*b^9*c^3*d + 6*a^6*b^6*c^2*d^2 - 4*a^9*b^3*c*d^3 + a^12*d^4)/(c^2*d^5))^(1/3)*(I*sqrt
(3) + 1)))*sqrt(-(64*a*b^7*c^2 - 128*a^4*b^4*c*d + 64*a^7*b*d^2 - 24*(12*a^2*b^2/d - 2*(1/2)^(2/3)*(36*a^4*b^4
/d^2 - (4*a*b^7*c^2 + 19*a^4*b^4*c*d + 4*a^7*b*d^2)/(c*d^3))*(-I*sqrt(3) + 1)/(432*a^6*b^6/d^3 - 18*(4*a*b^7*c
^2 + 19*a^4*b^4*c*d + 4*a^7*b*d^2)*a^2*b^2/(c*d^4) - (b^12*c^4 + 52*a^3*b^9*c^3*d - 52*a^9*b^3*c*d^3 - a^12*d^
4)/(c^2*d^5) + (b^12*c^4 - 4*a^3*b^9*c^3*d + 6*a^6*b^6*c^2*d^2 - 4*a^9*b^3*c*d^3 + a^12*d^4)/(c^2*d^5))^(1/3)
- (1/2)^(1/3)*(432*a^6*b^6/d^3 - 18*(4*a*b^7*c^2 + 19*a^4*b^4*c*d + 4*a^7*b*d^2)*a^2*b^2/(c*d^4) - (b^12*c^4 +
52*a^3*b^9*c^3*d - 52*a^9*b^3*c*d^3 - a^12*d^4)/(c^2*d^5) + (b^12*c^4 - 4*a^3*b^9*c^3*d + 6*a^6*b^6*c^2*d^2 -
4*a^9*b^3*c*d^3 + a^12*d^4)/(c^2*d^5))^(1/3)*(I*sqrt(3) + 1))*a^2*b^2*c*d^2 + (12*a^2*b^2/d - 2*(1/2)^(2/3)*(
36*a^4*b^4/d^2 - (4*a*b^7*c^2 + 19*a^4*b^4*c*d + 4*a^7*b*d^2)/(c*d^3))*(-I*sqrt(3) + 1)/(432*a^6*b^6/d^3 - 18*
(4*a*b^7*c^2 + 19*a^4*b^4*c*d + 4*a^7*b*d^2)*a^2*b^2/(c*d^4) - (b^12*c^4 + 52*a^3*b^9*c^3*d - 52*a^9*b^3*c*d^3
- a^12*d^4)/(c^2*d^5) + (b^12*c^4 - 4*a^3*b^9*c^3*d + 6*a^6*b^6*c^2*d^2 - 4*a^9*b^3*c*d^3 + a^12*d^4)/(c^2*d^
5))^(1/3) - (1/2)^(1/3)*(432*a^6*b^6/d^3 - 18*(4*a*b^7*c^2 + 19*a^4*b^4*c*d + 4*a^7*b*d^2)*a^2*b^2/(c*d^4) - (
b^12*c^4 + 52*a^3*b^9*c^3*d - 52*a^9*b^3*c*d^3 - a^12*d^4)/(c^2*d^5) + (b^12*c^4 - 4*a^3*b^9*c^3*d + 6*a^6*b^6
*c^2*d^2 - 4*a^9*b^3*c*d^3 + a^12*d^4)/(c^2*d^5))^(1/3)*(I*sqrt(3) + 1))^2*c*d^3)/(c*d^3))))/d

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Sympy [A]  time = 5.47972, size = 325, normalized size = 1.15 \begin{align*} \frac{4 a b^{3} x}{d} + \frac{b^{4} x^{2}}{2 d} + \operatorname{RootSum}{\left (27 t^{3} c^{2} d^{5} - 162 t^{2} a^{2} b^{2} c^{2} d^{4} + t \left (36 a^{7} b c d^{4} + 171 a^{4} b^{4} c^{2} d^{3} + 36 a b^{7} c^{3} d^{2}\right ) - a^{12} d^{4} + 4 a^{9} b^{3} c d^{3} - 6 a^{6} b^{6} c^{2} d^{2} + 4 a^{3} b^{9} c^{3} d - b^{12} c^{4}, \left ( t \mapsto t \log{\left (x + \frac{36 t^{2} a^{3} b c^{2} d^{4} - 9 t^{2} b^{4} c^{3} d^{3} + 3 t a^{8} c d^{4} - 168 t a^{5} b^{3} c^{2} d^{3} + 84 t a^{2} b^{6} c^{3} d^{2} + 26 a^{10} b^{2} c d^{3} + 48 a^{7} b^{5} c^{2} d^{2} - 66 a^{4} b^{8} c^{3} d - 8 a b^{11} c^{4}}{a^{12} d^{4} + 52 a^{9} b^{3} c d^{3} - 52 a^{3} b^{9} c^{3} d - b^{12} c^{4}} \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*a*b**3*x/d + b**4*x**2/(2*d) + RootSum(27*_t**3*c**2*d**5 - 162*_t**2*a**2*b**2*c**2*d**4 + _t*(36*a**7*b*c*
d**4 + 171*a**4*b**4*c**2*d**3 + 36*a*b**7*c**3*d**2) - a**12*d**4 + 4*a**9*b**3*c*d**3 - 6*a**6*b**6*c**2*d**
2 + 4*a**3*b**9*c**3*d - b**12*c**4, Lambda(_t, _t*log(x + (36*_t**2*a**3*b*c**2*d**4 - 9*_t**2*b**4*c**3*d**3
+ 3*_t*a**8*c*d**4 - 168*_t*a**5*b**3*c**2*d**3 + 84*_t*a**2*b**6*c**3*d**2 + 26*a**10*b**2*c*d**3 + 48*a**7*
b**5*c**2*d**2 - 66*a**4*b**8*c**3*d - 8*a*b**11*c**4)/(a**12*d**4 + 52*a**9*b**3*c*d**3 - 52*a**3*b**9*c**3*d
- b**12*c**4))))

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Giac [A]  time = 1.08479, size = 427, normalized size = 1.51 \begin{align*} \frac{2 \, a^{2} b^{2} \log \left ({\left | d x^{3} + c \right |}\right )}{d} + \frac{b^{4} d x^{2} + 8 \, a b^{3} d x}{2 \, d^{2}} - \frac{\sqrt{3}{\left (4 \, \left (-c d^{2}\right )^{\frac{1}{3}} a b^{3} c d - \left (-c d^{2}\right )^{\frac{1}{3}} a^{4} d^{2} - \left (-c d^{2}\right )^{\frac{2}{3}} b^{4} c + 4 \, \left (-c d^{2}\right )^{\frac{2}{3}} a^{3} b d\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 (4 \, \left (-c d^{2}\right )^{\frac{1}{3}} a b^{3} c d - \left (-c d^{2}\right )^{\frac{1}{3}} a^{4} d^{2} + \left (-c d^{2}\right )^{\frac{2}{3}} b^{4} c - 4 \, \left (-c d^{2}\right )^{\frac{2}{3}} a^{3} b d\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^{4} c d^{4} \left (-\frac{c}{d}\right )^{\frac{1}{3}} - 4 \, a^{3} b d^{5} \left (-\frac{c}{d}\right )^{\frac{1}{3}} + 4 \, a b^{3} c d^{4} - a^{4} d^{5}\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^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a^2*b^2*log(abs(d*x^3 + c))/d + 1/2*(b^4*d*x^2 + 8*a*b^3*d*x)/d^2 - 1/3*sqrt(3)*(4*(-c*d^2)^(1/3)*a*b^3*c*d
- (-c*d^2)^(1/3)*a^4*d^2 - (-c*d^2)^(2/3)*b^4*c + 4*(-c*d^2)^(2/3)*a^3*b*d)*arctan(1/3*sqrt(3)*(2*x + (-c/d)^(
1/3))/(-c/d)^(1/3))/(c*d^3) - 1/6*(4*(-c*d^2)^(1/3)*a*b^3*c*d - (-c*d^2)^(1/3)*a^4*d^2 + (-c*d^2)^(2/3)*b^4*c
- 4*(-c*d^2)^(2/3)*a^3*b*d)*log(x^2 + x*(-c/d)^(1/3) + (-c/d)^(2/3))/(c*d^3) + 1/3*(b^4*c*d^4*(-c/d)^(1/3) - 4
*a^3*b*d^5*(-c/d)^(1/3) + 4*a*b^3*c*d^4 - a^4*d^5)*(-c/d)^(1/3)*log(abs(x - (-c/d)^(1/3)))/(c*d^5)