3.197 \(\int \frac{\log (c (a+b x^3)^p)}{(d+e x)^3} \, dx\)

Optimal. Leaf size=391 \[ -\frac{\sqrt [3]{a} b^{2/3} p \left (3 \sqrt [3]{a} b^{2/3} d^2 e+a e^3+2 b d^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{4 \left (b d^3-a e^3\right )^2}+\frac{\sqrt [3]{a} b^{2/3} p \left (3 \sqrt [3]{a} b^{2/3} d^2 e+a e^3+2 b d^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 \left (b d^3-a e^3\right )^2}-\frac{\sqrt{3} \sqrt [3]{a} b^{2/3} p \left (-3 \sqrt [3]{a} b^{2/3} d^2 e+a e^3+2 b d^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{2 \left (b d^3-a e^3\right )^2}-\frac{\log \left (c \left (a+b x^3\right )^p\right )}{2 e (d+e x)^2}+\frac{b d p \left (2 a e^3+b d^3\right ) \log \left (a+b x^3\right )}{2 e \left (b d^3-a e^3\right )^2}+\frac{3 b d^2 p}{2 e (d+e x) \left (b d^3-a e^3\right )}-\frac{3 b d p \left (2 a e^3+b d^3\right ) \log (d+e x)}{2 e \left (b d^3-a e^3\right )^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.712848, antiderivative size = 391, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2463, 6725, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ -\frac{\sqrt [3]{a} b^{2/3} p \left (3 \sqrt [3]{a} b^{2/3} d^2 e+a e^3+2 b d^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{4 \left (b d^3-a e^3\right )^2}+\frac{\sqrt [3]{a} b^{2/3} p \left (3 \sqrt [3]{a} b^{2/3} d^2 e+a e^3+2 b d^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 \left (b d^3-a e^3\right )^2}-\frac{\sqrt{3} \sqrt [3]{a} b^{2/3} p \left (-3 \sqrt [3]{a} b^{2/3} d^2 e+a e^3+2 b d^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{2 \left (b d^3-a e^3\right )^2}-\frac{\log \left (c \left (a+b x^3\right )^p\right )}{2 e (d+e x)^2}+\frac{b d p \left (2 a e^3+b d^3\right ) \log \left (a+b x^3\right )}{2 e \left (b d^3-a e^3\right )^2}+\frac{3 b d^2 p}{2 e (d+e x) \left (b d^3-a e^3\right )}-\frac{3 b d p \left (2 a e^3+b d^3\right ) \log (d+e x)}{2 e \left (b d^3-a e^3\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(a + b*x^3)^p]/(d + e*x)^3,x]

[Out]

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

Rule 2463

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.) + (g_.)*(x_))^(r_.), x_Symbol] :> Simp[((
f + g*x)^(r + 1)*(a + b*Log[c*(d + e*x^n)^p]))/(g*(r + 1)), x] - Dist[(b*e*n*p)/(g*(r + 1)), Int[(x^(n - 1)*(f
+ g*x)^(r + 1))/(d + e*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, r}, x] && (IGtQ[r, 0] || RationalQ[n
]) && NeQ[r, -1]

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

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

Mathematica [C]  time = 0.706256, size = 303, normalized size = 0.77 \[ \frac{\frac{b^{2/3} p (d+e x) \left (-\sqrt [3]{a} e (d+e x) \left (a e^3+2 b d^3\right ) \left (\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )\right )-9 b^{4/3} d^2 e^2 x^2 (d+e x) \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};-\frac{b x^3}{a}\right )+2 \sqrt [3]{b} d (d+e x) \left (2 a e^3+b d^3\right ) \log \left (a+b x^3\right )-6 \sqrt [3]{b} d (d+e x) \left (2 a e^3+b d^3\right ) \log (d+e x)+2 \sqrt [3]{a} e (d+e x) \left (a e^3+2 b d^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+6 \sqrt [3]{b} d^2 \left (b d^3-a e^3\right )\right )}{\left (b d^3-a e^3\right )^2}-2 \log \left (c \left (a+b x^3\right )^p\right )}{4 e (d+e x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b^(2/3)*p*(d + e*x)*(6*b^(1/3)*d^2*(b*d^3 - a*e^3) - 9*b^(4/3)*d^2*e^2*x^2*(d + e*x)*Hypergeometric2F1[2/3,
1, 5/3, -((b*x^3)/a)] + 2*a^(1/3)*e*(2*b*d^3 + a*e^3)*(d + e*x)*Log[a^(1/3) + b^(1/3)*x] - 6*b^(1/3)*d*(b*d^3
+ 2*a*e^3)*(d + e*x)*Log[d + e*x] - a^(1/3)*e*(2*b*d^3 + a*e^3)*(d + e*x)*(2*Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*x)
/a^(1/3))/Sqrt[3]] + Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]) + 2*b^(1/3)*d*(b*d^3 + 2*a*e^3)*(d + e*x)
*Log[a + b*x^3]))/(b*d^3 - a*e^3)^2 - 2*Log[c*(a + b*x^3)^p])/(4*e*(d + e*x)^2)

________________________________________________________________________________________

Maple [C]  time = 0.591, size = 4085, normalized size = 10.5 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*(b*x^3+a)^p)/(e*x+d)^3,x)

[Out]

-1/2/e/(e*x+d)^2*ln((b*x^3+a)^p)+1/4*(6*b^2*d^6*p+2*sum(_R*ln(((-4*a^3*e^13+6*a^2*b*d^3*e^10-2*b^3*d^9*e^4)*_R
^3+(14*a^2*b*d*e^9*p-10*a*b^2*d^4*e^6*p-4*b^3*d^7*e^3*p)*_R^2+(3*a*b^2*d^2*e^5*p^2+6*b^3*d^5*e^2*p^2)*_R+3*a*b
^2*e^4*p^3)*x+(-5*a^3*d*e^12+9*a^2*b*d^4*e^9-3*a*b^2*d^7*e^6-b^3*d^10*e^3)*_R^3+(8*a^2*b*d^2*e^8*p-7*a*b^2*d^5
*e^5*p-b^3*d^8*e^2*p)*_R^2+(-a^2*b*e^7*p^2+5*a*b^2*d^3*e^4*p^2+5*b^3*d^6*e*p^2)*_R-3*a*b^2*d*e^3*p^3-3*b^3*d^4
*p^3),_R=RootOf((a^2*e^9-2*a*b*d^3*e^6+b^2*d^6*e^3)*_Z^3+(-6*a*b*d*e^5*p-3*b^2*d^4*e^2*p)*_Z^2+3*b^2*d^2*e*p^2
*_Z-b^2*p^3))*a^2*e^9*x^2+2*sum(_R*ln(((-4*a^3*e^13+6*a^2*b*d^3*e^10-2*b^3*d^9*e^4)*_R^3+(14*a^2*b*d*e^9*p-10*
a*b^2*d^4*e^6*p-4*b^3*d^7*e^3*p)*_R^2+(3*a*b^2*d^2*e^5*p^2+6*b^3*d^5*e^2*p^2)*_R+3*a*b^2*e^4*p^3)*x+(-5*a^3*d*
e^12+9*a^2*b*d^4*e^9-3*a*b^2*d^7*e^6-b^3*d^10*e^3)*_R^3+(8*a^2*b*d^2*e^8*p-7*a*b^2*d^5*e^5*p-b^3*d^8*e^2*p)*_R
^2+(-a^2*b*e^7*p^2+5*a*b^2*d^3*e^4*p^2+5*b^3*d^6*e*p^2)*_R-3*a*b^2*d*e^3*p^3-3*b^3*d^4*p^3),_R=RootOf((a^2*e^9
-2*a*b*d^3*e^6+b^2*d^6*e^3)*_Z^3+(-6*a*b*d*e^5*p-3*b^2*d^4*e^2*p)*_Z^2+3*b^2*d^2*e*p^2*_Z-b^2*p^3))*a^2*d^2*e^
7+2*sum(_R*ln(((-4*a^3*e^13+6*a^2*b*d^3*e^10-2*b^3*d^9*e^4)*_R^3+(14*a^2*b*d*e^9*p-10*a*b^2*d^4*e^6*p-4*b^3*d^
7*e^3*p)*_R^2+(3*a*b^2*d^2*e^5*p^2+6*b^3*d^5*e^2*p^2)*_R+3*a*b^2*e^4*p^3)*x+(-5*a^3*d*e^12+9*a^2*b*d^4*e^9-3*a
*b^2*d^7*e^6-b^3*d^10*e^3)*_R^3+(8*a^2*b*d^2*e^8*p-7*a*b^2*d^5*e^5*p-b^3*d^8*e^2*p)*_R^2+(-a^2*b*e^7*p^2+5*a*b
^2*d^3*e^4*p^2+5*b^3*d^6*e*p^2)*_R-3*a*b^2*d*e^3*p^3-3*b^3*d^4*p^3),_R=RootOf((a^2*e^9-2*a*b*d^3*e^6+b^2*d^6*e
^3)*_Z^3+(-6*a*b*d*e^5*p-3*b^2*d^4*e^2*p)*_Z^2+3*b^2*d^2*e*p^2*_Z-b^2*p^3))*b^2*d^8*e-6*ln(e*x+d)*b^2*d^6*p-6*
a*d^3*e^3*b*p-2*I*Pi*a*b*d^3*e^3*csgn(I*c*(b*x^3+a)^p)^3+I*Pi*a^2*e^6*csgn(I*(b*x^3+a)^p)*csgn(I*c*(b*x^3+a)^p
)*csgn(I*c)+2*I*Pi*a*b*d^3*e^3*csgn(I*(b*x^3+a)^p)*csgn(I*c*(b*x^3+a)^p)^2+2*I*Pi*a*b*d^3*e^3*csgn(I*c*(b*x^3+
a)^p)^2*csgn(I*c)-2*ln(c)*b^2*d^6-2*ln(c)*a^2*e^6+6*b^2*d^5*e*p*x+I*Pi*a^2*e^6*csgn(I*c*(b*x^3+a)^p)^3+I*Pi*b^
2*d^6*csgn(I*c*(b*x^3+a)^p)^3+2*sum(_R*ln(((-4*a^3*e^13+6*a^2*b*d^3*e^10-2*b^3*d^9*e^4)*_R^3+(14*a^2*b*d*e^9*p
-10*a*b^2*d^4*e^6*p-4*b^3*d^7*e^3*p)*_R^2+(3*a*b^2*d^2*e^5*p^2+6*b^3*d^5*e^2*p^2)*_R+3*a*b^2*e^4*p^3)*x+(-5*a^
3*d*e^12+9*a^2*b*d^4*e^9-3*a*b^2*d^7*e^6-b^3*d^10*e^3)*_R^3+(8*a^2*b*d^2*e^8*p-7*a*b^2*d^5*e^5*p-b^3*d^8*e^2*p
)*_R^2+(-a^2*b*e^7*p^2+5*a*b^2*d^3*e^4*p^2+5*b^3*d^6*e*p^2)*_R-3*a*b^2*d*e^3*p^3-3*b^3*d^4*p^3),_R=RootOf((a^2
*e^9-2*a*b*d^3*e^6+b^2*d^6*e^3)*_Z^3+(-6*a*b*d*e^5*p-3*b^2*d^4*e^2*p)*_Z^2+3*b^2*d^2*e*p^2*_Z-b^2*p^3))*b^2*d^
6*e^3*x^2+4*sum(_R*ln(((-4*a^3*e^13+6*a^2*b*d^3*e^10-2*b^3*d^9*e^4)*_R^3+(14*a^2*b*d*e^9*p-10*a*b^2*d^4*e^6*p-
4*b^3*d^7*e^3*p)*_R^2+(3*a*b^2*d^2*e^5*p^2+6*b^3*d^5*e^2*p^2)*_R+3*a*b^2*e^4*p^3)*x+(-5*a^3*d*e^12+9*a^2*b*d^4
*e^9-3*a*b^2*d^7*e^6-b^3*d^10*e^3)*_R^3+(8*a^2*b*d^2*e^8*p-7*a*b^2*d^5*e^5*p-b^3*d^8*e^2*p)*_R^2+(-a^2*b*e^7*p
^2+5*a*b^2*d^3*e^4*p^2+5*b^3*d^6*e*p^2)*_R-3*a*b^2*d*e^3*p^3-3*b^3*d^4*p^3),_R=RootOf((a^2*e^9-2*a*b*d^3*e^6+b
^2*d^6*e^3)*_Z^3+(-6*a*b*d*e^5*p-3*b^2*d^4*e^2*p)*_Z^2+3*b^2*d^2*e*p^2*_Z-b^2*p^3))*a^2*d*e^8*x+4*sum(_R*ln(((
-4*a^3*e^13+6*a^2*b*d^3*e^10-2*b^3*d^9*e^4)*_R^3+(14*a^2*b*d*e^9*p-10*a*b^2*d^4*e^6*p-4*b^3*d^7*e^3*p)*_R^2+(3
*a*b^2*d^2*e^5*p^2+6*b^3*d^5*e^2*p^2)*_R+3*a*b^2*e^4*p^3)*x+(-5*a^3*d*e^12+9*a^2*b*d^4*e^9-3*a*b^2*d^7*e^6-b^3
*d^10*e^3)*_R^3+(8*a^2*b*d^2*e^8*p-7*a*b^2*d^5*e^5*p-b^3*d^8*e^2*p)*_R^2+(-a^2*b*e^7*p^2+5*a*b^2*d^3*e^4*p^2+5
*b^3*d^6*e*p^2)*_R-3*a*b^2*d*e^3*p^3-3*b^3*d^4*p^3),_R=RootOf((a^2*e^9-2*a*b*d^3*e^6+b^2*d^6*e^3)*_Z^3+(-6*a*b
*d*e^5*p-3*b^2*d^4*e^2*p)*_Z^2+3*b^2*d^2*e*p^2*_Z-b^2*p^3))*b^2*d^7*e^2*x-4*sum(_R*ln(((-4*a^3*e^13+6*a^2*b*d^
3*e^10-2*b^3*d^9*e^4)*_R^3+(14*a^2*b*d*e^9*p-10*a*b^2*d^4*e^6*p-4*b^3*d^7*e^3*p)*_R^2+(3*a*b^2*d^2*e^5*p^2+6*b
^3*d^5*e^2*p^2)*_R+3*a*b^2*e^4*p^3)*x+(-5*a^3*d*e^12+9*a^2*b*d^4*e^9-3*a*b^2*d^7*e^6-b^3*d^10*e^3)*_R^3+(8*a^2
*b*d^2*e^8*p-7*a*b^2*d^5*e^5*p-b^3*d^8*e^2*p)*_R^2+(-a^2*b*e^7*p^2+5*a*b^2*d^3*e^4*p^2+5*b^3*d^6*e*p^2)*_R-3*a
*b^2*d*e^3*p^3-3*b^3*d^4*p^3),_R=RootOf((a^2*e^9-2*a*b*d^3*e^6+b^2*d^6*e^3)*_Z^3+(-6*a*b*d*e^5*p-3*b^2*d^4*e^2
*p)*_Z^2+3*b^2*d^2*e*p^2*_Z-b^2*p^3))*a*b*d^5*e^4+4*ln(c)*a*b*d^3*e^3-8*sum(_R*ln(((-4*a^3*e^13+6*a^2*b*d^3*e^
10-2*b^3*d^9*e^4)*_R^3+(14*a^2*b*d*e^9*p-10*a*b^2*d^4*e^6*p-4*b^3*d^7*e^3*p)*_R^2+(3*a*b^2*d^2*e^5*p^2+6*b^3*d
^5*e^2*p^2)*_R+3*a*b^2*e^4*p^3)*x+(-5*a^3*d*e^12+9*a^2*b*d^4*e^9-3*a*b^2*d^7*e^6-b^3*d^10*e^3)*_R^3+(8*a^2*b*d
^2*e^8*p-7*a*b^2*d^5*e^5*p-b^3*d^8*e^2*p)*_R^2+(-a^2*b*e^7*p^2+5*a*b^2*d^3*e^4*p^2+5*b^3*d^6*e*p^2)*_R-3*a*b^2
*d*e^3*p^3-3*b^3*d^4*p^3),_R=RootOf((a^2*e^9-2*a*b*d^3*e^6+b^2*d^6*e^3)*_Z^3+(-6*a*b*d*e^5*p-3*b^2*d^4*e^2*p)*
_Z^2+3*b^2*d^2*e*p^2*_Z-b^2*p^3))*a*b*d^4*e^5*x-6*ln(e*x+d)*b^2*d^4*e^2*p*x^2-12*ln(e*x+d)*b^2*d^5*e*p*x-12*ln
(e*x+d)*a*b*d^3*e^3*p-12*ln(e*x+d)*a*b*d*e^5*p*x^2-24*ln(e*x+d)*a*b*d^2*e^4*p*x-I*Pi*b^2*d^6*csgn(I*(b*x^3+a)^
p)*csgn(I*c*(b*x^3+a)^p)^2-I*Pi*b^2*d^6*csgn(I*c*(b*x^3+a)^p)^2*csgn(I*c)-6*a*b*d^2*e^4*p*x-2*I*Pi*a*b*d^3*e^3
*csgn(I*(b*x^3+a)^p)*csgn(I*c*(b*x^3+a)^p)*csgn(I*c)+I*Pi*b^2*d^6*csgn(I*(b*x^3+a)^p)*csgn(I*c*(b*x^3+a)^p)*cs
gn(I*c)-I*Pi*a^2*e^6*csgn(I*(b*x^3+a)^p)*csgn(I*c*(b*x^3+a)^p)^2-I*Pi*a^2*e^6*csgn(I*c*(b*x^3+a)^p)^2*csgn(I*c
)-4*sum(_R*ln(((-4*a^3*e^13+6*a^2*b*d^3*e^10-2*b^3*d^9*e^4)*_R^3+(14*a^2*b*d*e^9*p-10*a*b^2*d^4*e^6*p-4*b^3*d^
7*e^3*p)*_R^2+(3*a*b^2*d^2*e^5*p^2+6*b^3*d^5*e^2*p^2)*_R+3*a*b^2*e^4*p^3)*x+(-5*a^3*d*e^12+9*a^2*b*d^4*e^9-3*a
*b^2*d^7*e^6-b^3*d^10*e^3)*_R^3+(8*a^2*b*d^2*e^8*p-7*a*b^2*d^5*e^5*p-b^3*d^8*e^2*p)*_R^2+(-a^2*b*e^7*p^2+5*a*b
^2*d^3*e^4*p^2+5*b^3*d^6*e*p^2)*_R-3*a*b^2*d*e^3*p^3-3*b^3*d^4*p^3),_R=RootOf((a^2*e^9-2*a*b*d^3*e^6+b^2*d^6*e
^3)*_Z^3+(-6*a*b*d*e^5*p-3*b^2*d^4*e^2*p)*_Z^2+3*b^2*d^2*e*p^2*_Z-b^2*p^3))*a*b*d^3*e^6*x^2)/(e*x+d)^2/(-a*e^3
+b*d^3)^2/e

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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(log(c*(b*x^3+a)^p)/(e*x+d)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [C]  time = 40.5713, size = 26734, normalized size = 68.37 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(24*(b^2*d^5*e - a*b*d^2*e^4)*p*x + 2*(b^2*d^8*e - 2*a*b*d^5*e^4 + a^2*d^2*e^7 + (b^2*d^6*e^3 - 2*a*b*d^3
*e^6 + a^2*e^9)*x^2 + 2*(b^2*d^7*e^2 - 2*a*b*d^4*e^5 + a^2*d*e^8)*x)*((b^2*d^2*p^2/(b^2*d^6*e^2 - 2*a*b*d^3*e^
5 + a^2*e^8) - (b^2*d^4*p + 2*a*b*d*e^3*p)^2/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^2)*(-I*sqrt(3) + 1)/(-3/16*
(b^2*d^4*p + 2*a*b*d*e^3*p)*b^2*d^2*p^2/((b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8)*(b^2*d^6*e - 2*a*b*d^3*e^4 +
a^2*e^7)) + 1/16*b^2*p^3/(b^2*d^6*e^3 - 2*a*b*d^3*e^6 + a^2*e^9) + 1/16*(8*b*d^3 + a*e^3)*a*b^2*p^3/(b*d^3 - a
*e^3)^4 + 1/8*(b^2*d^4*p + 2*a*b*d*e^3*p)^3/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^3)^(1/3) - 4*(-3/16*(b^2*d^4
*p + 2*a*b*d*e^3*p)*b^2*d^2*p^2/((b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8)*(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)
) + 1/16*b^2*p^3/(b^2*d^6*e^3 - 2*a*b*d^3*e^6 + a^2*e^9) + 1/16*(8*b*d^3 + a*e^3)*a*b^2*p^3/(b*d^3 - a*e^3)^4
+ 1/8*(b^2*d^4*p + 2*a*b*d*e^3*p)^3/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^3)^(1/3)*(I*sqrt(3) + 1) + 4*(b^2*d^
4*p + 2*a*b*d*e^3*p)/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7))*log((8*b^2*d^3*e + a*b*e^4)*p^2*x - 3/16*(b^2*d^8*
e^2 - 2*a*b*d^5*e^5 + a^2*d^2*e^8)*((b^2*d^2*p^2/(b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8) - (b^2*d^4*p + 2*a*b*
d*e^3*p)^2/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^2)*(-I*sqrt(3) + 1)/(-3/16*(b^2*d^4*p + 2*a*b*d*e^3*p)*b^2*d^
2*p^2/((b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8)*(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)) + 1/16*b^2*p^3/(b^2*d^6*
e^3 - 2*a*b*d^3*e^6 + a^2*e^9) + 1/16*(8*b*d^3 + a*e^3)*a*b^2*p^3/(b*d^3 - a*e^3)^4 + 1/8*(b^2*d^4*p + 2*a*b*d
*e^3*p)^3/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^3)^(1/3) - 4*(-3/16*(b^2*d^4*p + 2*a*b*d*e^3*p)*b^2*d^2*p^2/((
b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8)*(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)) + 1/16*b^2*p^3/(b^2*d^6*e^3 - 2*
a*b*d^3*e^6 + a^2*e^9) + 1/16*(8*b*d^3 + a*e^3)*a*b^2*p^3/(b*d^3 - a*e^3)^4 + 1/8*(b^2*d^4*p + 2*a*b*d*e^3*p)^
3/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^3)^(1/3)*(I*sqrt(3) + 1) + 4*(b^2*d^4*p + 2*a*b*d*e^3*p)/(b^2*d^6*e -
2*a*b*d^3*e^4 + a^2*e^7))^2 + 1/4*(10*b^2*d^6*e + 16*a*b*d^3*e^4 + a^2*e^7)*((b^2*d^2*p^2/(b^2*d^6*e^2 - 2*a*b
*d^3*e^5 + a^2*e^8) - (b^2*d^4*p + 2*a*b*d*e^3*p)^2/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^2)*(-I*sqrt(3) + 1)/
(-3/16*(b^2*d^4*p + 2*a*b*d*e^3*p)*b^2*d^2*p^2/((b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8)*(b^2*d^6*e - 2*a*b*d^3
*e^4 + a^2*e^7)) + 1/16*b^2*p^3/(b^2*d^6*e^3 - 2*a*b*d^3*e^6 + a^2*e^9) + 1/16*(8*b*d^3 + a*e^3)*a*b^2*p^3/(b*
d^3 - a*e^3)^4 + 1/8*(b^2*d^4*p + 2*a*b*d*e^3*p)^3/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^3)^(1/3) - 4*(-3/16*(
b^2*d^4*p + 2*a*b*d*e^3*p)*b^2*d^2*p^2/((b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8)*(b^2*d^6*e - 2*a*b*d^3*e^4 + a
^2*e^7)) + 1/16*b^2*p^3/(b^2*d^6*e^3 - 2*a*b*d^3*e^6 + a^2*e^9) + 1/16*(8*b*d^3 + a*e^3)*a*b^2*p^3/(b*d^3 - a*
e^3)^4 + 1/8*(b^2*d^4*p + 2*a*b*d*e^3*p)^3/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^3)^(1/3)*(I*sqrt(3) + 1) + 4*
(b^2*d^4*p + 2*a*b*d*e^3*p)/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7))*p - (7*b^2*d^4 + 2*a*b*d*e^3)*p^2) - 8*(b^2
*d^6 - 2*a*b*d^3*e^3 + a^2*e^6)*p*log(b*x^3 + a) + 24*(b^2*d^6 - a*b*d^3*e^3)*p + (12*(b^2*d^4*e^2 + 2*a*b*d*e
^5)*p*x^2 + 24*(b^2*d^5*e + 2*a*b*d^2*e^4)*p*x - (b^2*d^8*e - 2*a*b*d^5*e^4 + a^2*d^2*e^7 + (b^2*d^6*e^3 - 2*a
*b*d^3*e^6 + a^2*e^9)*x^2 + 2*(b^2*d^7*e^2 - 2*a*b*d^4*e^5 + a^2*d*e^8)*x)*((b^2*d^2*p^2/(b^2*d^6*e^2 - 2*a*b*
d^3*e^5 + a^2*e^8) - (b^2*d^4*p + 2*a*b*d*e^3*p)^2/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^2)*(-I*sqrt(3) + 1)/(
-3/16*(b^2*d^4*p + 2*a*b*d*e^3*p)*b^2*d^2*p^2/((b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8)*(b^2*d^6*e - 2*a*b*d^3*
e^4 + a^2*e^7)) + 1/16*b^2*p^3/(b^2*d^6*e^3 - 2*a*b*d^3*e^6 + a^2*e^9) + 1/16*(8*b*d^3 + a*e^3)*a*b^2*p^3/(b*d
^3 - a*e^3)^4 + 1/8*(b^2*d^4*p + 2*a*b*d*e^3*p)^3/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^3)^(1/3) - 4*(-3/16*(b
^2*d^4*p + 2*a*b*d*e^3*p)*b^2*d^2*p^2/((b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8)*(b^2*d^6*e - 2*a*b*d^3*e^4 + a^
2*e^7)) + 1/16*b^2*p^3/(b^2*d^6*e^3 - 2*a*b*d^3*e^6 + a^2*e^9) + 1/16*(8*b*d^3 + a*e^3)*a*b^2*p^3/(b*d^3 - a*e
^3)^4 + 1/8*(b^2*d^4*p + 2*a*b*d*e^3*p)^3/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^3)^(1/3)*(I*sqrt(3) + 1) + 4*(
b^2*d^4*p + 2*a*b*d*e^3*p)/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)) + 12*(b^2*d^6 + 2*a*b*d^3*e^3)*p + sqrt(3)*(
b^2*d^8*e - 2*a*b*d^5*e^4 + a^2*d^2*e^7 + (b^2*d^6*e^3 - 2*a*b*d^3*e^6 + a^2*e^9)*x^2 + 2*(b^2*d^7*e^2 - 2*a*b
*d^4*e^5 + a^2*d*e^8)*x)*sqrt(-((b^4*d^12*e^2 - 4*a*b^3*d^9*e^5 + 6*a^2*b^2*d^6*e^8 - 4*a^3*b*d^3*e^11 + a^4*e
^14)*((b^2*d^2*p^2/(b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8) - (b^2*d^4*p + 2*a*b*d*e^3*p)^2/(b^2*d^6*e - 2*a*b*
d^3*e^4 + a^2*e^7)^2)*(-I*sqrt(3) + 1)/(-3/16*(b^2*d^4*p + 2*a*b*d*e^3*p)*b^2*d^2*p^2/((b^2*d^6*e^2 - 2*a*b*d^
3*e^5 + a^2*e^8)*(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)) + 1/16*b^2*p^3/(b^2*d^6*e^3 - 2*a*b*d^3*e^6 + a^2*e^9)
+ 1/16*(8*b*d^3 + a*e^3)*a*b^2*p^3/(b*d^3 - a*e^3)^4 + 1/8*(b^2*d^4*p + 2*a*b*d*e^3*p)^3/(b^2*d^6*e - 2*a*b*d
^3*e^4 + a^2*e^7)^3)^(1/3) - 4*(-3/16*(b^2*d^4*p + 2*a*b*d*e^3*p)*b^2*d^2*p^2/((b^2*d^6*e^2 - 2*a*b*d^3*e^5 +
a^2*e^8)*(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)) + 1/16*b^2*p^3/(b^2*d^6*e^3 - 2*a*b*d^3*e^6 + a^2*e^9) + 1/16*
(8*b*d^3 + a*e^3)*a*b^2*p^3/(b*d^3 - a*e^3)^4 + 1/8*(b^2*d^4*p + 2*a*b*d*e^3*p)^3/(b^2*d^6*e - 2*a*b*d^3*e^4 +
a^2*e^7)^3)^(1/3)*(I*sqrt(3) + 1) + 4*(b^2*d^4*p + 2*a*b*d*e^3*p)/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7))^2 -
8*(b^4*d^10*e - 3*a^2*b^2*d^4*e^7 + 2*a^3*b*d*e^10)*((b^2*d^2*p^2/(b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8) - (b
^2*d^4*p + 2*a*b*d*e^3*p)^2/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^2)*(-I*sqrt(3) + 1)/(-3/16*(b^2*d^4*p + 2*a*
b*d*e^3*p)*b^2*d^2*p^2/((b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8)*(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)) + 1/16*
b^2*p^3/(b^2*d^6*e^3 - 2*a*b*d^3*e^6 + a^2*e^9) + 1/16*(8*b*d^3 + a*e^3)*a*b^2*p^3/(b*d^3 - a*e^3)^4 + 1/8*(b^
2*d^4*p + 2*a*b*d*e^3*p)^3/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^3)^(1/3) - 4*(-3/16*(b^2*d^4*p + 2*a*b*d*e^3*
p)*b^2*d^2*p^2/((b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8)*(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)) + 1/16*b^2*p^3/
(b^2*d^6*e^3 - 2*a*b*d^3*e^6 + a^2*e^9) + 1/16*(8*b*d^3 + a*e^3)*a*b^2*p^3/(b*d^3 - a*e^3)^4 + 1/8*(b^2*d^4*p
+ 2*a*b*d*e^3*p)^3/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^3)^(1/3)*(I*sqrt(3) + 1) + 4*(b^2*d^4*p + 2*a*b*d*e^3
*p)/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7))*p + 16*(b^4*d^8 - 20*a*b^3*d^5*e^3 - 8*a^2*b^2*d^2*e^6)*p^2)/(b^4*d
^12*e^2 - 4*a*b^3*d^9*e^5 + 6*a^2*b^2*d^6*e^8 - 4*a^3*b*d^3*e^11 + a^4*e^14)))*log(2*(8*b^2*d^3*e + a*b*e^4)*p
^2*x + 3/16*(b^2*d^8*e^2 - 2*a*b*d^5*e^5 + a^2*d^2*e^8)*((b^2*d^2*p^2/(b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8)
- (b^2*d^4*p + 2*a*b*d*e^3*p)^2/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^2)*(-I*sqrt(3) + 1)/(-3/16*(b^2*d^4*p +
2*a*b*d*e^3*p)*b^2*d^2*p^2/((b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8)*(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)) + 1
/16*b^2*p^3/(b^2*d^6*e^3 - 2*a*b*d^3*e^6 + a^2*e^9) + 1/16*(8*b*d^3 + a*e^3)*a*b^2*p^3/(b*d^3 - a*e^3)^4 + 1/8
*(b^2*d^4*p + 2*a*b*d*e^3*p)^3/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^3)^(1/3) - 4*(-3/16*(b^2*d^4*p + 2*a*b*d*
e^3*p)*b^2*d^2*p^2/((b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8)*(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)) + 1/16*b^2*
p^3/(b^2*d^6*e^3 - 2*a*b*d^3*e^6 + a^2*e^9) + 1/16*(8*b*d^3 + a*e^3)*a*b^2*p^3/(b*d^3 - a*e^3)^4 + 1/8*(b^2*d^
4*p + 2*a*b*d*e^3*p)^3/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^3)^(1/3)*(I*sqrt(3) + 1) + 4*(b^2*d^4*p + 2*a*b*d
*e^3*p)/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7))^2 - 1/4*(10*b^2*d^6*e + 16*a*b*d^3*e^4 + a^2*e^7)*((b^2*d^2*p^2
/(b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8) - (b^2*d^4*p + 2*a*b*d*e^3*p)^2/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)
^2)*(-I*sqrt(3) + 1)/(-3/16*(b^2*d^4*p + 2*a*b*d*e^3*p)*b^2*d^2*p^2/((b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8)*(
b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)) + 1/16*b^2*p^3/(b^2*d^6*e^3 - 2*a*b*d^3*e^6 + a^2*e^9) + 1/16*(8*b*d^3 +
a*e^3)*a*b^2*p^3/(b*d^3 - a*e^3)^4 + 1/8*(b^2*d^4*p + 2*a*b*d*e^3*p)^3/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^
3)^(1/3) - 4*(-3/16*(b^2*d^4*p + 2*a*b*d*e^3*p)*b^2*d^2*p^2/((b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8)*(b^2*d^6*
e - 2*a*b*d^3*e^4 + a^2*e^7)) + 1/16*b^2*p^3/(b^2*d^6*e^3 - 2*a*b*d^3*e^6 + a^2*e^9) + 1/16*(8*b*d^3 + a*e^3)*
a*b^2*p^3/(b*d^3 - a*e^3)^4 + 1/8*(b^2*d^4*p + 2*a*b*d*e^3*p)^3/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^3)^(1/3)
*(I*sqrt(3) + 1) + 4*(b^2*d^4*p + 2*a*b*d*e^3*p)/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7))*p + (7*b^2*d^4 + 2*a*b
*d*e^3)*p^2 + 1/16*sqrt(3)*(3*(b^2*d^8*e^2 - 2*a*b*d^5*e^5 + a^2*d^2*e^8)*((b^2*d^2*p^2/(b^2*d^6*e^2 - 2*a*b*d
^3*e^5 + a^2*e^8) - (b^2*d^4*p + 2*a*b*d*e^3*p)^2/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^2)*(-I*sqrt(3) + 1)/(-
3/16*(b^2*d^4*p + 2*a*b*d*e^3*p)*b^2*d^2*p^2/((b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8)*(b^2*d^6*e - 2*a*b*d^3*e
^4 + a^2*e^7)) + 1/16*b^2*p^3/(b^2*d^6*e^3 - 2*a*b*d^3*e^6 + a^2*e^9) + 1/16*(8*b*d^3 + a*e^3)*a*b^2*p^3/(b*d^
3 - a*e^3)^4 + 1/8*(b^2*d^4*p + 2*a*b*d*e^3*p)^3/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^3)^(1/3) - 4*(-3/16*(b^
2*d^4*p + 2*a*b*d*e^3*p)*b^2*d^2*p^2/((b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8)*(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2
*e^7)) + 1/16*b^2*p^3/(b^2*d^6*e^3 - 2*a*b*d^3*e^6 + a^2*e^9) + 1/16*(8*b*d^3 + a*e^3)*a*b^2*p^3/(b*d^3 - a*e^
3)^4 + 1/8*(b^2*d^4*p + 2*a*b*d*e^3*p)^3/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^3)^(1/3)*(I*sqrt(3) + 1) + 4*(b
^2*d^4*p + 2*a*b*d*e^3*p)/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)) + 4*(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)*p)*
sqrt(-((b^4*d^12*e^2 - 4*a*b^3*d^9*e^5 + 6*a^2*b^2*d^6*e^8 - 4*a^3*b*d^3*e^11 + a^4*e^14)*((b^2*d^2*p^2/(b^2*d
^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8) - (b^2*d^4*p + 2*a*b*d*e^3*p)^2/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^2)*(-I
*sqrt(3) + 1)/(-3/16*(b^2*d^4*p + 2*a*b*d*e^3*p)*b^2*d^2*p^2/((b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8)*(b^2*d^6
*e - 2*a*b*d^3*e^4 + a^2*e^7)) + 1/16*b^2*p^3/(b^2*d^6*e^3 - 2*a*b*d^3*e^6 + a^2*e^9) + 1/16*(8*b*d^3 + a*e^3)
*a*b^2*p^3/(b*d^3 - a*e^3)^4 + 1/8*(b^2*d^4*p + 2*a*b*d*e^3*p)^3/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^3)^(1/3
) - 4*(-3/16*(b^2*d^4*p + 2*a*b*d*e^3*p)*b^2*d^2*p^2/((b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8)*(b^2*d^6*e - 2*a
*b*d^3*e^4 + a^2*e^7)) + 1/16*b^2*p^3/(b^2*d^6*e^3 - 2*a*b*d^3*e^6 + a^2*e^9) + 1/16*(8*b*d^3 + a*e^3)*a*b^2*p
^3/(b*d^3 - a*e^3)^4 + 1/8*(b^2*d^4*p + 2*a*b*d*e^3*p)^3/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^3)^(1/3)*(I*sqr
t(3) + 1) + 4*(b^2*d^4*p + 2*a*b*d*e^3*p)/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7))^2 - 8*(b^4*d^10*e - 3*a^2*b^2
*d^4*e^7 + 2*a^3*b*d*e^10)*((b^2*d^2*p^2/(b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8) - (b^2*d^4*p + 2*a*b*d*e^3*p)
^2/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^2)*(-I*sqrt(3) + 1)/(-3/16*(b^2*d^4*p + 2*a*b*d*e^3*p)*b^2*d^2*p^2/((
b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8)*(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)) + 1/16*b^2*p^3/(b^2*d^6*e^3 - 2*
a*b*d^3*e^6 + a^2*e^9) + 1/16*(8*b*d^3 + a*e^3)*a*b^2*p^3/(b*d^3 - a*e^3)^4 + 1/8*(b^2*d^4*p + 2*a*b*d*e^3*p)^
3/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^3)^(1/3) - 4*(-3/16*(b^2*d^4*p + 2*a*b*d*e^3*p)*b^2*d^2*p^2/((b^2*d^6*
e^2 - 2*a*b*d^3*e^5 + a^2*e^8)*(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)) + 1/16*b^2*p^3/(b^2*d^6*e^3 - 2*a*b*d^3*
e^6 + a^2*e^9) + 1/16*(8*b*d^3 + a*e^3)*a*b^2*p^3/(b*d^3 - a*e^3)^4 + 1/8*(b^2*d^4*p + 2*a*b*d*e^3*p)^3/(b^2*d
^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^3)^(1/3)*(I*sqrt(3) + 1) + 4*(b^2*d^4*p + 2*a*b*d*e^3*p)/(b^2*d^6*e - 2*a*b*d^
3*e^4 + a^2*e^7))*p + 16*(b^4*d^8 - 20*a*b^3*d^5*e^3 - 8*a^2*b^2*d^2*e^6)*p^2)/(b^4*d^12*e^2 - 4*a*b^3*d^9*e^5
+ 6*a^2*b^2*d^6*e^8 - 4*a^3*b*d^3*e^11 + a^4*e^14))) + (12*(b^2*d^4*e^2 + 2*a*b*d*e^5)*p*x^2 + 24*(b^2*d^5*e
+ 2*a*b*d^2*e^4)*p*x - (b^2*d^8*e - 2*a*b*d^5*e^4 + a^2*d^2*e^7 + (b^2*d^6*e^3 - 2*a*b*d^3*e^6 + a^2*e^9)*x^2
+ 2*(b^2*d^7*e^2 - 2*a*b*d^4*e^5 + a^2*d*e^8)*x)*((b^2*d^2*p^2/(b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8) - (b^2*
d^4*p + 2*a*b*d*e^3*p)^2/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^2)*(-I*sqrt(3) + 1)/(-3/16*(b^2*d^4*p + 2*a*b*d
*e^3*p)*b^2*d^2*p^2/((b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8)*(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)) + 1/16*b^2
*p^3/(b^2*d^6*e^3 - 2*a*b*d^3*e^6 + a^2*e^9) + 1/16*(8*b*d^3 + a*e^3)*a*b^2*p^3/(b*d^3 - a*e^3)^4 + 1/8*(b^2*d
^4*p + 2*a*b*d*e^3*p)^3/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^3)^(1/3) - 4*(-3/16*(b^2*d^4*p + 2*a*b*d*e^3*p)*
b^2*d^2*p^2/((b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8)*(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)) + 1/16*b^2*p^3/(b^
2*d^6*e^3 - 2*a*b*d^3*e^6 + a^2*e^9) + 1/16*(8*b*d^3 + a*e^3)*a*b^2*p^3/(b*d^3 - a*e^3)^4 + 1/8*(b^2*d^4*p + 2
*a*b*d*e^3*p)^3/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^3)^(1/3)*(I*sqrt(3) + 1) + 4*(b^2*d^4*p + 2*a*b*d*e^3*p)
/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)) + 12*(b^2*d^6 + 2*a*b*d^3*e^3)*p - sqrt(3)*(b^2*d^8*e - 2*a*b*d^5*e^4
+ a^2*d^2*e^7 + (b^2*d^6*e^3 - 2*a*b*d^3*e^6 + a^2*e^9)*x^2 + 2*(b^2*d^7*e^2 - 2*a*b*d^4*e^5 + a^2*d*e^8)*x)*s
qrt(-((b^4*d^12*e^2 - 4*a*b^3*d^9*e^5 + 6*a^2*b^2*d^6*e^8 - 4*a^3*b*d^3*e^11 + a^4*e^14)*((b^2*d^2*p^2/(b^2*d^
6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8) - (b^2*d^4*p + 2*a*b*d*e^3*p)^2/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^2)*(-I*
sqrt(3) + 1)/(-3/16*(b^2*d^4*p + 2*a*b*d*e^3*p)*b^2*d^2*p^2/((b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8)*(b^2*d^6*
e - 2*a*b*d^3*e^4 + a^2*e^7)) + 1/16*b^2*p^3/(b^2*d^6*e^3 - 2*a*b*d^3*e^6 + a^2*e^9) + 1/16*(8*b*d^3 + a*e^3)*
a*b^2*p^3/(b*d^3 - a*e^3)^4 + 1/8*(b^2*d^4*p + 2*a*b*d*e^3*p)^3/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^3)^(1/3)
- 4*(-3/16*(b^2*d^4*p + 2*a*b*d*e^3*p)*b^2*d^2*p^2/((b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8)*(b^2*d^6*e - 2*a*
b*d^3*e^4 + a^2*e^7)) + 1/16*b^2*p^3/(b^2*d^6*e^3 - 2*a*b*d^3*e^6 + a^2*e^9) + 1/16*(8*b*d^3 + a*e^3)*a*b^2*p^
3/(b*d^3 - a*e^3)^4 + 1/8*(b^2*d^4*p + 2*a*b*d*e^3*p)^3/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^3)^(1/3)*(I*sqrt
(3) + 1) + 4*(b^2*d^4*p + 2*a*b*d*e^3*p)/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7))^2 - 8*(b^4*d^10*e - 3*a^2*b^2*
d^4*e^7 + 2*a^3*b*d*e^10)*((b^2*d^2*p^2/(b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8) - (b^2*d^4*p + 2*a*b*d*e^3*p)^
2/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^2)*(-I*sqrt(3) + 1)/(-3/16*(b^2*d^4*p + 2*a*b*d*e^3*p)*b^2*d^2*p^2/((b
^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8)*(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)) + 1/16*b^2*p^3/(b^2*d^6*e^3 - 2*a
*b*d^3*e^6 + a^2*e^9) + 1/16*(8*b*d^3 + a*e^3)*a*b^2*p^3/(b*d^3 - a*e^3)^4 + 1/8*(b^2*d^4*p + 2*a*b*d*e^3*p)^3
/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^3)^(1/3) - 4*(-3/16*(b^2*d^4*p + 2*a*b*d*e^3*p)*b^2*d^2*p^2/((b^2*d^6*e
^2 - 2*a*b*d^3*e^5 + a^2*e^8)*(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)) + 1/16*b^2*p^3/(b^2*d^6*e^3 - 2*a*b*d^3*e
^6 + a^2*e^9) + 1/16*(8*b*d^3 + a*e^3)*a*b^2*p^3/(b*d^3 - a*e^3)^4 + 1/8*(b^2*d^4*p + 2*a*b*d*e^3*p)^3/(b^2*d^
6*e - 2*a*b*d^3*e^4 + a^2*e^7)^3)^(1/3)*(I*sqrt(3) + 1) + 4*(b^2*d^4*p + 2*a*b*d*e^3*p)/(b^2*d^6*e - 2*a*b*d^3
*e^4 + a^2*e^7))*p + 16*(b^4*d^8 - 20*a*b^3*d^5*e^3 - 8*a^2*b^2*d^2*e^6)*p^2)/(b^4*d^12*e^2 - 4*a*b^3*d^9*e^5
+ 6*a^2*b^2*d^6*e^8 - 4*a^3*b*d^3*e^11 + a^4*e^14)))*log(2*(8*b^2*d^3*e + a*b*e^4)*p^2*x + 3/16*(b^2*d^8*e^2 -
2*a*b*d^5*e^5 + a^2*d^2*e^8)*((b^2*d^2*p^2/(b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8) - (b^2*d^4*p + 2*a*b*d*e^3
*p)^2/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^2)*(-I*sqrt(3) + 1)/(-3/16*(b^2*d^4*p + 2*a*b*d*e^3*p)*b^2*d^2*p^2
/((b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8)*(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)) + 1/16*b^2*p^3/(b^2*d^6*e^3 -
2*a*b*d^3*e^6 + a^2*e^9) + 1/16*(8*b*d^3 + a*e^3)*a*b^2*p^3/(b*d^3 - a*e^3)^4 + 1/8*(b^2*d^4*p + 2*a*b*d*e^3*
p)^3/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^3)^(1/3) - 4*(-3/16*(b^2*d^4*p + 2*a*b*d*e^3*p)*b^2*d^2*p^2/((b^2*d
^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8)*(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)) + 1/16*b^2*p^3/(b^2*d^6*e^3 - 2*a*b*d
^3*e^6 + a^2*e^9) + 1/16*(8*b*d^3 + a*e^3)*a*b^2*p^3/(b*d^3 - a*e^3)^4 + 1/8*(b^2*d^4*p + 2*a*b*d*e^3*p)^3/(b^
2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^3)^(1/3)*(I*sqrt(3) + 1) + 4*(b^2*d^4*p + 2*a*b*d*e^3*p)/(b^2*d^6*e - 2*a*b
*d^3*e^4 + a^2*e^7))^2 - 1/4*(10*b^2*d^6*e + 16*a*b*d^3*e^4 + a^2*e^7)*((b^2*d^2*p^2/(b^2*d^6*e^2 - 2*a*b*d^3*
e^5 + a^2*e^8) - (b^2*d^4*p + 2*a*b*d*e^3*p)^2/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^2)*(-I*sqrt(3) + 1)/(-3/1
6*(b^2*d^4*p + 2*a*b*d*e^3*p)*b^2*d^2*p^2/((b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8)*(b^2*d^6*e - 2*a*b*d^3*e^4
+ a^2*e^7)) + 1/16*b^2*p^3/(b^2*d^6*e^3 - 2*a*b*d^3*e^6 + a^2*e^9) + 1/16*(8*b*d^3 + a*e^3)*a*b^2*p^3/(b*d^3 -
a*e^3)^4 + 1/8*(b^2*d^4*p + 2*a*b*d*e^3*p)^3/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^3)^(1/3) - 4*(-3/16*(b^2*d
^4*p + 2*a*b*d*e^3*p)*b^2*d^2*p^2/((b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8)*(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^
7)) + 1/16*b^2*p^3/(b^2*d^6*e^3 - 2*a*b*d^3*e^6 + a^2*e^9) + 1/16*(8*b*d^3 + a*e^3)*a*b^2*p^3/(b*d^3 - a*e^3)^
4 + 1/8*(b^2*d^4*p + 2*a*b*d*e^3*p)^3/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^3)^(1/3)*(I*sqrt(3) + 1) + 4*(b^2*
d^4*p + 2*a*b*d*e^3*p)/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7))*p + (7*b^2*d^4 + 2*a*b*d*e^3)*p^2 - 1/16*sqrt(3)
*(3*(b^2*d^8*e^2 - 2*a*b*d^5*e^5 + a^2*d^2*e^8)*((b^2*d^2*p^2/(b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8) - (b^2*d
^4*p + 2*a*b*d*e^3*p)^2/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^2)*(-I*sqrt(3) + 1)/(-3/16*(b^2*d^4*p + 2*a*b*d*
e^3*p)*b^2*d^2*p^2/((b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8)*(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)) + 1/16*b^2*
p^3/(b^2*d^6*e^3 - 2*a*b*d^3*e^6 + a^2*e^9) + 1/16*(8*b*d^3 + a*e^3)*a*b^2*p^3/(b*d^3 - a*e^3)^4 + 1/8*(b^2*d^
4*p + 2*a*b*d*e^3*p)^3/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^3)^(1/3) - 4*(-3/16*(b^2*d^4*p + 2*a*b*d*e^3*p)*b
^2*d^2*p^2/((b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8)*(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)) + 1/16*b^2*p^3/(b^2
*d^6*e^3 - 2*a*b*d^3*e^6 + a^2*e^9) + 1/16*(8*b*d^3 + a*e^3)*a*b^2*p^3/(b*d^3 - a*e^3)^4 + 1/8*(b^2*d^4*p + 2*
a*b*d*e^3*p)^3/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^3)^(1/3)*(I*sqrt(3) + 1) + 4*(b^2*d^4*p + 2*a*b*d*e^3*p)/
(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)) + 4*(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)*p)*sqrt(-((b^4*d^12*e^2 - 4*a
*b^3*d^9*e^5 + 6*a^2*b^2*d^6*e^8 - 4*a^3*b*d^3*e^11 + a^4*e^14)*((b^2*d^2*p^2/(b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a
^2*e^8) - (b^2*d^4*p + 2*a*b*d*e^3*p)^2/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^2)*(-I*sqrt(3) + 1)/(-3/16*(b^2*
d^4*p + 2*a*b*d*e^3*p)*b^2*d^2*p^2/((b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8)*(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e
^7)) + 1/16*b^2*p^3/(b^2*d^6*e^3 - 2*a*b*d^3*e^6 + a^2*e^9) + 1/16*(8*b*d^3 + a*e^3)*a*b^2*p^3/(b*d^3 - a*e^3)
^4 + 1/8*(b^2*d^4*p + 2*a*b*d*e^3*p)^3/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^3)^(1/3) - 4*(-3/16*(b^2*d^4*p +
2*a*b*d*e^3*p)*b^2*d^2*p^2/((b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8)*(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)) + 1
/16*b^2*p^3/(b^2*d^6*e^3 - 2*a*b*d^3*e^6 + a^2*e^9) + 1/16*(8*b*d^3 + a*e^3)*a*b^2*p^3/(b*d^3 - a*e^3)^4 + 1/8
*(b^2*d^4*p + 2*a*b*d*e^3*p)^3/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)^3)^(1/3)*(I*sqrt(3) + 1) + 4*(b^2*d^4*p +
2*a*b*d*e^3*p)/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7))^2 - 8*(b^4*d^10*e - 3*a^2*b^2*d^4*e^7 + 2*a^3*b*d*e^10)
*((b^2*d^2*p^2/(b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*e^8) - (b^2*d^4*p + 2*a*b*d*e^3*p)^2/(b^2*d^6*e - 2*a*b*d^3*
e^4 + a^2*e^7)^2)*(-I*sqrt(3) + 1)/(-3/16*(b^2*d^4*p + 2*a*b*d*e^3*p)*b^2*d^2*p^2/((b^2*d^6*e^2 - 2*a*b*d^3*e^
5 + a^2*e^8)*(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)) + 1/16*b^2*p^3/(b^2*d^6*e^3 - 2*a*b*d^3*e^6 + a^2*e^9) + 1
/16*(8*b*d^3 + a*e^3)*a*b^2*p^3/(b*d^3 - a*e^3)^4 + 1/8*(b^2*d^4*p + 2*a*b*d*e^3*p)^3/(b^2*d^6*e - 2*a*b*d^3*e
^4 + a^2*e^7)^3)^(1/3) - 4*(-3/16*(b^2*d^4*p + 2*a*b*d*e^3*p)*b^2*d^2*p^2/((b^2*d^6*e^2 - 2*a*b*d^3*e^5 + a^2*
e^8)*(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)) + 1/16*b^2*p^3/(b^2*d^6*e^3 - 2*a*b*d^3*e^6 + a^2*e^9) + 1/16*(8*b
*d^3 + a*e^3)*a*b^2*p^3/(b*d^3 - a*e^3)^4 + 1/8*(b^2*d^4*p + 2*a*b*d*e^3*p)^3/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2
*e^7)^3)^(1/3)*(I*sqrt(3) + 1) + 4*(b^2*d^4*p + 2*a*b*d*e^3*p)/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7))*p + 16*(
b^4*d^8 - 20*a*b^3*d^5*e^3 - 8*a^2*b^2*d^2*e^6)*p^2)/(b^4*d^12*e^2 - 4*a*b^3*d^9*e^5 + 6*a^2*b^2*d^6*e^8 - 4*a
^3*b*d^3*e^11 + a^4*e^14))) - 24*((b^2*d^4*e^2 + 2*a*b*d*e^5)*p*x^2 + 2*(b^2*d^5*e + 2*a*b*d^2*e^4)*p*x + (b^2
*d^6 + 2*a*b*d^3*e^3)*p)*log(e*x + d) - 8*(b^2*d^6 - 2*a*b*d^3*e^3 + a^2*e^6)*log(c))/(b^2*d^8*e - 2*a*b*d^5*e
^4 + a^2*d^2*e^7 + (b^2*d^6*e^3 - 2*a*b*d^3*e^6 + a^2*e^9)*x^2 + 2*(b^2*d^7*e^2 - 2*a*b*d^4*e^5 + a^2*d*e^8)*x
)

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Sympy [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(ln(c*(b*x**3+a)**p)/(e*x+d)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.75418, size = 1079, normalized size = 2.76 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*a*b^5*d^9*p*e^2 - 3*a*b^5*d^8*p*(-a/b)^(1/3)*e^3 - 3*a^2*b^4*d^6*p*e^5 + 6*a^2*b^4*d^5*p*(-a/b)^(1/3)*
e^6 - 3*a^3*b^3*d^2*p*(-a/b)^(1/3)*e^9 + a^4*b^2*p*e^11)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^5*d^12*e
^2 - 4*a^2*b^4*d^9*e^5 + 6*a^3*b^3*d^6*e^8 - 4*a^4*b^2*d^3*e^11 + a^5*b*e^14) + 1/2*(2*sqrt(3)*(-a*b^2)^(1/3)*
a*b^2*d*p - sqrt(3)*(-a*b^2)^(2/3)*a*b*p*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a*b^3*d^4 -
2*(-a*b^2)^(1/3)*a*b^2*d^3*e + 2*a^2*b^2*d*e^3 + 3*(-a*b^2)^(2/3)*a*b*d^2*e^2 - (-a*b^2)^(1/3)*a^2*b*e^4) + 1
/4*(2*(-a*b^2)^(1/3)*b*d^3*p - 3*(-a*b^2)^(2/3)*d^2*p*e + (-a*b^2)^(1/3)*a*p*e^3)*log(x^2 + x*(-a/b)^(1/3) + (
-a/b)^(2/3))/(b^2*d^6 - 2*a*b*d^3*e^3 + a^2*e^6) + 1/2*(b^2*d^4*p + 2*a*b*d*p*e^3)*log(abs(b*x^3 + a))/(b^2*d^
6*e - 2*a*b*d^3*e^4 + a^2*e^7) - 1/2*(3*b^2*d^4*p*x^2*e^2*log(x*e + d) + 6*b^2*d^5*p*x*e*log(x*e + d) - 3*b^2*
d^5*p*x*e + b^2*d^6*p*log(b*x^3 + a) + 3*b^2*d^6*p*log(x*e + d) - 3*b^2*d^6*p + b^2*d^6*log(c) - 2*a*b*d^3*p*e
^3*log(b*x^3 + a) + 6*a*b*d*p*x^2*e^5*log(x*e + d) + 12*a*b*d^2*p*x*e^4*log(x*e + d) + 6*a*b*d^3*p*e^3*log(x*e
+ d) + 3*a*b*d^2*p*x*e^4 + 3*a*b*d^3*p*e^3 - 2*a*b*d^3*e^3*log(c) + a^2*p*e^6*log(b*x^3 + a) + a^2*e^6*log(c)
)/(b^2*d^6*x^2*e^3 + 2*b^2*d^7*x*e^2 + b^2*d^8*e - 2*a*b*d^3*x^2*e^6 - 4*a*b*d^4*x*e^5 - 2*a*b*d^5*e^4 + a^2*x
^2*e^9 + 2*a^2*d*x*e^8 + a^2*d^2*e^7)