3.2.97 \(\int \frac {\log (c (a+b x^3)^p)}{(d+e x)^3} \, dx\) [197]
Optimal. Leaf size=391 \[ \frac {3 b d^2 p}{2 e \left (b d^3-a e^3\right ) (d+e x)}-\frac {\sqrt {3} \sqrt [3]{a} b^{2/3} \left (2 b d^3-3 \sqrt [3]{a} b^{2/3} d^2 e+a e^3\right ) p \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{2 \left (b d^3-a e^3\right )^2}+\frac {\sqrt [3]{a} b^{2/3} \left (2 b d^3+3 \sqrt [3]{a} b^{2/3} d^2 e+a e^3\right ) p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 \left (b d^3-a e^3\right )^2}-\frac {3 b d \left (b d^3+2 a e^3\right ) p \log (d+e x)}{2 e \left (b d^3-a e^3\right )^2}-\frac {\sqrt [3]{a} b^{2/3} \left (2 b d^3+3 \sqrt [3]{a} b^{2/3} d^2 e+a e^3\right ) p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{4 \left (b d^3-a e^3\right )^2}+\frac {b d \left (b d^3+2 a e^3\right ) p \log \left (a+b x^3\right )}{2 e \left (b d^3-a e^3\right )^2}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{2 e (d+e x)^2} \]
[Out]
3/2*b*d^2*p/e/(-a*e^3+b*d^3)/(e*x+d)+1/2*a^(1/3)*b^(2/3)*(2*b*d^3+3*a^(1/3)*b^(2/3)*d^2*e+a*e^3)*p*ln(a^(1/3)+
b^(1/3)*x)/(-a*e^3+b*d^3)^2-3/2*b*d*(2*a*e^3+b*d^3)*p*ln(e*x+d)/e/(-a*e^3+b*d^3)^2-1/4*a^(1/3)*b^(2/3)*(2*b*d^
3+3*a^(1/3)*b^(2/3)*d^2*e+a*e^3)*p*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(-a*e^3+b*d^3)^2+1/2*b*d*(2*a*e^3
+b*d^3)*p*ln(b*x^3+a)/e/(-a*e^3+b*d^3)^2-1/2*ln(c*(b*x^3+a)^p)/e/(e*x+d)^2-1/2*a^(1/3)*b^(2/3)*(2*b*d^3-3*a^(1
/3)*b^(2/3)*d^2*e+a*e^3)*p*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))*3^(1/2)/(-a*e^3+b*d^3)^2
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Rubi [A]
time = 0.48, antiderivative size = 391, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2513, 6857,
1885, 1874, 31, 648, 631, 210, 642, 266} \begin {gather*} -\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} \sqrt [3]{a} b^{2/3} p \text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-3 \sqrt [3]{a} b^{2/3} d^2 e+a e^3+2 b d^3\right )}{2 \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 {\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 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}+\frac {3 b d^2 p}{2 e (d+e x) \left (b d^3-a e^3\right )} \end {gather*}
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 31
Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]
Rule 210
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
Rule 266
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]
Rule 631
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 642
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 648
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 1874
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 1885
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 2513
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 6857
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]
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 ) \text {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*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.45, size = 303, normalized size = 0.77 \begin {gather*} \frac {\frac {b^{2/3} p (d+e x) \left (6 \sqrt [3]{b} d^2 \left (b d^3-a e^3\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]{a} e \left (2 b d^3+a e^3\right ) (d+e x) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-6 \sqrt [3]{b} d \left (b d^3+2 a e^3\right ) (d+e x) \log (d+e x)-\sqrt [3]{a} e \left (2 b d^3+a e^3\right ) (d+e x) \left (2 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )\right )+2 \sqrt [3]{b} d \left (b d^3+2 a e^3\right ) (d+e x) \log \left (a+b x^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} \end {gather*}
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)
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.88, size = 4085, normalized size = 10.45
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| method |
result |
size |
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| risch |
\(\text {Expression too large to display}\) |
\(4085\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(ln(c*(b*x^3+a)^p)/(e*x+d)^3,x,method=_RETURNVERBOSE)
[Out]
-1/2/e/(e*x+d)^2*ln((b*x^3+a)^p)+1/4*(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=Roo
tOf((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-I*Pi*a^2*e^6*csgn(I*c*(b*x^3+a)^
p)^2*csgn(I*c)-I*Pi*b^2*d^6*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)+6*b^2*d^6*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+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*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=Roo
tOf((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*l
n(((-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*(b*x^3+a)^p)*csgn(I*c*(b*x^3+a)^p)*csgn(I*c)-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+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+6*b^2*d^5*e*p*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...
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Maxima [A]
time = 0.56, size = 500, normalized size = 1.28 \begin {gather*} \frac {1}{4} \, {\left (\frac {2 \, \sqrt {3} {\left (2 \, a b d^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}} e - 3 \, a b d^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} e^{2} + a^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}} e^{4}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{{\left (b^{3} d^{6} \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a b^{2} d^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}} e^{3} + a^{2} b \left (\frac {a}{b}\right )^{\frac {2}{3}} e^{6}\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {6 \, d^{2}}{b d^{4} - a d e^{3} + {\left (b d^{3} e - a e^{4}\right )} x} + \frac {{\left (2 \, b^{2} d^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a b d^{3} e - 3 \, a b d^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}} e^{2} + 4 \, a b d \left (\frac {a}{b}\right )^{\frac {2}{3}} e^{3} - a^{2} e^{4}\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{b^{3} d^{6} \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a b^{2} d^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}} e^{3} + a^{2} b \left (\frac {a}{b}\right )^{\frac {2}{3}} e^{6}} - \frac {6 \, {\left (b d^{4} + 2 \, a d e^{3}\right )} \log \left (x e + d\right )}{b^{2} d^{6} - 2 \, a b d^{3} e^{3} + a^{2} e^{6}} + \frac {2 \, {\left (b^{2} d^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}} + 2 \, a b d^{3} e + 3 \, a b d^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}} e^{2} + 2 \, a b d \left (\frac {a}{b}\right )^{\frac {2}{3}} e^{3} + a^{2} e^{4}\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{b^{3} d^{6} \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a b^{2} d^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}} e^{3} + a^{2} b \left (\frac {a}{b}\right )^{\frac {2}{3}} e^{6}}\right )} b p e^{\left (-1\right )} - \frac {e^{\left (-1\right )} \log \left ({\left (b x^{3} + a\right )}^{p} c\right )}{2 \, {\left (x e + d\right )}^{2}} \end {gather*}
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]
1/4*(2*sqrt(3)*(2*a*b*d^3*(a/b)^(1/3)*e - 3*a*b*d^2*(a/b)^(2/3)*e^2 + a^2*(a/b)^(1/3)*e^4)*arctan(1/3*sqrt(3)*
(2*x - (a/b)^(1/3))/(a/b)^(1/3))/((b^3*d^6*(a/b)^(2/3) - 2*a*b^2*d^3*(a/b)^(2/3)*e^3 + a^2*b*(a/b)^(2/3)*e^6)*
(a/b)^(1/3)) + 6*d^2/(b*d^4 - a*d*e^3 + (b*d^3*e - a*e^4)*x) + (2*b^2*d^4*(a/b)^(2/3) - 2*a*b*d^3*e - 3*a*b*d^
2*(a/b)^(1/3)*e^2 + 4*a*b*d*(a/b)^(2/3)*e^3 - a^2*e^4)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(b^3*d^6*(a/b)^(
2/3) - 2*a*b^2*d^3*(a/b)^(2/3)*e^3 + a^2*b*(a/b)^(2/3)*e^6) - 6*(b*d^4 + 2*a*d*e^3)*log(x*e + d)/(b^2*d^6 - 2*
a*b*d^3*e^3 + a^2*e^6) + 2*(b^2*d^4*(a/b)^(2/3) + 2*a*b*d^3*e + 3*a*b*d^2*(a/b)^(1/3)*e^2 + 2*a*b*d*(a/b)^(2/3
)*e^3 + a^2*e^4)*log(x + (a/b)^(1/3))/(b^3*d^6*(a/b)^(2/3) - 2*a*b^2*d^3*(a/b)^(2/3)*e^3 + a^2*b*(a/b)^(2/3)*e
^6))*b*p*e^(-1) - 1/2*e^(-1)*log((b*x^3 + a)^p*c)/(x*e + d)^2
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Fricas [C] Result contains complex when optimal does not.
time = 4.72, size = 12591, normalized size = 32.20 \begin {gather*} \text {Too large to display} \end {gather*}
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*p*x*e + 24*b^2*d^6*p - 24*a*b*d^2*p*x*e^4 - 24*a*b*d^3*p*e^3 + 2*(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)*((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*p*e^3)^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*p*e^3)*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*p*e^3)^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*p*e^3)*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*p*e^3)^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*p*e^3)/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7))*l
og(8*b^2*d^3*p^2*x*e - 7*b^2*d^4*p^2 + a*b*p^2*x*e^4 - 2*a*b*d*p^2*e^3 - 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*p*e^3)^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*p*e^3)*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*p*e^3)^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*p*e^3)*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*p*e^3)^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*p*e^3)/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7
))^2 + 1/4*(10*b^2*d^6*p*e + 16*a*b*d^3*p*e^4 + a^2*p*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*p*e^3)^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*p*e^3)*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*p*e^3)^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*p*e^3)*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*p*e^3)^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*p*e^3)/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7))) + (12*b^2*d^4*p*x^2*e^2 + 24*b^2*d^5*p*x*e + 12*b^2*d^6*p +
24*a*b*d*p*x^2*e^5 + 48*a*b*d^2*p*x*e^4 + 24*a*b*d^3*p*e^3 - (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)*((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*p*e^3)^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*p*e^3)*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*p*e^3)^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*p*e^3)*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*p*e^3)^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*p*e^3)/(b^2*d^6*e - 2*a*b*d^3*e^4 + a^2*e^7)) + sqrt(3)*(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)*sqrt(-(16*b^4*d^8*p^2 - 320*a*b^3*d^5*p^2*e^3 - 128*a^2*b^2*d^2*p^2*e^6 + (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*p*e^3)^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*p*e^3)*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*p*e^3)^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*p*e^3)*b^2*d^2*p^2/((b^2*d^6*e^...
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 790 vs.
\(2 (325) = 650\).
time = 3.25, size = 790, normalized size = 2.02 \begin {gather*} -\frac {{\left (2 \, a b^{5} d^{9} p e^{2} - 3 \, a b^{5} d^{8} p \left (-\frac {a}{b}\right )^{\frac {1}{3}} e^{3} - 3 \, a^{2} b^{4} d^{6} p e^{5} + 6 \, a^{2} b^{4} d^{5} p \left (-\frac {a}{b}\right )^{\frac {1}{3}} e^{6} - 3 \, a^{3} b^{3} d^{2} p \left (-\frac {a}{b}\right )^{\frac {1}{3}} e^{9} + a^{4} b^{2} p e^{11}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{2 \, {\left (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}\right )}} + \frac {3 \, {\left (2 \, \left (-a b^{2}\right )^{\frac {1}{3}} b d p - \left (-a b^{2}\right )^{\frac {2}{3}} p e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{2 \, {\left (\sqrt {3} b^{2} d^{4} - 2 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} b d^{3} e + 2 \, \sqrt {3} a b d e^{3} + 3 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} d^{2} e^{2} - \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} a e^{4}\right )}} + \frac {{\left (2 \, \left (-a b^{2}\right )^{\frac {1}{3}} b d^{3} p - 3 \, \left (-a b^{2}\right )^{\frac {2}{3}} d^{2} p e + \left (-a b^{2}\right )^{\frac {1}{3}} a p e^{3}\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{4 \, {\left (b^{2} d^{6} - 2 \, a b d^{3} e^{3} + a^{2} e^{6}\right )}} + \frac {{\left (b^{2} d^{4} p + 2 \, a b d p e^{3}\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{2 \, {\left (b^{2} d^{6} e - 2 \, a b d^{3} e^{4} + a^{2} e^{7}\right )}} - \frac {3 \, b^{2} d^{4} p x^{2} e^{2} \log \left (x e + d\right ) + 6 \, b^{2} d^{5} p x e \log \left (x e + d\right ) - 3 \, b^{2} d^{5} p x e + b^{2} d^{6} p \log \left (b x^{3} + a\right ) + 3 \, b^{2} d^{6} p \log \left (x e + d\right ) - 3 \, b^{2} d^{6} p + b^{2} d^{6} \log \left (c\right ) - 2 \, a b d^{3} p e^{3} \log \left (b x^{3} + a\right ) + 6 \, a b d p x^{2} e^{5} \log \left (x e + d\right ) + 12 \, a b d^{2} p x e^{4} \log \left (x e + d\right ) + 6 \, a b d^{3} p e^{3} \log \left (x e + d\right ) + 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 \left (c\right ) + a^{2} p e^{6} \log \left (b x^{3} + a\right ) + a^{2} e^{6} \log \left (c\right )}{2 \, {\left (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}\right )}} \end {gather*}
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) + 3/2*(2*(-a*b^2)^(1/3)*b*d*p -
(-a*b^2)^(2/3)*p*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(sqrt(3)*b^2*d^4 - 2*sqrt(3)*(-a*b^2
)^(1/3)*b*d^3*e + 2*sqrt(3)*a*b*d*e^3 + 3*sqrt(3)*(-a*b^2)^(2/3)*d^2*e^2 - sqrt(3)*(-a*b^2)^(1/3)*a*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)
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Mupad [B]
time = 0.88, size = 2227, normalized size = 5.70 \begin {gather*} \left (\sum _{k=1}^3\ln \left (-\frac {27\,a\,b^6\,d^4\,p^3+{\mathrm {root}\left (16\,a\,b\,d^3\,e^6\,z^3-8\,b^2\,d^6\,e^3\,z^3-8\,a^2\,e^9\,z^3+24\,a\,b\,d\,e^5\,p\,z^2+12\,b^2\,d^4\,e^2\,p\,z^2-6\,b^2\,d^2\,e\,p^2\,z+b^2\,p^3,z,k\right )}^3\,a^2\,b^5\,d^7\,e^6\,216-{\mathrm {root}\left (16\,a\,b\,d^3\,e^6\,z^3-8\,b^2\,d^6\,e^3\,z^3-8\,a^2\,e^9\,z^3+24\,a\,b\,d\,e^5\,p\,z^2+12\,b^2\,d^4\,e^2\,p\,z^2-6\,b^2\,d^2\,e\,p^2\,z+b^2\,p^3,z,k\right )}^3\,a^3\,b^4\,d^4\,e^9\,648+{\mathrm {root}\left (16\,a\,b\,d^3\,e^6\,z^3-8\,b^2\,d^6\,e^3\,z^3-8\,a^2\,e^9\,z^3+24\,a\,b\,d\,e^5\,p\,z^2+12\,b^2\,d^4\,e^2\,p\,z^2-6\,b^2\,d^2\,e\,p^2\,z+b^2\,p^3,z,k\right )}^3\,a\,b^6\,d^{10}\,e^3\,72+{\mathrm {root}\left (16\,a\,b\,d^3\,e^6\,z^3-8\,b^2\,d^6\,e^3\,z^3-8\,a^2\,e^9\,z^3+24\,a\,b\,d\,e^5\,p\,z^2+12\,b^2\,d^4\,e^2\,p\,z^2-6\,b^2\,d^2\,e\,p^2\,z+b^2\,p^3,z,k\right )}^3\,a^4\,b^3\,d\,e^{12}\,360+\mathrm {root}\left (16\,a\,b\,d^3\,e^6\,z^3-8\,b^2\,d^6\,e^3\,z^3-8\,a^2\,e^9\,z^3+24\,a\,b\,d\,e^5\,p\,z^2+12\,b^2\,d^4\,e^2\,p\,z^2-6\,b^2\,d^2\,e\,p^2\,z+b^2\,p^3,z,k\right )\,a^3\,b^4\,e^7\,p^2\,18+{\mathrm {root}\left (16\,a\,b\,d^3\,e^6\,z^3-8\,b^2\,d^6\,e^3\,z^3-8\,a^2\,e^9\,z^3+24\,a\,b\,d\,e^5\,p\,z^2+12\,b^2\,d^4\,e^2\,p\,z^2-6\,b^2\,d^2\,e\,p^2\,z+b^2\,p^3,z,k\right )}^3\,a^4\,b^3\,e^{13}\,x\,288+27\,a^2\,b^5\,d\,e^3\,p^3-27\,a^2\,b^5\,e^4\,p^3\,x+{\mathrm {root}\left (16\,a\,b\,d^3\,e^6\,z^3-8\,b^2\,d^6\,e^3\,z^3-8\,a^2\,e^9\,z^3+24\,a\,b\,d\,e^5\,p\,z^2+12\,b^2\,d^4\,e^2\,p\,z^2-6\,b^2\,d^2\,e\,p^2\,z+b^2\,p^3,z,k\right )}^2\,a\,b^6\,d^8\,e^2\,p\,36+{\mathrm {root}\left (16\,a\,b\,d^3\,e^6\,z^3-8\,b^2\,d^6\,e^3\,z^3-8\,a^2\,e^9\,z^3+24\,a\,b\,d\,e^5\,p\,z^2+12\,b^2\,d^4\,e^2\,p\,z^2-6\,b^2\,d^2\,e\,p^2\,z+b^2\,p^3,z,k\right )}^3\,a\,b^6\,d^9\,e^4\,x\,144-\mathrm {root}\left (16\,a\,b\,d^3\,e^6\,z^3-8\,b^2\,d^6\,e^3\,z^3-8\,a^2\,e^9\,z^3+24\,a\,b\,d\,e^5\,p\,z^2+12\,b^2\,d^4\,e^2\,p\,z^2-6\,b^2\,d^2\,e\,p^2\,z+b^2\,p^3,z,k\right )\,a^2\,b^5\,d^3\,e^4\,p^2\,90+{\mathrm {root}\left (16\,a\,b\,d^3\,e^6\,z^3-8\,b^2\,d^6\,e^3\,z^3-8\,a^2\,e^9\,z^3+24\,a\,b\,d\,e^5\,p\,z^2+12\,b^2\,d^4\,e^2\,p\,z^2-6\,b^2\,d^2\,e\,p^2\,z+b^2\,p^3,z,k\right )}^2\,a^2\,b^5\,d^5\,e^5\,p\,252-{\mathrm {root}\left (16\,a\,b\,d^3\,e^6\,z^3-8\,b^2\,d^6\,e^3\,z^3-8\,a^2\,e^9\,z^3+24\,a\,b\,d\,e^5\,p\,z^2+12\,b^2\,d^4\,e^2\,p\,z^2-6\,b^2\,d^2\,e\,p^2\,z+b^2\,p^3,z,k\right )}^2\,a^3\,b^4\,d^2\,e^8\,p\,288-{\mathrm {root}\left (16\,a\,b\,d^3\,e^6\,z^3-8\,b^2\,d^6\,e^3\,z^3-8\,a^2\,e^9\,z^3+24\,a\,b\,d\,e^5\,p\,z^2+12\,b^2\,d^4\,e^2\,p\,z^2-6\,b^2\,d^2\,e\,p^2\,z+b^2\,p^3,z,k\right )}^3\,a^3\,b^4\,d^3\,e^{10}\,x\,432-\mathrm {root}\left (16\,a\,b\,d^3\,e^6\,z^3-8\,b^2\,d^6\,e^3\,z^3-8\,a^2\,e^9\,z^3+24\,a\,b\,d\,e^5\,p\,z^2+12\,b^2\,d^4\,e^2\,p\,z^2-6\,b^2\,d^2\,e\,p^2\,z+b^2\,p^3,z,k\right )\,a\,b^6\,d^6\,e\,p^2\,90-\mathrm {root}\left (16\,a\,b\,d^3\,e^6\,z^3-8\,b^2\,d^6\,e^3\,z^3-8\,a^2\,e^9\,z^3+24\,a\,b\,d\,e^5\,p\,z^2+12\,b^2\,d^4\,e^2\,p\,z^2-6\,b^2\,d^2\,e\,p^2\,z+b^2\,p^3,z,k\right )\,a^2\,b^5\,d^2\,e^5\,p^2\,x\,54+{\mathrm {root}\left (16\,a\,b\,d^3\,e^6\,z^3-8\,b^2\,d^6\,e^3\,z^3-8\,a^2\,e^9\,z^3+24\,a\,b\,d\,e^5\,p\,z^2+12\,b^2\,d^4\,e^2\,p\,z^2-6\,b^2\,d^2\,e\,p^2\,z+b^2\,p^3,z,k\right )}^2\,a^2\,b^5\,d^4\,e^6\,p\,x\,360-\mathrm {root}\left (16\,a\,b\,d^3\,e^6\,z^3-8\,b^2\,d^6\,e^3\,z^3-8\,a^2\,e^9\,z^3+24\,a\,b\,d\,e^5\,p\,z^2+12\,b^2\,d^4\,e^2\,p\,z^2-6\,b^2\,d^2\,e\,p^2\,z+b^2\,p^3,z,k\right )\,a\,b^6\,d^5\,e^2\,p^2\,x\,108+{\mathrm {root}\left (16\,a\,b\,d^3\,e^6\,z^3-8\,b^2\,d^6\,e^3\,z^3-8\,a^2\,e^9\,z^3+24\,a\,b\,d\,e^5\,p\,z^2+12\,b^2\,d^4\,e^2\,p\,z^2-6\,b^2\,d^2\,e\,p^2\,z+b^2\,p^3,z,k\right )}^2\,a\,b^6\,d^7\,e^3\,p\,x\,144-{\mathrm {root}\left (16\,a\,b\,d^3\,e^6\,z^3-8\,b^2\,d^6\,e^3\,z^3-8\,a^2\,e^9\,z^3+24\,a\,b\,d\,e^5\,p\,z^2+12\,b^2\,d^4\,e^2\,p\,z^2-6\,b^2\,d^2\,e\,p^2\,z+b^2\,p^3,z,k\right )}^2\,a^3\,b^4\,d\,e^9\,p\,x\,504}{8\,a^2\,e^8-16\,a\,b\,d^3\,e^5+8\,b^2\,d^6\,e^2}\right )\,\mathrm {root}\left (16\,a\,b\,d^3\,e^6\,z^3-8\,b^2\,d^6\,e^3\,z^3-8\,a^2\,e^9\,z^3+24\,a\,b\,d\,e^5\,p\,z^2+12\,b^2\,d^4\,e^2\,p\,z^2-6\,b^2\,d^2\,e\,p^2\,z+b^2\,p^3,z,k\right )\right )-\frac {\ln \left (c\,{\left (b\,x^3+a\right )}^p\right )}{2\,\left (d^2\,e+2\,d\,e^2\,x+e^3\,x^2\right )}-\frac {3\,b\,d^2\,p}{-2\,b\,d^4\,e-2\,b\,x\,d^3\,e^2+2\,a\,d\,e^4+2\,a\,x\,e^5}-\frac {3\,b^2\,d^4\,p\,\ln \left (d+e\,x\right )}{2\,a^2\,e^7-4\,a\,b\,d^3\,e^4+2\,b^2\,d^6\,e}-\frac {6\,a\,b\,d\,e^3\,p\,\ln \left (d+e\,x\right )}{2\,a^2\,e^7-4\,a\,b\,d^3\,e^4+2\,b^2\,d^6\,e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(log(c*(a + b*x^3)^p)/(d + e*x)^3,x)
[Out]
symsum(log(-(27*a*b^6*d^4*p^3 + 216*root(16*a*b*d^3*e^6*z^3 - 8*b^2*d^6*e^3*z^3 - 8*a^2*e^9*z^3 + 24*a*b*d*e^5
*p*z^2 + 12*b^2*d^4*e^2*p*z^2 - 6*b^2*d^2*e*p^2*z + b^2*p^3, z, k)^3*a^2*b^5*d^7*e^6 - 648*root(16*a*b*d^3*e^6
*z^3 - 8*b^2*d^6*e^3*z^3 - 8*a^2*e^9*z^3 + 24*a*b*d*e^5*p*z^2 + 12*b^2*d^4*e^2*p*z^2 - 6*b^2*d^2*e*p^2*z + b^2
*p^3, z, k)^3*a^3*b^4*d^4*e^9 + 72*root(16*a*b*d^3*e^6*z^3 - 8*b^2*d^6*e^3*z^3 - 8*a^2*e^9*z^3 + 24*a*b*d*e^5*
p*z^2 + 12*b^2*d^4*e^2*p*z^2 - 6*b^2*d^2*e*p^2*z + b^2*p^3, z, k)^3*a*b^6*d^10*e^3 + 360*root(16*a*b*d^3*e^6*z
^3 - 8*b^2*d^6*e^3*z^3 - 8*a^2*e^9*z^3 + 24*a*b*d*e^5*p*z^2 + 12*b^2*d^4*e^2*p*z^2 - 6*b^2*d^2*e*p^2*z + b^2*p
^3, z, k)^3*a^4*b^3*d*e^12 + 18*root(16*a*b*d^3*e^6*z^3 - 8*b^2*d^6*e^3*z^3 - 8*a^2*e^9*z^3 + 24*a*b*d*e^5*p*z
^2 + 12*b^2*d^4*e^2*p*z^2 - 6*b^2*d^2*e*p^2*z + b^2*p^3, z, k)*a^3*b^4*e^7*p^2 + 288*root(16*a*b*d^3*e^6*z^3 -
8*b^2*d^6*e^3*z^3 - 8*a^2*e^9*z^3 + 24*a*b*d*e^5*p*z^2 + 12*b^2*d^4*e^2*p*z^2 - 6*b^2*d^2*e*p^2*z + b^2*p^3,
z, k)^3*a^4*b^3*e^13*x + 27*a^2*b^5*d*e^3*p^3 - 27*a^2*b^5*e^4*p^3*x + 36*root(16*a*b*d^3*e^6*z^3 - 8*b^2*d^6*
e^3*z^3 - 8*a^2*e^9*z^3 + 24*a*b*d*e^5*p*z^2 + 12*b^2*d^4*e^2*p*z^2 - 6*b^2*d^2*e*p^2*z + b^2*p^3, z, k)^2*a*b
^6*d^8*e^2*p + 144*root(16*a*b*d^3*e^6*z^3 - 8*b^2*d^6*e^3*z^3 - 8*a^2*e^9*z^3 + 24*a*b*d*e^5*p*z^2 + 12*b^2*d
^4*e^2*p*z^2 - 6*b^2*d^2*e*p^2*z + b^2*p^3, z, k)^3*a*b^6*d^9*e^4*x - 90*root(16*a*b*d^3*e^6*z^3 - 8*b^2*d^6*e
^3*z^3 - 8*a^2*e^9*z^3 + 24*a*b*d*e^5*p*z^2 + 12*b^2*d^4*e^2*p*z^2 - 6*b^2*d^2*e*p^2*z + b^2*p^3, z, k)*a^2*b^
5*d^3*e^4*p^2 + 252*root(16*a*b*d^3*e^6*z^3 - 8*b^2*d^6*e^3*z^3 - 8*a^2*e^9*z^3 + 24*a*b*d*e^5*p*z^2 + 12*b^2*
d^4*e^2*p*z^2 - 6*b^2*d^2*e*p^2*z + b^2*p^3, z, k)^2*a^2*b^5*d^5*e^5*p - 288*root(16*a*b*d^3*e^6*z^3 - 8*b^2*d
^6*e^3*z^3 - 8*a^2*e^9*z^3 + 24*a*b*d*e^5*p*z^2 + 12*b^2*d^4*e^2*p*z^2 - 6*b^2*d^2*e*p^2*z + b^2*p^3, z, k)^2*
a^3*b^4*d^2*e^8*p - 432*root(16*a*b*d^3*e^6*z^3 - 8*b^2*d^6*e^3*z^3 - 8*a^2*e^9*z^3 + 24*a*b*d*e^5*p*z^2 + 12*
b^2*d^4*e^2*p*z^2 - 6*b^2*d^2*e*p^2*z + b^2*p^3, z, k)^3*a^3*b^4*d^3*e^10*x - 90*root(16*a*b*d^3*e^6*z^3 - 8*b
^2*d^6*e^3*z^3 - 8*a^2*e^9*z^3 + 24*a*b*d*e^5*p*z^2 + 12*b^2*d^4*e^2*p*z^2 - 6*b^2*d^2*e*p^2*z + b^2*p^3, z, k
)*a*b^6*d^6*e*p^2 - 54*root(16*a*b*d^3*e^6*z^3 - 8*b^2*d^6*e^3*z^3 - 8*a^2*e^9*z^3 + 24*a*b*d*e^5*p*z^2 + 12*b
^2*d^4*e^2*p*z^2 - 6*b^2*d^2*e*p^2*z + b^2*p^3, z, k)*a^2*b^5*d^2*e^5*p^2*x + 360*root(16*a*b*d^3*e^6*z^3 - 8*
b^2*d^6*e^3*z^3 - 8*a^2*e^9*z^3 + 24*a*b*d*e^5*p*z^2 + 12*b^2*d^4*e^2*p*z^2 - 6*b^2*d^2*e*p^2*z + b^2*p^3, z,
k)^2*a^2*b^5*d^4*e^6*p*x - 108*root(16*a*b*d^3*e^6*z^3 - 8*b^2*d^6*e^3*z^3 - 8*a^2*e^9*z^3 + 24*a*b*d*e^5*p*z^
2 + 12*b^2*d^4*e^2*p*z^2 - 6*b^2*d^2*e*p^2*z + b^2*p^3, z, k)*a*b^6*d^5*e^2*p^2*x + 144*root(16*a*b*d^3*e^6*z^
3 - 8*b^2*d^6*e^3*z^3 - 8*a^2*e^9*z^3 + 24*a*b*d*e^5*p*z^2 + 12*b^2*d^4*e^2*p*z^2 - 6*b^2*d^2*e*p^2*z + b^2*p^
3, z, k)^2*a*b^6*d^7*e^3*p*x - 504*root(16*a*b*d^3*e^6*z^3 - 8*b^2*d^6*e^3*z^3 - 8*a^2*e^9*z^3 + 24*a*b*d*e^5*
p*z^2 + 12*b^2*d^4*e^2*p*z^2 - 6*b^2*d^2*e*p^2*z + b^2*p^3, z, k)^2*a^3*b^4*d*e^9*p*x)/(8*a^2*e^8 + 8*b^2*d^6*
e^2 - 16*a*b*d^3*e^5))*root(16*a*b*d^3*e^6*z^3 - 8*b^2*d^6*e^3*z^3 - 8*a^2*e^9*z^3 + 24*a*b*d*e^5*p*z^2 + 12*b
^2*d^4*e^2*p*z^2 - 6*b^2*d^2*e*p^2*z + b^2*p^3, z, k), k, 1, 3) - log(c*(a + b*x^3)^p)/(2*(d^2*e + e^3*x^2 + 2
*d*e^2*x)) - (3*b*d^2*p)/(2*a*d*e^4 - 2*b*d^4*e + 2*a*e^5*x - 2*b*d^3*e^2*x) - (3*b^2*d^4*p*log(d + e*x))/(2*a
^2*e^7 + 2*b^2*d^6*e - 4*a*b*d^3*e^4) - (6*a*b*d*e^3*p*log(d + e*x))/(2*a^2*e^7 + 2*b^2*d^6*e - 4*a*b*d^3*e^4)
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