\(\int \frac {\log (c (a+b x^3)^p)}{(d+e x)^3} \, dx\) [197]

Optimal result

Integrand size = 20, antiderivative size = 391 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{(d+e x)^3} \, dx=\frac {3 b d^2 p}{2 e \left (b d^3-a e^3\right ) (d+e x)}-\frac {\sqrt {3} \sqrt [3]{a} b^{2/3} \left (2 b d^3-3 \sqrt [3]{a} b^{2/3} d^2 e+a e^3\right ) p \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{2 \left (b d^3-a e^3\right )^2}+\frac {\sqrt [3]{a} b^{2/3} \left (2 b d^3+3 \sqrt [3]{a} b^{2/3} d^2 e+a e^3\right ) p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 \left (b d^3-a e^3\right )^2}-\frac {3 b d \left (b d^3+2 a e^3\right ) p \log (d+e x)}{2 e \left (b d^3-a e^3\right )^2}-\frac {\sqrt [3]{a} b^{2/3} \left (2 b d^3+3 \sqrt [3]{a} b^{2/3} d^2 e+a e^3\right ) p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{4 \left (b d^3-a e^3\right )^2}+\frac {b d \left (b d^3+2 a e^3\right ) p \log \left (a+b x^3\right )}{2 e \left (b d^3-a e^3\right )^2}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{2 e (d+e x)^2} \]

[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

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2513, 6857, 1885, 1874, 31, 648, 631, 210, 642, 266} \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{(d+e x)^3} \, dx=-\frac {\sqrt [3]{a} b^{2/3} p \left (3 \sqrt [3]{a} b^{2/3} d^2 e+a e^3+2 b d^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{4 \left (b d^3-a e^3\right )^2}-\frac {\sqrt {3} \sqrt [3]{a} b^{2/3} p \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-3 \sqrt [3]{a} b^{2/3} d^2 e+a e^3+2 b d^3\right )}{2 \left (b d^3-a e^3\right )^2}+\frac {\sqrt [3]{a} b^{2/3} p \left (3 \sqrt [3]{a} b^{2/3} d^2 e+a e^3+2 b d^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 \left (b d^3-a e^3\right )^2}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{2 e (d+e x)^2}+\frac {b d p \left (2 a e^3+b d^3\right ) \log \left (a+b x^3\right )}{2 e \left (b d^3-a e^3\right )^2}-\frac {3 b d p \left (2 a e^3+b d^3\right ) \log (d+e x)}{2 e \left (b d^3-a e^3\right )^2}+\frac {3 b d^2 p}{2 e (d+e x) \left (b d^3-a e^3\right )} \]

[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*} \text {integral}& = -\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*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.39 (sec) , antiderivative size = 325, normalized size of antiderivative = 0.83 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{(d+e x)^3} \, dx=\frac {\frac {b^{2/3} p (d+e x) \left (6 \sqrt [3]{b} d^2 \left (b d^3-a e^3\right )-2 \sqrt {3} \sqrt [3]{a} e \left (2 b d^3+a e^3\right ) (d+e x) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )-9 b^{4/3} d^2 e^2 x^2 (d+e x) \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {5}{3},-\frac {b x^3}{a}\right )+2 \sqrt [3]{a} e \left (2 b d^3+a e^3\right ) (d+e x) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-6 \sqrt [3]{b} d \left (b d^3+2 a e^3\right ) (d+e x) \log (d+e x)-\sqrt [3]{a} e \left (2 b d^3+a e^3\right ) (d+e x) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+2 \sqrt [3]{b} d \left (b d^3+2 a e^3\right ) (d+e x) \log \left (a+b x^3\right )\right )}{\left (b d^3-a e^3\right )^2}-2 \log \left (c \left (a+b x^3\right )^p\right )}{4 e (d+e x)^2} \]

[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) - 2*Sqrt[3]*a^(1/3)*e*(2*b*d^3 + a*e^3)*(d + e*x)*ArcTan[
(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[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)*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 [A] (verified)

Time = 2.62 (sec) , antiderivative size = 347, normalized size of antiderivative = 0.89

method	result	size
parts	\(-\frac {\ln \left (c \left (b \,x^{3}+a \right )^{p}\right )}{2 e \left (e x +d \right )^{2}}+\frac {3 p b \left (-\frac {d^{2}}{\left (a \,e^{3}-b \,d^{3}\right ) \left (e x +d \right )}-\frac {d \left (2 a \,e^{3}+b \,d^{3}\right ) \ln \left (e x +d \right )}{\left (a \,e^{3}-b \,d^{3}\right )^{2}}+\frac {\left (a^{2} e^{4}+2 a b \,d^{3} e \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )-3 a b \,d^{2} e^{2} \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )+\frac {\left (2 a b d \,e^{3}+b^{2} d^{4}\right ) \ln \left (b \,x^{3}+a \right )}{3 b}}{\left (a \,e^{3}-b \,d^{3}\right )^{2}}\right )}{2 e}\)	\(347\)
risch	\(\text {Expression too large to display}\)	\(4085\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 9.42 (sec) , antiderivative size = 13236, normalized size of antiderivative = 33.85 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{(d+e x)^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{(d+e x)^3} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 517, normalized size of antiderivative = 1.32 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{(d+e x)^3} \, dx=-\frac {{\left (\frac {2 \, \sqrt {3} {\left (3 \, a b d^{2} e^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a b d^{3} e \left (\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} e^{4} \left (\frac {a}{b}\right )^{\frac {1}{3}}\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} e^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}} + a^{2} b e^{6} \left (\frac {a}{b}\right )^{\frac {2}{3}}\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 {6 \, {\left (b d^{4} + 2 \, a d e^{3}\right )} \log \left (e x + d\right )}{b^{2} d^{6} - 2 \, a b d^{3} e^{3} + a^{2} e^{6}} - \frac {{\left (2 \, b^{2} d^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}} + 4 \, a b d e^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}} - 3 \, a b d^{2} e^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, a b d^{3} e - 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} e^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}} + a^{2} b e^{6} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {2 \, {\left (b^{2} d^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}} + 2 \, a b d e^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}} + 3 \, a b d^{2} e^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, a b d^{3} e + 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} e^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}} + a^{2} b e^{6} \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )} b p}{4 \, e} - \frac {\log \left ({\left (b x^{3} + a\right )}^{p} c\right )}{2 \, {\left (e x + d\right )}^{2} e} \]

[In]

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

[Out]

-1/4*(2*sqrt(3)*(3*a*b*d^2*e^2*(a/b)^(2/3) - 2*a*b*d^3*e*(a/b)^(1/3) - a^2*e^4*(a/b)^(1/3))*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*e^3*(a/b)^(2/3) + a^2*b*e^6*(a/b)^(2/3))
*(a/b)^(1/3)) - 6*d^2/(b*d^4 - a*d*e^3 + (b*d^3*e - a*e^4)*x) + 6*(b*d^4 + 2*a*d*e^3)*log(e*x + 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) + 4*a*b*d*e^3*(a/b)^(2/3) - 3*a*b*d^2*e^2*(a/b)^(1/3) - 2*a*
b*d^3*e - 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*e^3*(a/b)^(2/3) +
a^2*b*e^6*(a/b)^(2/3)) - 2*(b^2*d^4*(a/b)^(2/3) + 2*a*b*d*e^3*(a/b)^(2/3) + 3*a*b*d^2*e^2*(a/b)^(1/3) + 2*a*b
*d^3*e + a^2*e^4)*log(x + (a/b)^(1/3))/(b^3*d^6*(a/b)^(2/3) - 2*a*b^2*d^3*e^3*(a/b)^(2/3) + a^2*b*e^6*(a/b)^(2
/3)))*b*p/e - 1/2*log((b*x^3 + a)^p*c)/((e*x + d)^2*e)

Giac [A] (verification not implemented)

none

Time = 0.46 (sec) , antiderivative size = 660, normalized size of antiderivative = 1.69 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{(d+e x)^3} \, dx=\frac {{\left (3 \, a b^{5} d^{8} e^{3} p \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 6 \, a^{2} b^{4} d^{5} e^{6} p \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 3 \, a^{3} b^{3} d^{2} e^{9} p \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, a b^{5} d^{9} e^{2} p + 3 \, a^{2} b^{4} d^{6} e^{5} p - a^{4} b^{2} e^{11} p\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}} e p\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} a b d e^{3} - 2 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} b d^{3} e - \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} a e^{4} + 3 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} d^{2} e^{2}\right )}} - \frac {p \log \left (b x^{3} + a\right )}{2 \, {\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} - \frac {3 \, {\left (b^{2} d^{4} p + 2 \, a b d e^{3} p\right )} \log \left (e x + d\right )}{2 \, {\left (b^{2} d^{6} e - 2 \, a b d^{3} e^{4} + a^{2} e^{7}\right )}} + \frac {{\left (2 \, \left (-a b^{2}\right )^{\frac {1}{3}} b d^{3} p + \left (-a b^{2}\right )^{\frac {1}{3}} a e^{3} p - 3 \, \left (-a b^{2}\right )^{\frac {2}{3}} d^{2} e p\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 e^{3} p\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 d^{2} e p x + 3 \, b d^{3} p - b d^{3} \log \left (c\right ) + a e^{3} \log \left (c\right )}{2 \, {\left (b d^{3} e^{3} x^{2} - a e^{6} x^{2} + 2 \, b d^{4} e^{2} x - 2 \, a d e^{5} x + b d^{5} e - a d^{2} e^{4}\right )}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 1.92 (sec) , antiderivative size = 2227, normalized size of antiderivative = 5.70 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{(d+e x)^3} \, dx=\text {Too large to display} \]

[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)