\(\int (d+e x) \log (c (a+b x^3)^p) \, dx\) [193]

Optimal result

Integrand size = 18, antiderivative size = 229 \[ \int (d+e x) \log \left (c \left (a+b x^3\right )^p\right ) \, dx=-3 d p x-\frac {3}{4} e p x^2-\frac {\sqrt {3} \sqrt [3]{a} \left (2 \sqrt [3]{b} d+\sqrt [3]{a} e\right ) p \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{2 b^{2/3}}+\frac {\sqrt [3]{a} \left (2 \sqrt [3]{b} d-\sqrt [3]{a} e\right ) p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 b^{2/3}}-\frac {\sqrt [3]{a} \left (2 \sqrt [3]{b} d-\sqrt [3]{a} e\right ) p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{4 b^{2/3}}-\frac {d^2 p \log \left (a+b x^3\right )}{2 e}+\frac {(d+e x)^2 \log \left (c \left (a+b x^3\right )^p\right )}{2 e} \]

[Out]

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

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {2513, 1901, 1885, 1874, 31, 648, 631, 210, 642, 266} \[ \int (d+e x) \log \left (c \left (a+b x^3\right )^p\right ) \, dx=-\frac {\sqrt [3]{a} p \left (2 \sqrt [3]{b} d-\sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{4 b^{2/3}}-\frac {\sqrt {3} \sqrt [3]{a} p \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (\sqrt [3]{a} e+2 \sqrt [3]{b} d\right )}{2 b^{2/3}}+\frac {\sqrt [3]{a} p \left (2 \sqrt [3]{b} d-\sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 b^{2/3}}+\frac {(d+e x)^2 \log \left (c \left (a+b x^3\right )^p\right )}{2 e}-\frac {d^2 p \log \left (a+b x^3\right )}{2 e}-3 d p x-\frac {3}{4} e p x^2 \]

[In]

Int[(d + e*x)*Log[c*(a + b*x^3)^p],x]

[Out]

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

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 1901

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

Rule 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]

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

Mathematica [C] (verified)

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

Time = 0.05 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.89 \[ \int (d+e x) \log \left (c \left (a+b x^3\right )^p\right ) \, dx=-3 d p x-\frac {3}{4} e p x^2+\frac {\sqrt {3} \sqrt [3]{a} d p \arctan \left (\frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{b}}\right )}{\sqrt [3]{b}}+\frac {3}{4} e p x^2 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {5}{3},-\frac {b x^3}{a}\right )+\frac {\sqrt [3]{a} d p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}-\frac {\sqrt [3]{a} d p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}+d x \log \left (c \left (a+b x^3\right )^p\right )+\frac {1}{2} e x^2 \log \left (c \left (a+b x^3\right )^p\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.71 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.08

method	result	size
parts	\(\frac {\ln \left (c \left (b \,x^{3}+a \right )^{p}\right ) e \,x^{2}}{2}+d \ln \left (c \left (b \,x^{3}+a \right )^{p}\right ) x -\frac {3 p b \left (\frac {\frac {1}{2} e \,x^{2}+2 d x}{b}-\frac {\left (2 d \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+e \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )\right ) a}{b}\right )}{2}\)	\(247\)
risch	\(\left (\frac {1}{2} e \,x^{2}+d x \right ) \ln \left (\left (b \,x^{3}+a \right )^{p}\right )+\frac {i {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i \left (b \,x^{3}+a \right )^{p}\right ) x^{2} e \pi }{4}-\frac {i \pi e \,x^{2} \operatorname {csgn}\left (i \left (b \,x^{3}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{4}-\frac {i \pi e \,x^{2} {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{3}}{4}+\frac {i \operatorname {csgn}\left (i c \right ) {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{2} x^{2} e \pi }{4}+\frac {i x \pi d \,\operatorname {csgn}\left (i \left (b \,x^{3}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{2}}{2}-\frac {i x \pi d \,\operatorname {csgn}\left (i \left (b \,x^{3}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i x \pi d {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{3}}{2}+\frac {i x \pi d {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\frac {\ln \left (c \right ) e \,x^{2}}{2}-\frac {3 e p \,x^{2}}{4}+x \ln \left (c \right ) d -3 d p x +\frac {a p \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b +a \right )}{\sum }\frac {\left (e \textit {\_R} +2 d \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}\right )}{2 b}\)	\(335\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.13 (sec) , antiderivative size = 2284, normalized size of antiderivative = 9.97 \[ \int (d+e x) \log \left (c \left (a+b x^3\right )^p\right ) \, dx=\text {Too large to display} \]

[In]

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

[Out]

-3/4*e*p*x^2 - 3*d*p*x + 1/4*(4*(1/2)^(2/3)*a*d*e*p^2*(-I*sqrt(3) + 1)/(((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*
d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*b) - (1/2)^(1/3)*((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3
)/b^2)^(1/3)*(I*sqrt(3) + 1))*log(4*a*d*e^2*p^2 + 2*(4*(1/2)^(2/3)*a*d*e*p^2*(-I*sqrt(3) + 1)/(((8*b*d^3 + a*e
^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*b) - (1/2)^(1/3)*((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*
b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*(I*sqrt(3) + 1))*b*d^2*p + 1/4*(4*(1/2)^(2/3)*a*d*e*p^2*(-I*sqrt(3) + 1)/(
((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*b) - (1/2)^(1/3)*((8*b*d^3 + a*e^3)*a*
p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*(I*sqrt(3) + 1))^2*b*e + (8*b*d^3 + a*e^3)*p^2*x) - 1/8*(4*
(1/2)^(2/3)*a*d*e*p^2*(-I*sqrt(3) + 1)/(((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3
)*b) - (1/2)^(1/3)*((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*(I*sqrt(3) + 1) - s
qrt(3)*sqrt(-(32*a*d*e*p^2 + (4*(1/2)^(2/3)*a*d*e*p^2*(-I*sqrt(3) + 1)/(((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*
d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*b) - (1/2)^(1/3)*((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3
)/b^2)^(1/3)*(I*sqrt(3) + 1))^2*b)/b))*log(-2*a*d*e^2*p^2 - (4*(1/2)^(2/3)*a*d*e*p^2*(-I*sqrt(3) + 1)/(((8*b*d
^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*b) - (1/2)^(1/3)*((8*b*d^3 + a*e^3)*a*p^3/b^2
+ (8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*(I*sqrt(3) + 1))*b*d^2*p - 1/8*(4*(1/2)^(2/3)*a*d*e*p^2*(-I*sqrt(3
) + 1)/(((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*b) - (1/2)^(1/3)*((8*b*d^3 + a
*e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*(I*sqrt(3) + 1))^2*b*e + (8*b*d^3 + a*e^3)*p^2*x +
1/8*sqrt(3)*(8*b*d^2*p - (4*(1/2)^(2/3)*a*d*e*p^2*(-I*sqrt(3) + 1)/(((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3*
p^3 - a^2*e^3*p^3)/b^2)^(1/3)*b) - (1/2)^(1/3)*((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^
2)^(1/3)*(I*sqrt(3) + 1))*b*e)*sqrt(-(32*a*d*e*p^2 + (4*(1/2)^(2/3)*a*d*e*p^2*(-I*sqrt(3) + 1)/(((8*b*d^3 + a*
e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*b) - (1/2)^(1/3)*((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a
*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*(I*sqrt(3) + 1))^2*b)/b)) - 1/8*(4*(1/2)^(2/3)*a*d*e*p^2*(-I*sqrt(3) + 1)
/(((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*b) - (1/2)^(1/3)*((8*b*d^3 + a*e^3)*
a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*(I*sqrt(3) + 1) + sqrt(3)*sqrt(-(32*a*d*e*p^2 + (4*(1/2)^
(2/3)*a*d*e*p^2*(-I*sqrt(3) + 1)/(((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*b) -
(1/2)^(1/3)*((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*(I*sqrt(3) + 1))^2*b)/b))
*log(-2*a*d*e^2*p^2 - (4*(1/2)^(2/3)*a*d*e*p^2*(-I*sqrt(3) + 1)/(((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3
- a^2*e^3*p^3)/b^2)^(1/3)*b) - (1/2)^(1/3)*((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^
(1/3)*(I*sqrt(3) + 1))*b*d^2*p - 1/8*(4*(1/2)^(2/3)*a*d*e*p^2*(-I*sqrt(3) + 1)/(((8*b*d^3 + a*e^3)*a*p^3/b^2 +
(8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*b) - (1/2)^(1/3)*((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2
*e^3*p^3)/b^2)^(1/3)*(I*sqrt(3) + 1))^2*b*e + (8*b*d^3 + a*e^3)*p^2*x - 1/8*sqrt(3)*(8*b*d^2*p - (4*(1/2)^(2/3
)*a*d*e*p^2*(-I*sqrt(3) + 1)/(((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*b) - (1/
2)^(1/3)*((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*(I*sqrt(3) + 1))*b*e)*sqrt(-(
32*a*d*e*p^2 + (4*(1/2)^(2/3)*a*d*e*p^2*(-I*sqrt(3) + 1)/(((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*
e^3*p^3)/b^2)^(1/3)*b) - (1/2)^(1/3)*((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*(
I*sqrt(3) + 1))^2*b)/b)) + 1/2*(e*p*x^2 + 2*d*p*x)*log(b*x^3 + a) + 1/2*(e*x^2 + 2*d*x)*log(c)

Sympy [A] (verification not implemented)

Time = 10.94 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.49 \[ \int (d+e x) \log \left (c \left (a+b x^3\right )^p\right ) \, dx=3 a d p \operatorname {RootSum} {\left (27 t^{3} a^{2} b - 1, \left ( t \mapsto t \log {\left (3 t a + x \right )} \right )\right )} + \frac {3 a e p \operatorname {RootSum} {\left (27 t^{3} a b^{2} + 1, \left ( t \mapsto t \log {\left (9 t^{2} a b + x \right )} \right )\right )}}{2} - 3 d p x + d x \log {\left (c \left (a + b x^{3}\right )^{p} \right )} - \frac {3 e p x^{2}}{4} + \frac {e x^{2} \log {\left (c \left (a + b x^{3}\right )^{p} \right )}}{2} \]

[In]

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

[Out]

3*a*d*p*RootSum(27*_t**3*a**2*b - 1, Lambda(_t, _t*log(3*_t*a + x))) + 3*a*e*p*RootSum(27*_t**3*a*b**2 + 1, La
mbda(_t, _t*log(9*_t**2*a*b + x)))/2 - 3*d*p*x + d*x*log(c*(a + b*x**3)**p) - 3*e*p*x**2/4 + e*x**2*log(c*(a +
b*x**3)**p)/2

Maxima [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.82 \[ \int (d+e x) \log \left (c \left (a+b x^3\right )^p\right ) \, dx=-\frac {1}{4} \, b p {\left (\frac {3 \, {\left (e x^{2} + 4 \, d x\right )}}{b} - \frac {2 \, \sqrt {3} {\left (a e \left (\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, a d\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (a e \left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, a d\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {2 \, {\left (a e \left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, a d\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )} + \frac {1}{2} \, {\left (e x^{2} + 2 \, d x\right )} \log \left ({\left (b x^{3} + a\right )}^{p} c\right ) \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.93 \[ \int (d+e x) \log \left (c \left (a+b x^3\right )^p\right ) \, dx=-\frac {1}{4} \, {\left (3 \, e p - 2 \, e \log \left (c\right )\right )} x^{2} - {\left (3 \, d p - d \log \left (c\right )\right )} x + \frac {1}{2} \, {\left (e p x^{2} + 2 \, d p x\right )} \log \left (b x^{3} + a\right ) - \frac {{\left (a e p \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, a d p\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{2 \, a} + \frac {{\left (2 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} b d p - \sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} e p\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{2 \, b^{2}} + \frac {{\left (2 \, \left (-a b^{2}\right )^{\frac {1}{3}} b d p + \left (-a b^{2}\right )^{\frac {2}{3}} e p\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{4 \, b^{2}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 1.31 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.92 \[ \int (d+e x) \log \left (c \left (a+b x^3\right )^p\right ) \, dx=\left (\sum _{k=1}^3\ln \left (\mathrm {root}\left (8\,b^2\,c^3+12\,a\,b\,c\,d\,e\,p^2-8\,a\,b\,d^3\,p^3+a^2\,e^3\,p^3,c,k\right )\,\left (\mathrm {root}\left (8\,b^2\,c^3+12\,a\,b\,c\,d\,e\,p^2-8\,a\,b\,d^3\,p^3+a^2\,e^3\,p^3,c,k\right )\,a\,b^2\,9+9\,a\,b^2\,d\,p\,x\right )+\frac {9\,a^2\,b\,d\,e\,p^2}{2}+\frac {9\,a^2\,b\,e^2\,p^2\,x}{4}\right )\,\mathrm {root}\left (8\,b^2\,c^3+12\,a\,b\,c\,d\,e\,p^2-8\,a\,b\,d^3\,p^3+a^2\,e^3\,p^3,c,k\right )\right )+\ln \left (c\,{\left (b\,x^3+a\right )}^p\right )\,\left (\frac {e\,x^2}{2}+d\,x\right )-\frac {3\,e\,p\,x^2}{4}-3\,d\,p\,x \]

[In]

int(log(c*(a + b*x^3)^p)*(d + e*x),x)

[Out]

symsum(log(root(8*b^2*c^3 + 12*a*b*c*d*e*p^2 - 8*a*b*d^3*p^3 + a^2*e^3*p^3, c, k)*(9*root(8*b^2*c^3 + 12*a*b*c
*d*e*p^2 - 8*a*b*d^3*p^3 + a^2*e^3*p^3, c, k)*a*b^2 + 9*a*b^2*d*p*x) + (9*a^2*b*d*e*p^2)/2 + (9*a^2*b*e^2*p^2*
x)/4)*root(8*b^2*c^3 + 12*a*b*c*d*e*p^2 - 8*a*b*d^3*p^3 + a^2*e^3*p^3, c, k), k, 1, 3) + log(c*(a + b*x^3)^p)*
(d*x + (e*x^2)/2) - (3*e*p*x^2)/4 - 3*d*p*x