\(\int \frac {x^3 \log (c (a+\frac {b}{x^3})^p)}{d+e x} \, dx\) [254]

Optimal result

Integrand size = 23, antiderivative size = 714 \[ \int \frac {x^3 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d+e x} \, dx=-\frac {\sqrt {3} \sqrt [3]{b} d^2 p \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt [3]{a} e^3}+\frac {\sqrt {3} b^{2/3} d p \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{2 a^{2/3} e^2}+\frac {d^2 x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e^3}-\frac {d x^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{2 e^2}+\frac {x^3 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{3 e}+\frac {\sqrt [3]{b} d^2 p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} e^3}+\frac {b^{2/3} d p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{2 a^{2/3} e^2}-\frac {d^3 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{e^4}-\frac {3 d^3 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^4}+\frac {d^3 p \log \left (-\frac {e \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right ) \log (d+e x)}{e^4}+\frac {d^3 p \log \left (-\frac {e \left ((-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right ) \log (d+e x)}{e^4}+\frac {d^3 p \log \left (\frac {\sqrt [3]{-1} e \left (\sqrt [3]{b}+(-1)^{2/3} \sqrt [3]{a} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right ) \log (d+e x)}{e^4}-\frac {\sqrt [3]{b} d^2 p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{2 \sqrt [3]{a} e^3}-\frac {b^{2/3} d p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{4 a^{2/3} e^2}+\frac {b p \log \left (b+a x^3\right )}{3 a e}+\frac {d^3 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{e^4}+\frac {d^3 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{e^4}+\frac {d^3 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right )}{e^4}-\frac {3 d^3 p \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{e^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 714, normalized size of antiderivative = 1.00, number of steps used = 37, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.783, Rules used = {2516, 2498, 269, 206, 31, 648, 631, 210, 642, 2505, 298, 266, 2512, 2463, 2441, 2352, 2440, 2438} \[ \int \frac {x^3 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d+e x} \, dx=\frac {\sqrt {3} b^{2/3} d p \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{2 a^{2/3} e^2}-\frac {\sqrt [3]{b} d^2 p \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{2 \sqrt [3]{a} e^3}-\frac {b^{2/3} d p \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{4 a^{2/3} e^2}+\frac {b^{2/3} d p \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{2 a^{2/3} e^2}-\frac {\sqrt {3} \sqrt [3]{b} d^2 p \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt [3]{a} e^3}-\frac {d^3 \log (d+e x) \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e^4}+\frac {d^2 x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e^3}-\frac {d x^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{2 e^2}+\frac {x^3 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{3 e}+\frac {d^3 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{e^4}+\frac {d^3 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{e^4}+\frac {d^3 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right )}{e^4}+\frac {d^3 p \log (d+e x) \log \left (-\frac {e \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{e^4}+\frac {d^3 p \log (d+e x) \log \left (-\frac {e \left (\sqrt [3]{a} x+(-1)^{2/3} \sqrt [3]{b}\right )}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right )}{e^4}+\frac {d^3 p \log (d+e x) \log \left (\frac {\sqrt [3]{-1} e \left ((-1)^{2/3} \sqrt [3]{a} x+\sqrt [3]{b}\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{e^4}+\frac {\sqrt [3]{b} d^2 p \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{\sqrt [3]{a} e^3}+\frac {b p \log \left (a x^3+b\right )}{3 a e}-\frac {3 d^3 p \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )}{e^4}-\frac {3 d^3 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^4} \]

[In]

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

[Out]

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

Rule 31

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

Rule 206

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], 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 269

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rule 298

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Dist[-(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3
]^2*x^2), x], x] /; FreeQ[{a, b}, x]

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 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2440

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + c*e*(x/g)])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2441

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

Rule 2463

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2498

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]

Rule 2505

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

Rule 2512

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

Rule 2516

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_.) + (g_.)*(x_))^(r_.), x_S
ymbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, x^m*(f + g*x)^r, x], x] /; FreeQ[{a, b, c, d, e,
f, g, n, p, q}, x] && IntegerQ[m] && IntegerQ[r]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e^3}-\frac {d x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e^2}+\frac {x^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e}-\frac {d^3 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e^3 (d+e x)}\right ) \, dx \\ & = \frac {d^2 \int \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \, dx}{e^3}-\frac {d^3 \int \frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d+e x} \, dx}{e^3}-\frac {d \int x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \, dx}{e^2}+\frac {\int x^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \, dx}{e} \\ & = \frac {d^2 x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e^3}-\frac {d x^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{2 e^2}+\frac {x^3 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{3 e}-\frac {d^3 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{e^4}-\frac {\left (3 b d^3 p\right ) \int \frac {\log (d+e x)}{\left (a+\frac {b}{x^3}\right ) x^4} \, dx}{e^4}+\frac {\left (3 b d^2 p\right ) \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^3} \, dx}{e^3}-\frac {(3 b d p) \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^2} \, dx}{2 e^2}+\frac {(b p) \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x} \, dx}{e} \\ & = \frac {d^2 x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e^3}-\frac {d x^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{2 e^2}+\frac {x^3 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{3 e}-\frac {d^3 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{e^4}-\frac {\left (3 b d^3 p\right ) \int \left (\frac {\log (d+e x)}{b x}-\frac {a x^2 \log (d+e x)}{b \left (b+a x^3\right )}\right ) \, dx}{e^4}+\frac {\left (3 b d^2 p\right ) \int \frac {1}{b+a x^3} \, dx}{e^3}-\frac {(3 b d p) \int \frac {x}{b+a x^3} \, dx}{2 e^2}+\frac {(b p) \int \frac {x^2}{b+a x^3} \, dx}{e} \\ & = \frac {d^2 x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e^3}-\frac {d x^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{2 e^2}+\frac {x^3 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{3 e}-\frac {d^3 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{e^4}+\frac {b p \log \left (b+a x^3\right )}{3 a e}-\frac {\left (3 d^3 p\right ) \int \frac {\log (d+e x)}{x} \, dx}{e^4}+\frac {\left (3 a d^3 p\right ) \int \frac {x^2 \log (d+e x)}{b+a x^3} \, dx}{e^4}+\frac {\left (\sqrt [3]{b} d^2 p\right ) \int \frac {1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{e^3}+\frac {\left (\sqrt [3]{b} d^2 p\right ) \int \frac {2 \sqrt [3]{b}-\sqrt [3]{a} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{e^3}+\frac {\left (b^{2/3} d p\right ) \int \frac {1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{2 \sqrt [3]{a} e^2}-\frac {\left (b^{2/3} d p\right ) \int \frac {\sqrt [3]{b}+\sqrt [3]{a} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{2 \sqrt [3]{a} e^2} \\ & = \frac {d^2 x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e^3}-\frac {d x^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{2 e^2}+\frac {x^3 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{3 e}+\frac {\sqrt [3]{b} d^2 p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} e^3}+\frac {b^{2/3} d p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{2 a^{2/3} e^2}-\frac {d^3 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{e^4}-\frac {3 d^3 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^4}+\frac {b p \log \left (b+a x^3\right )}{3 a e}+\frac {\left (3 a d^3 p\right ) \int \left (\frac {\log (d+e x)}{3 a^{2/3} \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}+\frac {\log (d+e x)}{3 a^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{b}+\sqrt [3]{a} x\right )}+\frac {\log (d+e x)}{3 a^{2/3} \left ((-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} x\right )}\right ) \, dx}{e^4}-\frac {\left (\sqrt [3]{b} d^2 p\right ) \int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 a^{2/3} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{2 \sqrt [3]{a} e^3}+\frac {\left (3 b^{2/3} d^2 p\right ) \int \frac {1}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{2 e^3}+\frac {\left (3 d^3 p\right ) \int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx}{e^3}-\frac {\left (b^{2/3} d p\right ) \int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 a^{2/3} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{4 a^{2/3} e^2}-\frac {(3 b d p) \int \frac {1}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{4 \sqrt [3]{a} e^2} \\ & = \frac {d^2 x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e^3}-\frac {d x^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{2 e^2}+\frac {x^3 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{3 e}+\frac {\sqrt [3]{b} d^2 p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} e^3}+\frac {b^{2/3} d p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{2 a^{2/3} e^2}-\frac {d^3 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{e^4}-\frac {3 d^3 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^4}-\frac {\sqrt [3]{b} d^2 p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{2 \sqrt [3]{a} e^3}-\frac {b^{2/3} d p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{4 a^{2/3} e^2}+\frac {b p \log \left (b+a x^3\right )}{3 a e}-\frac {3 d^3 p \text {Li}_2\left (1+\frac {e x}{d}\right )}{e^4}+\frac {\left (\sqrt [3]{a} d^3 p\right ) \int \frac {\log (d+e x)}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{e^4}+\frac {\left (\sqrt [3]{a} d^3 p\right ) \int \frac {\log (d+e x)}{-\sqrt [3]{-1} \sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{e^4}+\frac {\left (\sqrt [3]{a} d^3 p\right ) \int \frac {\log (d+e x)}{(-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{e^4}+\frac {\left (3 \sqrt [3]{b} d^2 p\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{\sqrt [3]{a} e^3}-\frac {\left (3 b^{2/3} d p\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{2 a^{2/3} e^2} \\ & = -\frac {\sqrt {3} \sqrt [3]{b} d^2 p \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt [3]{a} e^3}+\frac {\sqrt {3} b^{2/3} d p \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{2 a^{2/3} e^2}+\frac {d^2 x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e^3}-\frac {d x^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{2 e^2}+\frac {x^3 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{3 e}+\frac {\sqrt [3]{b} d^2 p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} e^3}+\frac {b^{2/3} d p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{2 a^{2/3} e^2}-\frac {d^3 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{e^4}-\frac {3 d^3 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^4}+\frac {d^3 p \log \left (-\frac {e \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right ) \log (d+e x)}{e^4}+\frac {d^3 p \log \left (-\frac {e \left ((-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right ) \log (d+e x)}{e^4}+\frac {d^3 p \log \left (\frac {\sqrt [3]{-1} e \left (\sqrt [3]{b}+(-1)^{2/3} \sqrt [3]{a} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right ) \log (d+e x)}{e^4}-\frac {\sqrt [3]{b} d^2 p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{2 \sqrt [3]{a} e^3}-\frac {b^{2/3} d p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{4 a^{2/3} e^2}+\frac {b p \log \left (b+a x^3\right )}{3 a e}-\frac {3 d^3 p \text {Li}_2\left (1+\frac {e x}{d}\right )}{e^4}-\frac {\left (d^3 p\right ) \int \frac {\log \left (\frac {e \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{-\sqrt [3]{a} d+\sqrt [3]{b} e}\right )}{d+e x} \, dx}{e^3}-\frac {\left (d^3 p\right ) \int \frac {\log \left (\frac {e \left (-\sqrt [3]{-1} \sqrt [3]{b}+\sqrt [3]{a} x\right )}{-\sqrt [3]{a} d-\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{d+e x} \, dx}{e^3}-\frac {\left (d^3 p\right ) \int \frac {\log \left (\frac {e \left ((-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} x\right )}{-\sqrt [3]{a} d+(-1)^{2/3} \sqrt [3]{b} e}\right )}{d+e x} \, dx}{e^3} \\ & = -\frac {\sqrt {3} \sqrt [3]{b} d^2 p \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt [3]{a} e^3}+\frac {\sqrt {3} b^{2/3} d p \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{2 a^{2/3} e^2}+\frac {d^2 x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e^3}-\frac {d x^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{2 e^2}+\frac {x^3 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{3 e}+\frac {\sqrt [3]{b} d^2 p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} e^3}+\frac {b^{2/3} d p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{2 a^{2/3} e^2}-\frac {d^3 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{e^4}-\frac {3 d^3 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^4}+\frac {d^3 p \log \left (-\frac {e \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right ) \log (d+e x)}{e^4}+\frac {d^3 p \log \left (-\frac {e \left ((-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right ) \log (d+e x)}{e^4}+\frac {d^3 p \log \left (\frac {\sqrt [3]{-1} e \left (\sqrt [3]{b}+(-1)^{2/3} \sqrt [3]{a} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right ) \log (d+e x)}{e^4}-\frac {\sqrt [3]{b} d^2 p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{2 \sqrt [3]{a} e^3}-\frac {b^{2/3} d p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{4 a^{2/3} e^2}+\frac {b p \log \left (b+a x^3\right )}{3 a e}-\frac {3 d^3 p \text {Li}_2\left (1+\frac {e x}{d}\right )}{e^4}-\frac {\left (d^3 p\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt [3]{a} x}{-\sqrt [3]{a} d+\sqrt [3]{b} e}\right )}{x} \, dx,x,d+e x\right )}{e^4}-\frac {\left (d^3 p\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt [3]{a} x}{-\sqrt [3]{a} d-\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{x} \, dx,x,d+e x\right )}{e^4}-\frac {\left (d^3 p\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt [3]{a} x}{-\sqrt [3]{a} d+(-1)^{2/3} \sqrt [3]{b} e}\right )}{x} \, dx,x,d+e x\right )}{e^4} \\ & = -\frac {\sqrt {3} \sqrt [3]{b} d^2 p \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt [3]{a} e^3}+\frac {\sqrt {3} b^{2/3} d p \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{2 a^{2/3} e^2}+\frac {d^2 x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e^3}-\frac {d x^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{2 e^2}+\frac {x^3 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{3 e}+\frac {\sqrt [3]{b} d^2 p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} e^3}+\frac {b^{2/3} d p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{2 a^{2/3} e^2}-\frac {d^3 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{e^4}-\frac {3 d^3 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^4}+\frac {d^3 p \log \left (-\frac {e \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right ) \log (d+e x)}{e^4}+\frac {d^3 p \log \left (-\frac {e \left ((-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right ) \log (d+e x)}{e^4}+\frac {d^3 p \log \left (\frac {\sqrt [3]{-1} e \left (\sqrt [3]{b}+(-1)^{2/3} \sqrt [3]{a} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right ) \log (d+e x)}{e^4}-\frac {\sqrt [3]{b} d^2 p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{2 \sqrt [3]{a} e^3}-\frac {b^{2/3} d p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{4 a^{2/3} e^2}+\frac {b p \log \left (b+a x^3\right )}{3 a e}+\frac {d^3 p \text {Li}_2\left (\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{e^4}+\frac {d^3 p \text {Li}_2\left (\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{e^4}+\frac {d^3 p \text {Li}_2\left (\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right )}{e^4}-\frac {3 d^3 p \text {Li}_2\left (1+\frac {e x}{d}\right )}{e^4} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.21 (sec) , antiderivative size = 536, normalized size of antiderivative = 0.75 \[ \int \frac {x^3 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d+e x} \, dx=\frac {3 b d p \operatorname {Hypergeometric2F1}\left (\frac {1}{3},1,\frac {4}{3},-\frac {b}{a x^3}\right )}{2 a e^2 x}-\frac {3 b d^2 p \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {5}{3},-\frac {b}{a x^3}\right )}{2 a e^3 x^2}+\frac {b p \log \left (a+\frac {b}{x^3}\right )}{3 a e}+\frac {d^2 x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e^3}-\frac {d x^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{2 e^2}+\frac {x^3 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{3 e}+\frac {b p \log (x)}{a e}-\frac {d^3 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{e^4}-\frac {3 d^3 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^4}+\frac {d^3 p \log \left (-\frac {e \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right ) \log (d+e x)}{e^4}+\frac {d^3 p \log \left (-\frac {(-1)^{2/3} e \left (\sqrt [3]{b}-\sqrt [3]{-1} \sqrt [3]{a} x\right )}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right ) \log (d+e x)}{e^4}+\frac {d^3 p \log \left (\frac {\sqrt [3]{-1} e \left (\sqrt [3]{b}+(-1)^{2/3} \sqrt [3]{a} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right ) \log (d+e x)}{e^4}-\frac {3 d^3 p \operatorname {PolyLog}\left (2,\frac {d+e x}{d}\right )}{e^4}+\frac {d^3 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{e^4}+\frac {d^3 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{e^4}+\frac {d^3 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right )}{e^4} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 2.47 (sec) , antiderivative size = 301, normalized size of antiderivative = 0.42

method	result	size
parts	\(\frac {x^{3} \ln \left (c \left (a +\frac {b}{x^{3}}\right )^{p}\right )}{3 e}-\frac {d \,x^{2} \ln \left (c \left (a +\frac {b}{x^{3}}\right )^{p}\right )}{2 e^{2}}+\frac {d^{2} x \ln \left (c \left (a +\frac {b}{x^{3}}\right )^{p}\right )}{e^{3}}-\frac {d^{3} \ln \left (c \left (a +\frac {b}{x^{3}}\right )^{p}\right ) \ln \left (e x +d \right )}{e^{4}}+3 p b \,e^{3} \left (-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} a -3 \textit {\_Z}^{2} a d +3 \textit {\_Z} a \,d^{2}-a \,d^{3}+e^{3} b \right )}{\sum }\frac {\left (2 \textit {\_R}^{2}-7 \textit {\_R} d +11 d^{2}\right ) \ln \left (e x -\textit {\_R} +d \right )}{-\textit {\_R}^{2}+2 \textit {\_R} d -d^{2}}}{18 e^{4} a}-\frac {d^{3} \left (\frac {\operatorname {dilog}\left (-\frac {e x}{d}\right )+\ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{b \,e^{3}}-\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\textit {\_Z}^{3} a -3 \textit {\_Z}^{2} a d +3 \textit {\_Z} a \,d^{2}-a \,d^{3}+e^{3} b \right )}{\sum }\left (\ln \left (e x +d \right ) \ln \left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )\right )}{3 b \,e^{3}}\right )}{e^{4}}\right )\)	\(301\)

[In]

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

[Out]

1/3*x^3*ln(c*(a+b/x^3)^p)/e-1/2*d*x^2*ln(c*(a+b/x^3)^p)/e^2+d^2*x*ln(c*(a+b/x^3)^p)/e^3-d^3*ln(c*(a+b/x^3)^p)*
ln(e*x+d)/e^4+3*p*b*e^3*(-1/18/e^4/a*sum((2*_R^2-7*_R*d+11*d^2)/(-_R^2+2*_R*d-d^2)*ln(e*x-_R+d),_R=RootOf(_Z^3
*a-3*_Z^2*a*d+3*_Z*a*d^2-a*d^3+b*e^3))-1/e^4*d^3*(1/b/e^3*(dilog(-e*x/d)+ln(e*x+d)*ln(-e*x/d))-1/3/b/e^3*sum(l
n(e*x+d)*ln((-e*x+_R1-d)/_R1)+dilog((-e*x+_R1-d)/_R1),_R1=RootOf(_Z^3*a-3*_Z^2*a*d+3*_Z*a*d^2-a*d^3+b*e^3))))

Fricas [F]

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

[In]

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

[Out]

integral(x^3*log(c*((a*x^3 + b)/x^3)^p)/(e*x + d), x)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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