\(\int \frac {x^3 \log (c+d x)}{a+b x^3} \, dx\) [288]

Optimal result

Integrand size = 19, antiderivative size = 383 \[ \int \frac {x^3 \log (c+d x)}{a+b x^3} \, dx=-\frac {x}{b}+\frac {(c+d x) \log (c+d x)}{b d}-\frac {\sqrt [3]{a} \log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right ) \log (c+d x)}{3 b^{4/3}}-\frac {(-1)^{2/3} \sqrt [3]{a} \log \left (\frac {d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right ) \log (c+d x)}{3 b^{4/3}}+\frac {\sqrt [3]{-1} \sqrt [3]{a} \log \left (-\frac {d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right ) \log (c+d x)}{3 b^{4/3}}-\frac {\sqrt [3]{a} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b^{4/3}}+\frac {\sqrt [3]{-1} \sqrt [3]{a} \operatorname {PolyLog}\left (2,\frac {(-1)^{2/3} \sqrt [3]{b} (c+d x)}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b^{4/3}}-\frac {(-1)^{2/3} \sqrt [3]{a} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{b} (c+d x)}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right )}{3 b^{4/3}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.737, Rules used = {327, 206, 31, 648, 631, 210, 642, 2463, 2436, 2332, 2456, 2441, 2440, 2438} \[ \int \frac {x^3 \log (c+d x)}{a+b x^3} \, dx=-\frac {\sqrt [3]{a} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b^{4/3}}+\frac {\sqrt [3]{-1} \sqrt [3]{a} \operatorname {PolyLog}\left (2,\frac {(-1)^{2/3} \sqrt [3]{b} (c+d x)}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b^{4/3}}-\frac {(-1)^{2/3} \sqrt [3]{a} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{b} (c+d x)}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right )}{3 b^{4/3}}-\frac {\sqrt [3]{a} \log (c+d x) \log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b^{4/3}}-\frac {(-1)^{2/3} \sqrt [3]{a} \log (c+d x) \log \left (\frac {d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} c}\right )}{3 b^{4/3}}+\frac {\sqrt [3]{-1} \sqrt [3]{a} \log (c+d x) \log \left (-\frac {d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b^{4/3}}+\frac {(c+d x) \log (c+d x)}{b d}-\frac {x}{b} \]

[In]

Int[(x^3*Log[c + d*x])/(a + b*x^3),x]

[Out]

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

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 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, 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 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2436

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

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 2456

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

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]

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 369, normalized size of antiderivative = 0.96 \[ \int \frac {x^3 \log (c+d x)}{a+b x^3} \, dx=\frac {-3 \sqrt [3]{b} d x+3 \sqrt [3]{b} c \log (c+d x)+3 \sqrt [3]{b} d x \log (c+d x)-\sqrt [3]{a} d \log \left (\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{-\sqrt [3]{b} c+\sqrt [3]{a} d}\right ) \log (c+d x)-(-1)^{2/3} \sqrt [3]{a} d \log \left (\frac {d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right ) \log (c+d x)+\sqrt [3]{-1} \sqrt [3]{a} d \log \left (\frac {d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{-(-1)^{2/3} \sqrt [3]{b} c+\sqrt [3]{a} d}\right ) \log (c+d x)-\sqrt [3]{a} d \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )+\sqrt [3]{-1} \sqrt [3]{a} d \operatorname {PolyLog}\left (2,\frac {(-1)^{2/3} \sqrt [3]{b} (c+d x)}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )-(-1)^{2/3} \sqrt [3]{a} d \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{b} (c+d x)}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right )}{3 b^{4/3} d} \]

[In]

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

[Out]

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

Maple [C] (verified)

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

Time = 0.57 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.33

method	result	size
derivativedivides	\(\frac {\frac {d^{3} \left (\left (d x +c \right ) \ln \left (d x +c \right )-d x -c \right )}{b}-\frac {a \,d^{6} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{3 b^{2}}}{d^{4}}\)	\(127\)
default	\(\frac {\frac {d^{3} \left (\left (d x +c \right ) \ln \left (d x +c \right )-d x -c \right )}{b}-\frac {a \,d^{6} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{3 b^{2}}}{d^{4}}\)	\(127\)
risch	\(\frac {x \ln \left (d x +c \right )}{b}+\frac {\ln \left (d x +c \right ) c}{d b}-\frac {x}{b}-\frac {c}{d b}-\frac {d^{2} a \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{3 b^{2}}\)	\(136\)

[In]

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

[Out]

1/d^4*(d^3/b*((d*x+c)*ln(d*x+c)-d*x-c)-1/3*a*d^6/b^2*sum(1/(_R1^2-2*_R1*c+c^2)*(ln(d*x+c)*ln((-d*x+_R1-c)/_R1)
+dilog((-d*x+_R1-c)/_R1)),_R1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3)))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int((x^3*log(c + d*x))/(a + b*x^3),x)

[Out]

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