\(\int \frac {\log (c+d x)}{x^2 (a+b x^3)} \, dx\) [291]

Optimal result

Integrand size = 19, antiderivative size = 398 \[ \int \frac {\log (c+d x)}{x^2 \left (a+b x^3\right )} \, dx=\frac {d \log (x)}{a c}-\frac {d \log (c+d x)}{a c}-\frac {\log (c+d x)}{a x}+\frac {\sqrt [3]{b} \log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^{4/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{b} \log \left (\frac {d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^{4/3}}+\frac {(-1)^{2/3} \sqrt [3]{b} \log \left (-\frac {d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^{4/3}}+\frac {\sqrt [3]{b} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{4/3}}+\frac {(-1)^{2/3} \sqrt [3]{b} \operatorname {PolyLog}\left (2,\frac {(-1)^{2/3} \sqrt [3]{b} (c+d x)}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{4/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{b} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{b} (c+d x)}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right )}{3 a^{4/3}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.737, Rules used = {331, 298, 31, 648, 631, 210, 642, 2463, 2442, 36, 29, 2441, 2440, 2438} \[ \int \frac {\log (c+d x)}{x^2 \left (a+b x^3\right )} \, dx=\frac {\sqrt [3]{b} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{4/3}}+\frac {(-1)^{2/3} \sqrt [3]{b} \operatorname {PolyLog}\left (2,\frac {(-1)^{2/3} \sqrt [3]{b} (c+d x)}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{4/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{b} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{b} (c+d x)}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right )}{3 a^{4/3}}+\frac {\sqrt [3]{b} \log (c+d x) \log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{4/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{b} \log (c+d x) \log \left (\frac {d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} c}\right )}{3 a^{4/3}}+\frac {(-1)^{2/3} \sqrt [3]{b} \log (c+d x) \log \left (-\frac {d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{4/3}}+\frac {d \log (x)}{a c}-\frac {d \log (c+d x)}{a c}-\frac {\log (c+d x)}{a x} \]

[In]

Int[Log[c + d*x]/(x^2*(a + b*x^3)),x]

[Out]

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

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

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

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 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 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && 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 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 2442

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[(f + g*
x)^(q + 1)*((a + b*Log[c*(d + e*x)^n])/(g*(q + 1))), x] - Dist[b*e*(n/(g*(q + 1))), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -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)}{a x^2}-\frac {b x \log (c+d x)}{a \left (a+b x^3\right )}\right ) \, dx \\ & = \frac {\int \frac {\log (c+d x)}{x^2} \, dx}{a}-\frac {b \int \frac {x \log (c+d x)}{a+b x^3} \, dx}{a} \\ & = -\frac {\log (c+d x)}{a x}-\frac {b \int \left (-\frac {\log (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {(-1)^{2/3} \log (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}+\frac {\sqrt [3]{-1} \log (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}\right ) \, dx}{a}+\frac {d \int \frac {1}{x (c+d x)} \, dx}{a} \\ & = -\frac {\log (c+d x)}{a x}+\frac {b^{2/3} \int \frac {\log (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^{4/3}}-\frac {\left (\sqrt [3]{-1} b^{2/3}\right ) \int \frac {\log (c+d x)}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{3 a^{4/3}}+\frac {\left ((-1)^{2/3} b^{2/3}\right ) \int \frac {\log (c+d x)}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{3 a^{4/3}}+\frac {d \int \frac {1}{x} \, dx}{a c}-\frac {d^2 \int \frac {1}{c+d x} \, dx}{a c} \\ & = \frac {d \log (x)}{a c}-\frac {d \log (c+d x)}{a c}-\frac {\log (c+d x)}{a x}+\frac {\sqrt [3]{b} \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 a^{4/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{b} \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 a^{4/3}}+\frac {(-1)^{2/3} \sqrt [3]{b} \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 a^{4/3}}-\frac {\left (\sqrt [3]{b} 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 a^{4/3}}+\frac {\left (\sqrt [3]{-1} \sqrt [3]{b} 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 a^{4/3}}-\frac {\left ((-1)^{2/3} \sqrt [3]{b} 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 a^{4/3}} \\ & = \frac {d \log (x)}{a c}-\frac {d \log (c+d x)}{a c}-\frac {\log (c+d x)}{a x}+\frac {\sqrt [3]{b} \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 a^{4/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{b} \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 a^{4/3}}+\frac {(-1)^{2/3} \sqrt [3]{b} \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 a^{4/3}}-\frac {\sqrt [3]{b} \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 a^{4/3}}+\frac {\left (\sqrt [3]{-1} \sqrt [3]{b}\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 a^{4/3}}-\frac {\left ((-1)^{2/3} \sqrt [3]{b}\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 a^{4/3}} \\ & = \frac {d \log (x)}{a c}-\frac {d \log (c+d x)}{a c}-\frac {\log (c+d x)}{a x}+\frac {\sqrt [3]{b} \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 a^{4/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{b} \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 a^{4/3}}+\frac {(-1)^{2/3} \sqrt [3]{b} \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 a^{4/3}}+\frac {\sqrt [3]{b} \text {Li}_2\left (\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{4/3}}+\frac {(-1)^{2/3} \sqrt [3]{b} \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 a^{4/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{b} \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 a^{4/3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 378, normalized size of antiderivative = 0.95 \[ \int \frac {\log (c+d x)}{x^2 \left (a+b x^3\right )} \, dx=\frac {3 \sqrt [3]{a} d x \log (x)-3 \sqrt [3]{a} c \log (c+d x)-3 \sqrt [3]{a} d x \log (c+d x)+\sqrt [3]{b} c x \log \left (\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{-\sqrt [3]{b} c+\sqrt [3]{a} d}\right ) \log (c+d x)-\sqrt [3]{-1} \sqrt [3]{b} c x \log \left (\frac {d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right ) \log (c+d x)+(-1)^{2/3} \sqrt [3]{b} c x \log \left (\frac {d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{-(-1)^{2/3} \sqrt [3]{b} c+\sqrt [3]{a} d}\right ) \log (c+d x)+\sqrt [3]{b} c x \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )+(-1)^{2/3} \sqrt [3]{b} c x \operatorname {PolyLog}\left (2,\frac {(-1)^{2/3} \sqrt [3]{b} (c+d x)}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )-\sqrt [3]{-1} \sqrt [3]{b} c x \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{b} (c+d x)}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right )}{3 a^{4/3} c x} \]

[In]

Integrate[Log[c + d*x]/(x^2*(a + b*x^3)),x]

[Out]

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

Maple [C] (verified)

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

Time = 0.64 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.31

method	result	size
derivativedivides	\(d \left (\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{-\textit {\_R1} +c}}{3 a}+\frac {\frac {\ln \left (-d x \right )}{c}-\frac {\ln \left (d x +c \right ) \left (d x +c \right )}{c d x}}{a}\right )\)	\(124\)
default	\(d \left (\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{-\textit {\_R1} +c}}{3 a}+\frac {\frac {\ln \left (-d x \right )}{c}-\frac {\ln \left (d x +c \right ) \left (d x +c \right )}{c d x}}{a}\right )\)	\(124\)
risch	\(\frac {d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{-\textit {\_R1} +c}\right )}{3 a}+\frac {d \ln \left (-d x \right )}{a c}-\frac {d \ln \left (d x +c \right )}{a c}-\frac {\ln \left (d x +c \right )}{a x}\)	\(129\)

[In]

int(ln(d*x+c)/x^2/(b*x^3+a),x,method=_RETURNVERBOSE)

[Out]

d*(1/3/a*sum(1/(-_R1+c)*(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))+1/a*(1/c*ln(-d*x)-ln(d*x+c)*(d*x+c)/c/d/x))

Fricas [F]

\[ \int \frac {\log (c+d x)}{x^2 \left (a+b x^3\right )} \, dx=\int { \frac {\log \left (d x + c\right )}{{\left (b x^{3} + a\right )} x^{2}} \,d x } \]

[In]

integrate(log(d*x+c)/x^2/(b*x^3+a),x, algorithm="fricas")

[Out]

integral(log(d*x + c)/(b*x^5 + a*x^2), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\log (c+d x)}{x^2 \left (a+b x^3\right )} \, dx=\text {Timed out} \]

[In]

integrate(ln(d*x+c)/x**2/(b*x**3+a),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\log (c+d x)}{x^2 \left (a+b x^3\right )} \, dx=\int { \frac {\log \left (d x + c\right )}{{\left (b x^{3} + a\right )} x^{2}} \,d x } \]

[In]

integrate(log(d*x+c)/x^2/(b*x^3+a),x, algorithm="maxima")

[Out]

integrate(log(d*x + c)/((b*x^3 + a)*x^2), x)

Giac [F]

\[ \int \frac {\log (c+d x)}{x^2 \left (a+b x^3\right )} \, dx=\int { \frac {\log \left (d x + c\right )}{{\left (b x^{3} + a\right )} x^{2}} \,d x } \]

[In]

integrate(log(d*x+c)/x^2/(b*x^3+a),x, algorithm="giac")

[Out]

integrate(log(d*x + c)/((b*x^3 + a)*x^2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\log (c+d x)}{x^2 \left (a+b x^3\right )} \, dx=\int \frac {\ln \left (c+d\,x\right )}{x^2\,\left (b\,x^3+a\right )} \,d x \]

[In]

int(log(c + d*x)/(x^2*(a + b*x^3)),x)

[Out]

int(log(c + d*x)/(x^2*(a + b*x^3)), x)