3.286 \(\int \frac{\log (c+d x)}{x^4 (a+b x^3)} \, dx\)

Optimal. Leaf size=414 \[ \frac{b \text{PolyLog}\left (2,\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^2}+\frac{b \text{PolyLog}\left (2,\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{-1} \sqrt [3]{a} d+\sqrt [3]{b} c}\right )}{3 a^2}+\frac{b \text{PolyLog}\left (2,\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 a^2}-\frac{b \text{PolyLog}\left (2,\frac{d x}{c}+1\right )}{a^2}-\frac{b \log \left (-\frac{d x}{c}\right ) \log (c+d x)}{a^2}+\frac{b \log (c+d x) \log \left (-\frac{d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^2}+\frac{b \log (c+d x) \log \left (-\frac{d \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 a^2}+\frac{b \log (c+d x) \log \left (\frac{\sqrt [3]{-1} d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{a} d+\sqrt [3]{b} c}\right )}{3 a^2}+\frac{d^2}{3 a c^2 x}+\frac{d^3 \log (x)}{3 a c^3}-\frac{d^3 \log (c+d x)}{3 a c^3}-\frac{d}{6 a c x^2}-\frac{\log (c+d x)}{3 a x^3} \]

[Out]

-d/(6*a*c*x^2) + d^2/(3*a*c^2*x) + (d^3*Log[x])/(3*a*c^3) - (d^3*Log[c + d*x])/(3*a*c^3) - Log[c + d*x]/(3*a*x
^3) - (b*Log[-((d*x)/c)]*Log[c + d*x])/a^2 + (b*Log[-((d*(a^(1/3) + b^(1/3)*x))/(b^(1/3)*c - a^(1/3)*d))]*Log[
c + d*x])/(3*a^2) + (b*Log[-((d*((-1)^(2/3)*a^(1/3) + b^(1/3)*x))/(b^(1/3)*c - (-1)^(2/3)*a^(1/3)*d))]*Log[c +
d*x])/(3*a^2) + (b*Log[((-1)^(1/3)*d*(a^(1/3) + (-1)^(2/3)*b^(1/3)*x))/(b^(1/3)*c + (-1)^(1/3)*a^(1/3)*d)]*Lo
g[c + d*x])/(3*a^2) + (b*PolyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c - a^(1/3)*d)])/(3*a^2) + (b*PolyLog[2, (b^(
1/3)*(c + d*x))/(b^(1/3)*c + (-1)^(1/3)*a^(1/3)*d)])/(3*a^2) + (b*PolyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c -
(-1)^(2/3)*a^(1/3)*d)])/(3*a^2) - (b*PolyLog[2, 1 + (d*x)/c])/a^2

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Rubi [A]  time = 0.495631, antiderivative size = 414, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474, Rules used = {266, 44, 2416, 2395, 2394, 2315, 260, 2393, 2391} \[ \frac{b \text{PolyLog}\left (2,\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^2}+\frac{b \text{PolyLog}\left (2,\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{-1} \sqrt [3]{a} d+\sqrt [3]{b} c}\right )}{3 a^2}+\frac{b \text{PolyLog}\left (2,\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 a^2}-\frac{b \text{PolyLog}\left (2,\frac{d x}{c}+1\right )}{a^2}-\frac{b \log \left (-\frac{d x}{c}\right ) \log (c+d x)}{a^2}+\frac{b \log (c+d x) \log \left (-\frac{d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^2}+\frac{b \log (c+d x) \log \left (-\frac{d \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 a^2}+\frac{b \log (c+d x) \log \left (\frac{\sqrt [3]{-1} d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{a} d+\sqrt [3]{b} c}\right )}{3 a^2}+\frac{d^2}{3 a c^2 x}+\frac{d^3 \log (x)}{3 a c^3}-\frac{d^3 \log (c+d x)}{3 a c^3}-\frac{d}{6 a c x^2}-\frac{\log (c+d x)}{3 a x^3} \]

Antiderivative was successfully verified.

[In]

Int[Log[c + d*x]/(x^4*(a + b*x^3)),x]

[Out]

-d/(6*a*c*x^2) + d^2/(3*a*c^2*x) + (d^3*Log[x])/(3*a*c^3) - (d^3*Log[c + d*x])/(3*a*c^3) - Log[c + d*x]/(3*a*x
^3) - (b*Log[-((d*x)/c)]*Log[c + d*x])/a^2 + (b*Log[-((d*(a^(1/3) + b^(1/3)*x))/(b^(1/3)*c - a^(1/3)*d))]*Log[
c + d*x])/(3*a^2) + (b*Log[-((d*((-1)^(2/3)*a^(1/3) + b^(1/3)*x))/(b^(1/3)*c - (-1)^(2/3)*a^(1/3)*d))]*Log[c +
d*x])/(3*a^2) + (b*Log[((-1)^(1/3)*d*(a^(1/3) + (-1)^(2/3)*b^(1/3)*x))/(b^(1/3)*c + (-1)^(1/3)*a^(1/3)*d)]*Lo
g[c + d*x])/(3*a^2) + (b*PolyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c - a^(1/3)*d)])/(3*a^2) + (b*PolyLog[2, (b^(
1/3)*(c + d*x))/(b^(1/3)*c + (-1)^(1/3)*a^(1/3)*d)])/(3*a^2) + (b*PolyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c -
(-1)^(2/3)*a^(1/3)*d)])/(3*a^2) - (b*PolyLog[2, 1 + (d*x)/c])/a^2

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 2416

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 2395

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 2394

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 2315

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

Rule 260

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 2393

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 2391

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]

Rubi steps

\begin{align*} \int \frac{\log (c+d x)}{x^4 \left (a+b x^3\right )} \, dx &=\int \left (\frac{\log (c+d x)}{a x^4}-\frac{b \log (c+d x)}{a^2 x}+\frac{b^2 x^2 \log (c+d x)}{a^2 \left (a+b x^3\right )}\right ) \, dx\\ &=\frac{\int \frac{\log (c+d x)}{x^4} \, dx}{a}-\frac{b \int \frac{\log (c+d x)}{x} \, dx}{a^2}+\frac{b^2 \int \frac{x^2 \log (c+d x)}{a+b x^3} \, dx}{a^2}\\ &=-\frac{\log (c+d x)}{3 a x^3}-\frac{b \log \left (-\frac{d x}{c}\right ) \log (c+d x)}{a^2}+\frac{b^2 \int \left (\frac{\log (c+d x)}{3 b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{\log (c+d x)}{3 b^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{\log (c+d x)}{3 b^{2/3} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}\right ) \, dx}{a^2}+\frac{d \int \frac{1}{x^3 (c+d x)} \, dx}{3 a}+\frac{(b d) \int \frac{\log \left (-\frac{d x}{c}\right )}{c+d x} \, dx}{a^2}\\ &=-\frac{\log (c+d x)}{3 a x^3}-\frac{b \log \left (-\frac{d x}{c}\right ) \log (c+d x)}{a^2}-\frac{b \text{Li}_2\left (1+\frac{d x}{c}\right )}{a^2}+\frac{b^{4/3} \int \frac{\log (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^2}+\frac{b^{4/3} \int \frac{\log (c+d x)}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^2}+\frac{b^{4/3} \int \frac{\log (c+d x)}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^2}+\frac{d \int \left (\frac{1}{c x^3}-\frac{d}{c^2 x^2}+\frac{d^2}{c^3 x}-\frac{d^3}{c^3 (c+d x)}\right ) \, dx}{3 a}\\ &=-\frac{d}{6 a c x^2}+\frac{d^2}{3 a c^2 x}+\frac{d^3 \log (x)}{3 a c^3}-\frac{d^3 \log (c+d x)}{3 a c^3}-\frac{\log (c+d x)}{3 a x^3}-\frac{b \log \left (-\frac{d x}{c}\right ) \log (c+d x)}{a^2}+\frac{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^2}+\frac{b \log \left (-\frac{d \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^2}+\frac{b \log \left (\frac{\sqrt [3]{-1} d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^2}-\frac{b \text{Li}_2\left (1+\frac{d x}{c}\right )}{a^2}-\frac{(b d) \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^2}-\frac{(b d) \int \frac{\log \left (\frac{d \left (-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{-\sqrt [3]{b} c-\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{c+d x} \, dx}{3 a^2}-\frac{(b d) \int \frac{\log \left (\frac{d \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{-\sqrt [3]{b} c+(-1)^{2/3} \sqrt [3]{a} d}\right )}{c+d x} \, dx}{3 a^2}\\ &=-\frac{d}{6 a c x^2}+\frac{d^2}{3 a c^2 x}+\frac{d^3 \log (x)}{3 a c^3}-\frac{d^3 \log (c+d x)}{3 a c^3}-\frac{\log (c+d x)}{3 a x^3}-\frac{b \log \left (-\frac{d x}{c}\right ) \log (c+d x)}{a^2}+\frac{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^2}+\frac{b \log \left (-\frac{d \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^2}+\frac{b \log \left (\frac{\sqrt [3]{-1} d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^2}-\frac{b \text{Li}_2\left (1+\frac{d x}{c}\right )}{a^2}-\frac{b \operatorname{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^2}-\frac{b \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt [3]{b} x}{-\sqrt [3]{b} c-\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{x} \, dx,x,c+d x\right )}{3 a^2}-\frac{b \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt [3]{b} x}{-\sqrt [3]{b} c+(-1)^{2/3} \sqrt [3]{a} d}\right )}{x} \, dx,x,c+d x\right )}{3 a^2}\\ &=-\frac{d}{6 a c x^2}+\frac{d^2}{3 a c^2 x}+\frac{d^3 \log (x)}{3 a c^3}-\frac{d^3 \log (c+d x)}{3 a c^3}-\frac{\log (c+d x)}{3 a x^3}-\frac{b \log \left (-\frac{d x}{c}\right ) \log (c+d x)}{a^2}+\frac{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^2}+\frac{b \log \left (-\frac{d \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^2}+\frac{b \log \left (\frac{\sqrt [3]{-1} d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^2}+\frac{b \text{Li}_2\left (\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^2}+\frac{b \text{Li}_2\left (\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{3 a^2}+\frac{b \text{Li}_2\left (\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 a^2}-\frac{b \text{Li}_2\left (1+\frac{d x}{c}\right )}{a^2}\\ \end{align*}

Mathematica [A]  time = 0.139216, size = 405, normalized size = 0.98 \[ -\frac{b \text{PolyLog}\left (2,\frac{c+d x}{c}\right )}{a^2}+\frac{b \text{PolyLog}\left (2,\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^2}+\frac{b \text{PolyLog}\left (2,\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{-1} \sqrt [3]{a} d+\sqrt [3]{b} c}\right )}{3 a^2}+\frac{b \text{PolyLog}\left (2,\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 a^2}-\frac{b \log \left (-\frac{d x}{c}\right ) \log (c+d x)}{a^2}+\frac{b \log (c+d x) \log \left (-\frac{d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^2}+\frac{b \log (c+d x) \log \left (-\frac{(-1)^{2/3} d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 a^2}+\frac{b \log (c+d x) \log \left (\frac{\sqrt [3]{-1} d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{a} d+\sqrt [3]{b} c}\right )}{3 a^2}-\frac{d \left (-\frac{2 d^2 \log (x)}{c^3}+\frac{2 d^2 \log (c+d x)}{c^3}-\frac{2 d}{c^2 x}+\frac{1}{c x^2}\right )}{6 a}-\frac{\log (c+d x)}{3 a x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c + d*x]/(x^4*(a + b*x^3)),x]

[Out]

-Log[c + d*x]/(3*a*x^3) - (b*Log[-((d*x)/c)]*Log[c + d*x])/a^2 + (b*Log[-((d*(a^(1/3) + b^(1/3)*x))/(b^(1/3)*c
- a^(1/3)*d))]*Log[c + d*x])/(3*a^2) + (b*Log[-(((-1)^(2/3)*d*(a^(1/3) - (-1)^(1/3)*b^(1/3)*x))/(b^(1/3)*c -
(-1)^(2/3)*a^(1/3)*d))]*Log[c + d*x])/(3*a^2) + (b*Log[((-1)^(1/3)*d*(a^(1/3) + (-1)^(2/3)*b^(1/3)*x))/(b^(1/3
)*c + (-1)^(1/3)*a^(1/3)*d)]*Log[c + d*x])/(3*a^2) - (d*(1/(c*x^2) - (2*d)/(c^2*x) - (2*d^2*Log[x])/c^3 + (2*d
^2*Log[c + d*x])/c^3))/(6*a) - (b*PolyLog[2, (c + d*x)/c])/a^2 + (b*PolyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c
- a^(1/3)*d)])/(3*a^2) + (b*PolyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c + (-1)^(1/3)*a^(1/3)*d)])/(3*a^2) + (b*P
olyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c - (-1)^(2/3)*a^(1/3)*d)])/(3*a^2)

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Maple [C]  time = 0.417, size = 185, normalized size = 0.5 \begin{align*} -{\frac{b\ln \left ( dx+c \right ) }{{a}^{2}}\ln \left ( -{\frac{dx}{c}} \right ) }-{\frac{b}{{a}^{2}}{\it dilog} \left ( -{\frac{dx}{c}} \right ) }+{\frac{b}{3\,{a}^{2}}\sum _{{\it \_R1}={\it RootOf} \left ( b{{\it \_Z}}^{3}-3\,{{\it \_Z}}^{2}bc+3\,{\it \_Z}\,b{c}^{2}+a{d}^{3}-b{c}^{3} \right ) }\ln \left ( dx+c \right ) \ln \left ({\frac{-dx+{\it \_R1}-c}{{\it \_R1}}} \right ) +{\it dilog} \left ({\frac{-dx+{\it \_R1}-c}{{\it \_R1}}} \right ) }+{\frac{{d}^{3}\ln \left ( dx \right ) }{3\,a{c}^{3}}}+{\frac{{d}^{2}}{3\,{c}^{2}ax}}-{\frac{d}{6\,ac{x}^{2}}}-{\frac{{d}^{3}\ln \left ( dx+c \right ) }{3\,a{c}^{3}}}-{\frac{\ln \left ( dx+c \right ) }{3\,a{x}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(d*x+c)/x^4/(b*x^3+a),x)

[Out]

-b*ln(-d*x/c)*ln(d*x+c)/a^2-b/a^2*dilog(-d*x/c)+1/3*b/a^2*sum(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))+1/3*d^3/a/c^3*ln(d*x)+1/3*d^2/a/c^2/x-1/6*d/a/c/x
^2-1/3*d^3*ln(d*x+c)/a/c^3-1/3*ln(d*x+c)/a/x^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (d x + c\right )}{{\left (b x^{3} + a\right )} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(d*x+c)/x^4/(b*x^3+a),x, algorithm="maxima")

[Out]

integrate(log(d*x + c)/((b*x^3 + a)*x^4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left (d x + c\right )}{b x^{7} + a x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(d*x+c)/x^4/(b*x^3+a),x, algorithm="fricas")

[Out]

integral(log(d*x + c)/(b*x^7 + a*x^4), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(d*x+c)/x**4/(b*x**3+a),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (d x + c\right )}{{\left (b x^{3} + a\right )} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(d*x+c)/x^4/(b*x^3+a),x, algorithm="giac")

[Out]

integrate(log(d*x + c)/((b*x^3 + a)*x^4), x)