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

Optimal. Leaf size=416 \[ \frac{a^{2/3} \text{PolyLog}\left (2,\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b^{5/3}}+\frac{(-1)^{2/3} a^{2/3} \text{PolyLog}\left (2,\frac{(-1)^{2/3} \sqrt [3]{b} (c+d x)}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b^{5/3}}-\frac{\sqrt [3]{-1} a^{2/3} \text{PolyLog}\left (2,\frac{\sqrt [3]{-1} \sqrt [3]{b} (c+d x)}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} c}\right )}{3 b^{5/3}}+\frac{a^{2/3} \log (c+d x) \log \left (-\frac{d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b^{5/3}}-\frac{\sqrt [3]{-1} a^{2/3} \log (c+d x) \log \left (\frac{d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} c}\right )}{3 b^{5/3}}+\frac{(-1)^{2/3} a^{2/3} \log (c+d x) \log \left (-\frac{d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b^{5/3}}-\frac{c^2 \log (c+d x)}{2 b d^2}+\frac{x^2 \log (c+d x)}{2 b}+\frac{c x}{2 b d}-\frac{x^2}{4 b} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.700564, antiderivative size = 416, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 13, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.684, Rules used = {321, 292, 31, 634, 617, 204, 628, 2416, 2395, 43, 2394, 2393, 2391} \[ \frac{a^{2/3} \text{PolyLog}\left (2,\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b^{5/3}}+\frac{(-1)^{2/3} a^{2/3} \text{PolyLog}\left (2,\frac{(-1)^{2/3} \sqrt [3]{b} (c+d x)}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b^{5/3}}-\frac{\sqrt [3]{-1} a^{2/3} \text{PolyLog}\left (2,\frac{\sqrt [3]{-1} \sqrt [3]{b} (c+d x)}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} c}\right )}{3 b^{5/3}}+\frac{a^{2/3} \log (c+d x) \log \left (-\frac{d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b^{5/3}}-\frac{\sqrt [3]{-1} a^{2/3} \log (c+d x) \log \left (\frac{d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} c}\right )}{3 b^{5/3}}+\frac{(-1)^{2/3} a^{2/3} \log (c+d x) \log \left (-\frac{d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b^{5/3}}-\frac{c^2 \log (c+d x)}{2 b d^2}+\frac{x^2 \log (c+d x)}{2 b}+\frac{c x}{2 b d}-\frac{x^2}{4 b} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*Log[c + d*x])/(a + b*x^3),x]

[Out]

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

Rule 321

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 292

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 31

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

Rule 634

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 617

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 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

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 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 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 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{x^4 \log (c+d x)}{a+b x^3} \, dx &=\int \left (\frac{x \log (c+d x)}{b}-\frac{a x \log (c+d x)}{b \left (a+b x^3\right )}\right ) \, dx\\ &=\frac{\int x \log (c+d x) \, dx}{b}-\frac{a \int \frac{x \log (c+d x)}{a+b x^3} \, dx}{b}\\ &=\frac{x^2 \log (c+d x)}{2 b}-\frac{a \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}{b}-\frac{d \int \frac{x^2}{c+d x} \, dx}{2 b}\\ &=\frac{x^2 \log (c+d x)}{2 b}+\frac{a^{2/3} \int \frac{\log (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^{4/3}}-\frac{\left (\sqrt [3]{-1} a^{2/3}\right ) \int \frac{\log (c+d x)}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{3 b^{4/3}}+\frac{\left ((-1)^{2/3} a^{2/3}\right ) \int \frac{\log (c+d x)}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{3 b^{4/3}}-\frac{d \int \left (-\frac{c}{d^2}+\frac{x}{d}+\frac{c^2}{d^2 (c+d x)}\right ) \, dx}{2 b}\\ &=\frac{c x}{2 b d}-\frac{x^2}{4 b}-\frac{c^2 \log (c+d x)}{2 b d^2}+\frac{x^2 \log (c+d x)}{2 b}+\frac{a^{2/3} \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^{5/3}}-\frac{\sqrt [3]{-1} a^{2/3} \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^{5/3}}+\frac{(-1)^{2/3} a^{2/3} \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^{5/3}}-\frac{\left (a^{2/3} 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^{5/3}}+\frac{\left (\sqrt [3]{-1} a^{2/3} 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^{5/3}}-\frac{\left ((-1)^{2/3} a^{2/3} 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^{5/3}}\\ &=\frac{c x}{2 b d}-\frac{x^2}{4 b}-\frac{c^2 \log (c+d x)}{2 b d^2}+\frac{x^2 \log (c+d x)}{2 b}+\frac{a^{2/3} \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^{5/3}}-\frac{\sqrt [3]{-1} a^{2/3} \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^{5/3}}+\frac{(-1)^{2/3} a^{2/3} \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^{5/3}}-\frac{a^{2/3} \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 b^{5/3}}+\frac{\left (\sqrt [3]{-1} a^{2/3}\right ) \operatorname{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^{5/3}}-\frac{\left ((-1)^{2/3} a^{2/3}\right ) \operatorname{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^{5/3}}\\ &=\frac{c x}{2 b d}-\frac{x^2}{4 b}-\frac{c^2 \log (c+d x)}{2 b d^2}+\frac{x^2 \log (c+d x)}{2 b}+\frac{a^{2/3} \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^{5/3}}-\frac{\sqrt [3]{-1} a^{2/3} \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^{5/3}}+\frac{(-1)^{2/3} a^{2/3} \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^{5/3}}+\frac{a^{2/3} \text{Li}_2\left (\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b^{5/3}}+\frac{(-1)^{2/3} a^{2/3} \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^{5/3}}-\frac{\sqrt [3]{-1} a^{2/3} \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^{5/3}}\\ \end{align*}

Mathematica [A]  time = 0.322644, size = 403, normalized size = 0.97 \[ \frac{4 a^{2/3} d^2 \text{PolyLog}\left (2,\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )+4 (-1)^{2/3} a^{2/3} d^2 \text{PolyLog}\left (2,\frac{(-1)^{2/3} \sqrt [3]{b} (c+d x)}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )-4 \sqrt [3]{-1} a^{2/3} d^2 \text{PolyLog}\left (2,\frac{\sqrt [3]{-1} \sqrt [3]{b} (c+d x)}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} c}\right )+4 a^{2/3} d^2 \log (c+d x) \log \left (\frac{d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a} d-\sqrt [3]{b} c}\right )-4 \sqrt [3]{-1} a^{2/3} d^2 \log (c+d x) \log \left (\frac{d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} c}\right )+4 (-1)^{2/3} a^{2/3} d^2 \log (c+d x) \log \left (\frac{d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} c}\right )-6 b^{2/3} c^2 \log (c+d x)+6 b^{2/3} d^2 x^2 \log (c+d x)+6 b^{2/3} c d x-3 b^{2/3} d^2 x^2}{12 b^{5/3} d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*b^(2/3)*c*d*x - 3*b^(2/3)*d^2*x^2 - 6*b^(2/3)*c^2*Log[c + d*x] + 6*b^(2/3)*d^2*x^2*Log[c + d*x] + 4*a^(2/3)
*d^2*Log[(d*(a^(1/3) + b^(1/3)*x))/(-(b^(1/3)*c) + a^(1/3)*d)]*Log[c + d*x] - 4*(-1)^(1/3)*a^(2/3)*d^2*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] + 4*(-1)^(2/3)*a^(2/3)*d^2*
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] + 4*a^(2/3)*d^2*P
olyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c - a^(1/3)*d)] + 4*(-1)^(2/3)*a^(2/3)*d^2*PolyLog[2, ((-1)^(2/3)*b^(1/
3)*(c + d*x))/((-1)^(2/3)*b^(1/3)*c - a^(1/3)*d)] - 4*(-1)^(1/3)*a^(2/3)*d^2*PolyLog[2, ((-1)^(1/3)*b^(1/3)*(c
+ d*x))/((-1)^(1/3)*b^(1/3)*c + a^(1/3)*d)])/(12*b^(5/3)*d^2)

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Maple [C]  time = 0.373, size = 148, normalized size = 0.4 \begin{align*}{\frac{{x}^{2}\ln \left ( dx+c \right ) }{2\,b}}-{\frac{{c}^{2}\ln \left ( dx+c \right ) }{2\,b{d}^{2}}}-{\frac{{x}^{2}}{4\,b}}+{\frac{cx}{2\,bd}}+{\frac{3\,{c}^{2}}{4\,b{d}^{2}}}-{\frac{ad}{3\,{b}^{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 ) }{\frac{1}{{\it \_R1}-c} \left ( \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 ) \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^4*log(d*x + c)/(b*x^3 + a), 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(x**4*ln(d*x+c)/(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{x^{4} \log \left (d x + c\right )}{b x^{3} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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