3.326 \(\int \frac{a+b \log (c x^n)}{(d+e x^3)^2} \, dx\)

Optimal. Leaf size=520 \[ \frac{2 b n \text{PolyLog}\left (2,-\frac{\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac{2 i \sqrt{3} b n \text{PolyLog}\left (2,\frac{\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}+\frac{2 b n \text{PolyLog}\left (2,-\frac{(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}+\frac{2 \log \left (\frac{\sqrt [3]{e} x}{\sqrt [3]{d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac{2 i \sqrt{3} \log \left (1-\frac{\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}+\frac{2 \log \left (\frac{(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}+\frac{x \left (a+b \log \left (c x^n\right )\right )}{9 d^{5/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}-\frac{\sqrt [3]{-1} x \left (a+b \log \left (c x^n\right )\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac{x \left (a+b \log \left (c x^n\right )\right )}{9 d^{5/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}+\frac{\sqrt [3]{-1} b n \log \left (-(-1)^{2/3} \sqrt [3]{d}-\sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}-\frac{b n \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{9 d^{5/3} \sqrt [3]{e}}+\frac{\sqrt [3]{-1} b n \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{9 d^{5/3} \sqrt [3]{e}} \]

[Out]

(x*(a + b*Log[c*x^n]))/(9*d^(5/3)*(d^(1/3) + e^(1/3)*x)) - ((-1)^(1/3)*x*(a + b*Log[c*x^n]))/((1 + (-1)^(1/3))
^4*d^(5/3)*((-1)^(2/3)*d^(1/3) + e^(1/3)*x)) + (x*(a + b*Log[c*x^n]))/(9*d^(5/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)
*x)) + ((-1)^(1/3)*b*n*Log[-((-1)^(2/3)*d^(1/3)) - e^(1/3)*x])/((1 + (-1)^(1/3))^4*d^(5/3)*e^(1/3)) - (b*n*Log
[d^(1/3) + e^(1/3)*x])/(9*d^(5/3)*e^(1/3)) + ((-1)^(1/3)*b*n*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x])/(9*d^(5/3)*e
^(1/3)) + (2*(a + b*Log[c*x^n])*Log[1 + (e^(1/3)*x)/d^(1/3)])/(9*d^(5/3)*e^(1/3)) - ((2*I)*Sqrt[3]*(a + b*Log[
c*x^n])*Log[1 - ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^5*d^(5/3)*e^(1/3)) + (2*(a + b*Log[c*x^n])*
Log[1 + ((-1)^(2/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^4*d^(5/3)*e^(1/3)) + (2*b*n*PolyLog[2, -((e^(1/3)*x
)/d^(1/3))])/(9*d^(5/3)*e^(1/3)) - ((2*I)*Sqrt[3]*b*n*PolyLog[2, ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(
1/3))^5*d^(5/3)*e^(1/3)) + (2*b*n*PolyLog[2, -(((-1)^(2/3)*e^(1/3)*x)/d^(1/3))])/((1 + (-1)^(1/3))^4*d^(5/3)*e
^(1/3))

________________________________________________________________________________________

Rubi [A]  time = 0.45713, antiderivative size = 520, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.55, Rules used = {199, 200, 31, 634, 617, 204, 628, 2330, 2314, 2317, 2391} \[ \frac{2 b n \text{PolyLog}\left (2,-\frac{\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac{2 i \sqrt{3} b n \text{PolyLog}\left (2,\frac{\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}+\frac{2 b n \text{PolyLog}\left (2,-\frac{(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}+\frac{2 \log \left (\frac{\sqrt [3]{e} x}{\sqrt [3]{d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac{2 i \sqrt{3} \log \left (1-\frac{\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}+\frac{2 \log \left (\frac{(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}+\frac{x \left (a+b \log \left (c x^n\right )\right )}{9 d^{5/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}-\frac{\sqrt [3]{-1} x \left (a+b \log \left (c x^n\right )\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac{x \left (a+b \log \left (c x^n\right )\right )}{9 d^{5/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}+\frac{\sqrt [3]{-1} b n \log \left (-(-1)^{2/3} \sqrt [3]{d}-\sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}-\frac{b n \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{9 d^{5/3} \sqrt [3]{e}}+\frac{\sqrt [3]{-1} b n \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{9 d^{5/3} \sqrt [3]{e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(a + b*Log[c*x^n]))/(9*d^(5/3)*(d^(1/3) + e^(1/3)*x)) - ((-1)^(1/3)*x*(a + b*Log[c*x^n]))/((1 + (-1)^(1/3))
^4*d^(5/3)*((-1)^(2/3)*d^(1/3) + e^(1/3)*x)) + (x*(a + b*Log[c*x^n]))/(9*d^(5/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)
*x)) + ((-1)^(1/3)*b*n*Log[-((-1)^(2/3)*d^(1/3)) - e^(1/3)*x])/((1 + (-1)^(1/3))^4*d^(5/3)*e^(1/3)) - (b*n*Log
[d^(1/3) + e^(1/3)*x])/(9*d^(5/3)*e^(1/3)) + ((-1)^(1/3)*b*n*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x])/(9*d^(5/3)*e
^(1/3)) + (2*(a + b*Log[c*x^n])*Log[1 + (e^(1/3)*x)/d^(1/3)])/(9*d^(5/3)*e^(1/3)) - ((2*I)*Sqrt[3]*(a + b*Log[
c*x^n])*Log[1 - ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^5*d^(5/3)*e^(1/3)) + (2*(a + b*Log[c*x^n])*
Log[1 + ((-1)^(2/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^4*d^(5/3)*e^(1/3)) + (2*b*n*PolyLog[2, -((e^(1/3)*x
)/d^(1/3))])/(9*d^(5/3)*e^(1/3)) - ((2*I)*Sqrt[3]*b*n*PolyLog[2, ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(
1/3))^5*d^(5/3)*e^(1/3)) + (2*b*n*PolyLog[2, -(((-1)^(2/3)*e^(1/3)*x)/d^(1/3))])/((1 + (-1)^(1/3))^4*d^(5/3)*e
^(1/3))

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 200

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 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 2330

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = Expand
Integrand[(a + b*Log[c*x^n])^p, (d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n, p, q, r}
, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[r]))

Rule 2314

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x_Symbol] :> Simp[(x*(d + e*x^r)^(q
+ 1)*(a + b*Log[c*x^n]))/d, x] - Dist[(b*n)/d, Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q,
r}, x] && EqQ[r*(q + 1) + 1, 0]

Rule 2317

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[1 + (e*x)/d]*(a +
b*Log[c*x^n])^p)/e, x] - Dist[(b*n*p)/e, Int[(Log[1 + (e*x)/d]*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 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{a+b \log \left (c x^n\right )}{\left (d+e x^3\right )^2} \, dx &=\int \left (\frac{a+b \log \left (c x^n\right )}{9 d^{4/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )^2}+\frac{2 \left (a+b \log \left (c x^n\right )\right )}{9 d^{5/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac{(-1)^{2/3} \left (a+b \log \left (c x^n\right )\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{4/3} \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right )^2}-\frac{2 (-1)^{5/6} \sqrt{3} \left (a+b \log \left (c x^n\right )\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right )}+\frac{a+b \log \left (c x^n\right )}{\left (-1+\sqrt [3]{-1}\right )^2 \left (1+\sqrt [3]{-1}\right )^4 d^{4/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )^2}+\frac{2 (-1)^{2/3} \left (a+b \log \left (c x^n\right )\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}\right ) \, dx\\ &=\frac{2 \int \frac{a+b \log \left (c x^n\right )}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{9 d^{5/3}}+\frac{2 \int \frac{a+b \log \left (c x^n\right )}{\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x} \, dx}{9 d^{5/3}}-\frac{\left (2 (-1)^{5/6} \sqrt{3}\right ) \int \frac{a+b \log \left (c x^n\right )}{-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x} \, dx}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3}}+\frac{\int \frac{a+b \log \left (c x^n\right )}{\left (\sqrt [3]{d}+\sqrt [3]{e} x\right )^2} \, dx}{9 d^{4/3}}+\frac{\int \frac{a+b \log \left (c x^n\right )}{\left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right )^2} \, dx}{9 d^{4/3}}+\frac{\int \frac{a+b \log \left (c x^n\right )}{\left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )^2} \, dx}{9 d^{4/3}}\\ &=\frac{x \left (a+b \log \left (c x^n\right )\right )}{9 d^{5/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac{(-1)^{2/3} x \left (a+b \log \left (c x^n\right )\right )}{9 d^{5/3} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac{x \left (a+b \log \left (c x^n\right )\right )}{9 d^{5/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}+\frac{2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac{2 i \sqrt{3} \left (a+b \log \left (c x^n\right )\right ) \log \left (1-\frac{\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}-\frac{2 \sqrt [3]{-1} \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac{(b n) \int \frac{1}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{9 d^{5/3}}+\frac{(b n) \int \frac{1}{-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x} \, dx}{9 d^{5/3}}-\frac{(b n) \int \frac{1}{\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x} \, dx}{9 d^{5/3}}-\frac{(2 b n) \int \frac{\log \left (1+\frac{\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{x} \, dx}{9 d^{5/3} \sqrt [3]{e}}+\frac{\left (2 \sqrt [3]{-1} b n\right ) \int \frac{\log \left (1+\frac{(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{x} \, dx}{9 d^{5/3} \sqrt [3]{e}}+\frac{\left (2 i \sqrt{3} b n\right ) \int \frac{\log \left (1-\frac{\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{x} \, dx}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}\\ &=\frac{x \left (a+b \log \left (c x^n\right )\right )}{9 d^{5/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac{(-1)^{2/3} x \left (a+b \log \left (c x^n\right )\right )}{9 d^{5/3} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac{x \left (a+b \log \left (c x^n\right )\right )}{9 d^{5/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}-\frac{(-1)^{2/3} b n \log \left (-(-1)^{2/3} \sqrt [3]{d}-\sqrt [3]{e} x\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac{b n \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{9 d^{5/3} \sqrt [3]{e}}+\frac{\sqrt [3]{-1} b n \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{9 d^{5/3} \sqrt [3]{e}}+\frac{2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac{2 i \sqrt{3} \left (a+b \log \left (c x^n\right )\right ) \log \left (1-\frac{\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}-\frac{2 \sqrt [3]{-1} \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}+\frac{2 b n \text{Li}_2\left (-\frac{\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac{2 i \sqrt{3} b n \text{Li}_2\left (\frac{\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}-\frac{2 \sqrt [3]{-1} b n \text{Li}_2\left (-\frac{(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}\\ \end{align*}

Mathematica [A]  time = 2.14886, size = 571, normalized size = 1.1 \[ \frac{\frac{3 b n \left (\frac{2 \sqrt [3]{-1} \left (\text{PolyLog}\left (2,-\frac{\sqrt [3]{e} x}{\sqrt [3]{d}}\right )+\log (x) \log \left (\frac{\sqrt [3]{e} x}{\sqrt [3]{d}}+1\right )\right )}{\sqrt [3]{e}}-\frac{2 \left (\text{PolyLog}\left (2,\frac{\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )+\log (x) \log \left (1-\frac{\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )\right )}{\sqrt [3]{e}}-\frac{2 \left (\sqrt [3]{-1}-1\right ) \left (\text{PolyLog}\left (2,-\frac{(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )+\log (x) \log \left (\frac{(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}+1\right )\right )}{\sqrt [3]{e}}+\frac{\left (\sqrt [3]{-1}-1\right ) \left (\left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (-(-1)^{2/3} \sqrt [3]{d}-\sqrt [3]{e} x\right )+\sqrt [3]{-1} \sqrt [3]{e} x \log (x)\right )}{(-1)^{2/3} \sqrt [3]{d} \sqrt [3]{e}+e^{2/3} x}+\frac{\left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )-(-1)^{2/3} \sqrt [3]{e} x \log (x)}{e^{2/3} x-\sqrt [3]{-1} \sqrt [3]{d} \sqrt [3]{e}}+\sqrt [3]{-1} \left (\frac{x \log (x)}{\sqrt [3]{d}+\sqrt [3]{e} x}-\frac{\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt [3]{e}}\right )\right )}{\left (1+\sqrt [3]{-1}\right )^2}-\frac{\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{\sqrt [3]{e}}+\frac{3 d^{2/3} x \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{d+e x^3}+\frac{2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{\sqrt [3]{e}}-\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{e} x}{\sqrt [3]{d}}}{\sqrt{3}}\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{\sqrt [3]{e}}}{9 d^{5/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((3*d^(2/3)*x*(a - b*n*Log[x] + b*Log[c*x^n]))/(d + e*x^3) - (2*Sqrt[3]*ArcTan[(1 - (2*e^(1/3)*x)/d^(1/3))/Sqr
t[3]]*(a - b*n*Log[x] + b*Log[c*x^n]))/e^(1/3) + (2*(a - b*n*Log[x] + b*Log[c*x^n])*Log[d^(1/3) + e^(1/3)*x])/
e^(1/3) - ((a - b*n*Log[x] + b*Log[c*x^n])*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/e^(1/3) + (3*b*n*((
(-1 + (-1)^(1/3))*((-1)^(1/3)*e^(1/3)*x*Log[x] + (d^(1/3) - (-1)^(1/3)*e^(1/3)*x)*Log[-((-1)^(2/3)*d^(1/3)) -
e^(1/3)*x]))/((-1)^(2/3)*d^(1/3)*e^(1/3) + e^(2/3)*x) + (-1)^(1/3)*((x*Log[x])/(d^(1/3) + e^(1/3)*x) - Log[d^(
1/3) + e^(1/3)*x]/e^(1/3)) + (-((-1)^(2/3)*e^(1/3)*x*Log[x]) + (d^(1/3) + (-1)^(2/3)*e^(1/3)*x)*Log[d^(1/3) +
(-1)^(2/3)*e^(1/3)*x])/(-((-1)^(1/3)*d^(1/3)*e^(1/3)) + e^(2/3)*x) + (2*(-1)^(1/3)*(Log[x]*Log[1 + (e^(1/3)*x)
/d^(1/3)] + PolyLog[2, -((e^(1/3)*x)/d^(1/3))]))/e^(1/3) - (2*(Log[x]*Log[1 - ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)]
+ PolyLog[2, ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)]))/e^(1/3) - (2*(-1 + (-1)^(1/3))*(Log[x]*Log[1 + ((-1)^(2/3)*e^(1
/3)*x)/d^(1/3)] + PolyLog[2, -(((-1)^(2/3)*e^(1/3)*x)/d^(1/3))]))/e^(1/3)))/(1 + (-1)^(1/3))^2)/(9*d^(5/3))

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Maple [C]  time = 0.352, size = 1388, normalized size = 2.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))/(e*x^3+d)^2,x)

[Out]

-1/9*b/d/e/(d/e)^(2/3)*ln(x^2-(d/e)^(1/3)*x+(d/e)^(2/3))*ln(x^n)+2/9*b/d/e/(d/e)^(2/3)*ln(x+(d/e)^(1/3))*ln(x^
n)+1/9*b/d/e/(d/e)^(2/3)*ln(x^2-(d/e)^(1/3)*x+(d/e)^(2/3))*n*ln(x)-1/9*b*n/d/e/(d/e)^(2/3)*3^(1/2)*arctan(1/3*
3^(1/2)*(2/(d/e)^(1/3)*x-1))+2/9*b*ln(c)/d/e/(d/e)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/3)*x-1))-2/9*b
/d/e/(d/e)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/3)*x-1))*n*ln(x)-1/9*I*b*Pi*csgn(I*c*x^n)^3/d/e/(d/e)^
(2/3)*ln(x+(d/e)^(1/3))+1/6*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)*x/d/(e*x^3+d)+1/18*I*b*Pi*csgn(I*c*x^n)^3/d/e/(d/
e)^(2/3)*ln(x^2-(d/e)^(1/3)*x+(d/e)^(2/3))+1/6*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2*x/d/(e*x^3+d)+1/3*a*x/d/(e*x
^3+d)-1/9*b*ln(c)/d/e/(d/e)^(2/3)*ln(x^2-(d/e)^(1/3)*x+(d/e)^(2/3))+2/9*a/d/e/(d/e)^(2/3)*3^(1/2)*arctan(1/3*3
^(1/2)*(2/(d/e)^(1/3)*x-1))+2/9*b*ln(c)/d/e/(d/e)^(2/3)*ln(x+(d/e)^(1/3))-1/9*b*n/d/e/(d/e)^(2/3)*ln(x+(d/e)^(
1/3))+1/18*b*n/d/e/(d/e)^(2/3)*ln(x^2-(d/e)^(1/3)*x+(d/e)^(2/3))-2/9*b/d/e/(d/e)^(2/3)*ln(x+(d/e)^(1/3))*n*ln(
x)-1/18*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d/e/(d/e)^(2/3)*ln(x^2-(d/e)^(1/3)*x+(d/e)^(2/3))+1/3*b*x/d/(e*x^3+
d)*ln(x^n)+2/9*a/d/e/(d/e)^(2/3)*ln(x+(d/e)^(1/3))-1/9*a/d/e/(d/e)^(2/3)*ln(x^2-(d/e)^(1/3)*x+(d/e)^(2/3))+1/3
*b*ln(c)*x/d/(e*x^3+d)-1/9*I*b*Pi*csgn(I*c*x^n)^3/d/e/(d/e)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/3)*x-
1))+1/9*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/d/e/(d/e)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/3)*x-1))-1/9*I
*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/d/e/(d/e)^(2/3)*ln(x+(d/e)^(1/3))+1/18*I*b*Pi*csgn(I*x^n)*csgn(I*c*x
^n)*csgn(I*c)/d/e/(d/e)^(2/3)*ln(x^2-(d/e)^(1/3)*x+(d/e)^(2/3))-1/6*I*b*Pi*csgn(I*c*x^n)^3*x/d/(e*x^3+d)+2/9*b
/d/e/(d/e)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/3)*x-1))*ln(x^n)+1/9*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^
2/d/e/(d/e)^(2/3)*ln(x+(d/e)^(1/3))-1/18*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/d/e/(d/e)^(2/3)*ln(x^2-(d/e)^(1/3)*x
+(d/e)^(2/3))+1/9*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/d/e/(d/e)^(2/3)*ln(x+(d/e)^(1/3))-1/6*I*b*Pi*csgn(I*x^n)*cs
gn(I*c*x^n)*csgn(I*c)*x/d/(e*x^3+d)+2/9*b*n/e/d*sum(1/_R1^2*(ln(x)*ln((_R1-x)/_R1)+dilog((_R1-x)/_R1)),_R1=Roo
tOf(_Z^3*e+d))+1/9*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d/e/(d/e)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/3
)*x-1))-1/9*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/d/e/(d/e)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/
3)*x-1))

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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((a+b*log(c*x^n))/(e*x^3+d)^2,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{b \log \left (c x^{n}\right ) + a}{e^{2} x^{6} + 2 \, d e x^{3} + d^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*log(c*x^n) + a)/(e^2*x^6 + 2*d*e*x^3 + d^2), 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((a+b*ln(c*x**n))/(e*x**3+d)**2,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \log \left (c x^{n}\right ) + a}{{\left (e x^{3} + d\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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