\(\int \frac {x^2 (a+b \log (c x^n))}{(d+e x^2)^3} \, dx\) [237]

Optimal result

Integrand size = 23, antiderivative size = 187 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx=\frac {b n x}{8 d e \left (d+e x^2\right )}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 e \left (d+e x^2\right )^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{8 d e \left (d+e x^2\right )}+\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 d^{3/2} e^{3/2}}-\frac {i b n \operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{3/2} e^{3/2}}+\frac {i b n \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{3/2} e^{3/2}} \]

[Out]

1/8*b*n*x/d/e/(e*x^2+d)-1/4*x*(a+b*ln(c*x^n))/e/(e*x^2+d)^2+1/8*x*(a+b*ln(c*x^n))/d/e/(e*x^2+d)+1/8*arctan(x*e
^(1/2)/d^(1/2))*(a+b*ln(c*x^n))/d^(3/2)/e^(3/2)-1/16*I*b*n*polylog(2,-I*x*e^(1/2)/d^(1/2))/d^(3/2)/e^(3/2)+1/1
6*I*b*n*polylog(2,I*x*e^(1/2)/d^(1/2))/d^(3/2)/e^(3/2)

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {294, 205, 211, 2393, 2360, 2361, 12, 4940, 2438} \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx=\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 d^{3/2} e^{3/2}}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{8 d e \left (d+e x^2\right )}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 e \left (d+e x^2\right )^2}-\frac {i b n \operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{3/2} e^{3/2}}+\frac {i b n \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{3/2} e^{3/2}}+\frac {b n x}{8 d e \left (d+e x^2\right )} \]

[In]

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

[Out]

(b*n*x)/(8*d*e*(d + e*x^2)) - (x*(a + b*Log[c*x^n]))/(4*e*(d + e*x^2)^2) + (x*(a + b*Log[c*x^n]))/(8*d*e*(d +
e*x^2)) + (ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(a + b*Log[c*x^n]))/(8*d^(3/2)*e^(3/2)) - ((I/16)*b*n*PolyLog[2, ((-I)*
Sqrt[e]*x)/Sqrt[d]])/(d^(3/2)*e^(3/2)) + ((I/16)*b*n*PolyLog[2, (I*Sqrt[e]*x)/Sqrt[d]])/(d^(3/2)*e^(3/2))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 205

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] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

Rule 211

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

Rule 294

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*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 2360

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Simp[(-x)*(d + e*x^2)^(q +
1)*((a + b*Log[c*x^n])/(2*d*(q + 1))), x] + (Dist[(2*q + 3)/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*(a + b*Log[
c*x^n]), x], x] + Dist[b*(n/(2*d*(q + 1))), Int[(d + e*x^2)^(q + 1), x], x]) /; FreeQ[{a, b, c, d, e, n}, x] &
& LtQ[q, -1]

Rule 2361

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> With[{u = IntHide[1/(d + e*x^2),
x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[u/x, x], x]] /; FreeQ[{a, b, c, d, e, n}, x]

Rule 2393

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = ExpandIntegrand[a + b*Log[c*x^n], (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c,
d, e, f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[m] && IntegerQ[r]))

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 4940

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (Dist[I*(b/2), Int[Log[1 - I*c*x
]/x, x], x] - Dist[I*(b/2), Int[Log[1 + I*c*x]/x, x], x]) /; FreeQ[{a, b, c}, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {d \left (a+b \log \left (c x^n\right )\right )}{e \left (d+e x^2\right )^3}+\frac {a+b \log \left (c x^n\right )}{e \left (d+e x^2\right )^2}\right ) \, dx \\ & = \frac {\int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^2} \, dx}{e}-\frac {d \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^3} \, dx}{e} \\ & = -\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 e \left (d+e x^2\right )^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{2 d e \left (d+e x^2\right )}-\frac {3 \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^2} \, dx}{4 e}+\frac {\int \frac {a+b \log \left (c x^n\right )}{d+e x^2} \, dx}{2 d e}+\frac {(b n) \int \frac {1}{\left (d+e x^2\right )^2} \, dx}{4 e}-\frac {(b n) \int \frac {1}{d+e x^2} \, dx}{2 d e} \\ & = \frac {b n x}{8 d e \left (d+e x^2\right )}-\frac {b n \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} e^{3/2}}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 e \left (d+e x^2\right )^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{8 d e \left (d+e x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^{3/2} e^{3/2}}-\frac {3 \int \frac {a+b \log \left (c x^n\right )}{d+e x^2} \, dx}{8 d e}+\frac {(b n) \int \frac {1}{d+e x^2} \, dx}{8 d e}+\frac {(3 b n) \int \frac {1}{d+e x^2} \, dx}{8 d e}-\frac {(b n) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e} x} \, dx}{2 d e} \\ & = \frac {b n x}{8 d e \left (d+e x^2\right )}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 e \left (d+e x^2\right )^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{8 d e \left (d+e x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 d^{3/2} e^{3/2}}-\frac {(b n) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{2 d^{3/2} e^{3/2}}+\frac {(3 b n) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e} x} \, dx}{8 d e} \\ & = \frac {b n x}{8 d e \left (d+e x^2\right )}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 e \left (d+e x^2\right )^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{8 d e \left (d+e x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 d^{3/2} e^{3/2}}-\frac {(i b n) \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{4 d^{3/2} e^{3/2}}+\frac {(i b n) \int \frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{4 d^{3/2} e^{3/2}}+\frac {(3 b n) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{8 d^{3/2} e^{3/2}} \\ & = \frac {b n x}{8 d e \left (d+e x^2\right )}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 e \left (d+e x^2\right )^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{8 d e \left (d+e x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 d^{3/2} e^{3/2}}-\frac {i b n \text {Li}_2\left (-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 d^{3/2} e^{3/2}}+\frac {i b n \text {Li}_2\left (\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 d^{3/2} e^{3/2}}+\frac {(3 i b n) \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{16 d^{3/2} e^{3/2}}-\frac {(3 i b n) \int \frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{16 d^{3/2} e^{3/2}} \\ & = \frac {b n x}{8 d e \left (d+e x^2\right )}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 e \left (d+e x^2\right )^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{8 d e \left (d+e x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 d^{3/2} e^{3/2}}-\frac {i b n \text {Li}_2\left (-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{3/2} e^{3/2}}+\frac {i b n \text {Li}_2\left (\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{3/2} e^{3/2}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(497\) vs. \(2(187)=374\).

Time = 0.61 (sec) , antiderivative size = 497, normalized size of antiderivative = 2.66 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx=\frac {\frac {d \left (a+b \log \left (c x^n\right )\right )}{(-d)^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )^2}+\frac {a+b \log \left (c x^n\right )}{\sqrt {-d} \left (\sqrt {-d}+\sqrt {e} x\right )^2}-\frac {a+b \log \left (c x^n\right )}{\sqrt {-d} d-d \sqrt {e} x}+\frac {a+b \log \left (c x^n\right )}{\sqrt {-d} d+d \sqrt {e} x}+\frac {b d n \left (\log (x)-\log \left (\sqrt {-d}-\sqrt {e} x\right )\right )}{(-d)^{5/2}}+\frac {b n \left (\log (x)-\log \left (\sqrt {-d}+\sqrt {e} x\right )\right )}{(-d)^{3/2}}+\frac {b n \left (d+\left (d-\sqrt {-d} \sqrt {e} x\right ) \log (x)+\left (-d+\sqrt {-d} \sqrt {e} x\right ) \log \left (\sqrt {-d}+\sqrt {e} x\right )\right )}{d^2 \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{(-d)^{3/2}}-\frac {b n \left (d+\left (d+\sqrt {-d} \sqrt {e} x\right ) \log (x)-\left (d+\sqrt {-d} \sqrt {e} x\right ) \log \left ((-d)^{3/2}+d \sqrt {e} x\right )\right )}{d^2 \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{(-d)^{5/2}}+\frac {b d n \operatorname {PolyLog}\left (2,\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{(-d)^{5/2}}+\frac {b n \operatorname {PolyLog}\left (2,\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{(-d)^{3/2}}}{16 e^{3/2}} \]

[In]

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

[Out]

((d*(a + b*Log[c*x^n]))/((-d)^(3/2)*(Sqrt[-d] - Sqrt[e]*x)^2) + (a + b*Log[c*x^n])/(Sqrt[-d]*(Sqrt[-d] + Sqrt[
e]*x)^2) - (a + b*Log[c*x^n])/(Sqrt[-d]*d - d*Sqrt[e]*x) + (a + b*Log[c*x^n])/(Sqrt[-d]*d + d*Sqrt[e]*x) + (b*
d*n*(Log[x] - Log[Sqrt[-d] - Sqrt[e]*x]))/(-d)^(5/2) + (b*n*(Log[x] - Log[Sqrt[-d] + Sqrt[e]*x]))/(-d)^(3/2) +
(b*n*(d + (d - Sqrt[-d]*Sqrt[e]*x)*Log[x] + (-d + Sqrt[-d]*Sqrt[e]*x)*Log[Sqrt[-d] + Sqrt[e]*x]))/(d^2*(Sqrt[
-d] + Sqrt[e]*x)) + ((a + b*Log[c*x^n])*Log[1 + (Sqrt[e]*x)/Sqrt[-d]])/(-d)^(3/2) - (b*n*(d + (d + Sqrt[-d]*Sq
rt[e]*x)*Log[x] - (d + Sqrt[-d]*Sqrt[e]*x)*Log[(-d)^(3/2) + d*Sqrt[e]*x]))/(d^2*(Sqrt[-d] - Sqrt[e]*x)) + (d*(
a + b*Log[c*x^n])*Log[1 + (d*Sqrt[e]*x)/(-d)^(3/2)])/(-d)^(5/2) + (b*d*n*PolyLog[2, (Sqrt[e]*x)/Sqrt[-d]])/(-d
)^(5/2) + (b*n*PolyLog[2, (d*Sqrt[e]*x)/(-d)^(3/2)])/(-d)^(3/2))/(16*e^(3/2))

Maple [C] (warning: unable to verify)

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

Time = 0.65 (sec) , antiderivative size = 826, normalized size of antiderivative = 4.42

method	result	size
risch	\(-\frac {b n \ln \left (x \right ) x^{3}}{2 d \left (e \,x^{2}+d \right )^{2}}+\frac {b \,x^{3} \ln \left (x^{n}\right )}{8 \left (e \,x^{2}+d \right )^{2} d}-\frac {b n \ln \left (x \right ) x}{2 e \left (e \,x^{2}+d \right )^{2}}-\frac {b x \ln \left (x^{n}\right )}{8 \left (e \,x^{2}+d \right )^{2} e}-\frac {b \arctan \left (\frac {x e}{\sqrt {d e}}\right ) n \ln \left (x \right )}{8 e d \sqrt {d e}}+\frac {b \arctan \left (\frac {x e}{\sqrt {d e}}\right ) \ln \left (x^{n}\right )}{8 e d \sqrt {d e}}+\frac {b n x}{8 d e \left (e \,x^{2}+d \right )}-\frac {3 b n e \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right ) x^{4}}{16 d \left (e \,x^{2}+d \right )^{2} \sqrt {-d e}}+\frac {3 b n e \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right ) x^{4}}{16 d \left (e \,x^{2}+d \right )^{2} \sqrt {-d e}}-\frac {3 b n \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right ) x^{2}}{8 \left (e \,x^{2}+d \right )^{2} \sqrt {-d e}}+\frac {3 b n \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right ) x^{2}}{8 \left (e \,x^{2}+d \right )^{2} \sqrt {-d e}}-\frac {3 b n d \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{16 e \left (e \,x^{2}+d \right )^{2} \sqrt {-d e}}+\frac {3 b n d \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{16 e \left (e \,x^{2}+d \right )^{2} \sqrt {-d e}}+\frac {b n \operatorname {dilog}\left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{16 d e \sqrt {-d e}}-\frac {b n \operatorname {dilog}\left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{16 d e \sqrt {-d e}}+\frac {b n \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right ) x^{2}}{4 d \left (e \,x^{2}+d \right ) \sqrt {-d e}}-\frac {b n \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right ) x^{2}}{4 d \left (e \,x^{2}+d \right ) \sqrt {-d e}}+\frac {b n \ln \left (x \right ) x}{2 e d \left (e \,x^{2}+d \right )}+\frac {b n \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{4 e \left (e \,x^{2}+d \right ) \sqrt {-d e}}-\frac {b n \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{4 e \left (e \,x^{2}+d \right ) \sqrt {-d e}}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \left (\frac {\frac {x^{3}}{8 d}-\frac {x}{8 e}}{\left (e \,x^{2}+d \right )^{2}}+\frac {\arctan \left (\frac {x e}{\sqrt {d e}}\right )}{8 e d \sqrt {d e}}\right )\)	\(826\)

[In]

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

[Out]

-1/2*b*n/d*ln(x)/(e*x^2+d)^2*x^3+1/8*b/(e*x^2+d)^2/d*x^3*ln(x^n)-1/2*b*n/e*ln(x)/(e*x^2+d)^2*x-1/8*b/(e*x^2+d)
^2*x/e*ln(x^n)-1/8*b/e/d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*n*ln(x)+1/8*b/e/d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1
/2))*ln(x^n)+1/8*b*n*x/d/e/(e*x^2+d)-3/16*b*n/d*e*ln(x)/(e*x^2+d)^2/(-d*e)^(1/2)*ln((-e*x+(-d*e)^(1/2))/(-d*e)
^(1/2))*x^4+3/16*b*n/d*e*ln(x)/(e*x^2+d)^2/(-d*e)^(1/2)*ln((e*x+(-d*e)^(1/2))/(-d*e)^(1/2))*x^4-3/8*b*n*ln(x)/
(e*x^2+d)^2/(-d*e)^(1/2)*ln((-e*x+(-d*e)^(1/2))/(-d*e)^(1/2))*x^2+3/8*b*n*ln(x)/(e*x^2+d)^2/(-d*e)^(1/2)*ln((e
*x+(-d*e)^(1/2))/(-d*e)^(1/2))*x^2-3/16*b*n*d/e*ln(x)/(e*x^2+d)^2/(-d*e)^(1/2)*ln((-e*x+(-d*e)^(1/2))/(-d*e)^(
1/2))+3/16*b*n*d/e*ln(x)/(e*x^2+d)^2/(-d*e)^(1/2)*ln((e*x+(-d*e)^(1/2))/(-d*e)^(1/2))+1/16*b*n/d/e/(-d*e)^(1/2
)*dilog((-e*x+(-d*e)^(1/2))/(-d*e)^(1/2))-1/16*b*n/d/e/(-d*e)^(1/2)*dilog((e*x+(-d*e)^(1/2))/(-d*e)^(1/2))+1/4
*b*n*ln(x)/d/(e*x^2+d)/(-d*e)^(1/2)*ln((-e*x+(-d*e)^(1/2))/(-d*e)^(1/2))*x^2-1/4*b*n*ln(x)/d/(e*x^2+d)/(-d*e)^
(1/2)*ln((e*x+(-d*e)^(1/2))/(-d*e)^(1/2))*x^2+1/2*b*n/e*ln(x)/d/(e*x^2+d)*x+1/4*b*n/e*ln(x)/(e*x^2+d)/(-d*e)^(
1/2)*ln((-e*x+(-d*e)^(1/2))/(-d*e)^(1/2))-1/4*b*n/e*ln(x)/(e*x^2+d)/(-d*e)^(1/2)*ln((e*x+(-d*e)^(1/2))/(-d*e)^
(1/2))+(-1/2*I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/2*I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+1/2*I*b*Pi*csgn(I
*x^n)*csgn(I*c*x^n)^2-1/2*I*b*Pi*csgn(I*c*x^n)^3+b*ln(c)+a)*((1/8/d*x^3-1/8*x/e)/(e*x^2+d)^2+1/8/e/d/(d*e)^(1/
2)*arctan(x*e/(d*e)^(1/2)))

Fricas [F]

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

[In]

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

[Out]

integral((b*x^2*log(c*x^n) + a*x^2)/(e^3*x^6 + 3*d*e^2*x^4 + 3*d^2*e*x^2 + d^3), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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