3.95 \(\int x^2 (a+b \log (c x^n)) \log (d (e+f x^2)^m) \, dx\)

Optimal. Leaf size=251 \[ \frac{i b e^{3/2} m n \text{PolyLog}\left (2,-\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{3 f^{3/2}}-\frac{i b e^{3/2} m n \text{PolyLog}\left (2,\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{3 f^{3/2}}+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac{2 e^{3/2} m \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^{3/2}}+\frac{2 e m x \left (a+b \log \left (c x^n\right )\right )}{3 f}-\frac{2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{1}{9} b n x^3 \log \left (d \left (e+f x^2\right )^m\right )+\frac{2 b e^{3/2} m n \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{9 f^{3/2}}-\frac{8 b e m n x}{9 f}+\frac{4}{27} b m n x^3 \]

[Out]

(-8*b*e*m*n*x)/(9*f) + (4*b*m*n*x^3)/27 + (2*b*e^(3/2)*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(9*f^(3/2)) + (2*e*m*x
*(a + b*Log[c*x^n]))/(3*f) - (2*m*x^3*(a + b*Log[c*x^n]))/9 - (2*e^(3/2)*m*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*(a + b*
Log[c*x^n]))/(3*f^(3/2)) - (b*n*x^3*Log[d*(e + f*x^2)^m])/9 + (x^3*(a + b*Log[c*x^n])*Log[d*(e + f*x^2)^m])/3
+ ((I/3)*b*e^(3/2)*m*n*PolyLog[2, ((-I)*Sqrt[f]*x)/Sqrt[e]])/f^(3/2) - ((I/3)*b*e^(3/2)*m*n*PolyLog[2, (I*Sqrt
[f]*x)/Sqrt[e]])/f^(3/2)

________________________________________________________________________________________

Rubi [A]  time = 0.184583, antiderivative size = 251, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2455, 302, 205, 2376, 4848, 2391} \[ \frac{i b e^{3/2} m n \text{PolyLog}\left (2,-\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{3 f^{3/2}}-\frac{i b e^{3/2} m n \text{PolyLog}\left (2,\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{3 f^{3/2}}+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac{2 e^{3/2} m \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^{3/2}}+\frac{2 e m x \left (a+b \log \left (c x^n\right )\right )}{3 f}-\frac{2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{1}{9} b n x^3 \log \left (d \left (e+f x^2\right )^m\right )+\frac{2 b e^{3/2} m n \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{9 f^{3/2}}-\frac{8 b e m n x}{9 f}+\frac{4}{27} b m n x^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*b*e*m*n*x)/(9*f) + (4*b*m*n*x^3)/27 + (2*b*e^(3/2)*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(9*f^(3/2)) + (2*e*m*x
*(a + b*Log[c*x^n]))/(3*f) - (2*m*x^3*(a + b*Log[c*x^n]))/9 - (2*e^(3/2)*m*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*(a + b*
Log[c*x^n]))/(3*f^(3/2)) - (b*n*x^3*Log[d*(e + f*x^2)^m])/9 + (x^3*(a + b*Log[c*x^n])*Log[d*(e + f*x^2)^m])/3
+ ((I/3)*b*e^(3/2)*m*n*PolyLog[2, ((-I)*Sqrt[f]*x)/Sqrt[e]])/f^(3/2) - ((I/3)*b*e^(3/2)*m*n*PolyLog[2, (I*Sqrt
[f]*x)/Sqrt[e]])/f^(3/2)

Rule 2455

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.)*(x_))^(m_.), x_Symbol] :> Simp[((f*x)^(m
+ 1)*(a + b*Log[c*(d + e*x^n)^p]))/(f*(m + 1)), x] - Dist[(b*e*n*p)/(f*(m + 1)), Int[(x^(n - 1)*(f*x)^(m + 1))
/(d + e*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && NeQ[m, -1]

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 205

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

Rule 2376

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((g_.)*(x_))^(q_.), x_Sym
bol] :> With[{u = IntHide[(g*x)^q*Log[d*(e + f*x^m)^r], x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[Dist
[1/x, u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && (IntegerQ[(q + 1)/m] || (RationalQ[m] &
& RationalQ[q])) && NeQ[q, -1]

Rule 4848

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]

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 x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right ) \, dx &=\frac{2 e m x \left (a+b \log \left (c x^n\right )\right )}{3 f}-\frac{2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{2 e^{3/2} m \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^{3/2}}+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-(b n) \int \left (\frac{2 e m}{3 f}-\frac{2 m x^2}{9}-\frac{2 e^{3/2} m \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{3 f^{3/2} x}+\frac{1}{3} x^2 \log \left (d \left (e+f x^2\right )^m\right )\right ) \, dx\\ &=-\frac{2 b e m n x}{3 f}+\frac{2}{27} b m n x^3+\frac{2 e m x \left (a+b \log \left (c x^n\right )\right )}{3 f}-\frac{2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{2 e^{3/2} m \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^{3/2}}+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac{1}{3} (b n) \int x^2 \log \left (d \left (e+f x^2\right )^m\right ) \, dx+\frac{\left (2 b e^{3/2} m n\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{x} \, dx}{3 f^{3/2}}\\ &=-\frac{2 b e m n x}{3 f}+\frac{2}{27} b m n x^3+\frac{2 e m x \left (a+b \log \left (c x^n\right )\right )}{3 f}-\frac{2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{2 e^{3/2} m \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^{3/2}}-\frac{1}{9} b n x^3 \log \left (d \left (e+f x^2\right )^m\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac{\left (i b e^{3/2} m n\right ) \int \frac{\log \left (1-\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{x} \, dx}{3 f^{3/2}}-\frac{\left (i b e^{3/2} m n\right ) \int \frac{\log \left (1+\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{x} \, dx}{3 f^{3/2}}+\frac{1}{9} (2 b f m n) \int \frac{x^4}{e+f x^2} \, dx\\ &=-\frac{2 b e m n x}{3 f}+\frac{2}{27} b m n x^3+\frac{2 e m x \left (a+b \log \left (c x^n\right )\right )}{3 f}-\frac{2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{2 e^{3/2} m \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^{3/2}}-\frac{1}{9} b n x^3 \log \left (d \left (e+f x^2\right )^m\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac{i b e^{3/2} m n \text{Li}_2\left (-\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{3 f^{3/2}}-\frac{i b e^{3/2} m n \text{Li}_2\left (\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{3 f^{3/2}}+\frac{1}{9} (2 b f m n) \int \left (-\frac{e}{f^2}+\frac{x^2}{f}+\frac{e^2}{f^2 \left (e+f x^2\right )}\right ) \, dx\\ &=-\frac{8 b e m n x}{9 f}+\frac{4}{27} b m n x^3+\frac{2 e m x \left (a+b \log \left (c x^n\right )\right )}{3 f}-\frac{2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{2 e^{3/2} m \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^{3/2}}-\frac{1}{9} b n x^3 \log \left (d \left (e+f x^2\right )^m\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac{i b e^{3/2} m n \text{Li}_2\left (-\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{3 f^{3/2}}-\frac{i b e^{3/2} m n \text{Li}_2\left (\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{3 f^{3/2}}+\frac{\left (2 b e^2 m n\right ) \int \frac{1}{e+f x^2} \, dx}{9 f}\\ &=-\frac{8 b e m n x}{9 f}+\frac{4}{27} b m n x^3+\frac{2 b e^{3/2} m n \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{9 f^{3/2}}+\frac{2 e m x \left (a+b \log \left (c x^n\right )\right )}{3 f}-\frac{2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{2 e^{3/2} m \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^{3/2}}-\frac{1}{9} b n x^3 \log \left (d \left (e+f x^2\right )^m\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac{i b e^{3/2} m n \text{Li}_2\left (-\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{3 f^{3/2}}-\frac{i b e^{3/2} m n \text{Li}_2\left (\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{3 f^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.12771, size = 389, normalized size = 1.55 \[ \frac{9 i b e^{3/2} m n \text{PolyLog}\left (2,-\frac{i \sqrt{f} x}{\sqrt{e}}\right )-9 i b e^{3/2} m n \text{PolyLog}\left (2,\frac{i \sqrt{f} x}{\sqrt{e}}\right )+9 a f^{3/2} x^3 \log \left (d \left (e+f x^2\right )^m\right )-18 a e^{3/2} m \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )+18 a e \sqrt{f} m x-6 a f^{3/2} m x^3+9 b f^{3/2} x^3 \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-18 b e^{3/2} m \log \left (c x^n\right ) \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )+18 b e \sqrt{f} m x \log \left (c x^n\right )-6 b f^{3/2} m x^3 \log \left (c x^n\right )-3 b f^{3/2} n x^3 \log \left (d \left (e+f x^2\right )^m\right )-9 i b e^{3/2} m n \log (x) \log \left (1-\frac{i \sqrt{f} x}{\sqrt{e}}\right )+9 i b e^{3/2} m n \log (x) \log \left (1+\frac{i \sqrt{f} x}{\sqrt{e}}\right )+6 b e^{3/2} m n \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )+18 b e^{3/2} m n \log (x) \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )-24 b e \sqrt{f} m n x+4 b f^{3/2} m n x^3}{27 f^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(18*a*e*Sqrt[f]*m*x - 24*b*e*Sqrt[f]*m*n*x - 6*a*f^(3/2)*m*x^3 + 4*b*f^(3/2)*m*n*x^3 - 18*a*e^(3/2)*m*ArcTan[(
Sqrt[f]*x)/Sqrt[e]] + 6*b*e^(3/2)*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]] + 18*b*e^(3/2)*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e
]]*Log[x] + 18*b*e*Sqrt[f]*m*x*Log[c*x^n] - 6*b*f^(3/2)*m*x^3*Log[c*x^n] - 18*b*e^(3/2)*m*ArcTan[(Sqrt[f]*x)/S
qrt[e]]*Log[c*x^n] - (9*I)*b*e^(3/2)*m*n*Log[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + (9*I)*b*e^(3/2)*m*n*Log[x]*Lo
g[1 + (I*Sqrt[f]*x)/Sqrt[e]] + 9*a*f^(3/2)*x^3*Log[d*(e + f*x^2)^m] - 3*b*f^(3/2)*n*x^3*Log[d*(e + f*x^2)^m] +
9*b*f^(3/2)*x^3*Log[c*x^n]*Log[d*(e + f*x^2)^m] + (9*I)*b*e^(3/2)*m*n*PolyLog[2, ((-I)*Sqrt[f]*x)/Sqrt[e]] -
(9*I)*b*e^(3/2)*m*n*PolyLog[2, (I*Sqrt[f]*x)/Sqrt[e]])/(27*f^(3/2))

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Maple [C]  time = 0.168, size = 2321, normalized size = 9.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3*ln(d)*a+1/6*I*x^3*Pi*ln(d)*b*csgn(I*x^n)*csgn(I*c*x^n)^2+2/3*e*a*m/f*x-2/3*m/f*e^2/(e*f)^(1/2)*arctan(
x*f/(e*f)^(1/2))*b*ln(c)+1/6*I*Pi*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)^2*x^3*a-1/6*I*Pi*csgn(I*d*(f*x^2+e
)^m)^3*x^3*b*ln(c)-1/6*I*Pi*csgn(I*d*(f*x^2+e)^m)^3*b*x^3*ln(x^n)-2/3*m/f*e^2/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/
2))*a+1/12*Pi^2*csgn(I*d*(f*x^2+e)^m)^3*x^3*b*csgn(I*x^n)*csgn(I*c*x^n)^2+1/9*I*m*x^3*Pi*b*csgn(I*c*x^n)^3+1/1
8*I*Pi*csgn(I*d*(f*x^2+e)^m)^3*b*x^3*n+1/6*I*Pi*csgn(I*d)*csgn(I*d*(f*x^2+e)^m)^2*x^3*a+(1/3*x^3*b*ln(x^n)+1/1
8*x^3*(-3*I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+3*I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+3*I*b*Pi*csgn(I*x^n)*c
sgn(I*c*x^n)^2-3*I*b*Pi*csgn(I*c*x^n)^3+6*b*ln(c)-2*b*n+6*a))*ln((f*x^2+e)^m)-8/9*b*e*m*n*x/f+4/27*b*m*n*x^3+1
/12*Pi^2*csgn(I*d)*csgn(I*d*(f*x^2+e)^m)^2*x^3*b*csgn(I*c*x^n)^3+1/12*Pi^2*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2
+e)^m)^2*x^3*b*csgn(I*c*x^n)^3+1/12*Pi^2*csgn(I*d*(f*x^2+e)^m)^3*x^3*b*csgn(I*c)*csgn(I*c*x^n)^2+1/3*x^3*ln(c)
*ln(d)*b-2/9*x^3*ln(c)*b*m-1/9*ln(d)*b*n*x^3+1/3*I*m/f*e^2/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*Pi*b*csgn(I*c*x
^n)^3+1/3*I*m/f*x*e*Pi*b*csgn(I*c)*csgn(I*c*x^n)^2-1/12*Pi^2*csgn(I*d)*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^
m)*x^3*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/3*I*m/f*x*e*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)^2+2/3*m/f*b*e^2/(e*f
)^(1/2)*arctan(x*f/(e*f)^(1/2))*n*ln(x)-1/3*m/f*b*n*e^2/(-e*f)^(1/2)*ln(x)*ln((-f*x+(-e*f)^(1/2))/(-e*f)^(1/2)
)+1/3*m/f*b*n*e^2/(-e*f)^(1/2)*ln(x)*ln((f*x+(-e*f)^(1/2))/(-e*f)^(1/2))+1/12*Pi^2*csgn(I*d)*csgn(I*(f*x^2+e)^
m)*csgn(I*d*(f*x^2+e)^m)*x^3*b*csgn(I*x^n)*csgn(I*c*x^n)^2+1/18*I*Pi*csgn(I*d)*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f
*x^2+e)^m)*b*x^3*n+1/12*Pi^2*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)^2*x^3*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*
x^n)+1/12*Pi^2*csgn(I*d)*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)*x^3*b*csgn(I*c)*csgn(I*c*x^n)^2+1/12*Pi^2*c
sgn(I*d)*csgn(I*d*(f*x^2+e)^m)^2*x^3*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/6*I*Pi*csgn(I*d)*csgn(I*(f*x^2+e)
^m)*csgn(I*d*(f*x^2+e)^m)*b*x^3*ln(x^n)+1/9*I*m*x^3*Pi*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/6*I*Pi*csgn(I*d
)*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)*x^3*b*ln(c)-1/3*I*m/f*x*e*Pi*b*csgn(I*c*x^n)^3-2/9*m*b*ln(x^n)*x^3
+1/3*ln(d)*b*x^3*ln(x^n)-1/3*m/f*b*n*e^2/(-e*f)^(1/2)*dilog((-f*x+(-e*f)^(1/2))/(-e*f)^(1/2))+1/3*m/f*b*n*e^2/
(-e*f)^(1/2)*dilog((f*x+(-e*f)^(1/2))/(-e*f)^(1/2))+2/9*m/f*e^2/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*b*n-1/3*I*
m/f*e^2/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*Pi*b*csgn(I*c)*csgn(I*c*x^n)^2+2/3/f*ln(c)*b*e*m*x-1/3*I*m/f*e^2/(
e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)^2-1/3*I*m/f*x*e*Pi*b*csgn(I*c)*csgn(I*x^n)*c
sgn(I*c*x^n)+1/6*I*x^3*Pi*ln(d)*b*csgn(I*c)*csgn(I*c*x^n)^2-2/9*x^3*a*m-1/6*I*Pi*csgn(I*d*(f*x^2+e)^m)^3*x^3*a
-1/12*Pi^2*csgn(I*d*(f*x^2+e)^m)^3*x^3*b*csgn(I*c*x^n)^3+2/3/f*m*b*ln(x^n)*x*e-1/12*Pi^2*csgn(I*d)*csgn(I*(f*x
^2+e)^m)*csgn(I*d*(f*x^2+e)^m)*x^3*b*csgn(I*c*x^n)^3-2/3*m/f*b*e^2/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*ln(x^n)
-1/12*Pi^2*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)^2*x^3*b*csgn(I*c)*csgn(I*c*x^n)^2-1/12*Pi^2*csgn(I*(f*x^2
+e)^m)*csgn(I*d*(f*x^2+e)^m)^2*x^3*b*csgn(I*x^n)*csgn(I*c*x^n)^2-1/12*Pi^2*csgn(I*d*(f*x^2+e)^m)^3*x^3*b*csgn(
I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/6*I*x^3*Pi*ln(d)*b*csgn(I*c*x^n)^3-1/6*I*Pi*csgn(I*d)*csgn(I*(f*x^2+e)^m)*csg
n(I*d*(f*x^2+e)^m)*x^3*a+1/6*I*Pi*csgn(I*d)*csgn(I*d*(f*x^2+e)^m)^2*x^3*b*ln(c)+1/6*I*Pi*csgn(I*d)*csgn(I*d*(f
*x^2+e)^m)^2*b*x^3*ln(x^n)-1/18*I*Pi*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)^2*b*x^3*n-1/9*I*m*x^3*Pi*b*csgn
(I*c)*csgn(I*c*x^n)^2-1/9*I*m*x^3*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)^2-1/18*I*Pi*csgn(I*d)*csgn(I*d*(f*x^2+e)^m)^2
*b*x^3*n+1/6*I*Pi*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)^2*x^3*b*ln(c)+1/6*I*Pi*csgn(I*(f*x^2+e)^m)*csgn(I*
d*(f*x^2+e)^m)^2*b*x^3*ln(x^n)-1/12*Pi^2*csgn(I*d)*csgn(I*d*(f*x^2+e)^m)^2*x^3*b*csgn(I*c)*csgn(I*c*x^n)^2-1/1
2*Pi^2*csgn(I*d)*csgn(I*d*(f*x^2+e)^m)^2*x^3*b*csgn(I*x^n)*csgn(I*c*x^n)^2-1/6*I*x^3*Pi*ln(d)*b*csgn(I*c)*csgn
(I*x^n)*csgn(I*c*x^n)+1/3*I*m/f*e^2/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*Pi*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^
n)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x^2*log(c*x^n) + a*x^2)*log((f*x^2 + e)^m*d), 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**2*(a+b*ln(c*x**n))*ln(d*(f*x**2+e)**m),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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