3.1.90 \(\int x^3 (a+b \log (c x^n)) \log (d (e+f x^2)^m) \, dx\) [90]

Optimal. Leaf size=221 \[ -\frac {3 b e m n x^2}{16 f}+\frac {1}{16} b m n x^4+\frac {e m x^2 \left (a+b \log \left (c x^n\right )\right )}{4 f}-\frac {1}{8} m x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {b e^2 m n \log \left (e+f x^2\right )}{16 f^2}+\frac {b e^2 m n \log \left (-\frac {f x^2}{e}\right ) \log \left (e+f x^2\right )}{8 f^2}-\frac {e^2 m \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^2\right )}{4 f^2}-\frac {1}{16} b n x^4 \log \left (d \left (e+f x^2\right )^m\right )+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac {b e^2 m n \text {Li}_2\left (1+\frac {f x^2}{e}\right )}{8 f^2} \]

[Out]

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

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Rubi [A]
time = 0.16, antiderivative size = 221, normalized size of antiderivative = 1.00, 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 = {2504, 2442, 45, 2423, 2441, 2352} \begin {gather*} \frac {b e^2 m n \text {PolyLog}\left (2,\frac {f x^2}{e}+1\right )}{8 f^2}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac {e^2 m \log \left (e+f x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{4 f^2}+\frac {e m x^2 \left (a+b \log \left (c x^n\right )\right )}{4 f}-\frac {1}{8} m x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{16} b n x^4 \log \left (d \left (e+f x^2\right )^m\right )+\frac {b e^2 m n \log \left (e+f x^2\right )}{16 f^2}+\frac {b e^2 m n \log \left (-\frac {f x^2}{e}\right ) \log \left (e+f x^2\right )}{8 f^2}-\frac {3 b e m n x^2}{16 f}+\frac {1}{16} b m n x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*b*e*m*n*x^2)/(16*f) + (b*m*n*x^4)/16 + (e*m*x^2*(a + b*Log[c*x^n]))/(4*f) - (m*x^4*(a + b*Log[c*x^n]))/8 +
(b*e^2*m*n*Log[e + f*x^2])/(16*f^2) + (b*e^2*m*n*Log[-((f*x^2)/e)]*Log[e + f*x^2])/(8*f^2) - (e^2*m*(a + b*Lo
g[c*x^n])*Log[e + f*x^2])/(4*f^2) - (b*n*x^4*Log[d*(e + f*x^2)^m])/16 + (x^4*(a + b*Log[c*x^n])*Log[d*(e + f*x
^2)^m])/4 + (b*e^2*m*n*PolyLog[2, 1 + (f*x^2)/e])/(8*f^2)

Rule 45

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 2352

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

Rule 2423

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 2441

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 2442

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 2504

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rubi steps

\begin {align*} \int x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right ) \, dx &=\frac {e m x^2 \left (a+b \log \left (c x^n\right )\right )}{4 f}-\frac {1}{8} m x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {e^2 m \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^2\right )}{4 f^2}+\frac {1}{4} x^4 \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 {e m x}{4 f}-\frac {m x^3}{8}-\frac {e^2 m \log \left (e+f x^2\right )}{4 f^2 x}+\frac {1}{4} x^3 \log \left (d \left (e+f x^2\right )^m\right )\right ) \, dx\\ &=-\frac {b e m n x^2}{8 f}+\frac {1}{32} b m n x^4+\frac {e m x^2 \left (a+b \log \left (c x^n\right )\right )}{4 f}-\frac {1}{8} m x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {e^2 m \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^2\right )}{4 f^2}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac {1}{4} (b n) \int x^3 \log \left (d \left (e+f x^2\right )^m\right ) \, dx+\frac {\left (b e^2 m n\right ) \int \frac {\log \left (e+f x^2\right )}{x} \, dx}{4 f^2}\\ &=-\frac {b e m n x^2}{8 f}+\frac {1}{32} b m n x^4+\frac {e m x^2 \left (a+b \log \left (c x^n\right )\right )}{4 f}-\frac {1}{8} m x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {e^2 m \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^2\right )}{4 f^2}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac {1}{8} (b n) \text {Subst}\left (\int x \log \left (d (e+f x)^m\right ) \, dx,x,x^2\right )+\frac {\left (b e^2 m n\right ) \text {Subst}\left (\int \frac {\log (e+f x)}{x} \, dx,x,x^2\right )}{8 f^2}\\ &=-\frac {b e m n x^2}{8 f}+\frac {1}{32} b m n x^4+\frac {e m x^2 \left (a+b \log \left (c x^n\right )\right )}{4 f}-\frac {1}{8} m x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {b e^2 m n \log \left (-\frac {f x^2}{e}\right ) \log \left (e+f x^2\right )}{8 f^2}-\frac {e^2 m \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^2\right )}{4 f^2}-\frac {1}{16} b n x^4 \log \left (d \left (e+f x^2\right )^m\right )+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac {\left (b e^2 m n\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {f x}{e}\right )}{e+f x} \, dx,x,x^2\right )}{8 f}+\frac {1}{16} (b f m n) \text {Subst}\left (\int \frac {x^2}{e+f x} \, dx,x,x^2\right )\\ &=-\frac {b e m n x^2}{8 f}+\frac {1}{32} b m n x^4+\frac {e m x^2 \left (a+b \log \left (c x^n\right )\right )}{4 f}-\frac {1}{8} m x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {b e^2 m n \log \left (-\frac {f x^2}{e}\right ) \log \left (e+f x^2\right )}{8 f^2}-\frac {e^2 m \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^2\right )}{4 f^2}-\frac {1}{16} b n x^4 \log \left (d \left (e+f x^2\right )^m\right )+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac {b e^2 m n \text {Li}_2\left (1+\frac {f x^2}{e}\right )}{8 f^2}+\frac {1}{16} (b f m n) \text {Subst}\left (\int \left (-\frac {e}{f^2}+\frac {x}{f}+\frac {e^2}{f^2 (e+f x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {3 b e m n x^2}{16 f}+\frac {1}{16} b m n x^4+\frac {e m x^2 \left (a+b \log \left (c x^n\right )\right )}{4 f}-\frac {1}{8} m x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {b e^2 m n \log \left (e+f x^2\right )}{16 f^2}+\frac {b e^2 m n \log \left (-\frac {f x^2}{e}\right ) \log \left (e+f x^2\right )}{8 f^2}-\frac {e^2 m \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^2\right )}{4 f^2}-\frac {1}{16} b n x^4 \log \left (d \left (e+f x^2\right )^m\right )+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac {b e^2 m n \text {Li}_2\left (1+\frac {f x^2}{e}\right )}{8 f^2}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.11, size = 324, normalized size = 1.47 \begin {gather*} -\frac {-4 a e f m x^2+3 b e f m n x^2+2 a f^2 m x^4-b f^2 m n x^4-4 b e f m x^2 \log \left (c x^n\right )+2 b f^2 m x^4 \log \left (c x^n\right )+4 b e^2 m n \log (x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+4 b e^2 m n \log (x) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )+4 a e^2 m \log \left (e+f x^2\right )-b e^2 m n \log \left (e+f x^2\right )-4 b e^2 m n \log (x) \log \left (e+f x^2\right )+4 b e^2 m \log \left (c x^n\right ) \log \left (e+f x^2\right )-4 a f^2 x^4 \log \left (d \left (e+f x^2\right )^m\right )+b f^2 n x^4 \log \left (d \left (e+f x^2\right )^m\right )-4 b f^2 x^4 \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+4 b e^2 m n \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+4 b e^2 m n \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{16 f^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/16*(-4*a*e*f*m*x^2 + 3*b*e*f*m*n*x^2 + 2*a*f^2*m*x^4 - b*f^2*m*n*x^4 - 4*b*e*f*m*x^2*Log[c*x^n] + 2*b*f^2*m
*x^4*Log[c*x^n] + 4*b*e^2*m*n*Log[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + 4*b*e^2*m*n*Log[x]*Log[1 + (I*Sqrt[f]*x)
/Sqrt[e]] + 4*a*e^2*m*Log[e + f*x^2] - b*e^2*m*n*Log[e + f*x^2] - 4*b*e^2*m*n*Log[x]*Log[e + f*x^2] + 4*b*e^2*
m*Log[c*x^n]*Log[e + f*x^2] - 4*a*f^2*x^4*Log[d*(e + f*x^2)^m] + b*f^2*n*x^4*Log[d*(e + f*x^2)^m] - 4*b*f^2*x^
4*Log[c*x^n]*Log[d*(e + f*x^2)^m] + 4*b*e^2*m*n*PolyLog[2, ((-I)*Sqrt[f]*x)/Sqrt[e]] + 4*b*e^2*m*n*PolyLog[2,
(I*Sqrt[f]*x)/Sqrt[e]])/f^2

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.69, size = 2270, normalized size = 10.27

method	result	size
risch	\(\text {Expression too large to display}\)	\(2270\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*I/f*Pi*b*e*m*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^2+1/8*I*e^2*m/f^2*ln(f*x^2+e)*Pi*b*csgn(I*c)*csgn(I*x^
n)*csgn(I*c*x^n)+1/16*b*m*n*x^4+1/4*b*e^2*m*n/f^2*ln(x)*ln(f*x^2+e)-1/16*Pi^2*csgn(I*d)*csgn(I*d*(f*x^2+e)^m)^
2*x^4*b*csgn(I*x^n)*csgn(I*c*x^n)^2+1/4/f*x^2*a*e*m+1/4/f*ln(c)*x^2*b*e*m+1/8*b*e^2*m*n/f^2-1/4*b*e^2*m*n/f^2*
dilog((-f*x+(-e*f)^(1/2))/(-e*f)^(1/2))-1/4*b*e^2*m*n/f^2*dilog((f*x+(-e*f)^(1/2))/(-e*f)^(1/2))-1/4*b*e^2*m*n
/f^2*ln(x)*ln((-f*x+(-e*f)^(1/2))/(-e*f)^(1/2))-1/4*b*e^2*m*n/f^2*ln(x)*ln((f*x+(-e*f)^(1/2))/(-e*f)^(1/2))-1/
4*m/f^2*b*ln(x^n)*e^2*ln(f*x^2+e)-1/16*Pi^2*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)^2*x^4*b*csgn(I*c)*csgn(I
*c*x^n)^2-1/16*Pi^2*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)^2*x^4*b*csgn(I*x^n)*csgn(I*c*x^n)^2-1/8*x^4*a*m+
1/16*b*e^2*m*n*ln(f*x^2+e)/f^2+1/8*I*Pi*x^4*a*csgn(I*d)*csgn(I*d*(f*x^2+e)^m)^2-1/8*I*x^4*ln(d)*Pi*b*csgn(I*c*
x^n)^3+1/8*I*Pi*x^4*a*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)^2+1/4*x^4*ln(d)*a-1/4*e^2*m/f^2*ln(f*x^2+e)*a-
1/8*I*x^4*ln(d)*Pi*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/16*Pi^2*csgn(I*d)*csgn(I*d*(f*x^2+e)^m)^2*x^4*b*csg
n(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/16*Pi^2*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)^2*x^4*b*csgn(I*c)*csgn(I*
x^n)*csgn(I*c*x^n)+1/16*Pi^2*csgn(I*d)*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)*x^4*b*csgn(I*c)*csgn(I*c*x^n)
^2-1/16*I*Pi*x^4*b*m*csgn(I*x^n)*csgn(I*c*x^n)^2-1/16*Pi^2*csgn(I*d)*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)
*x^4*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/8*I*Pi*csgn(I*d)*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)*b*x^4*
ln(x^n)+1/32*I*Pi*csgn(I*d)*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)*b*x^4*n-1/8*I*ln(c)*Pi*x^4*b*csgn(I*d*(f
*x^2+e)^m)^3-1/8*I*Pi*csgn(I*d*(f*x^2+e)^m)^3*b*x^4*ln(x^n)+1/16*Pi^2*csgn(I*d)*csgn(I*d*(f*x^2+e)^m)^2*x^4*b*
csgn(I*c*x^n)^3+1/16*Pi^2*csgn(I*d*(f*x^2+e)^m)^3*x^4*b*csgn(I*c)*csgn(I*c*x^n)^2+1/16*Pi^2*csgn(I*(f*x^2+e)^m
)*csgn(I*d*(f*x^2+e)^m)^2*x^4*b*csgn(I*c*x^n)^3+1/16*Pi^2*csgn(I*d*(f*x^2+e)^m)^3*x^4*b*csgn(I*x^n)*csgn(I*c*x
^n)^2+1/8*I*Pi*csgn(I*d)*csgn(I*d*(f*x^2+e)^m)^2*b*x^4*ln(x^n)-1/8*I*Pi*x^4*a*csgn(I*d*(f*x^2+e)^m)^3-1/8*I/f*
Pi*b*e*m*csgn(I*c*x^n)^3*x^2+1/8*I*e^2*m/f^2*ln(f*x^2+e)*Pi*b*csgn(I*c*x^n)^3+1/16*I*Pi*x^4*b*m*csgn(I*c)*csgn
(I*x^n)*csgn(I*c*x^n)-1/8*I*ln(c)*Pi*x^4*b*csgn(I*d)*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)-1/16*Pi^2*csgn(
I*d)*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)*x^4*b*csgn(I*c*x^n)^3+1/8*I*ln(c)*Pi*x^4*b*csgn(I*d)*csgn(I*d*(
f*x^2+e)^m)^2+1/8*I*ln(c)*Pi*x^4*b*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)^2+1/16*Pi^2*csgn(I*d)*csgn(I*(f*x
^2+e)^m)*csgn(I*d*(f*x^2+e)^m)*x^4*b*csgn(I*x^n)*csgn(I*c*x^n)^2-1/8*I*Pi*x^4*a*csgn(I*d)*csgn(I*(f*x^2+e)^m)*
csgn(I*d*(f*x^2+e)^m)-1/16*I*Pi*x^4*b*m*csgn(I*c)*csgn(I*c*x^n)^2-1/32*I*Pi*csgn(I*d)*csgn(I*d*(f*x^2+e)^m)^2*
b*x^4*n+1/4*x^4*ln(d)*ln(c)*b-1/8*x^4*ln(c)*b*m-1/16*ln(d)*b*n*x^4+1/8*I*Pi*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^
2+e)^m)^2*b*x^4*ln(x^n)+1/8*I/f*Pi*b*e*m*csgn(I*c)*csgn(I*c*x^n)^2*x^2+1/8*I/f*Pi*b*e*m*csgn(I*x^n)*csgn(I*c*x
^n)^2*x^2-1/8*I*e^2*m/f^2*ln(f*x^2+e)*Pi*b*csgn(I*c)*csgn(I*c*x^n)^2-1/8*I*e^2*m/f^2*ln(f*x^2+e)*Pi*b*csgn(I*x
^n)*csgn(I*c*x^n)^2+1/16*I*Pi*x^4*b*m*csgn(I*c*x^n)^3+1/32*I*Pi*csgn(I*d*(f*x^2+e)^m)^3*b*x^4*n-1/4*e^2*m/f^2*
ln(f*x^2+e)*b*ln(c)-3/16*b*e*m*n*x^2/f-1/16*Pi^2*csgn(I*d*(f*x^2+e)^m)^3*x^4*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*
x^n)+1/8*I*x^4*ln(d)*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)^2+1/8*I*x^4*ln(d)*Pi*b*csgn(I*c)*csgn(I*c*x^n)^2+(1/4*x^4*
b*ln(x^n)+1/16*x^4*(-2*I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+2*I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+2*I*b*Pi*
csgn(I*x^n)*csgn(I*c*x^n)^2-2*I*b*Pi*csgn(I*c*x^n)^3+4*b*ln(c)-b*n+4*a))*ln((f*x^2+e)^m)-1/8*m*b*ln(x^n)*x^4+1
/4*ln(d)*b*x^4*ln(x^n)-1/16*Pi^2*csgn(I*d*(f*x^2+e)^m)^3*x^4*b*csgn(I*c*x^n)^3-1/32*I*Pi*csgn(I*(f*x^2+e)^m)*c
sgn(I*d*(f*x^2+e)^m)^2*b*x^4*n-1/16*Pi^2*csgn(I*d)*csgn(I*d*(f*x^2+e)^m)^2*x^4*b*csgn(I*c)*csgn(I*c*x^n)^2+1/4
*m/f*b*ln(x^n)*e*x^2

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(4*b*m*x^4*log(x^n) - ((m*n - 4*m*log(c))*b - 4*a*m)*x^4)*log(f*x^2 + e) + integrate(-1/8*((4*(f*m - 2*f*
log(d))*a - (f*m*n - 4*(f*m - 2*f*log(d))*log(c))*b)*x^5 - 8*(b*log(c)*log(d) + a*log(d))*x^3*e + 4*((f*m - 2*
f*log(d))*b*x^5 - 2*b*x^3*e*log(d))*log(x^n))/(f*x^2 + e), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x^3*log(c*x^n) + a*x^3)*log((f*x^2 + e)^m*d), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(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^3*log((f*x^2 + e)^m*d), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^3\,\ln \left (d\,{\left (f\,x^2+e\right )}^m\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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