3.2.7 \(\int \frac {(a+b \log (c x^n))^2 \log (d (e+f x^2)^m)}{x^4} \, dx\) [107]

Optimal. Leaf size=571 \[ -\frac {52 b^2 f m n^2}{27 e x}-\frac {4 b^2 f^{3/2} m n^2 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{27 e^{3/2}}-\frac {16 b f m n \left (a+b \log \left (c x^n\right )\right )}{9 e x}-\frac {4 b f^{3/2} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 e^{3/2}}-\frac {2 f m \left (a+b \log \left (c x^n\right )\right )^2}{3 e x}+\frac {f^{3/2} m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 (-e)^{3/2}}-\frac {f^{3/2} m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 (-e)^{3/2}}-\frac {2 b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{27 x^3}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{3 x^3}-\frac {2 b f^{3/2} m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 (-e)^{3/2}}+\frac {2 b f^{3/2} m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 (-e)^{3/2}}+\frac {2 i b^2 f^{3/2} m n^2 \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{9 e^{3/2}}-\frac {2 i b^2 f^{3/2} m n^2 \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{9 e^{3/2}}+\frac {2 b^2 f^{3/2} m n^2 \text {Li}_3\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 (-e)^{3/2}}-\frac {2 b^2 f^{3/2} m n^2 \text {Li}_3\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 (-e)^{3/2}} \]

[Out]

-52/27*b^2*f*m*n^2/e/x-4/27*b^2*f^(3/2)*m*n^2*arctan(x*f^(1/2)/e^(1/2))/e^(3/2)-16/9*b*f*m*n*(a+b*ln(c*x^n))/e
/x-4/9*b*f^(3/2)*m*n*arctan(x*f^(1/2)/e^(1/2))*(a+b*ln(c*x^n))/e^(3/2)-2/3*f*m*(a+b*ln(c*x^n))^2/e/x-2/27*b^2*
n^2*ln(d*(f*x^2+e)^m)/x^3-2/9*b*n*(a+b*ln(c*x^n))*ln(d*(f*x^2+e)^m)/x^3-1/3*(a+b*ln(c*x^n))^2*ln(d*(f*x^2+e)^m
)/x^3+1/3*f^(3/2)*m*(a+b*ln(c*x^n))^2*ln(1-x*f^(1/2)/(-e)^(1/2))/(-e)^(3/2)-1/3*f^(3/2)*m*(a+b*ln(c*x^n))^2*ln
(1+x*f^(1/2)/(-e)^(1/2))/(-e)^(3/2)-2/3*b*f^(3/2)*m*n*(a+b*ln(c*x^n))*polylog(2,-x*f^(1/2)/(-e)^(1/2))/(-e)^(3
/2)+2/3*b*f^(3/2)*m*n*(a+b*ln(c*x^n))*polylog(2,x*f^(1/2)/(-e)^(1/2))/(-e)^(3/2)+2/9*I*b^2*f^(3/2)*m*n^2*polyl
og(2,-I*x*f^(1/2)/e^(1/2))/e^(3/2)-2/9*I*b^2*f^(3/2)*m*n^2*polylog(2,I*x*f^(1/2)/e^(1/2))/e^(3/2)+2/3*b^2*f^(3
/2)*m*n^2*polylog(3,-x*f^(1/2)/(-e)^(1/2))/(-e)^(3/2)-2/3*b^2*f^(3/2)*m*n^2*polylog(3,x*f^(1/2)/(-e)^(1/2))/(-
e)^(3/2)

________________________________________________________________________________________

Rubi [A]
time = 0.59, antiderivative size = 571, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 14, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2342, 2341, 2425, 331, 211, 2380, 2361, 12, 4940, 2438, 2367, 2354, 2421, 6724} \begin {gather*} -\frac {2 b f^{3/2} m n \text {PolyLog}\left (2,-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 (-e)^{3/2}}+\frac {2 b f^{3/2} m n \text {PolyLog}\left (2,\frac {\sqrt {f} x}{\sqrt {-e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 (-e)^{3/2}}+\frac {2 i b^2 f^{3/2} m n^2 \text {PolyLog}\left (2,-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{9 e^{3/2}}-\frac {2 i b^2 f^{3/2} m n^2 \text {PolyLog}\left (2,\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{9 e^{3/2}}+\frac {2 b^2 f^{3/2} m n^2 \text {PolyLog}\left (3,-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 (-e)^{3/2}}-\frac {2 b^2 f^{3/2} m n^2 \text {PolyLog}\left (3,\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 (-e)^{3/2}}-\frac {4 b f^{3/2} m n \text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 e^{3/2}}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{3 x^3}+\frac {f^{3/2} m \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{3 (-e)^{3/2}}-\frac {f^{3/2} m \log \left (\frac {\sqrt {f} x}{\sqrt {-e}}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{3 (-e)^{3/2}}-\frac {16 b f m n \left (a+b \log \left (c x^n\right )\right )}{9 e x}-\frac {2 f m \left (a+b \log \left (c x^n\right )\right )^2}{3 e x}-\frac {4 b^2 f^{3/2} m n^2 \text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{27 e^{3/2}}-\frac {2 b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{27 x^3}-\frac {52 b^2 f m n^2}{27 e x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*Log[c*x^n])^2*Log[d*(e + f*x^2)^m])/x^4,x]

[Out]

(-52*b^2*f*m*n^2)/(27*e*x) - (4*b^2*f^(3/2)*m*n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(27*e^(3/2)) - (16*b*f*m*n*(a +
b*Log[c*x^n]))/(9*e*x) - (4*b*f^(3/2)*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*(a + b*Log[c*x^n]))/(9*e^(3/2)) - (2*f*
m*(a + b*Log[c*x^n])^2)/(3*e*x) + (f^(3/2)*m*(a + b*Log[c*x^n])^2*Log[1 - (Sqrt[f]*x)/Sqrt[-e]])/(3*(-e)^(3/2)
) - (f^(3/2)*m*(a + b*Log[c*x^n])^2*Log[1 + (Sqrt[f]*x)/Sqrt[-e]])/(3*(-e)^(3/2)) - (2*b^2*n^2*Log[d*(e + f*x^
2)^m])/(27*x^3) - (2*b*n*(a + b*Log[c*x^n])*Log[d*(e + f*x^2)^m])/(9*x^3) - ((a + b*Log[c*x^n])^2*Log[d*(e + f
*x^2)^m])/(3*x^3) - (2*b*f^(3/2)*m*n*(a + b*Log[c*x^n])*PolyLog[2, -((Sqrt[f]*x)/Sqrt[-e])])/(3*(-e)^(3/2)) +
(2*b*f^(3/2)*m*n*(a + b*Log[c*x^n])*PolyLog[2, (Sqrt[f]*x)/Sqrt[-e]])/(3*(-e)^(3/2)) + (((2*I)/9)*b^2*f^(3/2)*
m*n^2*PolyLog[2, ((-I)*Sqrt[f]*x)/Sqrt[e]])/e^(3/2) - (((2*I)/9)*b^2*f^(3/2)*m*n^2*PolyLog[2, (I*Sqrt[f]*x)/Sq
rt[e]])/e^(3/2) + (2*b^2*f^(3/2)*m*n^2*PolyLog[3, -((Sqrt[f]*x)/Sqrt[-e])])/(3*(-e)^(3/2)) - (2*b^2*f^(3/2)*m*
n^2*PolyLog[3, (Sqrt[f]*x)/Sqrt[-e]])/(3*(-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 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 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2342

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Lo
g[c*x^n])^p/(d*(m + 1))), x] - Dist[b*n*(p/(m + 1)), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2354

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 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 2367

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 2380

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

Rule 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c*x^n])^p/m), x] + Dist[b*n*(p/m), Int[PolyLog[2, (-d)*f*x^m]*((a + b*L
og[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 2425

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((g_.)*(x_))^(q_.),
x_Symbol] :> With[{u = IntHide[(g*x)^q*(a + b*Log[c*x^n])^p, x]}, Dist[Log[d*(e + f*x^m)^r], u, x] - Dist[f*m
*r, Int[Dist[x^(m - 1)/(e + f*x^m), u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && IGtQ[p, 0
] && RationalQ[m] && RationalQ[q]

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]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin {align*} \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x^4} \, dx &=-\frac {2 b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{27 x^3}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{3 x^3}-(2 f m) \int \left (-\frac {2 b^2 n^2}{27 x^2 \left (e+f x^2\right )}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right )}{9 x^2 \left (e+f x^2\right )}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 x^2 \left (e+f x^2\right )}\right ) \, dx\\ &=-\frac {2 b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{27 x^3}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{3 x^3}+\frac {1}{3} (2 f m) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 \left (e+f x^2\right )} \, dx+\frac {1}{9} (4 b f m n) \int \frac {a+b \log \left (c x^n\right )}{x^2 \left (e+f x^2\right )} \, dx+\frac {1}{27} \left (4 b^2 f m n^2\right ) \int \frac {1}{x^2 \left (e+f x^2\right )} \, dx\\ &=-\frac {4 b^2 f m n^2}{27 e x}-\frac {2 b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{27 x^3}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{3 x^3}+\frac {1}{3} (2 f m) \int \left (\frac {\left (a+b \log \left (c x^n\right )\right )^2}{e x^2}-\frac {f \left (a+b \log \left (c x^n\right )\right )^2}{e \left (e+f x^2\right )}\right ) \, dx+\frac {1}{9} (4 b f m n) \int \left (\frac {a+b \log \left (c x^n\right )}{e x^2}-\frac {f \left (a+b \log \left (c x^n\right )\right )}{e \left (e+f x^2\right )}\right ) \, dx-\frac {\left (4 b^2 f^2 m n^2\right ) \int \frac {1}{e+f x^2} \, dx}{27 e}\\ &=-\frac {4 b^2 f m n^2}{27 e x}-\frac {4 b^2 f^{3/2} m n^2 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{27 e^{3/2}}-\frac {2 b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{27 x^3}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{3 x^3}+\frac {(2 f m) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx}{3 e}-\frac {\left (2 f^2 m\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{e+f x^2} \, dx}{3 e}+\frac {(4 b f m n) \int \frac {a+b \log \left (c x^n\right )}{x^2} \, dx}{9 e}-\frac {\left (4 b f^2 m n\right ) \int \frac {a+b \log \left (c x^n\right )}{e+f x^2} \, dx}{9 e}\\ &=-\frac {16 b^2 f m n^2}{27 e x}-\frac {4 b^2 f^{3/2} m n^2 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{27 e^{3/2}}-\frac {4 b f m n \left (a+b \log \left (c x^n\right )\right )}{9 e x}-\frac {4 b f^{3/2} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 e^{3/2}}-\frac {2 f m \left (a+b \log \left (c x^n\right )\right )^2}{3 e x}-\frac {2 b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{27 x^3}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{3 x^3}-\frac {\left (2 f^2 m\right ) \int \left (\frac {\sqrt {-e} \left (a+b \log \left (c x^n\right )\right )^2}{2 e \left (\sqrt {-e}-\sqrt {f} x\right )}+\frac {\sqrt {-e} \left (a+b \log \left (c x^n\right )\right )^2}{2 e \left (\sqrt {-e}+\sqrt {f} x\right )}\right ) \, dx}{3 e}+\frac {(4 b f m n) \int \frac {a+b \log \left (c x^n\right )}{x^2} \, dx}{3 e}+\frac {\left (4 b^2 f^2 m n^2\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} \sqrt {f} x} \, dx}{9 e}\\ &=-\frac {52 b^2 f m n^2}{27 e x}-\frac {4 b^2 f^{3/2} m n^2 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{27 e^{3/2}}-\frac {16 b f m n \left (a+b \log \left (c x^n\right )\right )}{9 e x}-\frac {4 b f^{3/2} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 e^{3/2}}-\frac {2 f m \left (a+b \log \left (c x^n\right )\right )^2}{3 e x}-\frac {2 b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{27 x^3}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{3 x^3}-\frac {\left (f^2 m\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {-e}-\sqrt {f} x} \, dx}{3 (-e)^{3/2}}-\frac {\left (f^2 m\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {-e}+\sqrt {f} x} \, dx}{3 (-e)^{3/2}}+\frac {\left (4 b^2 f^{3/2} m n^2\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{x} \, dx}{9 e^{3/2}}\\ &=-\frac {52 b^2 f m n^2}{27 e x}-\frac {4 b^2 f^{3/2} m n^2 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{27 e^{3/2}}-\frac {16 b f m n \left (a+b \log \left (c x^n\right )\right )}{9 e x}-\frac {4 b f^{3/2} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 e^{3/2}}-\frac {2 f m \left (a+b \log \left (c x^n\right )\right )^2}{3 e x}+\frac {f^{3/2} m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 (-e)^{3/2}}-\frac {f^{3/2} m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 (-e)^{3/2}}-\frac {2 b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{27 x^3}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{3 x^3}-\frac {\left (2 b f^{3/2} m n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{x} \, dx}{3 (-e)^{3/2}}+\frac {\left (2 b f^{3/2} m n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{x} \, dx}{3 (-e)^{3/2}}+\frac {\left (2 i b^2 f^{3/2} m n^2\right ) \int \frac {\log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{x} \, dx}{9 e^{3/2}}-\frac {\left (2 i b^2 f^{3/2} m n^2\right ) \int \frac {\log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{x} \, dx}{9 e^{3/2}}\\ &=-\frac {52 b^2 f m n^2}{27 e x}-\frac {4 b^2 f^{3/2} m n^2 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{27 e^{3/2}}-\frac {16 b f m n \left (a+b \log \left (c x^n\right )\right )}{9 e x}-\frac {4 b f^{3/2} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 e^{3/2}}-\frac {2 f m \left (a+b \log \left (c x^n\right )\right )^2}{3 e x}+\frac {f^{3/2} m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 (-e)^{3/2}}-\frac {f^{3/2} m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 (-e)^{3/2}}-\frac {2 b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{27 x^3}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{3 x^3}-\frac {2 b f^{3/2} m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 (-e)^{3/2}}+\frac {2 b f^{3/2} m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 (-e)^{3/2}}+\frac {2 i b^2 f^{3/2} m n^2 \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{9 e^{3/2}}-\frac {2 i b^2 f^{3/2} m n^2 \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{9 e^{3/2}}+\frac {\left (2 b^2 f^{3/2} m n^2\right ) \int \frac {\text {Li}_2\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{x} \, dx}{3 (-e)^{3/2}}-\frac {\left (2 b^2 f^{3/2} m n^2\right ) \int \frac {\text {Li}_2\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{x} \, dx}{3 (-e)^{3/2}}\\ &=-\frac {52 b^2 f m n^2}{27 e x}-\frac {4 b^2 f^{3/2} m n^2 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{27 e^{3/2}}-\frac {16 b f m n \left (a+b \log \left (c x^n\right )\right )}{9 e x}-\frac {4 b f^{3/2} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 e^{3/2}}-\frac {2 f m \left (a+b \log \left (c x^n\right )\right )^2}{3 e x}+\frac {f^{3/2} m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 (-e)^{3/2}}-\frac {f^{3/2} m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 (-e)^{3/2}}-\frac {2 b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{27 x^3}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{3 x^3}-\frac {2 b f^{3/2} m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 (-e)^{3/2}}+\frac {2 b f^{3/2} m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 (-e)^{3/2}}+\frac {2 i b^2 f^{3/2} m n^2 \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{9 e^{3/2}}-\frac {2 i b^2 f^{3/2} m n^2 \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{9 e^{3/2}}+\frac {2 b^2 f^{3/2} m n^2 \text {Li}_3\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 (-e)^{3/2}}-\frac {2 b^2 f^{3/2} m n^2 \text {Li}_3\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 (-e)^{3/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.25, size = 1083, normalized size = 1.90 \begin {gather*} \frac {-18 a^2 \sqrt {e} f m x^2-48 a b \sqrt {e} f m n x^2-52 b^2 \sqrt {e} f m n^2 x^2-18 a^2 f^{3/2} m x^3 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )-12 a b f^{3/2} m n x^3 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )-4 b^2 f^{3/2} m n^2 x^3 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )+36 a b f^{3/2} m n x^3 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log (x)+12 b^2 f^{3/2} m n^2 x^3 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log (x)-18 b^2 f^{3/2} m n^2 x^3 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log ^2(x)-36 a b \sqrt {e} f m x^2 \log \left (c x^n\right )-48 b^2 \sqrt {e} f m n x^2 \log \left (c x^n\right )-36 a b f^{3/2} m x^3 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log \left (c x^n\right )-12 b^2 f^{3/2} m n x^3 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log \left (c x^n\right )+36 b^2 f^{3/2} m n x^3 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log (x) \log \left (c x^n\right )-18 b^2 \sqrt {e} f m x^2 \log ^2\left (c x^n\right )-18 b^2 f^{3/2} m x^3 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log ^2\left (c x^n\right )-18 i a b f^{3/2} m n x^3 \log (x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-6 i b^2 f^{3/2} m n^2 x^3 \log (x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+9 i b^2 f^{3/2} m n^2 x^3 \log ^2(x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-18 i b^2 f^{3/2} m n x^3 \log (x) \log \left (c x^n\right ) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+18 i a b f^{3/2} m n x^3 \log (x) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )+6 i b^2 f^{3/2} m n^2 x^3 \log (x) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )-9 i b^2 f^{3/2} m n^2 x^3 \log ^2(x) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )+18 i b^2 f^{3/2} m n x^3 \log (x) \log \left (c x^n\right ) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )-9 a^2 e^{3/2} \log \left (d \left (e+f x^2\right )^m\right )-6 a b e^{3/2} n \log \left (d \left (e+f x^2\right )^m\right )-2 b^2 e^{3/2} n^2 \log \left (d \left (e+f x^2\right )^m\right )-18 a b e^{3/2} \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-6 b^2 e^{3/2} n \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-9 b^2 e^{3/2} \log ^2\left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+6 i b f^{3/2} m n x^3 \left (3 a+b n+3 b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-6 i b f^{3/2} m n x^3 \left (3 a+b n+3 b \log \left (c x^n\right )\right ) \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )-18 i b^2 f^{3/2} m n^2 x^3 \text {Li}_3\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+18 i b^2 f^{3/2} m n^2 x^3 \text {Li}_3\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{27 e^{3/2} x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*Log[c*x^n])^2*Log[d*(e + f*x^2)^m])/x^4,x]

[Out]

(-18*a^2*Sqrt[e]*f*m*x^2 - 48*a*b*Sqrt[e]*f*m*n*x^2 - 52*b^2*Sqrt[e]*f*m*n^2*x^2 - 18*a^2*f^(3/2)*m*x^3*ArcTan
[(Sqrt[f]*x)/Sqrt[e]] - 12*a*b*f^(3/2)*m*n*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]] - 4*b^2*f^(3/2)*m*n^2*x^3*ArcTan[(S
qrt[f]*x)/Sqrt[e]] + 36*a*b*f^(3/2)*m*n*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x] + 12*b^2*f^(3/2)*m*n^2*x^3*ArcT
an[(Sqrt[f]*x)/Sqrt[e]]*Log[x] - 18*b^2*f^(3/2)*m*n^2*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]^2 - 36*a*b*Sqrt[e
]*f*m*x^2*Log[c*x^n] - 48*b^2*Sqrt[e]*f*m*n*x^2*Log[c*x^n] - 36*a*b*f^(3/2)*m*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*
Log[c*x^n] - 12*b^2*f^(3/2)*m*n*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n] + 36*b^2*f^(3/2)*m*n*x^3*ArcTan[(Sq
rt[f]*x)/Sqrt[e]]*Log[x]*Log[c*x^n] - 18*b^2*Sqrt[e]*f*m*x^2*Log[c*x^n]^2 - 18*b^2*f^(3/2)*m*x^3*ArcTan[(Sqrt[
f]*x)/Sqrt[e]]*Log[c*x^n]^2 - (18*I)*a*b*f^(3/2)*m*n*x^3*Log[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] - (6*I)*b^2*f^(
3/2)*m*n^2*x^3*Log[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + (9*I)*b^2*f^(3/2)*m*n^2*x^3*Log[x]^2*Log[1 - (I*Sqrt[f]
*x)/Sqrt[e]] - (18*I)*b^2*f^(3/2)*m*n*x^3*Log[x]*Log[c*x^n]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + (18*I)*a*b*f^(3/2
)*m*n*x^3*Log[x]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + (6*I)*b^2*f^(3/2)*m*n^2*x^3*Log[x]*Log[1 + (I*Sqrt[f]*x)/Sqr
t[e]] - (9*I)*b^2*f^(3/2)*m*n^2*x^3*Log[x]^2*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + (18*I)*b^2*f^(3/2)*m*n*x^3*Log[x
]*Log[c*x^n]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - 9*a^2*e^(3/2)*Log[d*(e + f*x^2)^m] - 6*a*b*e^(3/2)*n*Log[d*(e +
f*x^2)^m] - 2*b^2*e^(3/2)*n^2*Log[d*(e + f*x^2)^m] - 18*a*b*e^(3/2)*Log[c*x^n]*Log[d*(e + f*x^2)^m] - 6*b^2*e^
(3/2)*n*Log[c*x^n]*Log[d*(e + f*x^2)^m] - 9*b^2*e^(3/2)*Log[c*x^n]^2*Log[d*(e + f*x^2)^m] + (6*I)*b*f^(3/2)*m*
n*x^3*(3*a + b*n + 3*b*Log[c*x^n])*PolyLog[2, ((-I)*Sqrt[f]*x)/Sqrt[e]] - (6*I)*b*f^(3/2)*m*n*x^3*(3*a + b*n +
3*b*Log[c*x^n])*PolyLog[2, (I*Sqrt[f]*x)/Sqrt[e]] - (18*I)*b^2*f^(3/2)*m*n^2*x^3*PolyLog[3, ((-I)*Sqrt[f]*x)/
Sqrt[e]] + (18*I)*b^2*f^(3/2)*m*n^2*x^3*PolyLog[3, (I*Sqrt[f]*x)/Sqrt[e]])/(27*e^(3/2)*x^3)

________________________________________________________________________________________

Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right )^{2} \ln \left (d \left (f \,x^{2}+e \right )^{m}\right )}{x^{4}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))^2*ln(d*(f*x^2+e)^m)/x^4,x)

[Out]

int((a+b*ln(c*x^n))^2*ln(d*(f*x^2+e)^m)/x^4,x)

________________________________________________________________________________________

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((a+b*log(c*x^n))^2*log(d*(f*x^2+e)^m)/x^4,x, algorithm="maxima")

[Out]

-1/27*(9*b^2*m*log(x^n)^2 + 6*(m*n + 3*m*log(c))*a*b + (2*m*n^2 + 6*m*n*log(c) + 9*m*log(c)^2)*b^2 + 9*a^2*m +
6*((m*n + 3*m*log(c))*b^2 + 3*a*b*m)*log(x^n))*log(f*x^2 + e)/x^3 + integrate(1/27*((9*(2*f*m + 3*f*log(d))*a
^2 + 6*(2*f*m*n + 3*(2*f*m + 3*f*log(d))*log(c))*a*b + (4*f*m*n^2 + 12*f*m*n*log(c) + 9*(2*f*m + 3*f*log(d))*l
og(c)^2)*b^2)*x^2 + 9*((2*f*m + 3*f*log(d))*b^2*x^2 + 3*b^2*e*log(d))*log(x^n)^2 + 27*(b^2*log(c)^2*log(d) + 2
*a*b*log(c)*log(d) + a^2*log(d))*e + 6*((3*(2*f*m + 3*f*log(d))*a*b + (2*f*m*n + 3*(2*f*m + 3*f*log(d))*log(c)
)*b^2)*x^2 + 9*(b^2*log(c)*log(d) + a*b*log(d))*e)*log(x^n))/(f*x^6 + x^4*e), x)

________________________________________________________________________________________

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((a+b*log(c*x^n))^2*log(d*(f*x^2+e)^m)/x^4,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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((a+b*ln(c*x**n))**2*ln(d*(f*x**2+e)**m)/x**4,x)

[Out]

Timed out

________________________________________________________________________________________

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((a+b*log(c*x^n))^2*log(d*(f*x^2+e)^m)/x^4,x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)^2*log((f*x^2 + e)^m*d)/x^4, x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\ln \left (d\,{\left (f\,x^2+e\right )}^m\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(d*(e + f*x^2)^m)*(a + b*log(c*x^n))^2)/x^4,x)

[Out]

int((log(d*(e + f*x^2)^m)*(a + b*log(c*x^n))^2)/x^4, x)

________________________________________________________________________________________