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

Optimal. Leaf size=227 \[ \frac{i b f^{3/2} m n \text{PolyLog}\left (2,-\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{3 e^{3/2}}-\frac{i b f^{3/2} m n \text{PolyLog}\left (2,\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{3 e^{3/2}}-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{3 x^3}-\frac{2 f^{3/2} m \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 e^{3/2}}-\frac{2 f m \left (a+b \log \left (c x^n\right )\right )}{3 e x}-\frac{b n \log \left (d \left (e+f x^2\right )^m\right )}{9 x^3}-\frac{2 b f^{3/2} m n \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{9 e^{3/2}}-\frac{8 b f m n}{9 e x} \]

[Out]

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

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Rubi [A]  time = 0.162726, antiderivative size = 227, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2455, 325, 205, 2376, 4848, 2391} \[ \frac{i b f^{3/2} m n \text{PolyLog}\left (2,-\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{3 e^{3/2}}-\frac{i b f^{3/2} m n \text{PolyLog}\left (2,\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{3 e^{3/2}}-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{3 x^3}-\frac{2 f^{3/2} m \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 e^{3/2}}-\frac{2 f m \left (a+b \log \left (c x^n\right )\right )}{3 e x}-\frac{b n \log \left (d \left (e+f x^2\right )^m\right )}{9 x^3}-\frac{2 b f^{3/2} m n \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{9 e^{3/2}}-\frac{8 b f m n}{9 e x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*b*f*m*n)/(9*e*x) - (2*b*f^(3/2)*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(9*e^(3/2)) - (2*f*m*(a + b*Log[c*x^n]))/
(3*e*x) - (2*f^(3/2)*m*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*(a + b*Log[c*x^n]))/(3*e^(3/2)) - (b*n*Log[d*(e + f*x^2)^m]
)/(9*x^3) - ((a + b*Log[c*x^n])*Log[d*(e + f*x^2)^m])/(3*x^3) + ((I/3)*b*f^(3/2)*m*n*PolyLog[2, ((-I)*Sqrt[f]*
x)/Sqrt[e]])/e^(3/2) - ((I/3)*b*f^(3/2)*m*n*PolyLog[2, (I*Sqrt[f]*x)/Sqrt[e]])/e^(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 325

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

Mathematica [C]  time = 0.107823, size = 362, normalized size = 1.59 \[ \frac{3 i b f^{3/2} m n x^3 \text{PolyLog}\left (2,-\frac{i \sqrt{f} x}{\sqrt{e}}\right )-3 i b f^{3/2} m n x^3 \text{PolyLog}\left (2,\frac{i \sqrt{f} x}{\sqrt{e}}\right )-3 a e^{3/2} \log \left (d \left (e+f x^2\right )^m\right )-6 a \sqrt{e} f m x^2 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};-\frac{f x^2}{e}\right )-3 b e^{3/2} \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-6 b f^{3/2} m x^3 \log \left (c x^n\right ) \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )-6 b \sqrt{e} f m x^2 \log \left (c x^n\right )-b e^{3/2} n \log \left (d \left (e+f x^2\right )^m\right )-3 i b f^{3/2} m n x^3 \log (x) \log \left (1-\frac{i \sqrt{f} x}{\sqrt{e}}\right )+3 i b f^{3/2} m n x^3 \log (x) \log \left (1+\frac{i \sqrt{f} x}{\sqrt{e}}\right )-2 b f^{3/2} m n x^3 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )+6 b f^{3/2} m n x^3 \log (x) \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )-8 b \sqrt{e} f m n x^2}{9 e^{3/2} x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*b*Sqrt[e]*f*m*n*x^2 - 2*b*f^(3/2)*m*n*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]] - 6*a*Sqrt[e]*f*m*x^2*Hypergeometric
2F1[-1/2, 1, 1/2, -((f*x^2)/e)] + 6*b*f^(3/2)*m*n*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x] - 6*b*Sqrt[e]*f*m*x^2
*Log[c*x^n] - 6*b*f^(3/2)*m*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n] - (3*I)*b*f^(3/2)*m*n*x^3*Log[x]*Log[1
- (I*Sqrt[f]*x)/Sqrt[e]] + (3*I)*b*f^(3/2)*m*n*x^3*Log[x]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - 3*a*e^(3/2)*Log[d*(
e + f*x^2)^m] - b*e^(3/2)*n*Log[d*(e + f*x^2)^m] - 3*b*e^(3/2)*Log[c*x^n]*Log[d*(e + f*x^2)^m] + (3*I)*b*f^(3/
2)*m*n*x^3*PolyLog[2, ((-I)*Sqrt[f]*x)/Sqrt[e]] - (3*I)*b*f^(3/2)*m*n*x^3*PolyLog[2, (I*Sqrt[f]*x)/Sqrt[e]])/(
9*e^(3/2)*x^3)

________________________________________________________________________________________

Maple [C]  time = 0.181, size = 2204, normalized size = 9.7 \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))*ln(d*(f*x^2+e)^m)/x^4,x)

[Out]

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

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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))*log(d*(f*x^2+e)^m)/x^4,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{{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )}{x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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