3.1.84 \(\int \frac {(a+b \log (c x^n))^2 \log (d (e+f x)^m)}{x^4} \, dx\) [84]

Optimal. Leaf size=420 \[ -\frac {19 b^2 f m n^2}{108 e x^2}+\frac {26 b^2 f^2 m n^2}{27 e^2 x}+\frac {2 b^2 f^3 m n^2 \log (x)}{27 e^3}-\frac {5 b f m n \left (a+b \log \left (c x^n\right )\right )}{18 e x^2}+\frac {8 b f^2 m n \left (a+b \log \left (c x^n\right )\right )}{9 e^2 x}-\frac {2 b f^3 m n \log \left (1+\frac {e}{f x}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 e^3}-\frac {f m \left (a+b \log \left (c x^n\right )\right )^2}{6 e x^2}+\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2 x}-\frac {f^3 m \log \left (1+\frac {e}{f x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{3 e^3}-\frac {2 b^2 f^3 m n^2 \log (e+f x)}{27 e^3}-\frac {2 b^2 n^2 \log \left (d (e+f x)^m\right )}{27 x^3}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{3 x^3}+\frac {2 b^2 f^3 m n^2 \text {Li}_2\left (-\frac {e}{f x}\right )}{9 e^3}+\frac {2 b f^3 m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e}{f x}\right )}{3 e^3}+\frac {2 b^2 f^3 m n^2 \text {Li}_3\left (-\frac {e}{f x}\right )}{3 e^3} \]

[Out]

-19/108*b^2*f*m*n^2/e/x^2+26/27*b^2*f^2*m*n^2/e^2/x+2/27*b^2*f^3*m*n^2*ln(x)/e^3-5/18*b*f*m*n*(a+b*ln(c*x^n))/
e/x^2+8/9*b*f^2*m*n*(a+b*ln(c*x^n))/e^2/x-2/9*b*f^3*m*n*ln(1+e/f/x)*(a+b*ln(c*x^n))/e^3-1/6*f*m*(a+b*ln(c*x^n)
)^2/e/x^2+1/3*f^2*m*(a+b*ln(c*x^n))^2/e^2/x-1/3*f^3*m*ln(1+e/f/x)*(a+b*ln(c*x^n))^2/e^3-2/27*b^2*f^3*m*n^2*ln(
f*x+e)/e^3-2/27*b^2*n^2*ln(d*(f*x+e)^m)/x^3-2/9*b*n*(a+b*ln(c*x^n))*ln(d*(f*x+e)^m)/x^3-1/3*(a+b*ln(c*x^n))^2*
ln(d*(f*x+e)^m)/x^3+2/9*b^2*f^3*m*n^2*polylog(2,-e/f/x)/e^3+2/3*b*f^3*m*n*(a+b*ln(c*x^n))*polylog(2,-e/f/x)/e^
3+2/3*b^2*f^3*m*n^2*polylog(3,-e/f/x)/e^3

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Rubi [A]
time = 0.50, antiderivative size = 420, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 9, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {2342, 2341, 2425, 46, 2380, 2379, 2438, 2421, 6724} \begin {gather*} \frac {2 b f^3 m n \text {PolyLog}\left (2,-\frac {e}{f x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {2 b^2 f^3 m n^2 \text {PolyLog}\left (2,-\frac {e}{f x}\right )}{9 e^3}+\frac {2 b^2 f^3 m n^2 \text {PolyLog}\left (3,-\frac {e}{f x}\right )}{3 e^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{3 x^3}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{9 x^3}-\frac {f^3 m \log \left (\frac {e}{f x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{3 e^3}-\frac {2 b f^3 m n \log \left (\frac {e}{f x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{9 e^3}+\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2 x}+\frac {8 b f^2 m n \left (a+b \log \left (c x^n\right )\right )}{9 e^2 x}-\frac {f m \left (a+b \log \left (c x^n\right )\right )^2}{6 e x^2}-\frac {5 b f m n \left (a+b \log \left (c x^n\right )\right )}{18 e x^2}-\frac {2 b^2 n^2 \log \left (d (e+f x)^m\right )}{27 x^3}+\frac {2 b^2 f^3 m n^2 \log (x)}{27 e^3}-\frac {2 b^2 f^3 m n^2 \log (e+f x)}{27 e^3}+\frac {26 b^2 f^2 m n^2}{27 e^2 x}-\frac {19 b^2 f m n^2}{108 e x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-19*b^2*f*m*n^2)/(108*e*x^2) + (26*b^2*f^2*m*n^2)/(27*e^2*x) + (2*b^2*f^3*m*n^2*Log[x])/(27*e^3) - (5*b*f*m*n
*(a + b*Log[c*x^n]))/(18*e*x^2) + (8*b*f^2*m*n*(a + b*Log[c*x^n]))/(9*e^2*x) - (2*b*f^3*m*n*Log[1 + e/(f*x)]*(
a + b*Log[c*x^n]))/(9*e^3) - (f*m*(a + b*Log[c*x^n])^2)/(6*e*x^2) + (f^2*m*(a + b*Log[c*x^n])^2)/(3*e^2*x) - (
f^3*m*Log[1 + e/(f*x)]*(a + b*Log[c*x^n])^2)/(3*e^3) - (2*b^2*f^3*m*n^2*Log[e + f*x])/(27*e^3) - (2*b^2*n^2*Lo
g[d*(e + f*x)^m])/(27*x^3) - (2*b*n*(a + b*Log[c*x^n])*Log[d*(e + f*x)^m])/(9*x^3) - ((a + b*Log[c*x^n])^2*Log
[d*(e + f*x)^m])/(3*x^3) + (2*b^2*f^3*m*n^2*PolyLog[2, -(e/(f*x))])/(9*e^3) + (2*b*f^3*m*n*(a + b*Log[c*x^n])*
PolyLog[2, -(e/(f*x))])/(3*e^3) + (2*b^2*f^3*m*n^2*PolyLog[3, -(e/(f*x))])/(3*e^3)

Rule 46

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

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 2379

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

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 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 (e+f x)^m\right )}{x^4} \, dx &=-\frac {2 b^2 n^2 \log \left (d (e+f x)^m\right )}{27 x^3}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{3 x^3}-(f m) \int \left (-\frac {2 b^2 n^2}{27 x^3 (e+f x)}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right )}{9 x^3 (e+f x)}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 x^3 (e+f x)}\right ) \, dx\\ &=-\frac {2 b^2 n^2 \log \left (d (e+f x)^m\right )}{27 x^3}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{3 x^3}+\frac {1}{3} (f m) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3 (e+f x)} \, dx+\frac {1}{9} (2 b f m n) \int \frac {a+b \log \left (c x^n\right )}{x^3 (e+f x)} \, dx+\frac {1}{27} \left (2 b^2 f m n^2\right ) \int \frac {1}{x^3 (e+f x)} \, dx\\ &=-\frac {2 b^2 n^2 \log \left (d (e+f x)^m\right )}{27 x^3}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{3 x^3}+\frac {1}{3} (f m) \int \left (\frac {\left (a+b \log \left (c x^n\right )\right )^2}{e x^3}-\frac {f \left (a+b \log \left (c x^n\right )\right )^2}{e^2 x^2}+\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^2}{e^3 x}-\frac {f^3 \left (a+b \log \left (c x^n\right )\right )^2}{e^3 (e+f x)}\right ) \, dx+\frac {1}{9} (2 b f m n) \int \left (\frac {a+b \log \left (c x^n\right )}{e x^3}-\frac {f \left (a+b \log \left (c x^n\right )\right )}{e^2 x^2}+\frac {f^2 \left (a+b \log \left (c x^n\right )\right )}{e^3 x}-\frac {f^3 \left (a+b \log \left (c x^n\right )\right )}{e^3 (e+f x)}\right ) \, dx+\frac {1}{27} \left (2 b^2 f m n^2\right ) \int \left (\frac {1}{e x^3}-\frac {f}{e^2 x^2}+\frac {f^2}{e^3 x}-\frac {f^3}{e^3 (e+f x)}\right ) \, dx\\ &=-\frac {b^2 f m n^2}{27 e x^2}+\frac {2 b^2 f^2 m n^2}{27 e^2 x}+\frac {2 b^2 f^3 m n^2 \log (x)}{27 e^3}-\frac {2 b^2 f^3 m n^2 \log (e+f x)}{27 e^3}-\frac {2 b^2 n^2 \log \left (d (e+f x)^m\right )}{27 x^3}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{3 x^3}+\frac {(f m) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx}{3 e}-\frac {\left (f^2 m\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx}{3 e^2}+\frac {\left (f^3 m\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx}{3 e^3}-\frac {\left (f^4 m\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{e+f x} \, dx}{3 e^3}+\frac {(2 b f m n) \int \frac {a+b \log \left (c x^n\right )}{x^3} \, dx}{9 e}-\frac {\left (2 b f^2 m n\right ) \int \frac {a+b \log \left (c x^n\right )}{x^2} \, dx}{9 e^2}+\frac {\left (2 b f^3 m n\right ) \int \frac {a+b \log \left (c x^n\right )}{x} \, dx}{9 e^3}-\frac {\left (2 b f^4 m n\right ) \int \frac {a+b \log \left (c x^n\right )}{e+f x} \, dx}{9 e^3}\\ &=-\frac {5 b^2 f m n^2}{54 e x^2}+\frac {8 b^2 f^2 m n^2}{27 e^2 x}+\frac {2 b^2 f^3 m n^2 \log (x)}{27 e^3}-\frac {b f m n \left (a+b \log \left (c x^n\right )\right )}{9 e x^2}+\frac {2 b f^2 m n \left (a+b \log \left (c x^n\right )\right )}{9 e^2 x}+\frac {f^3 m \left (a+b \log \left (c x^n\right )\right )^2}{9 e^3}-\frac {f m \left (a+b \log \left (c x^n\right )\right )^2}{6 e x^2}+\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2 x}-\frac {2 b^2 f^3 m n^2 \log (e+f x)}{27 e^3}-\frac {2 b^2 n^2 \log \left (d (e+f x)^m\right )}{27 x^3}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{3 x^3}-\frac {2 b f^3 m n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x}{e}\right )}{9 e^3}-\frac {f^3 m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x}{e}\right )}{3 e^3}+\frac {\left (f^3 m\right ) \text {Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )}{3 b e^3 n}+\frac {(b f m n) \int \frac {a+b \log \left (c x^n\right )}{x^3} \, dx}{3 e}-\frac {\left (2 b f^2 m n\right ) \int \frac {a+b \log \left (c x^n\right )}{x^2} \, dx}{3 e^2}+\frac {\left (2 b f^3 m n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x}{e}\right )}{x} \, dx}{3 e^3}+\frac {\left (2 b^2 f^3 m n^2\right ) \int \frac {\log \left (1+\frac {f x}{e}\right )}{x} \, dx}{9 e^3}\\ &=-\frac {19 b^2 f m n^2}{108 e x^2}+\frac {26 b^2 f^2 m n^2}{27 e^2 x}+\frac {2 b^2 f^3 m n^2 \log (x)}{27 e^3}-\frac {5 b f m n \left (a+b \log \left (c x^n\right )\right )}{18 e x^2}+\frac {8 b f^2 m n \left (a+b \log \left (c x^n\right )\right )}{9 e^2 x}+\frac {f^3 m \left (a+b \log \left (c x^n\right )\right )^2}{9 e^3}-\frac {f m \left (a+b \log \left (c x^n\right )\right )^2}{6 e x^2}+\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2 x}+\frac {f^3 m \left (a+b \log \left (c x^n\right )\right )^3}{9 b e^3 n}-\frac {2 b^2 f^3 m n^2 \log (e+f x)}{27 e^3}-\frac {2 b^2 n^2 \log \left (d (e+f x)^m\right )}{27 x^3}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{3 x^3}-\frac {2 b f^3 m n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x}{e}\right )}{9 e^3}-\frac {f^3 m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x}{e}\right )}{3 e^3}-\frac {2 b^2 f^3 m n^2 \text {Li}_2\left (-\frac {f x}{e}\right )}{9 e^3}-\frac {2 b f^3 m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x}{e}\right )}{3 e^3}+\frac {\left (2 b^2 f^3 m n^2\right ) \int \frac {\text {Li}_2\left (-\frac {f x}{e}\right )}{x} \, dx}{3 e^3}\\ &=-\frac {19 b^2 f m n^2}{108 e x^2}+\frac {26 b^2 f^2 m n^2}{27 e^2 x}+\frac {2 b^2 f^3 m n^2 \log (x)}{27 e^3}-\frac {5 b f m n \left (a+b \log \left (c x^n\right )\right )}{18 e x^2}+\frac {8 b f^2 m n \left (a+b \log \left (c x^n\right )\right )}{9 e^2 x}+\frac {f^3 m \left (a+b \log \left (c x^n\right )\right )^2}{9 e^3}-\frac {f m \left (a+b \log \left (c x^n\right )\right )^2}{6 e x^2}+\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2 x}+\frac {f^3 m \left (a+b \log \left (c x^n\right )\right )^3}{9 b e^3 n}-\frac {2 b^2 f^3 m n^2 \log (e+f x)}{27 e^3}-\frac {2 b^2 n^2 \log \left (d (e+f x)^m\right )}{27 x^3}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{3 x^3}-\frac {2 b f^3 m n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x}{e}\right )}{9 e^3}-\frac {f^3 m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x}{e}\right )}{3 e^3}-\frac {2 b^2 f^3 m n^2 \text {Li}_2\left (-\frac {f x}{e}\right )}{9 e^3}-\frac {2 b f^3 m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x}{e}\right )}{3 e^3}+\frac {2 b^2 f^3 m n^2 \text {Li}_3\left (-\frac {f x}{e}\right )}{3 e^3}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(909\) vs. \(2(420)=840\).
time = 0.26, size = 909, normalized size = 2.16 \begin {gather*} -\frac {18 a^2 e^2 f m x+30 a b e^2 f m n x+19 b^2 e^2 f m n^2 x-36 a^2 e f^2 m x^2-96 a b e f^2 m n x^2-104 b^2 e f^2 m n^2 x^2-36 a^2 f^3 m x^3 \log (x)-24 a b f^3 m n x^3 \log (x)-8 b^2 f^3 m n^2 x^3 \log (x)+36 a b f^3 m n x^3 \log ^2(x)+12 b^2 f^3 m n^2 x^3 \log ^2(x)-12 b^2 f^3 m n^2 x^3 \log ^3(x)+36 a b e^2 f m x \log \left (c x^n\right )+30 b^2 e^2 f m n x \log \left (c x^n\right )-72 a b e f^2 m x^2 \log \left (c x^n\right )-96 b^2 e f^2 m n x^2 \log \left (c x^n\right )-72 a b f^3 m x^3 \log (x) \log \left (c x^n\right )-24 b^2 f^3 m n x^3 \log (x) \log \left (c x^n\right )+36 b^2 f^3 m n x^3 \log ^2(x) \log \left (c x^n\right )+18 b^2 e^2 f m x \log ^2\left (c x^n\right )-36 b^2 e f^2 m x^2 \log ^2\left (c x^n\right )-36 b^2 f^3 m x^3 \log (x) \log ^2\left (c x^n\right )+36 a^2 f^3 m x^3 \log (e+f x)+24 a b f^3 m n x^3 \log (e+f x)+8 b^2 f^3 m n^2 x^3 \log (e+f x)-72 a b f^3 m n x^3 \log (x) \log (e+f x)-24 b^2 f^3 m n^2 x^3 \log (x) \log (e+f x)+36 b^2 f^3 m n^2 x^3 \log ^2(x) \log (e+f x)+72 a b f^3 m x^3 \log \left (c x^n\right ) \log (e+f x)+24 b^2 f^3 m n x^3 \log \left (c x^n\right ) \log (e+f x)-72 b^2 f^3 m n x^3 \log (x) \log \left (c x^n\right ) \log (e+f x)+36 b^2 f^3 m x^3 \log ^2\left (c x^n\right ) \log (e+f x)+36 a^2 e^3 \log \left (d (e+f x)^m\right )+24 a b e^3 n \log \left (d (e+f x)^m\right )+8 b^2 e^3 n^2 \log \left (d (e+f x)^m\right )+72 a b e^3 \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+24 b^2 e^3 n \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+36 b^2 e^3 \log ^2\left (c x^n\right ) \log \left (d (e+f x)^m\right )+72 a b f^3 m n x^3 \log (x) \log \left (1+\frac {f x}{e}\right )+24 b^2 f^3 m n^2 x^3 \log (x) \log \left (1+\frac {f x}{e}\right )-36 b^2 f^3 m n^2 x^3 \log ^2(x) \log \left (1+\frac {f x}{e}\right )+72 b^2 f^3 m n x^3 \log (x) \log \left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )+24 b f^3 m n x^3 \left (3 a+b n+3 b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x}{e}\right )-72 b^2 f^3 m n^2 x^3 \text {Li}_3\left (-\frac {f x}{e}\right )}{108 e^3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/108*(18*a^2*e^2*f*m*x + 30*a*b*e^2*f*m*n*x + 19*b^2*e^2*f*m*n^2*x - 36*a^2*e*f^2*m*x^2 - 96*a*b*e*f^2*m*n*x
^2 - 104*b^2*e*f^2*m*n^2*x^2 - 36*a^2*f^3*m*x^3*Log[x] - 24*a*b*f^3*m*n*x^3*Log[x] - 8*b^2*f^3*m*n^2*x^3*Log[x
] + 36*a*b*f^3*m*n*x^3*Log[x]^2 + 12*b^2*f^3*m*n^2*x^3*Log[x]^2 - 12*b^2*f^3*m*n^2*x^3*Log[x]^3 + 36*a*b*e^2*f
*m*x*Log[c*x^n] + 30*b^2*e^2*f*m*n*x*Log[c*x^n] - 72*a*b*e*f^2*m*x^2*Log[c*x^n] - 96*b^2*e*f^2*m*n*x^2*Log[c*x
^n] - 72*a*b*f^3*m*x^3*Log[x]*Log[c*x^n] - 24*b^2*f^3*m*n*x^3*Log[x]*Log[c*x^n] + 36*b^2*f^3*m*n*x^3*Log[x]^2*
Log[c*x^n] + 18*b^2*e^2*f*m*x*Log[c*x^n]^2 - 36*b^2*e*f^2*m*x^2*Log[c*x^n]^2 - 36*b^2*f^3*m*x^3*Log[x]*Log[c*x
^n]^2 + 36*a^2*f^3*m*x^3*Log[e + f*x] + 24*a*b*f^3*m*n*x^3*Log[e + f*x] + 8*b^2*f^3*m*n^2*x^3*Log[e + f*x] - 7
2*a*b*f^3*m*n*x^3*Log[x]*Log[e + f*x] - 24*b^2*f^3*m*n^2*x^3*Log[x]*Log[e + f*x] + 36*b^2*f^3*m*n^2*x^3*Log[x]
^2*Log[e + f*x] + 72*a*b*f^3*m*x^3*Log[c*x^n]*Log[e + f*x] + 24*b^2*f^3*m*n*x^3*Log[c*x^n]*Log[e + f*x] - 72*b
^2*f^3*m*n*x^3*Log[x]*Log[c*x^n]*Log[e + f*x] + 36*b^2*f^3*m*x^3*Log[c*x^n]^2*Log[e + f*x] + 36*a^2*e^3*Log[d*
(e + f*x)^m] + 24*a*b*e^3*n*Log[d*(e + f*x)^m] + 8*b^2*e^3*n^2*Log[d*(e + f*x)^m] + 72*a*b*e^3*Log[c*x^n]*Log[
d*(e + f*x)^m] + 24*b^2*e^3*n*Log[c*x^n]*Log[d*(e + f*x)^m] + 36*b^2*e^3*Log[c*x^n]^2*Log[d*(e + f*x)^m] + 72*
a*b*f^3*m*n*x^3*Log[x]*Log[1 + (f*x)/e] + 24*b^2*f^3*m*n^2*x^3*Log[x]*Log[1 + (f*x)/e] - 36*b^2*f^3*m*n^2*x^3*
Log[x]^2*Log[1 + (f*x)/e] + 72*b^2*f^3*m*n*x^3*Log[x]*Log[c*x^n]*Log[1 + (f*x)/e] + 24*b*f^3*m*n*x^3*(3*a + b*
n + 3*b*Log[c*x^n])*PolyLog[2, -((f*x)/e)] - 72*b^2*f^3*m*n^2*x^3*PolyLog[3, -((f*x)/e)])/(e^3*x^3)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.55, size = 13227, normalized size = 31.49

method	result	size
risch	\(\text {Expression too large to display}\)	\(13227\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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

[Out]

-1/54*(9*(2*b^2*f^3*m*x^3*log(f*x + e) - 2*b^2*f^3*m*x^3*log(x) - 2*b^2*f^2*m*x^2*e + b^2*f*m*x*e^2 + 2*b^2*e^
3*log(d))*log(x^n)^2 + 2*(9*b^2*e^3*log(x^n)^2 + 6*(b^2*(n + 3*log(c)) + 3*a*b)*e^3*log(x^n) + ((2*n^2 + 6*n*l
og(c) + 9*log(c)^2)*b^2 + 6*a*b*(n + 3*log(c)) + 9*a^2)*e^3)*log((f*x + e)^m))*e^(-3)/x^3 + integrate(1/27*((9
*(f*m + 3*f*log(d))*a^2 + 6*(f*m*n + 3*(f*m + 3*f*log(d))*log(c))*a*b + (2*f*m*n^2 + 6*f*m*n*log(c) + 9*(f*m +
3*f*log(d))*log(c)^2)*b^2)*x*e^3 + 27*(b^2*log(c)^2*log(d) + 2*a*b*log(c)*log(d) + a^2*log(d))*e^4 - 3*(6*b^2
*f^3*m*n*x^3*e + 3*b^2*f^2*m*n*x^2*e^2 - (6*(f*m + 3*f*log(d))*a*b + (5*f*m*n + 6*f*n*log(d) + 6*(f*m + 3*f*lo
g(d))*log(c))*b^2)*x*e^3 - 6*((n*log(d) + 3*log(c)*log(d))*b^2 + 3*a*b*log(d))*e^4 - 6*(b^2*f^4*m*n*x^4 + b^2*
f^3*m*n*x^3*e)*log(f*x + e) + 6*(b^2*f^4*m*n*x^4 + b^2*f^3*m*n*x^3*e)*log(x))*log(x^n))/(f*x^5*e^3 + x^4*e^4),
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((a+b*log(c*x^n))^2*log(d*(f*x+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 + e)^m*d)/x^4, 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((a+b*ln(c*x**n))**2*ln(d*(f*x+e)**m)/x**4,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((a+b*log(c*x^n))^2*log(d*(f*x+e)^m)/x^4,x, algorithm="giac")

[Out]

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

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

[Out]

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

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