3.1.83 \(\int \frac {(a+b \log (c x^n))^2 \log (d (e+f x)^m)}{x^3} \, dx\) [83]

Optimal. Leaf size=344 \[ -\frac {7 b^2 f m n^2}{4 e x}-\frac {b^2 f^2 m n^2 \log (x)}{4 e^2}-\frac {3 b f m n \left (a+b \log \left (c x^n\right )\right )}{2 e x}+\frac {b f^2 m n \log \left (1+\frac {e}{f x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac {f m \left (a+b \log \left (c x^n\right )\right )^2}{2 e x}+\frac {f^2 m \log \left (1+\frac {e}{f x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2}+\frac {b^2 f^2 m n^2 \log (e+f x)}{4 e^2}-\frac {b^2 n^2 \log \left (d (e+f x)^m\right )}{4 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{2 x^2}-\frac {b^2 f^2 m n^2 \text {Li}_2\left (-\frac {e}{f x}\right )}{2 e^2}-\frac {b f^2 m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e}{f x}\right )}{e^2}-\frac {b^2 f^2 m n^2 \text {Li}_3\left (-\frac {e}{f x}\right )}{e^2} \]

[Out]

-7/4*b^2*f*m*n^2/e/x-1/4*b^2*f^2*m*n^2*ln(x)/e^2-3/2*b*f*m*n*(a+b*ln(c*x^n))/e/x+1/2*b*f^2*m*n*ln(1+e/f/x)*(a+
b*ln(c*x^n))/e^2-1/2*f*m*(a+b*ln(c*x^n))^2/e/x+1/2*f^2*m*ln(1+e/f/x)*(a+b*ln(c*x^n))^2/e^2+1/4*b^2*f^2*m*n^2*l
n(f*x+e)/e^2-1/4*b^2*n^2*ln(d*(f*x+e)^m)/x^2-1/2*b*n*(a+b*ln(c*x^n))*ln(d*(f*x+e)^m)/x^2-1/2*(a+b*ln(c*x^n))^2
*ln(d*(f*x+e)^m)/x^2-1/2*b^2*f^2*m*n^2*polylog(2,-e/f/x)/e^2-b*f^2*m*n*(a+b*ln(c*x^n))*polylog(2,-e/f/x)/e^2-b
^2*f^2*m*n^2*polylog(3,-e/f/x)/e^2

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Rubi [A]
time = 0.35, antiderivative size = 344, normalized size of antiderivative = 1.00, number of steps used = 14, 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 {b f^2 m n \text {PolyLog}\left (2,-\frac {e}{f x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {b^2 f^2 m n^2 \text {PolyLog}\left (2,-\frac {e}{f x}\right )}{2 e^2}-\frac {b^2 f^2 m n^2 \text {PolyLog}\left (3,-\frac {e}{f x}\right )}{e^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{2 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{2 x^2}+\frac {f^2 m \log \left (\frac {e}{f x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2}+\frac {b f^2 m n \log \left (\frac {e}{f x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac {f m \left (a+b \log \left (c x^n\right )\right )^2}{2 e x}-\frac {3 b f m n \left (a+b \log \left (c x^n\right )\right )}{2 e x}-\frac {b^2 n^2 \log \left (d (e+f x)^m\right )}{4 x^2}-\frac {b^2 f^2 m n^2 \log (x)}{4 e^2}+\frac {b^2 f^2 m n^2 \log (e+f x)}{4 e^2}-\frac {7 b^2 f m n^2}{4 e x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-7*b^2*f*m*n^2)/(4*e*x) - (b^2*f^2*m*n^2*Log[x])/(4*e^2) - (3*b*f*m*n*(a + b*Log[c*x^n]))/(2*e*x) + (b*f^2*m*
n*Log[1 + e/(f*x)]*(a + b*Log[c*x^n]))/(2*e^2) - (f*m*(a + b*Log[c*x^n])^2)/(2*e*x) + (f^2*m*Log[1 + e/(f*x)]*
(a + b*Log[c*x^n])^2)/(2*e^2) + (b^2*f^2*m*n^2*Log[e + f*x])/(4*e^2) - (b^2*n^2*Log[d*(e + f*x)^m])/(4*x^2) -
(b*n*(a + b*Log[c*x^n])*Log[d*(e + f*x)^m])/(2*x^2) - ((a + b*Log[c*x^n])^2*Log[d*(e + f*x)^m])/(2*x^2) - (b^2
*f^2*m*n^2*PolyLog[2, -(e/(f*x))])/(2*e^2) - (b*f^2*m*n*(a + b*Log[c*x^n])*PolyLog[2, -(e/(f*x))])/e^2 - (b^2*
f^2*m*n^2*PolyLog[3, -(e/(f*x))])/e^2

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^3} \, dx &=-\frac {b^2 n^2 \log \left (d (e+f x)^m\right )}{4 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{2 x^2}-(f m) \int \left (-\frac {b^2 n^2}{4 x^2 (e+f x)}-\frac {b n \left (a+b \log \left (c x^n\right )\right )}{2 x^2 (e+f x)}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 x^2 (e+f x)}\right ) \, dx\\ &=-\frac {b^2 n^2 \log \left (d (e+f x)^m\right )}{4 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{2 x^2}+\frac {1}{2} (f m) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (e+f x)} \, dx+\frac {1}{2} (b f m n) \int \frac {a+b \log \left (c x^n\right )}{x^2 (e+f x)} \, dx+\frac {1}{4} \left (b^2 f m n^2\right ) \int \frac {1}{x^2 (e+f x)} \, dx\\ &=-\frac {b^2 n^2 \log \left (d (e+f x)^m\right )}{4 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{2 x^2}+\frac {1}{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^2 x}+\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^2}{e^2 (e+f x)}\right ) \, dx+\frac {1}{2} (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^2 x}+\frac {f^2 \left (a+b \log \left (c x^n\right )\right )}{e^2 (e+f x)}\right ) \, dx+\frac {1}{4} \left (b^2 f m n^2\right ) \int \left (\frac {1}{e x^2}-\frac {f}{e^2 x}+\frac {f^2}{e^2 (e+f x)}\right ) \, dx\\ &=-\frac {b^2 f m n^2}{4 e x}-\frac {b^2 f^2 m n^2 \log (x)}{4 e^2}+\frac {b^2 f^2 m n^2 \log (e+f x)}{4 e^2}-\frac {b^2 n^2 \log \left (d (e+f x)^m\right )}{4 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{2 x^2}+\frac {(f m) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx}{2 e}-\frac {\left (f^2 m\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx}{2 e^2}+\frac {\left (f^3 m\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{e+f x} \, dx}{2 e^2}+\frac {(b f m n) \int \frac {a+b \log \left (c x^n\right )}{x^2} \, dx}{2 e}-\frac {\left (b f^2 m n\right ) \int \frac {a+b \log \left (c x^n\right )}{x} \, dx}{2 e^2}+\frac {\left (b f^3 m n\right ) \int \frac {a+b \log \left (c x^n\right )}{e+f x} \, dx}{2 e^2}\\ &=-\frac {3 b^2 f m n^2}{4 e x}-\frac {b^2 f^2 m n^2 \log (x)}{4 e^2}-\frac {b f m n \left (a+b \log \left (c x^n\right )\right )}{2 e x}-\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^2}{4 e^2}-\frac {f m \left (a+b \log \left (c x^n\right )\right )^2}{2 e x}+\frac {b^2 f^2 m n^2 \log (e+f x)}{4 e^2}-\frac {b^2 n^2 \log \left (d (e+f x)^m\right )}{4 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{2 x^2}+\frac {b f^2 m n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x}{e}\right )}{2 e^2}+\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x}{e}\right )}{2 e^2}-\frac {\left (f^2 m\right ) \text {Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )}{2 b e^2 n}+\frac {(b f m n) \int \frac {a+b \log \left (c x^n\right )}{x^2} \, dx}{e}-\frac {\left (b f^2 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}{e^2}-\frac {\left (b^2 f^2 m n^2\right ) \int \frac {\log \left (1+\frac {f x}{e}\right )}{x} \, dx}{2 e^2}\\ &=-\frac {7 b^2 f m n^2}{4 e x}-\frac {b^2 f^2 m n^2 \log (x)}{4 e^2}-\frac {3 b f m n \left (a+b \log \left (c x^n\right )\right )}{2 e x}-\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^2}{4 e^2}-\frac {f m \left (a+b \log \left (c x^n\right )\right )^2}{2 e x}-\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^3}{6 b e^2 n}+\frac {b^2 f^2 m n^2 \log (e+f x)}{4 e^2}-\frac {b^2 n^2 \log \left (d (e+f x)^m\right )}{4 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{2 x^2}+\frac {b f^2 m n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x}{e}\right )}{2 e^2}+\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x}{e}\right )}{2 e^2}+\frac {b^2 f^2 m n^2 \text {Li}_2\left (-\frac {f x}{e}\right )}{2 e^2}+\frac {b f^2 m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x}{e}\right )}{e^2}-\frac {\left (b^2 f^2 m n^2\right ) \int \frac {\text {Li}_2\left (-\frac {f x}{e}\right )}{x} \, dx}{e^2}\\ &=-\frac {7 b^2 f m n^2}{4 e x}-\frac {b^2 f^2 m n^2 \log (x)}{4 e^2}-\frac {3 b f m n \left (a+b \log \left (c x^n\right )\right )}{2 e x}-\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^2}{4 e^2}-\frac {f m \left (a+b \log \left (c x^n\right )\right )^2}{2 e x}-\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^3}{6 b e^2 n}+\frac {b^2 f^2 m n^2 \log (e+f x)}{4 e^2}-\frac {b^2 n^2 \log \left (d (e+f x)^m\right )}{4 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{2 x^2}+\frac {b f^2 m n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x}{e}\right )}{2 e^2}+\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x}{e}\right )}{2 e^2}+\frac {b^2 f^2 m n^2 \text {Li}_2\left (-\frac {f x}{e}\right )}{2 e^2}+\frac {b f^2 m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x}{e}\right )}{e^2}-\frac {b^2 f^2 m n^2 \text {Li}_3\left (-\frac {f x}{e}\right )}{e^2}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(796\) vs. \(2(344)=688\).
time = 0.23, size = 796, normalized size = 2.31 \begin {gather*} -\frac {6 a^2 e f m x+18 a b e f m n x+21 b^2 e f m n^2 x+6 a^2 f^2 m x^2 \log (x)+6 a b f^2 m n x^2 \log (x)+3 b^2 f^2 m n^2 x^2 \log (x)-6 a b f^2 m n x^2 \log ^2(x)-3 b^2 f^2 m n^2 x^2 \log ^2(x)+2 b^2 f^2 m n^2 x^2 \log ^3(x)+12 a b e f m x \log \left (c x^n\right )+18 b^2 e f m n x \log \left (c x^n\right )+12 a b f^2 m x^2 \log (x) \log \left (c x^n\right )+6 b^2 f^2 m n x^2 \log (x) \log \left (c x^n\right )-6 b^2 f^2 m n x^2 \log ^2(x) \log \left (c x^n\right )+6 b^2 e f m x \log ^2\left (c x^n\right )+6 b^2 f^2 m x^2 \log (x) \log ^2\left (c x^n\right )-6 a^2 f^2 m x^2 \log (e+f x)-6 a b f^2 m n x^2 \log (e+f x)-3 b^2 f^2 m n^2 x^2 \log (e+f x)+12 a b f^2 m n x^2 \log (x) \log (e+f x)+6 b^2 f^2 m n^2 x^2 \log (x) \log (e+f x)-6 b^2 f^2 m n^2 x^2 \log ^2(x) \log (e+f x)-12 a b f^2 m x^2 \log \left (c x^n\right ) \log (e+f x)-6 b^2 f^2 m n x^2 \log \left (c x^n\right ) \log (e+f x)+12 b^2 f^2 m n x^2 \log (x) \log \left (c x^n\right ) \log (e+f x)-6 b^2 f^2 m x^2 \log ^2\left (c x^n\right ) \log (e+f x)+6 a^2 e^2 \log \left (d (e+f x)^m\right )+6 a b e^2 n \log \left (d (e+f x)^m\right )+3 b^2 e^2 n^2 \log \left (d (e+f x)^m\right )+12 a b e^2 \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+6 b^2 e^2 n \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+6 b^2 e^2 \log ^2\left (c x^n\right ) \log \left (d (e+f x)^m\right )-12 a b f^2 m n x^2 \log (x) \log \left (1+\frac {f x}{e}\right )-6 b^2 f^2 m n^2 x^2 \log (x) \log \left (1+\frac {f x}{e}\right )+6 b^2 f^2 m n^2 x^2 \log ^2(x) \log \left (1+\frac {f x}{e}\right )-12 b^2 f^2 m n x^2 \log (x) \log \left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )-6 b f^2 m n x^2 \left (2 a+b n+2 b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x}{e}\right )+12 b^2 f^2 m n^2 x^2 \text {Li}_3\left (-\frac {f x}{e}\right )}{12 e^2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/12*(6*a^2*e*f*m*x + 18*a*b*e*f*m*n*x + 21*b^2*e*f*m*n^2*x + 6*a^2*f^2*m*x^2*Log[x] + 6*a*b*f^2*m*n*x^2*Log[
x] + 3*b^2*f^2*m*n^2*x^2*Log[x] - 6*a*b*f^2*m*n*x^2*Log[x]^2 - 3*b^2*f^2*m*n^2*x^2*Log[x]^2 + 2*b^2*f^2*m*n^2*
x^2*Log[x]^3 + 12*a*b*e*f*m*x*Log[c*x^n] + 18*b^2*e*f*m*n*x*Log[c*x^n] + 12*a*b*f^2*m*x^2*Log[x]*Log[c*x^n] +
6*b^2*f^2*m*n*x^2*Log[x]*Log[c*x^n] - 6*b^2*f^2*m*n*x^2*Log[x]^2*Log[c*x^n] + 6*b^2*e*f*m*x*Log[c*x^n]^2 + 6*b
^2*f^2*m*x^2*Log[x]*Log[c*x^n]^2 - 6*a^2*f^2*m*x^2*Log[e + f*x] - 6*a*b*f^2*m*n*x^2*Log[e + f*x] - 3*b^2*f^2*m
*n^2*x^2*Log[e + f*x] + 12*a*b*f^2*m*n*x^2*Log[x]*Log[e + f*x] + 6*b^2*f^2*m*n^2*x^2*Log[x]*Log[e + f*x] - 6*b
^2*f^2*m*n^2*x^2*Log[x]^2*Log[e + f*x] - 12*a*b*f^2*m*x^2*Log[c*x^n]*Log[e + f*x] - 6*b^2*f^2*m*n*x^2*Log[c*x^
n]*Log[e + f*x] + 12*b^2*f^2*m*n*x^2*Log[x]*Log[c*x^n]*Log[e + f*x] - 6*b^2*f^2*m*x^2*Log[c*x^n]^2*Log[e + f*x
] + 6*a^2*e^2*Log[d*(e + f*x)^m] + 6*a*b*e^2*n*Log[d*(e + f*x)^m] + 3*b^2*e^2*n^2*Log[d*(e + f*x)^m] + 12*a*b*
e^2*Log[c*x^n]*Log[d*(e + f*x)^m] + 6*b^2*e^2*n*Log[c*x^n]*Log[d*(e + f*x)^m] + 6*b^2*e^2*Log[c*x^n]^2*Log[d*(
e + f*x)^m] - 12*a*b*f^2*m*n*x^2*Log[x]*Log[1 + (f*x)/e] - 6*b^2*f^2*m*n^2*x^2*Log[x]*Log[1 + (f*x)/e] + 6*b^2
*f^2*m*n^2*x^2*Log[x]^2*Log[1 + (f*x)/e] - 12*b^2*f^2*m*n*x^2*Log[x]*Log[c*x^n]*Log[1 + (f*x)/e] - 6*b*f^2*m*n
*x^2*(2*a + b*n + 2*b*Log[c*x^n])*PolyLog[2, -((f*x)/e)] + 12*b^2*f^2*m*n^2*x^2*PolyLog[3, -((f*x)/e)])/(e^2*x
^2)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.53, size = 12159, normalized size = 35.35

method	result	size
risch	\(\text {Expression too large to display}\)	\(12159\)

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^3,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^3,x, algorithm="maxima")

[Out]

1/4*(2*(b^2*f^2*m*x^2*log(f*x + e) - b^2*f^2*m*x^2*log(x) - b^2*f*m*x*e - b^2*e^2*log(d))*log(x^n)^2 - (2*b^2*
e^2*log(x^n)^2 + 2*(b^2*(n + 2*log(c)) + 2*a*b)*e^2*log(x^n) + ((n^2 + 2*n*log(c) + 2*log(c)^2)*b^2 + 2*a*b*(n
+ 2*log(c)) + 2*a^2)*e^2)*log((f*x + e)^m))*e^(-2)/x^2 - integrate(-1/4*((2*(f*m + 2*f*log(d))*a^2 + 2*(f*m*n
+ 2*(f*m + 2*f*log(d))*log(c))*a*b + (f*m*n^2 + 2*f*m*n*log(c) + 2*(f*m + 2*f*log(d))*log(c)^2)*b^2)*x*e^2 +
4*(b^2*log(c)^2*log(d) + 2*a*b*log(c)*log(d) + a^2*log(d))*e^3 + 2*(2*b^2*f^2*m*n*x^2*e + (2*(f*m + 2*f*log(d)
)*a*b + (3*f*m*n + 2*f*n*log(d) + 2*(f*m + 2*f*log(d))*log(c))*b^2)*x*e^2 + 2*((n*log(d) + 2*log(c)*log(d))*b^
2 + 2*a*b*log(d))*e^3 - 2*(b^2*f^3*m*n*x^3 + b^2*f^2*m*n*x^2*e)*log(f*x + e) + 2*(b^2*f^3*m*n*x^3 + b^2*f^2*m*
n*x^2*e)*log(x))*log(x^n))/(f*x^4*e^2 + x^3*e^3), 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^3,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^3, 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**3,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^3,x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)^2*log((f*x + e)^m*d)/x^3, 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^3} \,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^3,x)

[Out]

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

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