3.1.82 \(\int \frac {(a+b \log (c x^n))^2 \log (d (e+f x)^m)}{x^2} \, dx\) [82]

Optimal. Leaf size=248 \[ \frac {2 b^2 f m n^2 \log (x)}{e}-\frac {2 b f m n \log \left (1+\frac {e}{f x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {f m \log \left (1+\frac {e}{f x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac {2 b^2 f m n^2 \log (e+f x)}{e}-\frac {2 b^2 n^2 \log \left (d (e+f x)^m\right )}{x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x}+\frac {2 b^2 f m n^2 \text {Li}_2\left (-\frac {e}{f x}\right )}{e}+\frac {2 b f m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e}{f x}\right )}{e}+\frac {2 b^2 f m n^2 \text {Li}_3\left (-\frac {e}{f x}\right )}{e} \]

[Out]

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

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Rubi [A]
time = 0.20, antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2342, 2341, 2425, 36, 29, 31, 2379, 2438, 2421, 6724} \begin {gather*} \frac {2 b f m n \text {PolyLog}\left (2,-\frac {e}{f x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e}+\frac {2 b^2 f m n^2 \text {PolyLog}\left (2,-\frac {e}{f x}\right )}{e}+\frac {2 b^2 f m n^2 \text {PolyLog}\left (3,-\frac {e}{f x}\right )}{e}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x}-\frac {2 b f m n \log \left (\frac {e}{f x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {f m \log \left (\frac {e}{f x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac {2 b^2 n^2 \log \left (d (e+f x)^m\right )}{x}+\frac {2 b^2 f m n^2 \log (x)}{e}-\frac {2 b^2 f m n^2 \log (e+f x)}{e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 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 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^2} \, dx &=-\frac {2 b^2 n^2 \log \left (d (e+f x)^m\right )}{x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x}-(f m) \int \left (-\frac {2 b^2 n^2}{x (e+f x)}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right )}{x (e+f x)}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (e+f x)}\right ) \, dx\\ &=-\frac {2 b^2 n^2 \log \left (d (e+f x)^m\right )}{x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x}+(f m) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (e+f x)} \, dx+(2 b f m n) \int \frac {a+b \log \left (c x^n\right )}{x (e+f x)} \, dx+\left (2 b^2 f m n^2\right ) \int \frac {1}{x (e+f x)} \, dx\\ &=-\frac {2 b^2 n^2 \log \left (d (e+f x)^m\right )}{x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x}+\frac {(f m) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx}{e}-\frac {\left (f^2 m\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{e+f x} \, dx}{e}+\frac {(2 b f m n) \int \frac {a+b \log \left (c x^n\right )}{x} \, dx}{e}-\frac {\left (2 b f^2 m n\right ) \int \frac {a+b \log \left (c x^n\right )}{e+f x} \, dx}{e}+\frac {\left (2 b^2 f m n^2\right ) \int \frac {1}{x} \, dx}{e}-\frac {\left (2 b^2 f^2 m n^2\right ) \int \frac {1}{e+f x} \, dx}{e}\\ &=\frac {2 b^2 f m n^2 \log (x)}{e}+\frac {f m \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac {2 b^2 f m n^2 \log (e+f x)}{e}-\frac {2 b^2 n^2 \log \left (d (e+f x)^m\right )}{x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x}-\frac {2 b f m n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x}{e}\right )}{e}-\frac {f m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x}{e}\right )}{e}+\frac {(f m) \text {Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )}{b e n}+\frac {(2 b f m n) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x}{e}\right )}{x} \, dx}{e}+\frac {\left (2 b^2 f m n^2\right ) \int \frac {\log \left (1+\frac {f x}{e}\right )}{x} \, dx}{e}\\ &=\frac {2 b^2 f m n^2 \log (x)}{e}+\frac {f m \left (a+b \log \left (c x^n\right )\right )^2}{e}+\frac {f m \left (a+b \log \left (c x^n\right )\right )^3}{3 b e n}-\frac {2 b^2 f m n^2 \log (e+f x)}{e}-\frac {2 b^2 n^2 \log \left (d (e+f x)^m\right )}{x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x}-\frac {2 b f m n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x}{e}\right )}{e}-\frac {f m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x}{e}\right )}{e}-\frac {2 b^2 f m n^2 \text {Li}_2\left (-\frac {f x}{e}\right )}{e}-\frac {2 b f m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x}{e}\right )}{e}+\frac {\left (2 b^2 f m n^2\right ) \int \frac {\text {Li}_2\left (-\frac {f x}{e}\right )}{x} \, dx}{e}\\ &=\frac {2 b^2 f m n^2 \log (x)}{e}+\frac {f m \left (a+b \log \left (c x^n\right )\right )^2}{e}+\frac {f m \left (a+b \log \left (c x^n\right )\right )^3}{3 b e n}-\frac {2 b^2 f m n^2 \log (e+f x)}{e}-\frac {2 b^2 n^2 \log \left (d (e+f x)^m\right )}{x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x}-\frac {2 b f m n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x}{e}\right )}{e}-\frac {f m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x}{e}\right )}{e}-\frac {2 b^2 f m n^2 \text {Li}_2\left (-\frac {f x}{e}\right )}{e}-\frac {2 b f m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x}{e}\right )}{e}+\frac {2 b^2 f m n^2 \text {Li}_3\left (-\frac {f x}{e}\right )}{e}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(600\) vs. \(2(248)=496\).
time = 0.21, size = 600, normalized size = 2.42 \begin {gather*} -\frac {-3 a^2 f m x \log (x)-6 a b f m n x \log (x)-6 b^2 f m n^2 x \log (x)+3 a b f m n x \log ^2(x)+3 b^2 f m n^2 x \log ^2(x)-b^2 f m n^2 x \log ^3(x)-6 a b f m x \log (x) \log \left (c x^n\right )-6 b^2 f m n x \log (x) \log \left (c x^n\right )+3 b^2 f m n x \log ^2(x) \log \left (c x^n\right )-3 b^2 f m x \log (x) \log ^2\left (c x^n\right )+3 a^2 f m x \log (e+f x)+6 a b f m n x \log (e+f x)+6 b^2 f m n^2 x \log (e+f x)-6 a b f m n x \log (x) \log (e+f x)-6 b^2 f m n^2 x \log (x) \log (e+f x)+3 b^2 f m n^2 x \log ^2(x) \log (e+f x)+6 a b f m x \log \left (c x^n\right ) \log (e+f x)+6 b^2 f m n x \log \left (c x^n\right ) \log (e+f x)-6 b^2 f m n x \log (x) \log \left (c x^n\right ) \log (e+f x)+3 b^2 f m x \log ^2\left (c x^n\right ) \log (e+f x)+3 a^2 e \log \left (d (e+f x)^m\right )+6 a b e n \log \left (d (e+f x)^m\right )+6 b^2 e n^2 \log \left (d (e+f x)^m\right )+6 a b e \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+6 b^2 e n \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+3 b^2 e \log ^2\left (c x^n\right ) \log \left (d (e+f x)^m\right )+6 a b f m n x \log (x) \log \left (1+\frac {f x}{e}\right )+6 b^2 f m n^2 x \log (x) \log \left (1+\frac {f x}{e}\right )-3 b^2 f m n^2 x \log ^2(x) \log \left (1+\frac {f x}{e}\right )+6 b^2 f m n x \log (x) \log \left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )+6 b f m n x \left (a+b n+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x}{e}\right )-6 b^2 f m n^2 x \text {Li}_3\left (-\frac {f x}{e}\right )}{3 e x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.53, size = 10967, normalized size = 44.22

method	result	size
risch	\(\text {Expression too large to display}\)	\(10967\)

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^2,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^2,x, algorithm="maxima")

[Out]

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

[Out]

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

[Out]

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

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