3.1.81 \(\int \frac {(a+b \log (c x^n))^2 \log (d (e+f x)^m)}{x} \, dx\) [81]

Optimal. Leaf size=131 \[ \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{3 b n}-\frac {m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {f x}{e}\right )}{3 b n}-m \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {f x}{e}\right )+2 b m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {f x}{e}\right )-2 b^2 m n^2 \text {Li}_4\left (-\frac {f x}{e}\right ) \]

[Out]

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

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Rubi [A]
time = 0.10, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {2422, 2354, 2421, 2430, 6724} \begin {gather*} -m \text {PolyLog}\left (2,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2+2 b m n \text {PolyLog}\left (3,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )-2 b^2 m n^2 \text {PolyLog}\left (4,-\frac {f x}{e}\right )+\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{3 b n}-\frac {m \log \left (\frac {f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{3 b n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 2422

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

Rule 2430

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[PolyLo
g[k + 1, e*x^q]*((a + b*Log[c*x^n])^p/q), x] - Dist[b*n*(p/q), Int[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^(
p - 1)/x), x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]

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} \, dx &=\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{3 b n}-\frac {(f m) \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{e+f x} \, dx}{3 b n}\\ &=\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{3 b n}-\frac {m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {f x}{e}\right )}{3 b n}+m \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x}{e}\right )}{x} \, dx\\ &=\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{3 b n}-\frac {m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {f x}{e}\right )}{3 b n}-m \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {f x}{e}\right )+(2 b m n) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x}{e}\right )}{x} \, dx\\ &=\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{3 b n}-\frac {m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {f x}{e}\right )}{3 b n}-m \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {f x}{e}\right )+2 b m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {f x}{e}\right )-\left (2 b^2 m n^2\right ) \int \frac {\text {Li}_3\left (-\frac {f x}{e}\right )}{x} \, dx\\ &=\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{3 b n}-\frac {m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {f x}{e}\right )}{3 b n}-m \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {f x}{e}\right )+2 b m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {f x}{e}\right )-2 b^2 m n^2 \text {Li}_4\left (-\frac {f x}{e}\right )\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(329\) vs. \(2(131)=262\).
time = 0.09, size = 329, normalized size = 2.51 \begin {gather*} a^2 \log (x) \log \left (d (e+f x)^m\right )-a b n \log ^2(x) \log \left (d (e+f x)^m\right )+\frac {1}{3} b^2 n^2 \log ^3(x) \log \left (d (e+f x)^m\right )+2 a b \log (x) \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )-b^2 n \log ^2(x) \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+b^2 \log (x) \log ^2\left (c x^n\right ) \log \left (d (e+f x)^m\right )-a^2 m \log (x) \log \left (1+\frac {f x}{e}\right )+a b m n \log ^2(x) \log \left (1+\frac {f x}{e}\right )-\frac {1}{3} b^2 m n^2 \log ^3(x) \log \left (1+\frac {f x}{e}\right )-2 a b m \log (x) \log \left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )+b^2 m n \log ^2(x) \log \left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )-b^2 m \log (x) \log ^2\left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )-m \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {f x}{e}\right )+2 b m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {f x}{e}\right )-2 b^2 m n^2 \text {Li}_4\left (-\frac {f x}{e}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.55, size = 21792, normalized size = 166.35

method	result	size
risch	\(\text {Expression too large to display}\)	\(21792\)

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,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,x, algorithm="maxima")

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \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} \,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,x)

[Out]

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

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