3.1.87 \(\int \frac {(a+b \log (c x^n))^3 \log (d (e+f x)^m)}{x} \, dx\) [87]

Optimal. Leaf size=161 \[ \frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d (e+f x)^m\right )}{4 b n}-\frac {m \left (a+b \log \left (c x^n\right )\right )^4 \log \left (1+\frac {f x}{e}\right )}{4 b n}-m \left (a+b \log \left (c x^n\right )\right )^3 \text {Li}_2\left (-\frac {f x}{e}\right )+3 b m n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_3\left (-\frac {f x}{e}\right )-6 b^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_4\left (-\frac {f x}{e}\right )+6 b^3 m n^3 \text {Li}_5\left (-\frac {f x}{e}\right ) \]

[Out]

1/4*(a+b*ln(c*x^n))^4*ln(d*(f*x+e)^m)/b/n-1/4*m*(a+b*ln(c*x^n))^4*ln(1+f*x/e)/b/n-m*(a+b*ln(c*x^n))^3*polylog(
2,-f*x/e)+3*b*m*n*(a+b*ln(c*x^n))^2*polylog(3,-f*x/e)-6*b^2*m*n^2*(a+b*ln(c*x^n))*polylog(4,-f*x/e)+6*b^3*m*n^
3*polylog(5,-f*x/e)

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Rubi [A]
time = 0.13, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 6, 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*} -6 b^2 m n^2 \text {PolyLog}\left (4,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )-m \text {PolyLog}\left (2,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3+3 b m n \text {PolyLog}\left (3,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2+6 b^3 m n^3 \text {PolyLog}\left (5,-\frac {f x}{e}\right )+\frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d (e+f x)^m\right )}{4 b n}-\frac {m \log \left (\frac {f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^4}{4 b n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*Log[c*x^n])^4*Log[d*(e + f*x)^m])/(4*b*n) - (m*(a + b*Log[c*x^n])^4*Log[1 + (f*x)/e])/(4*b*n) - m*(a +
b*Log[c*x^n])^3*PolyLog[2, -((f*x)/e)] + 3*b*m*n*(a + b*Log[c*x^n])^2*PolyLog[3, -((f*x)/e)] - 6*b^2*m*n^2*(a
+ b*Log[c*x^n])*PolyLog[4, -((f*x)/e)] + 6*b^3*m*n^3*PolyLog[5, -((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 )^3 \log \left (d (e+f x)^m\right )}{x} \, dx &=\frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d (e+f x)^m\right )}{4 b n}-\frac {(f m) \int \frac {\left (a+b \log \left (c x^n\right )\right )^4}{e+f x} \, dx}{4 b n}\\ &=\frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d (e+f x)^m\right )}{4 b n}-\frac {m \left (a+b \log \left (c x^n\right )\right )^4 \log \left (1+\frac {f x}{e}\right )}{4 b n}+m \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {f x}{e}\right )}{x} \, dx\\ &=\frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d (e+f x)^m\right )}{4 b n}-\frac {m \left (a+b \log \left (c x^n\right )\right )^4 \log \left (1+\frac {f x}{e}\right )}{4 b n}-m \left (a+b \log \left (c x^n\right )\right )^3 \text {Li}_2\left (-\frac {f x}{e}\right )+(3 b m n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {f x}{e}\right )}{x} \, dx\\ &=\frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d (e+f x)^m\right )}{4 b n}-\frac {m \left (a+b \log \left (c x^n\right )\right )^4 \log \left (1+\frac {f x}{e}\right )}{4 b n}-m \left (a+b \log \left (c x^n\right )\right )^3 \text {Li}_2\left (-\frac {f x}{e}\right )+3 b m n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_3\left (-\frac {f x}{e}\right )-\left (6 b^2 m n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {f x}{e}\right )}{x} \, dx\\ &=\frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d (e+f x)^m\right )}{4 b n}-\frac {m \left (a+b \log \left (c x^n\right )\right )^4 \log \left (1+\frac {f x}{e}\right )}{4 b n}-m \left (a+b \log \left (c x^n\right )\right )^3 \text {Li}_2\left (-\frac {f x}{e}\right )+3 b m n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_3\left (-\frac {f x}{e}\right )-6 b^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_4\left (-\frac {f x}{e}\right )+\left (6 b^3 m n^3\right ) \int \frac {\text {Li}_4\left (-\frac {f x}{e}\right )}{x} \, dx\\ &=\frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d (e+f x)^m\right )}{4 b n}-\frac {m \left (a+b \log \left (c x^n\right )\right )^4 \log \left (1+\frac {f x}{e}\right )}{4 b n}-m \left (a+b \log \left (c x^n\right )\right )^3 \text {Li}_2\left (-\frac {f x}{e}\right )+3 b m n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_3\left (-\frac {f x}{e}\right )-6 b^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_4\left (-\frac {f x}{e}\right )+6 b^3 m n^3 \text {Li}_5\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. \(602\) vs. \(2(161)=322\).
time = 0.14, size = 602, normalized size = 3.74 \begin {gather*} a^3 \log (x) \log \left (d (e+f x)^m\right )-\frac {3}{2} a^2 b n \log ^2(x) \log \left (d (e+f x)^m\right )+a b^2 n^2 \log ^3(x) \log \left (d (e+f x)^m\right )-\frac {1}{4} b^3 n^3 \log ^4(x) \log \left (d (e+f x)^m\right )+3 a^2 b \log (x) \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )-3 a b^2 n \log ^2(x) \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+b^3 n^2 \log ^3(x) \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+3 a b^2 \log (x) \log ^2\left (c x^n\right ) \log \left (d (e+f x)^m\right )-\frac {3}{2} b^3 n \log ^2(x) \log ^2\left (c x^n\right ) \log \left (d (e+f x)^m\right )+b^3 \log (x) \log ^3\left (c x^n\right ) \log \left (d (e+f x)^m\right )-a^3 m \log (x) \log \left (1+\frac {f x}{e}\right )+\frac {3}{2} a^2 b m n \log ^2(x) \log \left (1+\frac {f x}{e}\right )-a b^2 m n^2 \log ^3(x) \log \left (1+\frac {f x}{e}\right )+\frac {1}{4} b^3 m n^3 \log ^4(x) \log \left (1+\frac {f x}{e}\right )-3 a^2 b m \log (x) \log \left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )+3 a b^2 m n \log ^2(x) \log \left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )-b^3 m n^2 \log ^3(x) \log \left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )-3 a b^2 m \log (x) \log ^2\left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )+\frac {3}{2} b^3 m n \log ^2(x) \log ^2\left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )-b^3 m \log (x) \log ^3\left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )-m \left (a+b \log \left (c x^n\right )\right )^3 \text {Li}_2\left (-\frac {f x}{e}\right )+3 b m n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_3\left (-\frac {f x}{e}\right )-6 a b^2 m n^2 \text {Li}_4\left (-\frac {f x}{e}\right )-6 b^3 m n^2 \log \left (c x^n\right ) \text {Li}_4\left (-\frac {f x}{e}\right )+6 b^3 m n^3 \text {Li}_5\left (-\frac {f x}{e}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a^3*Log[x]*Log[d*(e + f*x)^m] - (3*a^2*b*n*Log[x]^2*Log[d*(e + f*x)^m])/2 + a*b^2*n^2*Log[x]^3*Log[d*(e + f*x)
^m] - (b^3*n^3*Log[x]^4*Log[d*(e + f*x)^m])/4 + 3*a^2*b*Log[x]*Log[c*x^n]*Log[d*(e + f*x)^m] - 3*a*b^2*n*Log[x
]^2*Log[c*x^n]*Log[d*(e + f*x)^m] + b^3*n^2*Log[x]^3*Log[c*x^n]*Log[d*(e + f*x)^m] + 3*a*b^2*Log[x]*Log[c*x^n]
^2*Log[d*(e + f*x)^m] - (3*b^3*n*Log[x]^2*Log[c*x^n]^2*Log[d*(e + f*x)^m])/2 + b^3*Log[x]*Log[c*x^n]^3*Log[d*(
e + f*x)^m] - a^3*m*Log[x]*Log[1 + (f*x)/e] + (3*a^2*b*m*n*Log[x]^2*Log[1 + (f*x)/e])/2 - a*b^2*m*n^2*Log[x]^3
*Log[1 + (f*x)/e] + (b^3*m*n^3*Log[x]^4*Log[1 + (f*x)/e])/4 - 3*a^2*b*m*Log[x]*Log[c*x^n]*Log[1 + (f*x)/e] + 3
*a*b^2*m*n*Log[x]^2*Log[c*x^n]*Log[1 + (f*x)/e] - b^3*m*n^2*Log[x]^3*Log[c*x^n]*Log[1 + (f*x)/e] - 3*a*b^2*m*L
og[x]*Log[c*x^n]^2*Log[1 + (f*x)/e] + (3*b^3*m*n*Log[x]^2*Log[c*x^n]^2*Log[1 + (f*x)/e])/2 - b^3*m*Log[x]*Log[
c*x^n]^3*Log[1 + (f*x)/e] - m*(a + b*Log[c*x^n])^3*PolyLog[2, -((f*x)/e)] + 3*b*m*n*(a + b*Log[c*x^n])^2*PolyL
og[3, -((f*x)/e)] - 6*a*b^2*m*n^2*PolyLog[4, -((f*x)/e)] - 6*b^3*m*n^2*Log[c*x^n]*PolyLog[4, -((f*x)/e)] + 6*b
^3*m*n^3*PolyLog[5, -((f*x)/e)]

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 1.37, size = 60520, normalized size = 375.90

method	result	size
risch	\(\text {Expression too large to display}\)	\(60520\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))^3*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))^3*log(d*(f*x+e)^m)/x,x, algorithm="maxima")

[Out]

-1/4*(b^3*n^3*log(x)^4 - 4*b^3*log(x)*log(x^n)^3 - 4*(b^3*n^2*log(c) + a*b^2*n^2)*log(x)^3 + 6*(b^3*n*log(c)^2
+ 2*a*b^2*n*log(c) + a^2*b*n)*log(x)^2 + 6*(b^3*n*log(x)^2 - 2*(b^3*log(c) + a*b^2)*log(x))*log(x^n)^2 - 4*(b
^3*n^2*log(x)^3 - 3*(b^3*n*log(c) + a*b^2*n)*log(x)^2 + 3*(b^3*log(c)^2 + 2*a*b^2*log(c) + a^2*b)*log(x))*log(
x^n) - 4*(b^3*log(c)^3 + 3*a*b^2*log(c)^2 + 3*a^2*b*log(c) + a^3)*log(x))*log((f*x + e)^m) - integrate(-1/4*(b
^3*f*m*n^3*x*log(x)^4 - 4*(b^3*f*m*n^2*log(c) + a*b^2*f*m*n^2)*x*log(x)^3 + 6*(b^3*f*m*n*log(c)^2 + 2*a*b^2*f*
m*n*log(c) + a^2*b*f*m*n)*x*log(x)^2 - 4*(b^3*f*m*x*log(x) - b^3*f*x*log(d) - b^3*e*log(d))*log(x^n)^3 - 4*(b^
3*f*m*log(c)^3 + 3*a*b^2*f*m*log(c)^2 + 3*a^2*b*f*m*log(c) + a^3*f*m)*x*log(x) + 6*(b^3*f*m*n*x*log(x)^2 - 2*(
b^3*f*m*log(c) + a*b^2*f*m)*x*log(x) + 2*(b^3*f*log(c)*log(d) + a*b^2*f*log(d))*x + 2*(b^3*log(c)*log(d) + a*b
^2*log(d))*e)*log(x^n)^2 + 4*(b^3*f*log(c)^3*log(d) + 3*a*b^2*f*log(c)^2*log(d) + 3*a^2*b*f*log(c)*log(d) + a^
3*f*log(d))*x + 4*(b^3*log(c)^3*log(d) + 3*a*b^2*log(c)^2*log(d) + 3*a^2*b*log(c)*log(d) + a^3*log(d))*e - 4*(
b^3*f*m*n^2*x*log(x)^3 - 3*(b^3*f*m*n*log(c) + a*b^2*f*m*n)*x*log(x)^2 + 3*(b^3*f*m*log(c)^2 + 2*a*b^2*f*m*log
(c) + a^2*b*f*m)*x*log(x) - 3*(b^3*f*log(c)^2*log(d) + 2*a*b^2*f*log(c)*log(d) + a^2*b*f*log(d))*x - 3*(b^3*lo
g(c)^2*log(d) + 2*a*b^2*log(c)*log(d) + a^2*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))^3*log(d*(f*x+e)^m)/x,x, algorithm="fricas")

[Out]

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

[Out]

integrate((b*log(c*x^n) + a)^3*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 )}^3}{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))^3)/x,x)

[Out]

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

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