3.73 \(\int (a+b \log (c x^n)) \log (d (e+f x)^m) \, dx\)

Optimal. Leaf size=117 \[ \frac {(e+f x) \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{f}-m x \left (a+b \log \left (c x^n\right )\right )-\frac {b n (e+f x) \log \left (d (e+f x)^m\right )}{f}-\frac {b e n \log \left (-\frac {f x}{e}\right ) \log \left (d (e+f x)^m\right )}{f}-\frac {b e m n \text {Li}_2\left (\frac {f x}{e}+1\right )}{f}+2 b m n x \]

[Out]

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

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Rubi [A]  time = 0.15, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {2389, 2295, 2370, 2411, 43, 2351, 2317, 2391} \[ -\frac {b e m n \text {PolyLog}\left (2,\frac {f x}{e}+1\right )}{f}+\frac {(e+f x) \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{f}-m x \left (a+b \log \left (c x^n\right )\right )-\frac {b n (e+f x) \log \left (d (e+f x)^m\right )}{f}-\frac {b e n \log \left (-\frac {f x}{e}\right ) \log \left (d (e+f x)^m\right )}{f}+2 b m n x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2317

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 2351

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = ExpandIntegrand[a + b*Log[c*x^n], (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c,
d, e, f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[m] && IntegerQ[r]))

Rule 2370

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

Rule 2389

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

Rule 2391

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 2411

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((g*x)/e)^q*((e*h - d*i)/e + (i*x)/e)^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rubi steps

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

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Mathematica [A]  time = 0.07, size = 152, normalized size = 1.30 \[ \frac {a f x \log \left (d (e+f x)^m\right )+a e \log \left (d (e+f x)^m\right )-a f m x+b \log \left (c x^n\right ) \left (f x \left (\log \left (d (e+f x)^m\right )-m\right )+e m \log (e+f x)\right )-b f n x \log \left (d (e+f x)^m\right )+b e m n \text {Li}_2\left (-\frac {f x}{e}\right )-b e m n \log (e+f x)-b e m n \log (x) \log (e+f x)+b e m n \log (x) \log \left (\frac {f x}{e}+1\right )+2 b f m n x}{f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x + e\right )}^{m} d\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))*log(d*(f*x+e)^m),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x + e\right )}^{m} d\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.64, size = 1762, normalized size = 15.06 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-m*ln(c)*b*x+ln(c)*ln(d)*b*x-ln(d)*b*n*x+2*b*m*n*x-a*m*x+ln(d)*a*x+(b*x*ln(x^n)+1/2*(I*Pi*b*csgn(I*x^n)*csgn(I
*c*x^n)^2-I*Pi*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-I*Pi*b*csgn(I*c*x^n)^3+I*Pi*b*csgn(I*c)*csgn(I*c*x^n)^2+2
*b*ln(c)-2*b*n+2*a)*x)*ln((f*x+e)^m)+1/2*I*ln(c)*Pi*b*x*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)^2+1/2*I*Pi*ln(d)
*b*x*csgn(I*x^n)*csgn(I*c*x^n)^2+1/2*I*Pi*ln(x^n)*x*b*csgn(I*d)*csgn(I*d*(f*x+e)^m)^2-1/2*I/f*m*e*ln(f*x+e)*Pi
*b*csgn(I*c*x^n)^3-1/2*I*Pi*ln(x^n)*x*b*csgn(I*d)*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)+1/4*Pi^2*x*b*csgn(I*d)
*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*d*(f*x+e)^m)^2*csgn(I*c)+1/4*Pi^2*x*b*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*(f*x+
e)^m)*csgn(I*d*(f*x+e)^m)^2*csgn(I*c)+1/4*Pi^2*x*b*csgn(I*d)*csgn(I*x^n)*csgn(I*c*x^n)^2*csgn(I*(f*x+e)^m)*csg
n(I*d*(f*x+e)^m)+1/4*Pi^2*x*b*csgn(I*d)*csgn(I*c*x^n)^2*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)*csgn(I*c)-m*b*ln
(x^n)*x+ln(d)*ln(x^n)*x*b+1/2*I*Pi*b*n*x*csgn(I*d)*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)-1/2*I*Pi*ln(d)*b*x*cs
gn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+1/2*I*m*Pi*b*x*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-1/2*I*ln(c)*Pi*b*x*csgn(I
*d)*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)-1/2*I*Pi*a*x*csgn(I*d*(f*x+e)^m)^3-1/4*Pi^2*x*b*csgn(I*c*x^n)^3*csgn
(I*d*(f*x+e)^m)^3-n*b*e*m/f*dilog(-1/e*f*x)-1/4*Pi^2*x*b*csgn(I*d)*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*(f*x+e)^m)
*csgn(I*d*(f*x+e)^m)*csgn(I*c)+1/2*I/f*m*e*ln(f*x+e)*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)^2+1/2*I/f*m*e*ln(f*x+e)*Pi
*b*csgn(I*c*x^n)^2*csgn(I*c)+1/4*Pi^2*x*b*csgn(I*d)*csgn(I*c*x^n)^3*csgn(I*d*(f*x+e)^m)^2+1/4*Pi^2*x*b*csgn(I*
c*x^n)^3*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)^2+1/f*m*b*ln(x^n)*e*ln(f*x+e)+b*e*n*m/f-1/2*I*m*Pi*b*x*csgn(I*c
*x^n)^2*csgn(I*c)-1/2*I*Pi*a*x*csgn(I*d)*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)-1/2*I*Pi*ln(x^n)*x*b*csgn(I*d*(
f*x+e)^m)^3+1/2*I*Pi*b*n*x*csgn(I*d*(f*x+e)^m)^3+1/4*Pi^2*x*b*csgn(I*x^n)*csgn(I*c*x^n)^2*csgn(I*d*(f*x+e)^m)^
3+1/4*Pi^2*x*b*csgn(I*c*x^n)^2*csgn(I*d*(f*x+e)^m)^3*csgn(I*c)+1/2*I*ln(c)*Pi*b*x*csgn(I*d)*csgn(I*d*(f*x+e)^m
)^2-1/2*I*Pi*b*n*x*csgn(I*d)*csgn(I*d*(f*x+e)^m)^2-1/2*I*Pi*b*n*x*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)^2-1/2*
I*m*Pi*b*x*csgn(I*x^n)*csgn(I*c*x^n)^2-n*b*e*m/f*ln(f*x+e)*ln(-1/e*f*x)+a/f*m*e*ln(f*x+e)-b*e*n*m/f*ln(f*x+e)-
1/4*Pi^2*x*b*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*d*(f*x+e)^m)^3*csgn(I*c)-1/4*Pi^2*x*b*csgn(I*d)*csgn(I*c*x^n)^3*
csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)-1/4*Pi^2*x*b*csgn(I*d)*csgn(I*x^n)*csgn(I*c*x^n)^2*csgn(I*d*(f*x+e)^m)^2
+1/2*I*Pi*ln(x^n)*x*b*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)^2+1/2*I*Pi*ln(d)*b*x*csgn(I*c*x^n)^2*csgn(I*c)+1/2
*I*m*Pi*b*x*csgn(I*c*x^n)^3+1/f*m*e*ln(f*x+e)*b*ln(c)-1/4*Pi^2*x*b*csgn(I*d)*csgn(I*c*x^n)^2*csgn(I*d*(f*x+e)^
m)^2*csgn(I*c)-1/4*Pi^2*x*b*csgn(I*x^n)*csgn(I*c*x^n)^2*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)^2-1/4*Pi^2*x*b*c
sgn(I*c*x^n)^2*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)^2*csgn(I*c)-1/2*I/f*m*e*ln(f*x+e)*Pi*b*csgn(I*x^n)*csgn(I
*c*x^n)*csgn(I*c)+1/2*I*Pi*a*x*csgn(I*d)*csgn(I*d*(f*x+e)^m)^2+1/2*I*Pi*a*x*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)
^m)^2-1/2*I*ln(c)*Pi*b*x*csgn(I*d*(f*x+e)^m)^3-1/2*I*Pi*ln(d)*b*x*csgn(I*c*x^n)^3

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maxima [A]  time = 2.09, size = 188, normalized size = 1.61 \[ \frac {{\left (\log \left (\frac {f x}{e} + 1\right ) \log \relax (x) + {\rm Li}_2\left (-\frac {f x}{e}\right )\right )} b e m n}{f} + \frac {{\left (a e m - {\left (e m n - e m \log \relax (c)\right )} b\right )} \log \left (f x + e\right )}{f} - \frac {b e m n \log \left (f x + e\right ) \log \relax (x) + {\left ({\left (f m - f \log \relax (d)\right )} a - {\left (2 \, f m n - f n \log \relax (d) - {\left (f m - f \log \relax (d)\right )} \log \relax (c)\right )} b\right )} x - {\left (b f x \log \left (x^{n}\right ) - {\left ({\left (f n - f \log \relax (c)\right )} b - a f\right )} x\right )} \log \left ({\left (f x + e\right )}^{m}\right ) - {\left (b e m \log \left (f x + e\right ) - {\left (f m - f \log \relax (d)\right )} b x\right )} \log \left (x^{n}\right )}{f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(f*x/e + 1)*log(x) + dilog(-f*x/e))*b*e*m*n/f + (a*e*m - (e*m*n - e*m*log(c))*b)*log(f*x + e)/f - (b*e*m*n
*log(f*x + e)*log(x) + ((f*m - f*log(d))*a - (2*f*m*n - f*n*log(d) - (f*m - f*log(d))*log(c))*b)*x - (b*f*x*lo
g(x^n) - ((f*n - f*log(c))*b - a*f)*x)*log((f*x + e)^m) - (b*e*m*log(f*x + e) - (f*m - f*log(d))*b*x)*log(x^n)
)/f

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \ln \left (d\,{\left (e+f\,x\right )}^m\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))*ln(d*(f*x+e)**m),x)

[Out]

Timed out

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