∫ (a+b log(cxn)) log(d(e+fx)m)____________________

3.1.74 \(\int \frac {(a+b \log (c x^n)) \log (d (e+f x)^m)}{x} \, dx\) [74]

Optimal. Leaf size=100 \[ \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{2 b n}-\frac {m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x}{e}\right )}{2 b n}-m \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x}{e}\right )+b m n \text {Li}_3\left (-\frac {f x}{e}\right ) \]

[Out]

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

-f*x/e)+b*m*n*polylog(3,-f*x/e)

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Rubi [A]
time = 0.06, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2422, 2354, 2421, 6724} \begin {gather*} -m \text {PolyLog}\left (2,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )+b m n \text {PolyLog}\left (3,-\frac {f x}{e}\right )+\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{2 b n}-\frac {m \log \left (\frac {f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b n} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

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Mathematica [A]
time = 0.05, size = 147, normalized size = 1.47 \begin {gather*} -\frac {1}{2} b n \log ^2(x) \log \left (d (e+f x)^m\right )+a \log \left (-\frac {f x}{e}\right ) \log \left (d (e+f x)^m\right )+b \log (x) \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+\frac {1}{2} b m n \log ^2(x) \log \left (1+\frac {f x}{e}\right )-b m \log (x) \log \left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )-b m \log \left (c x^n\right ) \text {Li}_2\left (-\frac {f x}{e}\right )+a m \text {Li}_2\left (1+\frac {f x}{e}\right )+b m n \text {Li}_3\left (-\frac {f x}{e}\right ) \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

f*x)^m] + (b*m*n*Log[x]^2*Log[1 + (f*x)/e])/2 - b*m*Log[x]*Log[c*x^n]*Log[1 + (f*x)/e] - b*m*Log[c*x^n]*PolyL

og[2, -((f*x)/e)] + a*m*PolyLog[2, 1 + (f*x)/e] + b*m*n*PolyLog[3, -((f*x)/e)]

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.16, size = 1795, normalized size = 17.95


method	result	size

risch	\(\text {Expression too large to display}\)	\(1795\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*Pi^2*csgn(I*d)*csgn(I*d*(f*x+e)^m)^2*ln(x)*b*csgn(I*c*x^n)^3+1/4*Pi^2*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m

)^2*ln(x)*b*csgn(I*c*x^n)^3+1/4*Pi^2*csgn(I*d*(f*x+e)^m)^3*ln(x)*b*csgn(I*c)*csgn(I*c*x^n)^2+1/2*I*m*dilog((f*

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

d)*b/n*ln(x^n)^2+(b*ln(x)*ln(x^n)-1/2*b*n*ln(x)^2-1/2*I*Pi*ln(x)*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/2*I*P

i*ln(x)*b*csgn(I*c)*csgn(I*c*x^n)^2+1/2*I*Pi*ln(x)*b*csgn(I*x^n)*csgn(I*c*x^n)^2-1/2*I*Pi*ln(x)*b*csgn(I*c*x^n

)^3+ln(c)*ln(x)*b+ln(x)*a)*ln((f*x+e)^m)-1/2*I*Pi*csgn(I*d*(f*x+e)^m)^3*a*ln(x)-1/4*Pi^2*csgn(I*d*(f*x+e)^m)^3

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

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

i^2*csgn(I*d*(f*x+e)^m)^3*ln(x)*b*csgn(I*x^n)*csgn(I*c*x^n)^2+m*ln(x)^2*ln((f*x+e)/e)*b*n-m*b*n*ln(x)*polylog(

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

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

+e)/e)*b*ln(x^n)+b*m*n*polylog(3,-f*x/e)-1/2*I*m*dilog((f*x+e)/e)*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2-1/2*I*m*dilog

((f*x+e)/e)*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2+1/2*I*m*ln(x)*ln((f*x+e)/e)*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x

^n)-1/4*Pi^2*csgn(I*d)*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)*ln(x)*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/4*P

i^2*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)^2*ln(x)*b*csgn(I*x^n)*csgn(I*c*x^n)^2-1/4*Pi^2*csgn(I*d*(f*x+e)^m)^3

*ln(x)*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/4*I*Pi*csgn(I*d)*csgn(I*d*(f*x+e)^m)^2*b/n*ln(x^n)^2+1/2*I*ln(d

)*ln(x)*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2+1/2*I*ln(d)*ln(x)*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+1/2*I*Pi*csgn(I*(f*x

+e)^m)*csgn(I*d*(f*x+e)^m)^2*ln(x)*b*ln(c)-1/4*Pi^2*csgn(I*d)*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)*ln(x)*b*cs

gn(I*c*x^n)^3-1/2*I*Pi*csgn(I*d)*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)*a*ln(x)+1/2*I*Pi*csgn(I*d)*csgn(I*d*(f*

x+e)^m)^2*ln(x)*b*ln(c)+1/4*Pi^2*csgn(I*d)*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)*ln(x)*b*csgn(I*x^n)*csgn(I*c*

x^n)^2+1/4*Pi^2*csgn(I*d)*csgn(I*d*(f*x+e)^m)^2*ln(x)*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/4*I*Pi*csgn(I*d)

*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)*b/n*ln(x^n)^2-1/2*I*ln(d)*ln(x)*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n

)+1/4*I*Pi*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)^2*b/n*ln(x^n)^2+1/4*Pi^2*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m

)^2*ln(x)*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/4*Pi^2*csgn(I*d)*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)*ln(x)

*b*csgn(I*c)*csgn(I*c*x^n)^2-1/2*I*m*ln(x)*ln((f*x+e)/e)*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2-1/2*I*m*ln(x)*ln((f*x+

e)/e)*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2+1/2*I*m*dilog((f*x+e)/e)*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/2*I

*Pi*csgn(I*d)*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)*ln(x)*b*ln(c)-1/4*Pi^2*csgn(I*d)*csgn(I*d*(f*x+e)^m)^2*ln(

x)*b*csgn(I*c)*csgn(I*c*x^n)^2-1/4*Pi^2*csgn(I*d)*csgn(I*d*(f*x+e)^m)^2*ln(x)*b*csgn(I*x^n)*csgn(I*c*x^n)^2-1/

4*Pi^2*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)^2*ln(x)*b*csgn(I*c)*csgn(I*c*x^n)^2+1/2*I*m*ln(x)*ln((f*x+e)/e)*b

*Pi*csgn(I*c*x^n)^3

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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))*log(d*(f*x+e)^m)/x,x, algorithm="maxima")

[Out]

-1/2*(b*n*log(x)^2 - 2*b*log(x)*log(x^n) - 2*(b*log(c) + a)*log(x))*log((f*x + e)^m) - integrate(-1/2*(b*f*m*n

*x*log(x)^2 - 2*(b*f*m*log(c) + a*f*m)*x*log(x) + 2*(b*f*log(c)*log(d) + a*f*log(d))*x + 2*(b*log(c)*log(d) +

a*log(d))*e - 2*(b*f*m*x*log(x) - b*f*x*log(d) - b*e*log(d))*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))*log(d*(f*x+e)^m)/x,x, algorithm="fricas")

[Out]

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

[Out]

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

[Out]

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

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