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

Optimal. Leaf size=158 \[ -\frac{b d^2 m n \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{2 e^2}-\frac{1}{4} \left (m x^2-2 x^2 \log \left (f x^m\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{b d^2 n \log \left (\frac{e x}{d}+1\right ) \log \left (f x^m\right )}{2 e^2}+\frac{b d^2 m n \log (d+e x)}{4 e^2}+\frac{b d n x \log \left (f x^m\right )}{2 e}-\frac{3 b d m n x}{4 e}-\frac{1}{4} b n x^2 \log \left (f x^m\right )+\frac{1}{4} b m n x^2 \]

[Out]

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

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Rubi [A]  time = 0.13785, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {2426, 43, 2351, 2295, 2304, 2317, 2391} \[ -\frac{b d^2 m n \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{2 e^2}-\frac{1}{4} \left (m x^2-2 x^2 \log \left (f x^m\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{b d^2 n \log \left (\frac{e x}{d}+1\right ) \log \left (f x^m\right )}{2 e^2}+\frac{b d^2 m n \log (d+e x)}{4 e^2}+\frac{b d n x \log \left (f x^m\right )}{2 e}-\frac{3 b d m n x}{4 e}-\frac{1}{4} b n x^2 \log \left (f x^m\right )+\frac{1}{4} b m n x^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2426

Int[Log[(f_.)*(x_)^(m_.)]*((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((g_.)*(x_))^(q_.), x_Symbol] :
> -Simp[(((m*(g*x)^(q + 1))/(q + 1) - (g*x)^(q + 1)*Log[f*x^m])*(a + b*Log[c*(d + e*x)^n]))/(g*(q + 1)), x] +
(-Dist[(b*e*n)/(g*(q + 1)), Int[((g*x)^(q + 1)*Log[f*x^m])/(d + e*x), x], x] + Dist[(b*e*m*n)/(g*(q + 1)^2), I
nt[(g*x)^(q + 1)/(d + e*x), x], x]) /; FreeQ[{a, b, c, d, e, f, g, m, n, q}, x] && NeQ[q, -1]

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 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 2295

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

Rule 2304

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 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 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]

Rubi steps

\begin{align*} \int x \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx &=-\frac{1}{4} \left (m x^2-2 x^2 \log \left (f x^m\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{1}{2} (b e n) \int \frac{x^2 \log \left (f x^m\right )}{d+e x} \, dx+\frac{1}{4} (b e m n) \int \frac{x^2}{d+e x} \, dx\\ &=-\frac{1}{4} \left (m x^2-2 x^2 \log \left (f x^m\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{1}{2} (b e n) \int \left (-\frac{d \log \left (f x^m\right )}{e^2}+\frac{x \log \left (f x^m\right )}{e}+\frac{d^2 \log \left (f x^m\right )}{e^2 (d+e x)}\right ) \, dx+\frac{1}{4} (b e m n) \int \left (-\frac{d}{e^2}+\frac{x}{e}+\frac{d^2}{e^2 (d+e x)}\right ) \, dx\\ &=-\frac{b d m n x}{4 e}+\frac{1}{8} b m n x^2+\frac{b d^2 m n \log (d+e x)}{4 e^2}-\frac{1}{4} \left (m x^2-2 x^2 \log \left (f x^m\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{1}{2} (b n) \int x \log \left (f x^m\right ) \, dx+\frac{(b d n) \int \log \left (f x^m\right ) \, dx}{2 e}-\frac{\left (b d^2 n\right ) \int \frac{\log \left (f x^m\right )}{d+e x} \, dx}{2 e}\\ &=-\frac{3 b d m n x}{4 e}+\frac{1}{4} b m n x^2+\frac{b d n x \log \left (f x^m\right )}{2 e}-\frac{1}{4} b n x^2 \log \left (f x^m\right )+\frac{b d^2 m n \log (d+e x)}{4 e^2}-\frac{1}{4} \left (m x^2-2 x^2 \log \left (f x^m\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{b d^2 n \log \left (f x^m\right ) \log \left (1+\frac{e x}{d}\right )}{2 e^2}+\frac{\left (b d^2 m n\right ) \int \frac{\log \left (1+\frac{e x}{d}\right )}{x} \, dx}{2 e^2}\\ &=-\frac{3 b d m n x}{4 e}+\frac{1}{4} b m n x^2+\frac{b d n x \log \left (f x^m\right )}{2 e}-\frac{1}{4} b n x^2 \log \left (f x^m\right )+\frac{b d^2 m n \log (d+e x)}{4 e^2}-\frac{1}{4} \left (m x^2-2 x^2 \log \left (f x^m\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{b d^2 n \log \left (f x^m\right ) \log \left (1+\frac{e x}{d}\right )}{2 e^2}-\frac{b d^2 m n \text{Li}_2\left (-\frac{e x}{d}\right )}{2 e^2}\\ \end{align*}

Mathematica [A]  time = 0.128911, size = 164, normalized size = 1.04 \[ \frac{-2 b d^2 m n \text{PolyLog}\left (2,-\frac{e x}{d}\right )+m \left (-a e^2 x^2-b e^2 x^2 \log \left (c (d+e x)^n\right )+b d^2 n (2 \log (x)+1) \log (d+e x)-2 b d^2 n \log (x) \log \left (\frac{e x}{d}+1\right )-3 b d e n x+b e^2 n x^2\right )+\log \left (f x^m\right ) \left (e x \left (2 a e x+2 b e x \log \left (c (d+e x)^n\right )+2 b d n-b e n x\right )-2 b d^2 n \log (d+e x)\right )}{4 e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 1.031, size = 1994, normalized size = 12.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*I*x^2*Pi*b*n*csgn(I*x^m)*csgn(I*f*x^m)^2-1/4*I*x^2*Pi*a*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)+1/4*I*x^2*ln(
f)*Pi*b*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2-1/4*x^2*ln(c)*b*m+(1/2*b*x^2*ln(x^m)+1/4*b*x^2*(-I*Pi*csgn(I*f)*csgn(I
*x^m)*csgn(I*f*x^m)+I*Pi*csgn(I*f)*csgn(I*f*x^m)^2+I*Pi*csgn(I*x^m)*csgn(I*f*x^m)^2-I*Pi*csgn(I*f*x^m)^3+2*ln(
f)-m))*ln((e*x+d)^n)+1/4*I*x^2*Pi*a*csgn(I*x^m)*csgn(I*f*x^m)^2+1/8*b*Pi^2*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^
n)^2*x^2*csgn(I*f*x^m)^3+1/2/e*n*b*ln(x^m)*d*x-1/2/e^2*n*b*ln(x^m)*d^2*ln(e*x+d)-1/4*n*b*ln(x^m)*x^2+1/2*x^2*l
n(f)*ln(c)*b-1/4*x^2*ln(f)*b*n+1/4*I/e^2*b*d^2*n*ln(e*x+d)*Pi*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)-1/4*I*x^2*Pi
*a*csgn(I*f*x^m)^3-1/8*b*Pi^2*csgn(I*c*(e*x+d)^n)^3*x^2*csgn(I*f*x^m)^3-1/4*x^2*a*m+1/2*x^2*ln(f)*a-5/8*b*d^2*
m*n/e^2+1/2*a*x^2*ln(x^m)+1/2*b*ln(c)*x^2*ln(x^m)+1/4*I/e*Pi*b*d*n*csgn(I*f)*csgn(I*f*x^m)^2*x+1/8*I*x^2*Pi*b*
n*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)+1/4*b*m*n*x^2-1/8*b*Pi^2*csgn(I*c*(e*x+d)^n)^3*x^2*csgn(I*f)*csgn(I*x^m)
*csgn(I*f*x^m)+1/4*I*x^2*Pi*ln(c)*b*csgn(I*f)*csgn(I*f*x^m)^2+1/4*I*x^2*Pi*ln(c)*b*csgn(I*x^m)*csgn(I*f*x^m)^2
-1/8*b*Pi^2*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*x^2*csgn(I*f)*csgn(I*f*x^m)^2-1/8*b*Pi^2*csgn(I*c)*csgn(I*c*(e*x+d
)^n)^2*x^2*csgn(I*x^m)*csgn(I*f*x^m)^2-1/8*b*Pi^2*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2*x^2*csgn(I*f)*csgn(I
*f*x^m)^2-1/8*b*Pi^2*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2*x^2*csgn(I*x^m)*csgn(I*f*x^m)^2-1/8*I*x^2*Pi*b*n*
csgn(I*f)*csgn(I*f*x^m)^2+1/4*I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*x^2*ln(x^m)+1/4*I*b*Pi*csgn(I*(e*x+d)^n)*
csgn(I*c*(e*x+d)^n)^2*x^2*ln(x^m)-1/8*I*m*Pi*b*x^2*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2-1/8*I*m*Pi*b*x^2*csgn(I*(e*
x+d)^n)*csgn(I*c*(e*x+d)^n)^2-1/4*I*b*Pi*csgn(I*c*(e*x+d)^n)^3*x^2*ln(x^m)-1/4*I*x^2*ln(f)*Pi*b*csgn(I*c*(e*x+
d)^n)^3-1/4*I*x^2*Pi*ln(c)*b*csgn(I*f*x^m)^3+1/2*m/e^2*b*d^2*n*ln(e*x+d)*ln(-e*x/d)+1/8*b*Pi^2*csgn(I*c)*csgn(
I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*x^2*csgn(I*f)*csgn(I*f*x^m)^2+1/8*b*Pi^2*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c
*(e*x+d)^n)*x^2*csgn(I*x^m)*csgn(I*f*x^m)^2+1/8*b*Pi^2*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*x^2*csgn(I*f)*csgn(I*x^
m)*csgn(I*f*x^m)+1/8*b*Pi^2*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2*x^2*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)+1/
8*I*m*Pi*b*x^2*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-1/4*I/e*Pi*b*d*n*csgn(I*f)*csgn(I*x^m)*csgn(I*f
*x^m)*x-1/2/e^2*b*d^2*n*ln(e*x+d)*ln(f)+1/2/e*ln(f)*b*d*n*x-1/4*I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e
*x+d)^n)*x^2*ln(x^m)-1/4*I/e*Pi*b*d*n*csgn(I*f*x^m)^3*x+1/4*I/e^2*b*d^2*n*ln(e*x+d)*Pi*csgn(I*f*x^m)^3+1/8*b*P
i^2*csgn(I*c*(e*x+d)^n)^3*x^2*csgn(I*f)*csgn(I*f*x^m)^2+1/8*b*Pi^2*csgn(I*c*(e*x+d)^n)^3*x^2*csgn(I*x^m)*csgn(
I*f*x^m)^2+1/8*b*Pi^2*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*x^2*csgn(I*f*x^m)^3+1/8*I*m*Pi*b*x^2*csgn(I*c*(e*x+d)^n)
^3-1/4*I*x^2*ln(f)*Pi*b*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-1/4*I*x^2*Pi*ln(c)*b*csgn(I*f)*csgn(I*
x^m)*csgn(I*f*x^m)+1/8*I*x^2*Pi*b*n*csgn(I*f*x^m)^3+1/4*I*x^2*Pi*a*csgn(I*f)*csgn(I*f*x^m)^2-1/8*b*Pi^2*csgn(I
*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*x^2*csgn(I*f*x^m)^3+1/2*m/e^2*b*d^2*n*dilog(-e*x/d)+1/4*I/e*Pi*b*d*n
*csgn(I*x^m)*csgn(I*f*x^m)^2*x-1/4*I/e^2*b*d^2*n*ln(e*x+d)*Pi*csgn(I*f)*csgn(I*f*x^m)^2-1/8*b*Pi^2*csgn(I*c)*c
sgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*x^2*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)-1/4*I/e^2*b*d^2*n*ln(e*x+d)*Pi*cs
gn(I*x^m)*csgn(I*f*x^m)^2+1/4*I*x^2*ln(f)*Pi*b*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2+1/4*b*d^2*m*n*ln(e*x+d)
/e^2-3/4*b*d*m*n*x/e

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Maxima [A]  time = 1.25455, size = 240, normalized size = 1.52 \begin{align*} \frac{1}{4} \,{\left (\frac{2 \,{\left (\log \left (e x + d\right ) \log \left (-\frac{e x + d}{d} + 1\right ) +{\rm Li}_2\left (\frac{e x + d}{d}\right )\right )} b d^{2} n}{e^{2}} - \frac{b e^{2} x^{2} \log \left ({\left (e x + d\right )}^{n}\right ) + 3 \, b d e n x - b d^{2} n \log \left (e x + d\right ) +{\left (a e^{2} -{\left (e^{2} n - e^{2} \log \left (c\right )\right )} b\right )} x^{2}}{e^{2}}\right )} m - \frac{1}{4} \,{\left (b e n{\left (\frac{2 \, d^{2} \log \left (e x + d\right )}{e^{3}} + \frac{e x^{2} - 2 \, d x}{e^{2}}\right )} - 2 \, b x^{2} \log \left ({\left (e x + d\right )}^{n} c\right ) - 2 \, a x^{2}\right )} \log \left (f x^{m}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b x \log \left ({\left (e x + d\right )}^{n} c\right ) \log \left (f x^{m}\right ) + a x \log \left (f x^{m}\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} x \log \left (f x^{m}\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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