3.4.61 \(\int \log (f x^m) (a+b \log (c (d+e x)^n)) \, dx\) [361]

Optimal. Leaf size=99 \[ 2 b m n x-b n x \log \left (f x^m\right )-\frac {b d m n \log (d+e x)}{e}-x \left (m-\log \left (f x^m\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {b d n \log \left (f x^m\right ) \log \left (1+\frac {e x}{d}\right )}{e}+\frac {b d m n \text {Li}_2\left (-\frac {e x}{d}\right )}{e} \]

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2469, 45, 2393, 2332, 2354, 2438} \begin {gather*} \frac {b d m n \text {PolyLog}\left (2,-\frac {e x}{d}\right )}{e}-x \left (m-\log \left (f x^m\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {b d n \log \left (\frac {e x}{d}+1\right ) \log \left (f x^m\right )}{e}-\frac {b d m n \log (d+e x)}{e}-b n x \log \left (f x^m\right )+2 b m n x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 45

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 2332

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

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 2393

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 2438

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 2469

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

Rubi steps

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

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Mathematica [A]
time = 0.06, size = 116, normalized size = 1.17 \begin {gather*} \frac {\log \left (f x^m\right ) \left (b d n \log (d+e x)+e x \left (a-b n+b \log \left (c (d+e x)^n\right )\right )\right )-m \left (a e x-2 b e n x+b d n (1+\log (x)) \log (d+e x)+b e x \log \left (c (d+e x)^n\right )-b d n \log (x) \log \left (1+\frac {e x}{d}\right )\right )+b d m n \text {Li}_2\left (-\frac {e x}{d}\right )}{e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.68, size = 1724, normalized size = 17.41

method	result	size
risch	\(\text {Expression too large to display}\)	\(1724\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.33, size = 140, normalized size = 1.41 \begin {gather*} -{\left ({\left (\log \left (x e + d\right ) \log \left (-\frac {x e + d}{d} + 1\right ) + {\rm Li}_2\left (\frac {x e + d}{d}\right )\right )} b d n e^{\left (-1\right )} + {\left (b d n \log \left (x e + d\right ) + b x e \log \left ({\left (x e + d\right )}^{n}\right ) - {\left (b {\left (2 \, n - \log \left (c\right )\right )} - a\right )} x e\right )} e^{\left (-1\right )}\right )} m + {\left ({\left (d e^{\left (-2\right )} \log \left (x e + d\right ) - x e^{\left (-1\right )}\right )} b n e + b x \log \left ({\left (x e + d\right )}^{n} c\right ) + a x\right )} \log \left (f x^{m}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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