3.2.47 \(\int \frac {(a+b \log (c x^n)) \log (d (e+f x^m)^k)}{x} \, dx\) [147]

Optimal. Leaf size=114 \[ \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^m\right )^k\right )}{2 b n}-\frac {k \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x^m}{e}\right )}{2 b n}-\frac {k \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x^m}{e}\right )}{m}+\frac {b k n \text {Li}_3\left (-\frac {f x^m}{e}\right )}{m^2} \]

[Out]

1/2*(a+b*ln(c*x^n))^2*ln(d*(e+f*x^m)^k)/b/n-1/2*k*(a+b*ln(c*x^n))^2*ln(1+f*x^m/e)/b/n-k*(a+b*ln(c*x^n))*polylo
g(2,-f*x^m/e)/m+b*k*n*polylog(3,-f*x^m/e)/m^2

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Rubi [A]
time = 0.12, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2422, 2375, 2421, 6724} \begin {gather*} -\frac {k \text {PolyLog}\left (2,-\frac {f x^m}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{m}+\frac {b k n \text {PolyLog}\left (3,-\frac {f x^m}{e}\right )}{m^2}+\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^m\right )^k\right )}{2 b n}-\frac {k \log \left (\frac {f x^m}{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)^k])/x,x]

[Out]

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

Rule 2375

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

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(277\) vs. \(2(114)=228\).
time = 0.12, size = 277, normalized size = 2.43 \begin {gather*} -\frac {1}{6} b k m n \log ^3(x)-\frac {1}{2} b k n \log ^2(x) \log \left (1+\frac {e x^{-m}}{f}\right )+b k n \log ^2(x) \log \left (e+f x^m\right )-\frac {b k n \log (x) \log \left (-\frac {f x^m}{e}\right ) \log \left (e+f x^m\right )}{m}-b k \log (x) \log \left (c x^n\right ) \log \left (e+f x^m\right )+\frac {b k \log \left (-\frac {f x^m}{e}\right ) \log \left (c x^n\right ) \log \left (e+f x^m\right )}{m}-\frac {1}{2} b n \log ^2(x) \log \left (d \left (e+f x^m\right )^k\right )+\frac {a \log \left (-\frac {f x^m}{e}\right ) \log \left (d \left (e+f x^m\right )^k\right )}{m}+b \log (x) \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )+\frac {b k n \log (x) \text {Li}_2\left (-\frac {e x^{-m}}{f}\right )}{m}+\frac {k \left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \text {Li}_2\left (1+\frac {f x^m}{e}\right )}{m}+\frac {b k n \text {Li}_3\left (-\frac {e x^{-m}}{f}\right )}{m^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (d \left (e +f \,x^{m}\right )^{k}\right )}{x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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*(e+f*x^m)^k)/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^m + e)^k) - integrate(-1/2*((b*f*
k*m*n*log(x)^2 + 2*b*f*log(c)*log(d) + 2*a*f*log(d) - 2*(b*f*k*m*log(c) + a*f*k*m)*log(x))*x^m + 2*(b*log(c)*l
og(d) + a*log(d))*e + 2*(b*e*log(d) - (b*f*k*m*log(x) - b*f*log(d))*x^m)*log(x^n))/(f*x*x^m + x*e), x)

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Fricas [A]
time = 0.37, size = 173, normalized size = 1.52 \begin {gather*} \frac {b m^{2} n \log \left (d\right ) \log \left (x\right )^{2} + 2 \, b k n {\rm polylog}\left (3, -f x^{m} e^{\left (-1\right )}\right ) + 2 \, {\left (b m^{2} \log \left (c\right ) + a m^{2}\right )} \log \left (d\right ) \log \left (x\right ) - 2 \, {\left (b k m n \log \left (x\right ) + b k m \log \left (c\right ) + a k m\right )} {\rm Li}_2\left (-{\left (f x^{m} + e\right )} e^{\left (-1\right )} + 1\right ) + {\left (b k m^{2} n \log \left (x\right )^{2} + 2 \, {\left (b k m^{2} \log \left (c\right ) + a k m^{2}\right )} \log \left (x\right )\right )} \log \left (f x^{m} + e\right ) - {\left (b k m^{2} n \log \left (x\right )^{2} + 2 \, {\left (b k m^{2} \log \left (c\right ) + a k m^{2}\right )} \log \left (x\right )\right )} \log \left ({\left (f x^{m} + e\right )} e^{\left (-1\right )}\right )}{2 \, m^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> Invalid comparison of non-real zoo

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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*(e+f*x^m)^k)/x,x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)*log((f*x^m + e)^k*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^m\right )}^k\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)^k)*(a + b*log(c*x^n)))/x,x)

[Out]

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

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