3.141 \(\int \frac{(a+b \log (c x^n)) \log (d (e+f x^m)^r)}{x} \, dx\)

Optimal. Leaf size=114 \[ -\frac{r \text{PolyLog}\left (2,-\frac{f x^m}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{m}+\frac{b n r \text{PolyLog}\left (3,-\frac{f x^m}{e}\right )}{m^2}+\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^m\right )^r\right )}{2 b n}-\frac{r \log \left (\frac{f x^m}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b n} \]

[Out]

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

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Rubi [A]  time = 0.193429, antiderivative size = 114, normalized size of antiderivative = 1., 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 = {2375, 2337, 2374, 6589} \[ -\frac{r \text{PolyLog}\left (2,-\frac{f x^m}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{m}+\frac{b n r \text{PolyLog}\left (3,-\frac{f x^m}{e}\right )}{m^2}+\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^m\right )^r\right )}{2 b n}-\frac{r \log \left (\frac{f x^m}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2375

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 2337

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 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(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*Log
[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 6589

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

Mathematica [B]  time = 0.16502, size = 277, normalized size = 2.43 \[ \frac{r \text{PolyLog}\left (2,\frac{f x^m}{e}+1\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{m}+\frac{b n r \text{PolyLog}\left (3,-\frac{e x^{-m}}{f}\right )}{m^2}+\frac{b n r \log (x) \text{PolyLog}\left (2,-\frac{e x^{-m}}{f}\right )}{m}+\frac{a \log \left (-\frac{f x^m}{e}\right ) \log \left (d \left (e+f x^m\right )^r\right )}{m}+b \log (x) \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^r\right )-b r \log (x) \log \left (c x^n\right ) \log \left (e+f x^m\right )+\frac{b r \log \left (c x^n\right ) \log \left (-\frac{f x^m}{e}\right ) \log \left (e+f x^m\right )}{m}-\frac{1}{2} b n \log ^2(x) \log \left (d \left (e+f x^m\right )^r\right )-\frac{1}{2} b n r \log ^2(x) \log \left (\frac{e x^{-m}}{f}+1\right )+b n r \log ^2(x) \log \left (e+f x^m\right )-\frac{b n r \log (x) \log \left (-\frac{f x^m}{e}\right ) \log \left (e+f x^m\right )}{m}-\frac{1}{6} b m n r \log ^3(x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.085, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \ln \left ( d \left ( e+f{x}^{m} \right ) ^{r} \right ) }{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{2} \,{\left (b n \log \left (x\right )^{2} - 2 \, b \log \left (x\right ) \log \left (x^{n}\right ) - 2 \,{\left (b \log \left (c\right ) + a\right )} \log \left (x\right )\right )} \log \left ({\left (f x^{m} + e\right )}^{r}\right ) - \int -\frac{2 \, b e \log \left (c\right ) \log \left (d\right ) + 2 \, a e \log \left (d\right ) +{\left (b f m n r \log \left (x\right )^{2} + 2 \, b f \log \left (c\right ) \log \left (d\right ) + 2 \, a f \log \left (d\right ) - 2 \,{\left (b f m r \log \left (c\right ) + a f m r\right )} \log \left (x\right )\right )} x^{m} + 2 \,{\left (b e \log \left (d\right ) -{\left (b f m r \log \left (x\right ) - b f \log \left (d\right )\right )} x^{m}\right )} \log \left (x^{n}\right )}{2 \,{\left (f x x^{m} + e x\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Fricas [C]  time = 1.09761, size = 450, normalized size = 3.95 \begin{align*} \frac{b m^{2} n \log \left (d\right ) \log \left (x\right )^{2} + 2 \, b n r{\rm polylog}\left (3, -\frac{f x^{m}}{e}\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 m n r \log \left (x\right ) + b m r \log \left (c\right ) + a m r\right )}{\rm Li}_2\left (-\frac{f x^{m} + e}{e} + 1\right ) +{\left (b m^{2} n r \log \left (x\right )^{2} + 2 \,{\left (b m^{2} r \log \left (c\right ) + a m^{2} r\right )} \log \left (x\right )\right )} \log \left (f x^{m} + e\right ) -{\left (b m^{2} n r \log \left (x\right )^{2} + 2 \,{\left (b m^{2} r \log \left (c\right ) + a m^{2} r\right )} \log \left (x\right )\right )} \log \left (\frac{f x^{m} + e}{e}\right )}{2 \, m^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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