3.370 \(\int \frac{\log (f x^m) (a+b \log (c (d+e x)^n))^2}{x} \, dx\)

Optimal. Leaf size=823 \[ \text{result too large to display} \]

[Out]

(m*Log[x]^2*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2)/2 + Log[x]*(-(m*Log[x]) + Log[f*x^m])*(a - b*n*Lo
g[d + e*x] + b*Log[c*(d + e*x)^n])^2 + 2*b*n*(-(m*Log[x]) + Log[f*x^m])*(a - b*n*Log[d + e*x] + b*Log[c*(d + e
*x)^n])*(Log[x]*(Log[d + e*x] - Log[1 + (e*x)/d]) - PolyLog[2, -((e*x)/d)]) + 2*b*m*n*(a - b*n*Log[d + e*x] +
b*Log[c*(d + e*x)^n])*((Log[x]^2*(Log[d + e*x] - Log[1 + (e*x)/d]))/2 - Log[x]*PolyLog[2, -((e*x)/d)] + PolyLo
g[3, -((e*x)/d)]) - b^2*n^2*(m*Log[x] - Log[f*x^m])*(Log[-((e*x)/d)]*Log[d + e*x]^2 + 2*Log[d + e*x]*PolyLog[2
, 1 + (e*x)/d] - 2*PolyLog[3, 1 + (e*x)/d]) + (b^2*m*n^2*(Log[-((e*x)/d)]^4 + 6*Log[-((e*x)/d)]^2*Log[-((e*x)/
(d + e*x))]^2 - 4*(Log[-((e*x)/d)] + Log[d/(d + e*x)])*Log[-((e*x)/(d + e*x))]^3 + Log[-((e*x)/(d + e*x))]^4 +
6*Log[x]^2*Log[d + e*x]^2 + 4*(2*Log[-((e*x)/d)]^3 - 3*Log[x]^2*Log[d + e*x])*Log[1 + (e*x)/d] + 6*(Log[x] -
Log[-((e*x)/d)])*(Log[x] + 3*Log[-((e*x)/d)])*Log[1 + (e*x)/d]^2 - 4*Log[-((e*x)/d)]^2*Log[-((e*x)/(d + e*x))]
*(Log[-((e*x)/d)] + 3*Log[1 + (e*x)/d]) + 12*(Log[-((e*x)/d)]^2 - 2*Log[-((e*x)/d)]*(Log[-((e*x)/(d + e*x))] +
Log[1 + (e*x)/d]) + 2*Log[x]*(-Log[d + e*x] + Log[1 + (e*x)/d]))*PolyLog[2, -((e*x)/d)] - 12*Log[-((e*x)/(d +
e*x))]^2*PolyLog[2, (e*x)/(d + e*x)] + 12*(Log[-((e*x)/d)] - Log[-((e*x)/(d + e*x))])^2*PolyLog[2, 1 + (e*x)/
d] + 24*(Log[x] - Log[-((e*x)/d)])*Log[1 + (e*x)/d]*PolyLog[2, 1 + (e*x)/d] + 24*(Log[-((e*x)/(d + e*x))] + Lo
g[d + e*x])*PolyLog[3, -((e*x)/d)] + 24*Log[-((e*x)/(d + e*x))]*PolyLog[3, (e*x)/(d + e*x)] + 24*(-Log[x] + Lo
g[-((e*x)/(d + e*x))])*PolyLog[3, 1 + (e*x)/d] - 24*(PolyLog[4, -((e*x)/d)] + PolyLog[4, (e*x)/(d + e*x)] - Po
lyLog[4, 1 + (e*x)/d])))/12

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Rubi [F]  time = 0.0659637, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.386571, size = 823, normalized size = 1. \[ -b^2 \left (m \log (x)-\log \left (f x^m\right )\right ) \left (\log \left (-\frac{e x}{d}\right ) \log ^2(d+e x)+2 \text{PolyLog}\left (2,\frac{e x}{d}+1\right ) \log (d+e x)-2 \text{PolyLog}\left (3,\frac{e x}{d}+1\right )\right ) n^2+\frac{1}{12} b^2 m \left (\log ^4\left (-\frac{e x}{d}\right )+6 \log ^2\left (-\frac{e x}{d+e x}\right ) \log ^2\left (-\frac{e x}{d}\right )-4 \log \left (-\frac{e x}{d+e x}\right ) \left (\log \left (-\frac{e x}{d}\right )+3 \log \left (\frac{e x}{d}+1\right )\right ) \log ^2\left (-\frac{e x}{d}\right )+\log ^4\left (-\frac{e x}{d+e x}\right )-4 \left (\log \left (-\frac{e x}{d}\right )+\log \left (\frac{d}{d+e x}\right )\right ) \log ^3\left (-\frac{e x}{d+e x}\right )+6 \log ^2(x) \log ^2(d+e x)+6 \left (\log (x)-\log \left (-\frac{e x}{d}\right )\right ) \left (\log (x)+3 \log \left (-\frac{e x}{d}\right )\right ) \log ^2\left (\frac{e x}{d}+1\right )+4 \left (2 \log ^3\left (-\frac{e x}{d}\right )-3 \log ^2(x) \log (d+e x)\right ) \log \left (\frac{e x}{d}+1\right )+12 \left (\log ^2\left (-\frac{e x}{d}\right )-2 \left (\log \left (-\frac{e x}{d+e x}\right )+\log \left (\frac{e x}{d}+1\right )\right ) \log \left (-\frac{e x}{d}\right )+2 \log (x) \left (\log \left (\frac{e x}{d}+1\right )-\log (d+e x)\right )\right ) \text{PolyLog}\left (2,-\frac{e x}{d}\right )-12 \log ^2\left (-\frac{e x}{d+e x}\right ) \text{PolyLog}\left (2,\frac{e x}{d+e x}\right )+12 \left (\log \left (-\frac{e x}{d}\right )-\log \left (-\frac{e x}{d+e x}\right )\right )^2 \text{PolyLog}\left (2,\frac{e x}{d}+1\right )+24 \left (\log (x)-\log \left (-\frac{e x}{d}\right )\right ) \log \left (\frac{e x}{d}+1\right ) \text{PolyLog}\left (2,\frac{e x}{d}+1\right )+24 \left (\log \left (-\frac{e x}{d+e x}\right )+\log (d+e x)\right ) \text{PolyLog}\left (3,-\frac{e x}{d}\right )+24 \log \left (-\frac{e x}{d+e x}\right ) \text{PolyLog}\left (3,\frac{e x}{d+e x}\right )+24 \left (\log \left (-\frac{e x}{d+e x}\right )-\log (x)\right ) \text{PolyLog}\left (3,\frac{e x}{d}+1\right )-24 \left (\text{PolyLog}\left (4,-\frac{e x}{d}\right )+\text{PolyLog}\left (4,\frac{e x}{d+e x}\right )-\text{PolyLog}\left (4,\frac{e x}{d}+1\right )\right )\right ) n^2+2 b \left (\log \left (f x^m\right )-m \log (x)\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (\log (x) \left (\log (d+e x)-\log \left (\frac{e x}{d}+1\right )\right )-\text{PolyLog}\left (2,-\frac{e x}{d}\right )\right ) n+2 b m \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (\frac{1}{2} \left (\log (d+e x)-\log \left (\frac{e x}{d}+1\right )\right ) \log ^2(x)-\text{PolyLog}\left (2,-\frac{e x}{d}\right ) \log (x)+\text{PolyLog}\left (3,-\frac{e x}{d}\right )\right ) n+\frac{1}{2} m \log ^2(x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+\log (x) \left (\log \left (f x^m\right )-m \log (x)\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(m*Log[x]^2*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2)/2 + Log[x]*(-(m*Log[x]) + Log[f*x^m])*(a - b*n*Lo
g[d + e*x] + b*Log[c*(d + e*x)^n])^2 + 2*b*n*(-(m*Log[x]) + Log[f*x^m])*(a - b*n*Log[d + e*x] + b*Log[c*(d + e
*x)^n])*(Log[x]*(Log[d + e*x] - Log[1 + (e*x)/d]) - PolyLog[2, -((e*x)/d)]) + 2*b*m*n*(a - b*n*Log[d + e*x] +
b*Log[c*(d + e*x)^n])*((Log[x]^2*(Log[d + e*x] - Log[1 + (e*x)/d]))/2 - Log[x]*PolyLog[2, -((e*x)/d)] + PolyLo
g[3, -((e*x)/d)]) - b^2*n^2*(m*Log[x] - Log[f*x^m])*(Log[-((e*x)/d)]*Log[d + e*x]^2 + 2*Log[d + e*x]*PolyLog[2
, 1 + (e*x)/d] - 2*PolyLog[3, 1 + (e*x)/d]) + (b^2*m*n^2*(Log[-((e*x)/d)]^4 + 6*Log[-((e*x)/d)]^2*Log[-((e*x)/
(d + e*x))]^2 - 4*(Log[-((e*x)/d)] + Log[d/(d + e*x)])*Log[-((e*x)/(d + e*x))]^3 + Log[-((e*x)/(d + e*x))]^4 +
6*Log[x]^2*Log[d + e*x]^2 + 4*(2*Log[-((e*x)/d)]^3 - 3*Log[x]^2*Log[d + e*x])*Log[1 + (e*x)/d] + 6*(Log[x] -
Log[-((e*x)/d)])*(Log[x] + 3*Log[-((e*x)/d)])*Log[1 + (e*x)/d]^2 - 4*Log[-((e*x)/d)]^2*Log[-((e*x)/(d + e*x))]
*(Log[-((e*x)/d)] + 3*Log[1 + (e*x)/d]) + 12*(Log[-((e*x)/d)]^2 - 2*Log[-((e*x)/d)]*(Log[-((e*x)/(d + e*x))] +
Log[1 + (e*x)/d]) + 2*Log[x]*(-Log[d + e*x] + Log[1 + (e*x)/d]))*PolyLog[2, -((e*x)/d)] - 12*Log[-((e*x)/(d +
e*x))]^2*PolyLog[2, (e*x)/(d + e*x)] + 12*(Log[-((e*x)/d)] - Log[-((e*x)/(d + e*x))])^2*PolyLog[2, 1 + (e*x)/
d] + 24*(Log[x] - Log[-((e*x)/d)])*Log[1 + (e*x)/d]*PolyLog[2, 1 + (e*x)/d] + 24*(Log[-((e*x)/(d + e*x))] + Lo
g[d + e*x])*PolyLog[3, -((e*x)/d)] + 24*Log[-((e*x)/(d + e*x))]*PolyLog[3, (e*x)/(d + e*x)] + 24*(-Log[x] + Lo
g[-((e*x)/(d + e*x))])*PolyLog[3, 1 + (e*x)/d] - 24*(PolyLog[4, -((e*x)/d)] + PolyLog[4, (e*x)/(d + e*x)] - Po
lyLog[4, 1 + (e*x)/d])))/12

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Maple [F]  time = 1.754, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( f{x}^{m} \right ) \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{2}}{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(f*x^m)*(a+b*log(c*(e*x+d)^n))^2/x,x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b^2*log((e*x + d)^n*c)^2*log(f*x^m) + 2*a*b*log((e*x + d)^n*c)*log(f*x^m) + a^2*log(f*x^m))/x, 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(ln(f*x**m)*(a+b*ln(c*(e*x+d)**n))**2/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 ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} \log \left (f x^{m}\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(f*x^m)*(a+b*log(c*(e*x+d)^n))^2/x,x, algorithm="giac")

[Out]

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