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

Optimal. Leaf size=607 \[ \frac{b m n \left (e x \left (\log ^2(x)-2 \left (\text{PolyLog}\left (2,-\frac{e x}{d}\right )+\log (x) \log \left (\frac{e x}{d}+1\right )\right )\right )+2 e x \log \left (-\frac{e x}{d}\right )-2 (d+e x) \log (d+e x)-2 d \log (x) \log (d+e x)\right ) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )}{d x}-\frac{2 b^2 e n^2 \log \left (f x^m\right ) \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{d}+\frac{2 b^2 e m n^2 \text{PolyLog}\left (3,-\frac{e x}{d}\right )}{d}-\frac{2 b^2 e m n^2 \text{PolyLog}\left (3,\frac{e x}{d}+1\right )}{d}+\frac{2 b^2 e m n^2 (\log (d+e x)+1) \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{d}-\frac{2 b n \left (e x \log \left (-\frac{e x}{d}\right )-(d+e x) \log (d+e x)\right ) \left (m \log (x)-\log \left (f x^m\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )}{d x}-\frac{\left (\log \left (f x^m\right )+m (-\log (x))+m\right ) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )^2}{x}-\frac{m \log (x) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )^2}{x}-\frac{b^2 e n^2 \log ^2(d+e x) \log \left (f x^m\right )}{d}-\frac{b^2 n^2 \log ^2(d+e x) \log \left (f x^m\right )}{x}+\frac{2 b^2 e n^2 \log (x) \log (d+e x) \log \left (f x^m\right )}{d}-\frac{2 b^2 e n^2 \log (x) \log \left (\frac{e x}{d}+1\right ) \log \left (f x^m\right )}{d}-\frac{b^2 e m n^2 \log ^2(d+e x)}{d}+\frac{b^2 e m n^2 \log \left (-\frac{e x}{d}\right ) \log ^2(d+e x)}{d}-\frac{b^2 m n^2 \log ^2(d+e x)}{x}-\frac{b^2 e m n^2 \log ^2(x) \log (d+e x)}{d}+\frac{b^2 e m n^2 \log ^2(x) \log \left (\frac{e x}{d}+1\right )}{d}+\frac{2 b^2 e m n^2 \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{d} \]

[Out]

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

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Rubi [F]  time = 0.0289378, 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 = {} \[ \int \frac{\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.754481, size = 513, normalized size = 0.85 \[ \frac{-b m n \left (-e x \left (\log ^2(x)-2 \left (\text{PolyLog}\left (2,-\frac{e x}{d}\right )+\log (x) \log \left (\frac{e x}{d}+1\right )\right )\right )-2 e x \log \left (-\frac{e x}{d}\right )+2 (d+e x) \log (d+e x)+2 d \log (x) \log (d+e x)\right ) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )+b^2 n^2 \left (2 e x \text{PolyLog}\left (2,\frac{e x}{d}+1\right ) \left (m \log (d+e x)+\log \left (f x^m\right )+m (-\log (x))+m\right )+2 e m x \text{PolyLog}\left (3,-\frac{e x}{d}\right )-2 e m x \text{PolyLog}\left (3,\frac{e x}{d}+1\right )-2 e m x \log (x) \text{PolyLog}\left (2,-\frac{e x}{d}\right )-d \log ^2(d+e x) \log \left (f x^m\right )-e x \log ^2(d+e x) \log \left (f x^m\right )+2 e x \log \left (-\frac{e x}{d}\right ) \log (d+e x) \log \left (f x^m\right )+e m x \log ^2(x) \log (d+e x)-e m x \log ^2(x) \log \left (\frac{e x}{d}+1\right )-d m \log ^2(d+e x)-e m x \log ^2(d+e x)+e m x \log \left (-\frac{e x}{d}\right ) \log ^2(d+e x)-2 e m x \log (x) \log \left (-\frac{e x}{d}\right ) \log (d+e x)+2 e m x \log \left (-\frac{e x}{d}\right ) \log (d+e x)\right )+2 b n \left ((d+e x) \log (d+e x)-e x \log \left (-\frac{e x}{d}\right )\right ) \left (m \log (x)-\log \left (f x^m\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )+d \left (-\log \left (f x^m\right )+m \log (x)-m\right ) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )^2-d m \log (x) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )^2}{d x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 1.373, 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}^{2}}}\, 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^2,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (b^{2}{\left (m + \log \left (f\right )\right )} + b^{2} \log \left (x^{m}\right )\right )} \log \left ({\left (e x + d\right )}^{n}\right )^{2}}{x} + \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 + 2 \,{\left (b^{2} d \log \left (c\right ) \log \left (f\right ) + a b d \log \left (f\right ) +{\left (a b e \log \left (f\right ) +{\left (e \log \left (c\right ) \log \left (f\right ) +{\left (m n + n \log \left (f\right )\right )} e\right )} b^{2}\right )} x +{\left (b^{2} d \log \left (c\right ) + a b d +{\left ({\left (e n + e \log \left (c\right )\right )} b^{2} + 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^{3} + d x^{2}}\,{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^2,x, algorithm="maxima")

[Out]

-(b^2*(m + log(f)) + b^2*log(x^m))*log((e*x + d)^n)^2/x + integrate((b^2*d*log(c)^2*log(f) + 2*a*b*d*log(c)*lo
g(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*log(f))*x + 2*(b^2*d*log(c)*log(f
) + a*b*d*log(f) + (a*b*e*log(f) + (e*log(c)*log(f) + (m*n + n*log(f))*e)*b^2)*x + (b^2*d*log(c) + a*b*d + ((e
*n + e*log(c))*b^2 + a*b*e)*x)*log(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^3 + d*x^2), 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^{2}}, 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^2,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^2, 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**2,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^{2}}\,{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^2,x, algorithm="giac")

[Out]

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