\(\int \frac {\log (f x^m) (a+b \log (c (d+e x)^n))^2}{x^3} \, dx\) [372]

Optimal result

Integrand size = 26, antiderivative size = 939 \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3} \, dx=\frac {b^2 e^2 m n^2 \log (x)}{d^2}-\frac {b^2 e^2 m n^2 \log ^2(x)}{2 d^2}+\frac {b^2 e^2 m n^2 \log \left (-\frac {e x}{d}\right )}{2 d^2}+\frac {b^2 e^2 n^2 \log (x) \log \left (f x^m\right )}{d^2}-\frac {3 b^2 e^2 m n^2 \log (d+e x)}{2 d^2}-\frac {3 b^2 e m n^2 \log (d+e x)}{2 d x}+\frac {b^2 e^2 m n^2 \log (x) \log (d+e x)}{d^2}+\frac {b^2 e^2 m n^2 \log ^2(x) \log (d+e x)}{2 d^2}-\frac {b^2 e^2 m n^2 \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{2 d^2}-\frac {b^2 e^2 n^2 \log \left (f x^m\right ) \log (d+e x)}{d^2}-\frac {b^2 e n^2 \log \left (f x^m\right ) \log (d+e x)}{d x}-\frac {b^2 e^2 n^2 \log (x) \log \left (f x^m\right ) \log (d+e x)}{d^2}+\frac {b^2 e^2 m n^2 \log ^2(d+e x)}{4 d^2}-\frac {b^2 m n^2 \log ^2(d+e x)}{4 x^2}-\frac {b^2 e^2 m n^2 \log \left (-\frac {e x}{d}\right ) \log ^2(d+e x)}{2 d^2}+\frac {b^2 e^2 n^2 \log \left (f x^m\right ) \log ^2(d+e x)}{2 d^2}-\frac {b^2 n^2 \log \left (f x^m\right ) \log ^2(d+e x)}{2 x^2}+\frac {b n \left (m \log (x)-\log \left (f x^m\right )\right ) \left (e^2 x^2 \log \left (-\frac {e x}{d}\right )+(d+e x) (e x+(d-e x) \log (d+e x))\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )}{d^2 x^2}-\frac {m \log (x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2}{2 x^2}-\frac {\left (m-2 m \log (x)+2 \log \left (f x^m\right )\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2}{4 x^2}-\frac {b^2 e^2 m n^2 \log (x) \log \left (1+\frac {e x}{d}\right )}{d^2}-\frac {b^2 e^2 m n^2 \log ^2(x) \log \left (1+\frac {e x}{d}\right )}{2 d^2}+\frac {b^2 e^2 n^2 \log (x) \log \left (f x^m\right ) \log \left (1+\frac {e x}{d}\right )}{d^2}-\frac {b^2 e^2 n^2 \left (m-\log \left (f x^m\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^2}-\frac {b m n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (e x (d+e x)+e^2 x^2 \log \left (-\frac {e x}{d}\right )+\left (d^2-e^2 x^2\right ) \log (d+e x)+2 d^2 \log (x) \log (d+e x)+e x \left (e x \log ^2(x)+2 d (1+\log (x))-2 e x \left (\log (x) \log \left (1+\frac {e x}{d}\right )+\operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )\right )\right )\right )}{2 d^2 x^2}-\frac {b^2 e^2 m n^2 (1+2 \log (d+e x)) \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{2 d^2}-\frac {b^2 e^2 m n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{d^2}+\frac {b^2 e^2 m n^2 \operatorname {PolyLog}\left (3,1+\frac {e x}{d}\right )}{d^2} \]

[Out]

-3/2*b^2*e*m*n^2*ln(e*x+d)/d/x+1/2*b^2*e^2*m*n^2*ln(x)^2*ln(e*x+d)/d^2-1/2*b^2*e^2*m*n^2*ln(-e*x/d)*ln(e*x+d)/
d^2-1/2*b^2*e^2*m*n^2*ln(-e*x/d)*ln(e*x+d)^2/d^2-1/2*b^2*e^2*m*n^2*ln(x)^2*ln(1+e*x/d)/d^2-1/2*b*m*n*(a-b*n*ln
(e*x+d)+b*ln(c*(e*x+d)^n))*(e*x*(e*x+d)+e^2*x^2*ln(-e*x/d)+(-e^2*x^2+d^2)*ln(e*x+d)+2*d^2*ln(x)*ln(e*x+d)+e*x*
(e*x*ln(x)^2+2*d*(1+ln(x))-2*e*x*(ln(x)*ln(1+e*x/d)+polylog(2,-e*x/d))))/d^2/x^2-1/2*b^2*e^2*m*n^2*(1+2*ln(e*x
+d))*polylog(2,1+e*x/d)/d^2-1/4*b^2*m*n^2*ln(e*x+d)^2/x^2-1/2*b^2*n^2*ln(f*x^m)*ln(e*x+d)^2/x^2-1/2*b^2*e^2*m*
n^2*ln(x)^2/d^2+1/2*b^2*e^2*m*n^2*ln(-e*x/d)/d^2-3/2*b^2*e^2*m*n^2*ln(e*x+d)/d^2+1/4*b^2*e^2*m*n^2*ln(e*x+d)^2
/d^2+1/2*b^2*e^2*n^2*ln(f*x^m)*ln(e*x+d)^2/d^2+b^2*e^2*m*n^2*ln(x)*ln(e*x+d)/d^2-b^2*e^2*n^2*ln(x)*ln(f*x^m)*l
n(e*x+d)/d^2-b^2*e^2*m*n^2*ln(x)*ln(1+e*x/d)/d^2+b^2*e^2*n^2*ln(x)*ln(f*x^m)*ln(1+e*x/d)/d^2+b^2*e^2*m*n^2*pol
ylog(3,1+e*x/d)/d^2-b^2*e^2*n^2*ln(f*x^m)*ln(e*x+d)/d^2+b^2*e^2*m*n^2*ln(x)/d^2+b^2*e^2*n^2*ln(x)*ln(f*x^m)/d^
2-b^2*e^2*n^2*(m-ln(f*x^m))*polylog(2,-e*x/d)/d^2-b^2*e*n^2*ln(f*x^m)*ln(e*x+d)/d/x+b*n*(m*ln(x)-ln(f*x^m))*(e
^2*x^2*ln(-e*x/d)+(e*x+d)*(e*x+(-e*x+d)*ln(e*x+d)))*(a-b*n*ln(e*x+d)+b*ln(c*(e*x+d)^n))/d^2/x^2-1/2*m*ln(x)*(a
-b*n*ln(e*x+d)+b*ln(c*(e*x+d)^n))^2/x^2-1/4*(m-2*m*ln(x)+2*ln(f*x^m))*(a-b*n*ln(e*x+d)+b*ln(c*(e*x+d)^n))^2/x^
2-b^2*e^2*m*n^2*polylog(3,-e*x/d)/d^2

Rubi [F]

\[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3} \, dx=\int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3} \, dx \]

[In]

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

[Out]

Defer[Int][(Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])^2)/x^3, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.62 (sec) , antiderivative size = 781, normalized size of antiderivative = 0.83 \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3} \, dx=\frac {4 b n \left (m \log (x)-\log \left (f x^m\right )\right ) \left (e^2 x^2 \log \left (-\frac {e x}{d}\right )+(d+e x) (e x+(d-e x) \log (d+e x))\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )-2 d^2 m \log (x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+d^2 \left (-m+2 m \log (x)-2 \log \left (f x^m\right )\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2-2 b m n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (e x (d+e x)+e^2 x^2 \log \left (-\frac {e x}{d}\right )+\left (d^2-e^2 x^2\right ) \log (d+e x)+2 d^2 \log (x) \log (d+e x)+e x \left (e x \log ^2(x)+2 d (1+\log (x))-2 e x \left (\log (x) \log \left (1+\frac {e x}{d}\right )+\operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )\right )\right )\right )+b^2 n^2 \left (4 e^2 m x^2 \log (x)-2 e^2 m x^2 \log ^2(x)+2 e^2 m x^2 \log \left (-\frac {e x}{d}\right )+4 e^2 x^2 \log (x) \log \left (f x^m\right )-6 d e m x \log (d+e x)-6 e^2 m x^2 \log (d+e x)+4 e^2 m x^2 \log (x) \log (d+e x)-2 e^2 m x^2 \log ^2(x) \log (d+e x)-2 e^2 m x^2 \log \left (-\frac {e x}{d}\right ) \log (d+e x)+4 e^2 m x^2 \log (x) \log \left (-\frac {e x}{d}\right ) \log (d+e x)-4 d e x \log \left (f x^m\right ) \log (d+e x)-4 e^2 x^2 \log \left (f x^m\right ) \log (d+e x)-4 e^2 x^2 \log \left (-\frac {e x}{d}\right ) \log \left (f x^m\right ) \log (d+e x)-d^2 m \log ^2(d+e x)+e^2 m x^2 \log ^2(d+e x)-2 e^2 m x^2 \log \left (-\frac {e x}{d}\right ) \log ^2(d+e x)-2 d^2 \log \left (f x^m\right ) \log ^2(d+e x)+2 e^2 x^2 \log \left (f x^m\right ) \log ^2(d+e x)-4 e^2 m x^2 \log (x) \log \left (1+\frac {e x}{d}\right )+2 e^2 m x^2 \log ^2(x) \log \left (1+\frac {e x}{d}\right )+4 e^2 m x^2 (-1+\log (x)) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-2 e^2 x^2 \left (m-2 m \log (x)+2 \log \left (f x^m\right )+2 m \log (d+e x)\right ) \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )-4 e^2 m x^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )+4 e^2 m x^2 \operatorname {PolyLog}\left (3,1+\frac {e x}{d}\right )\right )}{4 d^2 x^2} \]

[In]

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

[Out]

(4*b*n*(m*Log[x] - Log[f*x^m])*(e^2*x^2*Log[-((e*x)/d)] + (d + e*x)*(e*x + (d - e*x)*Log[d + e*x]))*(a - b*n*L
og[d + e*x] + b*Log[c*(d + e*x)^n]) - 2*d^2*m*Log[x]*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2 + d^2*(-m
+ 2*m*Log[x] - 2*Log[f*x^m])*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2 - 2*b*m*n*(a - b*n*Log[d + e*x]
+ b*Log[c*(d + e*x)^n])*(e*x*(d + e*x) + e^2*x^2*Log[-((e*x)/d)] + (d^2 - e^2*x^2)*Log[d + e*x] + 2*d^2*Log[x]
*Log[d + e*x] + e*x*(e*x*Log[x]^2 + 2*d*(1 + Log[x]) - 2*e*x*(Log[x]*Log[1 + (e*x)/d] + PolyLog[2, -((e*x)/d)]
))) + b^2*n^2*(4*e^2*m*x^2*Log[x] - 2*e^2*m*x^2*Log[x]^2 + 2*e^2*m*x^2*Log[-((e*x)/d)] + 4*e^2*x^2*Log[x]*Log[
f*x^m] - 6*d*e*m*x*Log[d + e*x] - 6*e^2*m*x^2*Log[d + e*x] + 4*e^2*m*x^2*Log[x]*Log[d + e*x] - 2*e^2*m*x^2*Log
[x]^2*Log[d + e*x] - 2*e^2*m*x^2*Log[-((e*x)/d)]*Log[d + e*x] + 4*e^2*m*x^2*Log[x]*Log[-((e*x)/d)]*Log[d + e*x
] - 4*d*e*x*Log[f*x^m]*Log[d + e*x] - 4*e^2*x^2*Log[f*x^m]*Log[d + e*x] - 4*e^2*x^2*Log[-((e*x)/d)]*Log[f*x^m]
*Log[d + e*x] - d^2*m*Log[d + e*x]^2 + e^2*m*x^2*Log[d + e*x]^2 - 2*e^2*m*x^2*Log[-((e*x)/d)]*Log[d + e*x]^2 -
2*d^2*Log[f*x^m]*Log[d + e*x]^2 + 2*e^2*x^2*Log[f*x^m]*Log[d + e*x]^2 - 4*e^2*m*x^2*Log[x]*Log[1 + (e*x)/d] +
2*e^2*m*x^2*Log[x]^2*Log[1 + (e*x)/d] + 4*e^2*m*x^2*(-1 + Log[x])*PolyLog[2, -((e*x)/d)] - 2*e^2*x^2*(m - 2*m
*Log[x] + 2*Log[f*x^m] + 2*m*Log[d + e*x])*PolyLog[2, 1 + (e*x)/d] - 4*e^2*m*x^2*PolyLog[3, -((e*x)/d)] + 4*e^
2*m*x^2*PolyLog[3, 1 + (e*x)/d]))/(4*d^2*x^2)

Maple [F]

\[\int \frac {\ln \left (f \,x^{m}\right ) {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{2}}{x^{3}}d x\]

[In]

int(ln(f*x^m)*(a+b*ln(c*(e*x+d)^n))^2/x^3,x)

[Out]

int(ln(f*x^m)*(a+b*ln(c*(e*x+d)^n))^2/x^3,x)

Fricas [F]

\[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} \log \left (f x^{m}\right )}{x^{3}} \,d x } \]

[In]

integrate(log(f*x^m)*(a+b*log(c*(e*x+d)^n))^2/x^3,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^3, x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3} \, dx=\text {Timed out} \]

[In]

integrate(ln(f*x**m)*(a+b*ln(c*(e*x+d)**n))**2/x**3,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} \log \left (f x^{m}\right )}{x^{3}} \,d x } \]

[In]

integrate(log(f*x^m)*(a+b*log(c*(e*x+d)^n))^2/x^3,x, algorithm="maxima")

[Out]

-1/4*(b^2*(m + 2*log(f)) + 2*b^2*log(x^m))*log((e*x + d)^n)^2/x^2 + integrate(1/2*(2*b^2*d*log(c)^2*log(f) + 4
*a*b*d*log(c)*log(f) + 2*a^2*d*log(f) + 2*(b^2*e*log(c)^2*log(f) + 2*a*b*e*log(c)*log(f) + a^2*e*log(f))*x + (
4*b^2*d*log(c)*log(f) + 4*a*b*d*log(f) + (4*a*b*e*log(f) + (4*e*log(c)*log(f) + (m*n + 2*n*log(f))*e)*b^2)*x +
2*(2*b^2*d*log(c) + 2*a*b*d + ((e*n + 2*e*log(c))*b^2 + 2*a*b*e)*x)*log(x^m))*log((e*x + d)^n) + 2*(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^4 + d*x^3), x)

Giac [F]

\[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} \log \left (f x^{m}\right )}{x^{3}} \,d x } \]

[In]

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

[Out]

integrate((b*log((e*x + d)^n*c) + a)^2*log(f*x^m)/x^3, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3} \, dx=\int \frac {\ln \left (f\,x^m\right )\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}{x^3} \,d x \]

[In]

int((log(f*x^m)*(a + b*log(c*(d + e*x)^n))^2)/x^3,x)

[Out]

int((log(f*x^m)*(a + b*log(c*(d + e*x)^n))^2)/x^3, x)