3.1.99 \(\int \frac {(a+b \log (c x^n))^2}{x^4 (d+e x)} \, dx\) [99]

3.1.99.1 Optimal result

Integrand size = 23, antiderivative size = 273 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^4 (d+e x)} \, dx=-\frac {2 b^2 n^2}{27 d x^3}+\frac {b^2 e n^2}{4 d^2 x^2}-\frac {2 b^2 e^2 n^2}{d^3 x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right )}{9 d x^3}+\frac {b e n \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2}-\frac {2 b e^2 n \left (a+b \log \left (c x^n\right )\right )}{d^3 x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 d x^3}+\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 x^2}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^2}{d^3 x}+\frac {e^3 \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^4}-\frac {2 b e^3 n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d^4}-\frac {2 b^2 e^3 n^2 \operatorname {PolyLog}\left (3,-\frac {d}{e x}\right )}{d^4} \]

output

-2/27*b^2*n^2/d/x^3+1/4*b^2*e*n^2/d^2/x^2-2*b^2*e^2*n^2/d^3/x-2/9*b*n*(a+b 
*ln(c*x^n))/d/x^3+1/2*b*e*n*(a+b*ln(c*x^n))/d^2/x^2-2*b*e^2*n*(a+b*ln(c*x^ 
n))/d^3/x-1/3*(a+b*ln(c*x^n))^2/d/x^3+1/2*e*(a+b*ln(c*x^n))^2/d^2/x^2-e^2* 
(a+b*ln(c*x^n))^2/d^3/x+e^3*ln(1+d/e/x)*(a+b*ln(c*x^n))^2/d^4-2*b*e^3*n*(a 
+b*ln(c*x^n))*polylog(2,-d/e/x)/d^4-2*b^2*e^3*n^2*polylog(3,-d/e/x)/d^4
 

3.1.99.2 Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^4 (d+e x)} \, dx=\frac {-\frac {36 d^3 \left (a+b \log \left (c x^n\right )\right )^2}{x^3}+\frac {54 d^2 e \left (a+b \log \left (c x^n\right )\right )^2}{x^2}-\frac {108 d e^2 \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {36 e^3 \left (a+b \log \left (c x^n\right )\right )^3}{b n}-\frac {216 b d e^2 n \left (a+b n+b \log \left (c x^n\right )\right )}{x}+\frac {27 b d^2 e n \left (2 a+b n+2 b \log \left (c x^n\right )\right )}{x^2}-\frac {8 b d^3 n \left (3 a+b n+3 b \log \left (c x^n\right )\right )}{x^3}+108 e^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )+216 b e^3 n \left (\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-b n \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )\right )}{108 d^4} \]

input

Integrate[(a + b*Log[c*x^n])^2/(x^4*(d + e*x)),x]
 

output

((-36*d^3*(a + b*Log[c*x^n])^2)/x^3 + (54*d^2*e*(a + b*Log[c*x^n])^2)/x^2 
- (108*d*e^2*(a + b*Log[c*x^n])^2)/x - (36*e^3*(a + b*Log[c*x^n])^3)/(b*n) 
 - (216*b*d*e^2*n*(a + b*n + b*Log[c*x^n]))/x + (27*b*d^2*e*n*(2*a + b*n + 
 2*b*Log[c*x^n]))/x^2 - (8*b*d^3*n*(3*a + b*n + 3*b*Log[c*x^n]))/x^3 + 108 
*e^3*(a + b*Log[c*x^n])^2*Log[1 + (e*x)/d] + 216*b*e^3*n*((a + b*Log[c*x^n 
])*PolyLog[2, -((e*x)/d)] - b*n*PolyLog[3, -((e*x)/d)]))/(108*d^4)
 

3.1.99.3 Rubi [A] (verified)

Time = 1.51 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.93, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {2780, 2742, 2741, 2780, 2742, 2741, 2780, 2742, 2741, 2779, 2821, 7143}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^4 (d+e x)} \, dx\)

\(\Big \downarrow \) 2780

\(\displaystyle \frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^4}dx}{d}-\frac {e \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3 (d+e x)}dx}{d}\)

\(\Big \downarrow \) 2742

\(\displaystyle \frac {\frac {2}{3} b n \int \frac {a+b \log \left (c x^n\right )}{x^4}dx-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}}{d}-\frac {e \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3 (d+e x)}dx}{d}\)

\(\Big \downarrow \) 2741

\(\displaystyle \frac {\frac {2}{3} b n \left (-\frac {a+b \log \left (c x^n\right )}{3 x^3}-\frac {b n}{9 x^3}\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}}{d}-\frac {e \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3 (d+e x)}dx}{d}\)

\(\Big \downarrow \) 2780

\(\displaystyle \frac {\frac {2}{3} b n \left (-\frac {a+b \log \left (c x^n\right )}{3 x^3}-\frac {b n}{9 x^3}\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}}{d}-\frac {e \left (\frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3}dx}{d}-\frac {e \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)}dx}{d}\right )}{d}\)

\(\Big \downarrow \) 2742

\(\displaystyle \frac {\frac {2}{3} b n \left (-\frac {a+b \log \left (c x^n\right )}{3 x^3}-\frac {b n}{9 x^3}\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}}{d}-\frac {e \left (\frac {b n \int \frac {a+b \log \left (c x^n\right )}{x^3}dx-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}}{d}-\frac {e \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)}dx}{d}\right )}{d}\)

\(\Big \downarrow \) 2741

\(\displaystyle \frac {\frac {2}{3} b n \left (-\frac {a+b \log \left (c x^n\right )}{3 x^3}-\frac {b n}{9 x^3}\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}}{d}-\frac {e \left (\frac {b n \left (-\frac {a+b \log \left (c x^n\right )}{2 x^2}-\frac {b n}{4 x^2}\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}}{d}-\frac {e \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)}dx}{d}\right )}{d}\)

\(\Big \downarrow \) 2780

\(\displaystyle \frac {\frac {2}{3} b n \left (-\frac {a+b \log \left (c x^n\right )}{3 x^3}-\frac {b n}{9 x^3}\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}}{d}-\frac {e \left (\frac {b n \left (-\frac {a+b \log \left (c x^n\right )}{2 x^2}-\frac {b n}{4 x^2}\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}}{d}-\frac {e \left (\frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2}dx}{d}-\frac {e \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)}dx}{d}\right )}{d}\right )}{d}\)

\(\Big \downarrow \) 2742

\(\displaystyle \frac {\frac {2}{3} b n \left (-\frac {a+b \log \left (c x^n\right )}{3 x^3}-\frac {b n}{9 x^3}\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}}{d}-\frac {e \left (\frac {b n \left (-\frac {a+b \log \left (c x^n\right )}{2 x^2}-\frac {b n}{4 x^2}\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}}{d}-\frac {e \left (\frac {2 b n \int \frac {a+b \log \left (c x^n\right )}{x^2}dx-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{x}}{d}-\frac {e \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)}dx}{d}\right )}{d}\right )}{d}\)

\(\Big \downarrow \) 2741

\(\displaystyle \frac {\frac {2}{3} b n \left (-\frac {a+b \log \left (c x^n\right )}{3 x^3}-\frac {b n}{9 x^3}\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}}{d}-\frac {e \left (\frac {b n \left (-\frac {a+b \log \left (c x^n\right )}{2 x^2}-\frac {b n}{4 x^2}\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}}{d}-\frac {e \left (\frac {2 b n \left (-\frac {a+b \log \left (c x^n\right )}{x}-\frac {b n}{x}\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{x}}{d}-\frac {e \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)}dx}{d}\right )}{d}\right )}{d}\)

\(\Big \downarrow \) 2779

\(\displaystyle \frac {\frac {2}{3} b n \left (-\frac {a+b \log \left (c x^n\right )}{3 x^3}-\frac {b n}{9 x^3}\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}}{d}-\frac {e \left (\frac {b n \left (-\frac {a+b \log \left (c x^n\right )}{2 x^2}-\frac {b n}{4 x^2}\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}}{d}-\frac {e \left (\frac {2 b n \left (-\frac {a+b \log \left (c x^n\right )}{x}-\frac {b n}{x}\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{x}}{d}-\frac {e \left (\frac {2 b n \int \frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{x}dx}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d}\right )}{d}\right )}{d}\right )}{d}\)

\(\Big \downarrow \) 2821

\(\displaystyle \frac {\frac {2}{3} b n \left (-\frac {a+b \log \left (c x^n\right )}{3 x^3}-\frac {b n}{9 x^3}\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}}{d}-\frac {e \left (\frac {b n \left (-\frac {a+b \log \left (c x^n\right )}{2 x^2}-\frac {b n}{4 x^2}\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}}{d}-\frac {e \left (\frac {2 b n \left (-\frac {a+b \log \left (c x^n\right )}{x}-\frac {b n}{x}\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{x}}{d}-\frac {e \left (\frac {2 b n \left (\operatorname {PolyLog}\left (2,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \int \frac {\operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{x}dx\right )}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d}\right )}{d}\right )}{d}\right )}{d}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {\frac {2}{3} b n \left (-\frac {a+b \log \left (c x^n\right )}{3 x^3}-\frac {b n}{9 x^3}\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}}{d}-\frac {e \left (\frac {b n \left (-\frac {a+b \log \left (c x^n\right )}{2 x^2}-\frac {b n}{4 x^2}\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}}{d}-\frac {e \left (\frac {2 b n \left (-\frac {a+b \log \left (c x^n\right )}{x}-\frac {b n}{x}\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{x}}{d}-\frac {e \left (\frac {2 b n \left (\operatorname {PolyLog}\left (2,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )+b n \operatorname {PolyLog}\left (3,-\frac {d}{e x}\right )\right )}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d}\right )}{d}\right )}{d}\right )}{d}\)

input

Int[(a + b*Log[c*x^n])^2/(x^4*(d + e*x)),x]
 

output

(-1/3*(a + b*Log[c*x^n])^2/x^3 + (2*b*n*(-1/9*(b*n)/x^3 - (a + b*Log[c*x^n 
])/(3*x^3)))/3)/d - (e*((-1/2*(a + b*Log[c*x^n])^2/x^2 + b*n*(-1/4*(b*n)/x 
^2 - (a + b*Log[c*x^n])/(2*x^2)))/d - (e*((-((a + b*Log[c*x^n])^2/x) + 2*b 
*n*(-((b*n)/x) - (a + b*Log[c*x^n])/x))/d - (e*(-((Log[1 + d/(e*x)]*(a + b 
*Log[c*x^n])^2)/d) + (2*b*n*((a + b*Log[c*x^n])*PolyLog[2, -(d/(e*x))] + b 
*n*PolyLog[3, -(d/(e*x))]))/d))/d))/d))/d
 

3.1.99.3.1 Defintions of rubi rules used

rule 2741

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> 
Simp[(d*x)^(m + 1)*((a + b*Log[c*x^n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^( 
m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1]
 

rule 2742

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbo 
l] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^n])^p/(d*(m + 1))), x] - Simp[b*n* 
(p/(m + 1))   Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b 
, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]
 

rule 2779

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r 
_.))), x_Symbol] :> Simp[(-Log[1 + d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)) 
, x] + Simp[b*n*(p/(d*r))   Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^(p - 
 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]
 

rule 2780

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.))/((d_) + (e_.)* 
(x_)^(r_.)), x_Symbol] :> Simp[1/d   Int[x^m*(a + b*Log[c*x^n])^p, x], x] - 
 Simp[e/d   Int[(x^(m + r)*(a + b*Log[c*x^n])^p)/(d + e*x^r), x], x] /; Fre 
eQ[{a, b, c, d, e, m, n, r}, x] && IGtQ[p, 0] && IGtQ[r, 0] && ILtQ[m, -1]
 

rule 2821

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] + Simp[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 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> 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]
 

3.1.99.4 Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.53 (sec) , antiderivative size = 828, normalized size of antiderivative = 3.03

method	result	size
risch	\(\frac {b^{2} \ln \left (x^{n}\right )^{2} e^{3} \ln \left (e x +d \right )}{d^{4}}-\frac {b^{2} \ln \left (x^{n}\right )^{2}}{3 d \,x^{3}}-\frac {b^{2} \ln \left (x^{n}\right )^{2} e^{2}}{d^{3} x}+\frac {b^{2} \ln \left (x^{n}\right )^{2} e}{2 d^{2} x^{2}}-\frac {b^{2} \ln \left (x^{n}\right )^{2} e^{3} \ln \left (x \right )}{d^{4}}-\frac {2 b^{2} n \ln \left (x^{n}\right ) e^{2}}{d^{3} x}+\frac {b^{2} n \ln \left (x^{n}\right ) e}{2 d^{2} x^{2}}-\frac {2 b^{2} n \ln \left (x^{n}\right )}{9 d \,x^{3}}-\frac {2 b^{2} e^{2} n^{2}}{d^{3} x}+\frac {b^{2} e \,n^{2}}{4 d^{2} x^{2}}-\frac {2 b^{2} n^{2}}{27 d \,x^{3}}+\frac {b^{2} n \,e^{3} \ln \left (x^{n}\right ) \ln \left (x \right )^{2}}{d^{4}}-\frac {b^{2} e^{3} \ln \left (x \right )^{3} n^{2}}{3 d^{4}}+\frac {2 b^{2} e^{3} \ln \left (x \right ) \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right ) n^{2}}{d^{4}}+\frac {2 b^{2} e^{3} \ln \left (x \right ) \operatorname {dilog}\left (-\frac {e x}{d}\right ) n^{2}}{d^{4}}-\frac {2 b^{2} n \,e^{3} \ln \left (x^{n}\right ) \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{d^{4}}-\frac {2 b^{2} n \,e^{3} \ln \left (x^{n}\right ) \operatorname {dilog}\left (-\frac {e x}{d}\right )}{d^{4}}-\frac {b^{2} e^{3} n^{2} \ln \left (e x +d \right ) \ln \left (x \right )^{2}}{d^{4}}+\frac {b^{2} e^{3} n^{2} \ln \left (x \right )^{2} \ln \left (1+\frac {e x}{d}\right )}{d^{4}}+\frac {2 b^{2} e^{3} n^{2} \ln \left (x \right ) \operatorname {Li}_{2}\left (-\frac {e x}{d}\right )}{d^{4}}-\frac {2 b^{2} e^{3} n^{2} \operatorname {Li}_{3}\left (-\frac {e x}{d}\right )}{d^{4}}+\left (-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 a \right ) b \left (\frac {\ln \left (x^{n}\right ) e^{3} \ln \left (e x +d \right )}{d^{4}}-\frac {\ln \left (x^{n}\right )}{3 d \,x^{3}}-\frac {\ln \left (x^{n}\right ) e^{2}}{d^{3} x}+\frac {\ln \left (x^{n}\right ) e}{2 d^{2} x^{2}}-\frac {\ln \left (x^{n}\right ) e^{3} \ln \left (x \right )}{d^{4}}-\frac {n \left (-\frac {-\frac {6 e^{2}}{x}+\frac {3 d e}{2 x^{2}}-\frac {2 d^{2}}{3 x^{3}}}{d^{3}}-\frac {3 e^{3} \ln \left (x \right )^{2}}{d^{4}}+\frac {6 e^{3} \left (\operatorname {dilog}\left (-\frac {e x}{d}\right )+\ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )\right )}{d^{4}}\right )}{6}\right )+\frac {{\left (-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 a \right )}^{2} \left (\frac {e^{3} \ln \left (e x +d \right )}{d^{4}}-\frac {1}{3 d \,x^{3}}-\frac {e^{2}}{d^{3} x}+\frac {e}{2 d^{2} x^{2}}-\frac {e^{3} \ln \left (x \right )}{d^{4}}\right )}{4}\)	\(828\)

input

int((a+b*ln(c*x^n))^2/x^4/(e*x+d),x,method=_RETURNVERBOSE)
 

output

b^2*ln(x^n)^2*e^3/d^4*ln(e*x+d)-1/3*b^2*ln(x^n)^2/d/x^3-b^2*ln(x^n)^2*e^2/ 
d^3/x+1/2*b^2*ln(x^n)^2*e/d^2/x^2-b^2*ln(x^n)^2*e^3/d^4*ln(x)-2*b^2*n*ln(x 
^n)*e^2/d^3/x+1/2*b^2*n*ln(x^n)*e/d^2/x^2-2/9*b^2*n*ln(x^n)/d/x^3-2*b^2*e^ 
2*n^2/d^3/x+1/4*b^2*e*n^2/d^2/x^2-2/27*b^2*n^2/d/x^3+b^2*n*e^3/d^4*ln(x^n) 
*ln(x)^2-1/3*b^2*e^3/d^4*ln(x)^3*n^2+2*b^2*e^3/d^4*ln(x)*ln(e*x+d)*ln(-e*x 
/d)*n^2+2*b^2*e^3/d^4*ln(x)*dilog(-e*x/d)*n^2-2*b^2*n*e^3/d^4*ln(x^n)*ln(e 
*x+d)*ln(-e*x/d)-2*b^2*n*e^3/d^4*ln(x^n)*dilog(-e*x/d)-b^2*e^3/d^4*n^2*ln( 
e*x+d)*ln(x)^2+b^2*e^3/d^4*n^2*ln(x)^2*ln(1+e*x/d)+2*b^2*e^3/d^4*n^2*ln(x) 
*polylog(2,-e*x/d)-2*b^2*e^3/d^4*n^2*polylog(3,-e*x/d)+(-I*b*Pi*csgn(I*c)* 
csgn(I*x^n)*csgn(I*c*x^n)+I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+I*b*Pi*csgn(I*x 
^n)*csgn(I*c*x^n)^2-I*b*Pi*csgn(I*c*x^n)^3+2*b*ln(c)+2*a)*b*(ln(x^n)*e^3/d 
^4*ln(e*x+d)-1/3*ln(x^n)/d/x^3-ln(x^n)*e^2/d^3/x+1/2*ln(x^n)*e/d^2/x^2-ln( 
x^n)*e^3/d^4*ln(x)-1/6*n*(-1/d^3*(-6*e^2/x+3/2*d*e/x^2-2/3*d^2/x^3)-3*e^3/ 
d^4*ln(x)^2+6*e^3/d^4*(dilog(-e*x/d)+ln(e*x+d)*ln(-e*x/d))))+1/4*(-I*b*Pi* 
csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+I*b*P 
i*csgn(I*x^n)*csgn(I*c*x^n)^2-I*b*Pi*csgn(I*c*x^n)^3+2*b*ln(c)+2*a)^2*(e^3 
/d^4*ln(e*x+d)-1/3/d/x^3-e^2/d^3/x+1/2*e/d^2/x^2-e^3/d^4*ln(x))
 

3.1.99.5 Fricas [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^4 (d+e x)} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )} x^{4}} \,d x } \]

input

integrate((a+b*log(c*x^n))^2/x^4/(e*x+d),x, algorithm="fricas")
 

output

integral((b^2*log(c*x^n)^2 + 2*a*b*log(c*x^n) + a^2)/(e*x^5 + d*x^4), x)
 

3.1.99.6 Sympy [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^4 (d+e x)} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{2}}{x^{4} \left (d + e x\right )}\, dx \]

input

integrate((a+b*ln(c*x**n))**2/x**4/(e*x+d),x)
 

output

Integral((a + b*log(c*x**n))**2/(x**4*(d + e*x)), x)
 

3.1.99.7 Maxima [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^4 (d+e x)} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )} x^{4}} \,d x } \]

input

integrate((a+b*log(c*x^n))^2/x^4/(e*x+d),x, algorithm="maxima")
 

output

1/6*a^2*(6*e^3*log(e*x + d)/d^4 - 6*e^3*log(x)/d^4 - (6*e^2*x^2 - 3*d*e*x 
+ 2*d^2)/(d^3*x^3)) + integrate((b^2*log(c)^2 + b^2*log(x^n)^2 + 2*a*b*log 
(c) + 2*(b^2*log(c) + a*b)*log(x^n))/(e*x^5 + d*x^4), x)
 

3.1.99.8 Giac [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^4 (d+e x)} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )} x^{4}} \,d x } \]

input

integrate((a+b*log(c*x^n))^2/x^4/(e*x+d),x, algorithm="giac")
 

output

integrate((b*log(c*x^n) + a)^2/((e*x + d)*x^4), x)
 

3.1.99.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^4 (d+e x)} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x^4\,\left (d+e\,x\right )} \,d x \]

input

int((a + b*log(c*x^n))^2/(x^4*(d + e*x)),x)
 

output

int((a + b*log(c*x^n))^2/(x^4*(d + e*x)), x)