3.2.17 \(\int \frac {(a+b \log (c x^n))^2}{(d+e x)^4} \, dx\) [117]

3.2.17.1 Optimal result

Integrand size = 20, antiderivative size = 203 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx=-\frac {b^2 n^2}{3 d^2 e (d+e x)}-\frac {b^2 n^2 \log (x)}{3 d^3 e}+\frac {b n \left (a+b \log \left (c x^n\right )\right )}{3 d e (d+e x)^2}-\frac {2 b n x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 (d+e x)}-\frac {2 b n \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^3 e}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 e (d+e x)^3}+\frac {b^2 n^2 \log (d+e x)}{d^3 e}+\frac {2 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{3 d^3 e} \]

output

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

3.2.17.2 Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.04 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx=-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 e (d+e x)^3}+\frac {2 b n \left (\frac {a+b \log \left (c x^n\right )}{2 d (d+e x)^2}+\frac {a+b \log \left (c x^n\right )}{d^2 (d+e x)}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 b d^3 n}-\frac {b n \left (\frac {1}{d (d+e x)}+\frac {\log (x)}{d^2}-\frac {\log (d+e x)}{d^2}\right )}{2 d}-\frac {b n \left (\frac {\log (x)}{d}-\frac {\log (d+e x)}{d}\right )}{d^2}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (\frac {d+e x}{d}\right )}{d^3}-\frac {b n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^3}\right )}{3 e} \]

input

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

output

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

3.2.17.3 Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2756, 2789, 2756, 54, 2009, 2789, 2751, 16, 2779, 2838}

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.

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

\(\Big \downarrow \) 2756

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

\(\Big \downarrow \) 2789

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

\(\Big \downarrow \) 2756

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

\(\Big \downarrow \) 54

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 2789

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

\(\Big \downarrow \) 2751

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 2779

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

\(\Big \downarrow \) 2838

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

input

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

output

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

3.2.17.3.1 Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 54

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E 
xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && 
 ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2751

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x 
_Symbol] :> Simp[x*(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])/d), x] - Simp[b* 
(n/d)   Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q, r}, 
x] && EqQ[r*(q + 1) + 1, 0]
 

rule 2756

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), 
x_Symbol] :> Simp[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] 
- Simp[b*n*(p/(e*(q + 1)))   Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 
 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, 
 -1] && (EqQ[p, 1] || (IntegersQ[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] & 
& NeQ[q, 1]))
 

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 2789

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/ 
(x_), x_Symbol] :> Simp[1/d   Int[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x 
), x], x] - Simp[e/d   Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; Free 
Q[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

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

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

Time = 0.51 (sec) , antiderivative size = 495, normalized size of antiderivative = 2.44

method	result	size
risch	\(-\frac {b^{2} \ln \left (x^{n}\right )^{2}}{3 e \left (e x +d \right )^{3}}-\frac {2 b^{2} n \ln \left (x^{n}\right ) \ln \left (e x +d \right )}{3 e \,d^{3}}+\frac {2 b^{2} n \ln \left (x^{n}\right )}{3 e \,d^{2} \left (e x +d \right )}+\frac {b^{2} n \ln \left (x^{n}\right )}{3 e d \left (e x +d \right )^{2}}+\frac {2 b^{2} n \ln \left (x^{n}\right ) \ln \left (x \right )}{3 e \,d^{3}}-\frac {b^{2} n^{2}}{3 d^{2} e \left (e x +d \right )}+\frac {b^{2} n^{2} \ln \left (e x +d \right )}{d^{3} e}-\frac {b^{2} n^{2} \ln \left (x \right )}{d^{3} e}-\frac {b^{2} n^{2} \ln \left (x \right )^{2}}{3 e \,d^{3}}+\frac {2 b^{2} n^{2} \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{3 e \,d^{3}}+\frac {2 b^{2} n^{2} \operatorname {dilog}\left (-\frac {e x}{d}\right )}{3 e \,d^{3}}+\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 )}{3 e \left (e x +d \right )^{3}}+\frac {n \left (-\frac {\ln \left (e x +d \right )}{d^{3}}+\frac {1}{d^{2} \left (e x +d \right )}+\frac {1}{2 d \left (e x +d \right )^{2}}+\frac {\ln \left (x \right )}{d^{3}}\right )}{3 e}\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}}{12 \left (e x +d \right )^{3} e}\)	\(495\)

input

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

output

-1/3*b^2*ln(x^n)^2/e/(e*x+d)^3-2/3*b^2/e*n*ln(x^n)/d^3*ln(e*x+d)+2/3*b^2*n 
*ln(x^n)/e/d^2/(e*x+d)+1/3*b^2*n*ln(x^n)/e/d/(e*x+d)^2+2/3*b^2/e*n*ln(x^n) 
/d^3*ln(x)-1/3*b^2*n^2/d^2/e/(e*x+d)+b^2*n^2*ln(e*x+d)/d^3/e-b^2*n^2*ln(x) 
/d^3/e-1/3*b^2/e*n^2/d^3*ln(x)^2+2/3*b^2/e*n^2/d^3*ln(e*x+d)*ln(-e*x/d)+2/ 
3*b^2/e*n^2/d^3*dilog(-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*P 
i*csgn(I*c*x^n)^3+2*b*ln(c)+2*a)*b*(-1/3*ln(x^n)/e/(e*x+d)^3+1/3/e*n*(-1/d 
^3*ln(e*x+d)+1/d^2/(e*x+d)+1/2/d/(e*x+d)^2+1/d^3*ln(x)))-1/12*(-I*b*Pi*csg 
n(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*c 
sgn(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*x+d) 
^3/e
 

3.2.17.5 Fricas [F]

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

input

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

output

integral((b^2*log(c*x^n)^2 + 2*a*b*log(c*x^n) + a^2)/(e^4*x^4 + 4*d*e^3*x^ 
3 + 6*d^2*e^2*x^2 + 4*d^3*e*x + d^4), x)
 

3.2.17.6 Sympy [F]

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

input

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

output

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

3.2.17.7 Maxima [F]

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

input

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

output

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

3.2.17.8 Giac [F]

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

input

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

output

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

3.2.17.9 Mupad [F(-1)]

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

input

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

output

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