\(\int \frac {(a+b \log (c (d+\frac {e}{\sqrt {x}})^n))^2}{x^3} \, dx\) [434]

Optimal result

Integrand size = 24, antiderivative size = 341 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{x^3} \, dx=-\frac {3 b^2 d^2 n^2 \left (d+\frac {e}{\sqrt {x}}\right )^2}{2 e^4}+\frac {4 b^2 d n^2 \left (d+\frac {e}{\sqrt {x}}\right )^3}{9 e^4}-\frac {b^2 n^2 \left (d+\frac {e}{\sqrt {x}}\right )^4}{16 e^4}+\frac {4 b^2 d^3 n^2}{e^3 \sqrt {x}}-\frac {b^2 d^4 n^2 \log ^2\left (d+\frac {e}{\sqrt {x}}\right )}{2 e^4}-\frac {4 b d^3 n \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{e^4}+\frac {3 b d^2 n \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{e^4}-\frac {4 b d n \left (d+\frac {e}{\sqrt {x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{3 e^4}+\frac {b n \left (d+\frac {e}{\sqrt {x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{4 e^4}+\frac {b d^4 n \log \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{e^4}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 x^2} \] Output:

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.33 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.29 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{x^3} \, dx=\frac {-72 e^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2+b n \left (36 a e^4-9 b e^4 n-48 a d e^3 \sqrt {x}+28 b d e^3 n \sqrt {x}+72 a d^2 e^2 x-78 b d^2 e^2 n x-144 a d^3 e x^{3/2}+300 b d^3 e n x^{3/2}-300 b d^4 n x^2 \log \left (d+\frac {e}{\sqrt {x}}\right )+36 b e^4 \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )-48 b d e^3 \sqrt {x} \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )+72 b d^2 e^2 x \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )-144 b d^3 e x^{3/2} \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )+144 a d^4 x^2 \log \left (e+d \sqrt {x}\right )+144 b d^4 x^2 \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right ) \log \left (e+d \sqrt {x}\right )-72 b d^4 n x^2 \log ^2\left (e+d \sqrt {x}\right )+144 b d^4 x^2 \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right ) \log \left (-\frac {e}{d \sqrt {x}}\right )+144 b d^4 n x^2 \log \left (e+d \sqrt {x}\right ) \log \left (-\frac {d \sqrt {x}}{e}\right )-72 a d^4 x^2 \log (x)+144 b d^4 n x^2 \operatorname {PolyLog}\left (2,1+\frac {e}{d \sqrt {x}}\right )+144 b d^4 n x^2 \operatorname {PolyLog}\left (2,1+\frac {d \sqrt {x}}{e}\right )\right )}{144 e^4 x^2} \] Input:

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

Output:

(-72*e^4*(a + b*Log[c*(d + e/Sqrt[x])^n])^2 + b*n*(36*a*e^4 - 9*b*e^4*n - 
48*a*d*e^3*Sqrt[x] + 28*b*d*e^3*n*Sqrt[x] + 72*a*d^2*e^2*x - 78*b*d^2*e^2* 
n*x - 144*a*d^3*e*x^(3/2) + 300*b*d^3*e*n*x^(3/2) - 300*b*d^4*n*x^2*Log[d 
+ e/Sqrt[x]] + 36*b*e^4*Log[c*(d + e/Sqrt[x])^n] - 48*b*d*e^3*Sqrt[x]*Log[ 
c*(d + e/Sqrt[x])^n] + 72*b*d^2*e^2*x*Log[c*(d + e/Sqrt[x])^n] - 144*b*d^3 
*e*x^(3/2)*Log[c*(d + e/Sqrt[x])^n] + 144*a*d^4*x^2*Log[e + d*Sqrt[x]] + 1 
44*b*d^4*x^2*Log[c*(d + e/Sqrt[x])^n]*Log[e + d*Sqrt[x]] - 72*b*d^4*n*x^2* 
Log[e + d*Sqrt[x]]^2 + 144*b*d^4*x^2*Log[c*(d + e/Sqrt[x])^n]*Log[-(e/(d*S 
qrt[x]))] + 144*b*d^4*n*x^2*Log[e + d*Sqrt[x]]*Log[-((d*Sqrt[x])/e)] - 72* 
a*d^4*x^2*Log[x] + 144*b*d^4*n*x^2*PolyLog[2, 1 + e/(d*Sqrt[x])] + 144*b*d 
^4*n*x^2*PolyLog[2, 1 + (d*Sqrt[x])/e]))/(144*e^4*x^2)
 

Rubi [A] (warning: unable to verify)

Time = 0.88 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.67, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2904, 2845, 2858, 27, 2772, 2009}

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.

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

\(\Big \downarrow \) 2904

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

\(\Big \downarrow \) 2845

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

\(\Big \downarrow \) 2858

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2772

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

\(\Big \downarrow \) 2009

\(\displaystyle -2 \left (\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{4 x^2}-\frac {b n \left (d^4 \log \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c x^{-n/2}\right )\right )-4 d^3 \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c x^{-n/2}\right )\right )+\frac {3 d^2 \left (a+b \log \left (c x^{-n/2}\right )\right )}{x}-\frac {4 d \left (a+b \log \left (c x^{-n/2}\right )\right )}{3 x^{3/2}}+\frac {a+b \log \left (c x^{-n/2}\right )}{4 x^2}-b n \left (\frac {1}{2} d^4 \log ^2\left (d+\frac {e}{\sqrt {x}}\right )-4 d^3 \left (d+\frac {e}{\sqrt {x}}\right )+\frac {3 d^2}{2 x}-\frac {4 d}{9 x^{3/2}}+\frac {1}{16 x^2}\right )\right )}{2 e^4}\right )\)

Input:

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

Output:

-2*((a + b*Log[c*(d + e/Sqrt[x])^n])^2/(4*x^2) - (b*n*(-(b*n*(-4*d^3*(d + 
e/Sqrt[x]) + 1/(16*x^2) - (4*d)/(9*x^(3/2)) + (3*d^2)/(2*x) + (d^4*Log[d + 
 e/Sqrt[x]]^2)/2)) - 4*d^3*(d + e/Sqrt[x])*(a + b*Log[c/x^(n/2)]) + (a + b 
*Log[c/x^(n/2)])/(4*x^2) - (4*d*(a + b*Log[c/x^(n/2)]))/(3*x^(3/2)) + (3*d 
^2*(a + b*Log[c/x^(n/2)]))/x + d^4*Log[d + e/Sqrt[x]]*(a + b*Log[c/x^(n/2) 
])))/(2*e^4))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

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

rule 2772

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_ 
.))^(q_.), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^r)^q, x]}, Simp[(a + 
 b*Log[c*x^n])   u, x] - Simp[b*n   Int[SimplifyIntegrand[u/x, x], x], x]] 
/; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] &&  !(EqQ[q 
, 1] && EqQ[m, -1])
 

rule 2845

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

rule 2858

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_ 
.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))^(r_.), x_Symbol] :> Simp[1/e   Subst[In 
t[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x, d + 
e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - 
d*g, 0] && (IGtQ[p, 0] || IGtQ[r, 0]) && IntegerQ[2*r]
 

rule 2904

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

Maple [F]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{x^3} \, dx=-\frac {9 \, b^{2} e^{4} n^{2} + 72 \, b^{2} e^{4} \log \left (c\right )^{2} - 36 \, a b e^{4} n + 72 \, a^{2} e^{4} - 72 \, {\left (b^{2} d^{4} n^{2} x^{2} - b^{2} e^{4} n^{2}\right )} \log \left (\frac {d x + e \sqrt {x}}{x}\right )^{2} + 6 \, {\left (13 \, b^{2} d^{2} e^{2} n^{2} - 12 \, a b d^{2} e^{2} n\right )} x - 36 \, {\left (2 \, b^{2} d^{2} e^{2} n x + b^{2} e^{4} n - 4 \, a b e^{4}\right )} \log \left (c\right ) - 12 \, {\left (6 \, b^{2} d^{2} e^{2} n^{2} x + 3 \, b^{2} e^{4} n^{2} - 12 \, a b e^{4} n - {\left (25 \, b^{2} d^{4} n^{2} - 12 \, a b d^{4} n\right )} x^{2} + 12 \, {\left (b^{2} d^{4} n x^{2} - b^{2} e^{4} n\right )} \log \left (c\right ) - 4 \, {\left (3 \, b^{2} d^{3} e n^{2} x + b^{2} d e^{3} n^{2}\right )} \sqrt {x}\right )} \log \left (\frac {d x + e \sqrt {x}}{x}\right ) - 4 \, {\left (7 \, b^{2} d e^{3} n^{2} - 12 \, a b d e^{3} n + 3 \, {\left (25 \, b^{2} d^{3} e n^{2} - 12 \, a b d^{3} e n\right )} x - 12 \, {\left (3 \, b^{2} d^{3} e n x + b^{2} d e^{3} n\right )} \log \left (c\right )\right )} \sqrt {x}}{144 \, e^{4} x^{2}} \] Input:

integrate((a+b*log(c*(d+e/x^(1/2))^n))^2/x^3,x, algorithm="fricas")
 

Output:

-1/144*(9*b^2*e^4*n^2 + 72*b^2*e^4*log(c)^2 - 36*a*b*e^4*n + 72*a^2*e^4 - 
72*(b^2*d^4*n^2*x^2 - b^2*e^4*n^2)*log((d*x + e*sqrt(x))/x)^2 + 6*(13*b^2* 
d^2*e^2*n^2 - 12*a*b*d^2*e^2*n)*x - 36*(2*b^2*d^2*e^2*n*x + b^2*e^4*n - 4* 
a*b*e^4)*log(c) - 12*(6*b^2*d^2*e^2*n^2*x + 3*b^2*e^4*n^2 - 12*a*b*e^4*n - 
 (25*b^2*d^4*n^2 - 12*a*b*d^4*n)*x^2 + 12*(b^2*d^4*n*x^2 - b^2*e^4*n)*log( 
c) - 4*(3*b^2*d^3*e*n^2*x + b^2*d*e^3*n^2)*sqrt(x))*log((d*x + e*sqrt(x))/ 
x) - 4*(7*b^2*d*e^3*n^2 - 12*a*b*d*e^3*n + 3*(25*b^2*d^3*e*n^2 - 12*a*b*d^ 
3*e*n)*x - 12*(3*b^2*d^3*e*n*x + b^2*d*e^3*n)*log(c))*sqrt(x))/(e^4*x^2)
 

Sympy [F]

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

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

Output:

Integral((a + b*log(c*(d + e/sqrt(x))**n))**2/x**3, x)
 

Maxima [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 321, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{x^3} \, dx=\frac {1}{12} \, a b e n {\left (\frac {12 \, d^{4} \log \left (d \sqrt {x} + e\right )}{e^{5}} - \frac {6 \, d^{4} \log \left (x\right )}{e^{5}} - \frac {12 \, d^{3} x^{\frac {3}{2}} - 6 \, d^{2} e x + 4 \, d e^{2} \sqrt {x} - 3 \, e^{3}}{e^{4} x^{2}}\right )} + \frac {1}{144} \, {\left (12 \, e n {\left (\frac {12 \, d^{4} \log \left (d \sqrt {x} + e\right )}{e^{5}} - \frac {6 \, d^{4} \log \left (x\right )}{e^{5}} - \frac {12 \, d^{3} x^{\frac {3}{2}} - 6 \, d^{2} e x + 4 \, d e^{2} \sqrt {x} - 3 \, e^{3}}{e^{4} x^{2}}\right )} \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right ) - \frac {{\left (72 \, d^{4} x^{2} \log \left (d \sqrt {x} + e\right )^{2} + 18 \, d^{4} x^{2} \log \left (x\right )^{2} - 150 \, d^{4} x^{2} \log \left (x\right ) - 300 \, d^{3} e x^{\frac {3}{2}} + 78 \, d^{2} e^{2} x - 28 \, d e^{3} \sqrt {x} + 9 \, e^{4} - 12 \, {\left (6 \, d^{4} x^{2} \log \left (x\right ) - 25 \, d^{4} x^{2}\right )} \log \left (d \sqrt {x} + e\right )\right )} n^{2}}{e^{4} x^{2}}\right )} b^{2} - \frac {b^{2} \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right )^{2}}{2 \, x^{2}} - \frac {a b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right )}{x^{2}} - \frac {a^{2}}{2 \, x^{2}} \] Input:

integrate((a+b*log(c*(d+e/x^(1/2))^n))^2/x^3,x, algorithm="maxima")
 

Output:

1/12*a*b*e*n*(12*d^4*log(d*sqrt(x) + e)/e^5 - 6*d^4*log(x)/e^5 - (12*d^3*x 
^(3/2) - 6*d^2*e*x + 4*d*e^2*sqrt(x) - 3*e^3)/(e^4*x^2)) + 1/144*(12*e*n*( 
12*d^4*log(d*sqrt(x) + e)/e^5 - 6*d^4*log(x)/e^5 - (12*d^3*x^(3/2) - 6*d^2 
*e*x + 4*d*e^2*sqrt(x) - 3*e^3)/(e^4*x^2))*log(c*(d + e/sqrt(x))^n) - (72* 
d^4*x^2*log(d*sqrt(x) + e)^2 + 18*d^4*x^2*log(x)^2 - 150*d^4*x^2*log(x) - 
300*d^3*e*x^(3/2) + 78*d^2*e^2*x - 28*d*e^3*sqrt(x) + 9*e^4 - 12*(6*d^4*x^ 
2*log(x) - 25*d^4*x^2)*log(d*sqrt(x) + e))*n^2/(e^4*x^2))*b^2 - 1/2*b^2*lo 
g(c*(d + e/sqrt(x))^n)^2/x^2 - a*b*log(c*(d + e/sqrt(x))^n)/x^2 - 1/2*a^2/ 
x^2
 

Giac [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 578, normalized size of antiderivative = 1.70 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{x^3} \, dx=\frac {72 \, {\left (\frac {4 \, {\left (d \sqrt {x} + e\right )} b^{2} d^{3} n^{2}}{e^{3} \sqrt {x}} - \frac {6 \, {\left (d \sqrt {x} + e\right )}^{2} b^{2} d^{2} n^{2}}{e^{3} x} + \frac {4 \, {\left (d \sqrt {x} + e\right )}^{3} b^{2} d n^{2}}{e^{3} x^{\frac {3}{2}}} - \frac {{\left (d \sqrt {x} + e\right )}^{4} b^{2} n^{2}}{e^{3} x^{2}}\right )} \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )^{2} + 12 \, {\left (\frac {3 \, {\left (b^{2} n^{2} - 4 \, b^{2} n \log \left (c\right ) - 4 \, a b n\right )} {\left (d \sqrt {x} + e\right )}^{4}}{e^{3} x^{2}} - \frac {16 \, {\left (b^{2} d n^{2} - 3 \, b^{2} d n \log \left (c\right ) - 3 \, a b d n\right )} {\left (d \sqrt {x} + e\right )}^{3}}{e^{3} x^{\frac {3}{2}}} + \frac {36 \, {\left (b^{2} d^{2} n^{2} - 2 \, b^{2} d^{2} n \log \left (c\right ) - 2 \, a b d^{2} n\right )} {\left (d \sqrt {x} + e\right )}^{2}}{e^{3} x} - \frac {48 \, {\left (b^{2} d^{3} n^{2} - b^{2} d^{3} n \log \left (c\right ) - a b d^{3} n\right )} {\left (d \sqrt {x} + e\right )}}{e^{3} \sqrt {x}}\right )} \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right ) - \frac {9 \, {\left (b^{2} n^{2} - 4 \, b^{2} n \log \left (c\right ) + 8 \, b^{2} \log \left (c\right )^{2} - 4 \, a b n + 16 \, a b \log \left (c\right ) + 8 \, a^{2}\right )} {\left (d \sqrt {x} + e\right )}^{4}}{e^{3} x^{2}} + \frac {32 \, {\left (2 \, b^{2} d n^{2} - 6 \, b^{2} d n \log \left (c\right ) + 9 \, b^{2} d \log \left (c\right )^{2} - 6 \, a b d n + 18 \, a b d \log \left (c\right ) + 9 \, a^{2} d\right )} {\left (d \sqrt {x} + e\right )}^{3}}{e^{3} x^{\frac {3}{2}}} - \frac {216 \, {\left (b^{2} d^{2} n^{2} - 2 \, b^{2} d^{2} n \log \left (c\right ) + 2 \, b^{2} d^{2} \log \left (c\right )^{2} - 2 \, a b d^{2} n + 4 \, a b d^{2} \log \left (c\right ) + 2 \, a^{2} d^{2}\right )} {\left (d \sqrt {x} + e\right )}^{2}}{e^{3} x} + \frac {288 \, {\left (2 \, b^{2} d^{3} n^{2} - 2 \, b^{2} d^{3} n \log \left (c\right ) + b^{2} d^{3} \log \left (c\right )^{2} - 2 \, a b d^{3} n + 2 \, a b d^{3} \log \left (c\right ) + a^{2} d^{3}\right )} {\left (d \sqrt {x} + e\right )}}{e^{3} \sqrt {x}}}{144 \, e} \] Input:

integrate((a+b*log(c*(d+e/x^(1/2))^n))^2/x^3,x, algorithm="giac")
 

Output:

1/144*(72*(4*(d*sqrt(x) + e)*b^2*d^3*n^2/(e^3*sqrt(x)) - 6*(d*sqrt(x) + e) 
^2*b^2*d^2*n^2/(e^3*x) + 4*(d*sqrt(x) + e)^3*b^2*d*n^2/(e^3*x^(3/2)) - (d* 
sqrt(x) + e)^4*b^2*n^2/(e^3*x^2))*log((d*sqrt(x) + e)/sqrt(x))^2 + 12*(3*( 
b^2*n^2 - 4*b^2*n*log(c) - 4*a*b*n)*(d*sqrt(x) + e)^4/(e^3*x^2) - 16*(b^2* 
d*n^2 - 3*b^2*d*n*log(c) - 3*a*b*d*n)*(d*sqrt(x) + e)^3/(e^3*x^(3/2)) + 36 
*(b^2*d^2*n^2 - 2*b^2*d^2*n*log(c) - 2*a*b*d^2*n)*(d*sqrt(x) + e)^2/(e^3*x 
) - 48*(b^2*d^3*n^2 - b^2*d^3*n*log(c) - a*b*d^3*n)*(d*sqrt(x) + e)/(e^3*s 
qrt(x)))*log((d*sqrt(x) + e)/sqrt(x)) - 9*(b^2*n^2 - 4*b^2*n*log(c) + 8*b^ 
2*log(c)^2 - 4*a*b*n + 16*a*b*log(c) + 8*a^2)*(d*sqrt(x) + e)^4/(e^3*x^2) 
+ 32*(2*b^2*d*n^2 - 6*b^2*d*n*log(c) + 9*b^2*d*log(c)^2 - 6*a*b*d*n + 18*a 
*b*d*log(c) + 9*a^2*d)*(d*sqrt(x) + e)^3/(e^3*x^(3/2)) - 216*(b^2*d^2*n^2 
- 2*b^2*d^2*n*log(c) + 2*b^2*d^2*log(c)^2 - 2*a*b*d^2*n + 4*a*b*d^2*log(c) 
 + 2*a^2*d^2)*(d*sqrt(x) + e)^2/(e^3*x) + 288*(2*b^2*d^3*n^2 - 2*b^2*d^3*n 
*log(c) + b^2*d^3*log(c)^2 - 2*a*b*d^3*n + 2*a*b*d^3*log(c) + a^2*d^3)*(d* 
sqrt(x) + e)/(e^3*sqrt(x)))/e
 

Mupad [B] (verification not implemented)

Time = 14.80 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.24 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{x^3} \, dx=\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )\,\left (\frac {\frac {b\,d\,\left (4\,a-b\,n\right )}{3\,e}-\frac {4\,a\,b\,d}{3\,e}}{x^{3/2}}-\frac {b\,\left (4\,a-b\,n\right )}{4\,x^2}-\frac {d\,\left (\frac {b\,d\,\left (4\,a-b\,n\right )}{e}-\frac {4\,a\,b\,d}{e}\right )}{2\,e\,x}+\frac {d^2\,\left (\frac {b\,d\,\left (4\,a-b\,n\right )}{e}-\frac {4\,a\,b\,d}{e}\right )}{e^2\,\sqrt {x}}\right )+\frac {\frac {d\,\left (2\,a^2-a\,b\,n+\frac {b^2\,n^2}{4}\right )}{3\,e}-\frac {d\,\left (6\,a^2-b^2\,n^2\right )}{9\,e}}{x^{3/2}}-{\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )}^2\,\left (\frac {b^2}{2\,x^2}-\frac {b^2\,d^4}{2\,e^4}\right )-\frac {\frac {a^2}{2}-\frac {a\,b\,n}{4}+\frac {b^2\,n^2}{16}}{x^2}-\frac {\frac {d\,\left (\frac {d\,\left (2\,a^2-a\,b\,n+\frac {b^2\,n^2}{4}\right )}{e}-\frac {d\,\left (6\,a^2-b^2\,n^2\right )}{3\,e}\right )}{2\,e}+\frac {b^2\,d^2\,n^2}{4\,e^2}}{x}+\frac {\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (2\,a^2-a\,b\,n+\frac {b^2\,n^2}{4}\right )}{e}-\frac {d\,\left (6\,a^2-b^2\,n^2\right )}{3\,e}\right )}{e}+\frac {b^2\,d^2\,n^2}{2\,e^2}\right )}{e}+\frac {b^2\,d^3\,n^2}{e^3}}{\sqrt {x}}-\frac {\ln \left (d+\frac {e}{\sqrt {x}}\right )\,\left (25\,b^2\,d^4\,n^2-12\,a\,b\,d^4\,n\right )}{12\,e^4} \] Input:

int((a + b*log(c*(d + e/x^(1/2))^n))^2/x^3,x)
 

Output:

log(c*(d + e/x^(1/2))^n)*(((b*d*(4*a - b*n))/(3*e) - (4*a*b*d)/(3*e))/x^(3 
/2) - (b*(4*a - b*n))/(4*x^2) - (d*((b*d*(4*a - b*n))/e - (4*a*b*d)/e))/(2 
*e*x) + (d^2*((b*d*(4*a - b*n))/e - (4*a*b*d)/e))/(e^2*x^(1/2))) + ((d*(2* 
a^2 + (b^2*n^2)/4 - a*b*n))/(3*e) - (d*(6*a^2 - b^2*n^2))/(9*e))/x^(3/2) - 
 log(c*(d + e/x^(1/2))^n)^2*(b^2/(2*x^2) - (b^2*d^4)/(2*e^4)) - (a^2/2 + ( 
b^2*n^2)/16 - (a*b*n)/4)/x^2 - ((d*((d*(2*a^2 + (b^2*n^2)/4 - a*b*n))/e - 
(d*(6*a^2 - b^2*n^2))/(3*e)))/(2*e) + (b^2*d^2*n^2)/(4*e^2))/x + ((d*((d*( 
(d*(2*a^2 + (b^2*n^2)/4 - a*b*n))/e - (d*(6*a^2 - b^2*n^2))/(3*e)))/e + (b 
^2*d^2*n^2)/(2*e^2)))/e + (b^2*d^3*n^2)/e^3)/x^(1/2) - (log(d + e/x^(1/2)) 
*(25*b^2*d^4*n^2 - 12*a*b*d^4*n))/(12*e^4)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{x^3} \, dx=\frac {-144 \sqrt {x}\, \mathrm {log}\left (\frac {\left (\sqrt {x}\, d +e \right )^{n} c}{x^{\frac {n}{2}}}\right ) b^{2} d^{3} e n x -48 \sqrt {x}\, \mathrm {log}\left (\frac {\left (\sqrt {x}\, d +e \right )^{n} c}{x^{\frac {n}{2}}}\right ) b^{2} d \,e^{3} n -144 \sqrt {x}\, a b \,d^{3} e n x -48 \sqrt {x}\, a b d \,e^{3} n +300 \sqrt {x}\, b^{2} d^{3} e \,n^{2} x +28 \sqrt {x}\, b^{2} d \,e^{3} n^{2}+72 \mathrm {log}\left (\frac {\left (\sqrt {x}\, d +e \right )^{n} c}{x^{\frac {n}{2}}}\right )^{2} b^{2} d^{4} x^{2}-72 \mathrm {log}\left (\frac {\left (\sqrt {x}\, d +e \right )^{n} c}{x^{\frac {n}{2}}}\right )^{2} b^{2} e^{4}+144 \,\mathrm {log}\left (\frac {\left (\sqrt {x}\, d +e \right )^{n} c}{x^{\frac {n}{2}}}\right ) a b \,d^{4} x^{2}-144 \,\mathrm {log}\left (\frac {\left (\sqrt {x}\, d +e \right )^{n} c}{x^{\frac {n}{2}}}\right ) a b \,e^{4}-300 \,\mathrm {log}\left (\frac {\left (\sqrt {x}\, d +e \right )^{n} c}{x^{\frac {n}{2}}}\right ) b^{2} d^{4} n \,x^{2}+72 \,\mathrm {log}\left (\frac {\left (\sqrt {x}\, d +e \right )^{n} c}{x^{\frac {n}{2}}}\right ) b^{2} d^{2} e^{2} n x +36 \,\mathrm {log}\left (\frac {\left (\sqrt {x}\, d +e \right )^{n} c}{x^{\frac {n}{2}}}\right ) b^{2} e^{4} n -72 a^{2} e^{4}+72 a b \,d^{2} e^{2} n x +36 a b \,e^{4} n -78 b^{2} d^{2} e^{2} n^{2} x -9 b^{2} e^{4} n^{2}}{144 e^{4} x^{2}} \] Input:

int((a+b*log(c*(d+e/x^(1/2))^n))^2/x^3,x)
 

Output:

( - 144*sqrt(x)*log(((sqrt(x)*d + e)**n*c)/x**(n/2))*b**2*d**3*e*n*x - 48* 
sqrt(x)*log(((sqrt(x)*d + e)**n*c)/x**(n/2))*b**2*d*e**3*n - 144*sqrt(x)*a 
*b*d**3*e*n*x - 48*sqrt(x)*a*b*d*e**3*n + 300*sqrt(x)*b**2*d**3*e*n**2*x + 
 28*sqrt(x)*b**2*d*e**3*n**2 + 72*log(((sqrt(x)*d + e)**n*c)/x**(n/2))**2* 
b**2*d**4*x**2 - 72*log(((sqrt(x)*d + e)**n*c)/x**(n/2))**2*b**2*e**4 + 14 
4*log(((sqrt(x)*d + e)**n*c)/x**(n/2))*a*b*d**4*x**2 - 144*log(((sqrt(x)*d 
 + e)**n*c)/x**(n/2))*a*b*e**4 - 300*log(((sqrt(x)*d + e)**n*c)/x**(n/2))* 
b**2*d**4*n*x**2 + 72*log(((sqrt(x)*d + e)**n*c)/x**(n/2))*b**2*d**2*e**2* 
n*x + 36*log(((sqrt(x)*d + e)**n*c)/x**(n/2))*b**2*e**4*n - 72*a**2*e**4 + 
 72*a*b*d**2*e**2*n*x + 36*a*b*e**4*n - 78*b**2*d**2*e**2*n**2*x - 9*b**2* 
e**4*n**2)/(144*e**4*x**2)