\(\int \frac {(a+b \log (c (d+\frac {e}{\sqrt {x}})^n))^3}{x^4} \, dx\) [441]

Optimal result

Integrand size = 24, antiderivative size = 907 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^4} \, dx =\text {Too large to display} \] Output:

-1/18*b^2*n^2*(d+e/x^(1/2))^6*(a+b*ln(c*(d+e/x^(1/2))^n))/e^6+1/6*b*n*(d+e 
/x^(1/2))^6*(a+b*ln(c*(d+e/x^(1/2))^n))^2/e^6+15/4*b^3*d^4*n^3*(d+e/x^(1/2 
))^2/e^6-40/27*b^3*d^3*n^3*(d+e/x^(1/2))^3/e^6+15/32*b^3*d^2*n^3*(d+e/x^(1 
/2))^4/e^6-12/125*b^3*d*n^3*(d+e/x^(1/2))^5/e^6-12*b^3*d^5*n^3/e^5/x^(1/2) 
+12*a*b^2*d^5*n^2/e^5/x^(1/2)+12*b^3*d^5*n^2*(d+e/x^(1/2))*ln(c*(d+e/x^(1/ 
2))^n)/e^6-15/2*b^2*d^4*n^2*(d+e/x^(1/2))^2*(a+b*ln(c*(d+e/x^(1/2))^n))/e^ 
6+40/9*b^2*d^3*n^2*(d+e/x^(1/2))^3*(a+b*ln(c*(d+e/x^(1/2))^n))/e^6-15/8*b^ 
2*d^2*n^2*(d+e/x^(1/2))^4*(a+b*ln(c*(d+e/x^(1/2))^n))/e^6+12/25*b^2*d*n^2* 
(d+e/x^(1/2))^5*(a+b*ln(c*(d+e/x^(1/2))^n))/e^6-6*b*d^5*n*(d+e/x^(1/2))*(a 
+b*ln(c*(d+e/x^(1/2))^n))^2/e^6+15/2*b*d^4*n*(d+e/x^(1/2))^2*(a+b*ln(c*(d+ 
e/x^(1/2))^n))^2/e^6-20/3*b*d^3*n*(d+e/x^(1/2))^3*(a+b*ln(c*(d+e/x^(1/2))^ 
n))^2/e^6+15/4*b*d^2*n*(d+e/x^(1/2))^4*(a+b*ln(c*(d+e/x^(1/2))^n))^2/e^6-6 
/5*b*d*n*(d+e/x^(1/2))^5*(a+b*ln(c*(d+e/x^(1/2))^n))^2/e^6-1/3*(d+e/x^(1/2 
))^6*(a+b*ln(c*(d+e/x^(1/2))^n))^3/e^6+20/3*d^3*(d+e/x^(1/2))^3*(a+b*ln(c* 
(d+e/x^(1/2))^n))^3/e^6-5*d^2*(d+e/x^(1/2))^4*(a+b*ln(c*(d+e/x^(1/2))^n))^ 
3/e^6+2*d*(d+e/x^(1/2))^5*(a+b*ln(c*(d+e/x^(1/2))^n))^3/e^6+2*d^5*(d+e/x^( 
1/2))*(a+b*ln(c*(d+e/x^(1/2))^n))^3/e^6-5*d^4*(d+e/x^(1/2))^2*(a+b*ln(c*(d 
+e/x^(1/2))^n))^3/e^6+1/108*b^3*n^3*(d+e/x^(1/2))^6/e^6
 

Mathematica [A] (verified)

Time = 1.48 (sec) , antiderivative size = 950, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^4} \, dx =\text {Too large to display} \] Input:

Integrate[(a + b*Log[c*(d + e/Sqrt[x])^n])^3/x^4,x]
 

Output:

(-36000*a^3*e^6 + 18000*a^2*b*e^6*n - 6000*a*b^2*e^6*n^2 + 1000*b^3*e^6*n^ 
3 - 21600*a^2*b*d*e^5*n*Sqrt[x] + 15840*a*b^2*d*e^5*n^2*Sqrt[x] - 4368*b^3 
*d*e^5*n^3*Sqrt[x] + 27000*a^2*b*d^2*e^4*n*x - 33300*a*b^2*d^2*e^4*n^2*x + 
 13785*b^3*d^2*e^4*n^3*x - 36000*a^2*b*d^3*e^3*n*x^(3/2) + 68400*a*b^2*d^3 
*e^3*n^2*x^(3/2) - 41180*b^3*d^3*e^3*n^3*x^(3/2) + 54000*a^2*b*d^4*e^2*n*x 
^2 - 156600*a*b^2*d^4*e^2*n^2*x^2 + 140070*b^3*d^4*e^2*n^3*x^2 - 108000*a^ 
2*b*d^5*e*n*x^(5/2) + 529200*a*b^2*d^5*e*n^2*x^(5/2) - 809340*b^3*d^5*e*n^ 
3*x^(5/2) - 72000*b^3*d^6*n^3*x^3*Log[d + e/Sqrt[x]]^3 - 36000*b^3*e^6*Log 
[c*(d + e/Sqrt[x])^n]^3 + 108000*a^2*b*d^6*n*x^3*Log[e + d*Sqrt[x]] - 5292 
00*a*b^2*d^6*n^2*x^3*Log[e + d*Sqrt[x]] + 809340*b^3*d^6*n^3*x^3*Log[e + d 
*Sqrt[x]] + 5400*b^2*d^6*n^2*x^3*Log[d + e/Sqrt[x]]*(-20*a + 49*b*n - 20*b 
*Log[c*(d + e/Sqrt[x])^n])*(2*Log[e + d*Sqrt[x]] - Log[x]) - 54000*a^2*b*d 
^6*n*x^3*Log[x] + 264600*a*b^2*d^6*n^2*x^3*Log[x] - 404670*b^3*d^6*n^3*x^3 
*Log[x] + 5400*b^2*d^6*n^2*x^3*Log[d + e/Sqrt[x]]^2*(20*a - 49*b*n + 20*b* 
Log[c*(d + e/Sqrt[x])^n] + 20*b*n*Log[e + d*Sqrt[x]] - 10*b*n*Log[x]) + 18 
00*b^2*Log[c*(d + e/Sqrt[x])^n]^2*(e*(-60*a*e^5 + 10*b*e^5*n - 12*b*d*e^4* 
n*Sqrt[x] + 15*b*d^2*e^3*n*x - 20*b*d^3*e^2*n*x^(3/2) + 30*b*d^4*e*n*x^2 - 
 60*b*d^5*n*x^(5/2)) + 60*b*d^6*n*x^3*Log[e + d*Sqrt[x]] - 30*b*d^6*n*x^3* 
Log[x]) - 60*b*Log[c*(d + e/Sqrt[x])^n]*(1800*a^2*e^6 + b^2*e*n^2*(100*e^5 
 - 264*d*e^4*Sqrt[x] + 555*d^2*e^3*x - 1140*d^3*e^2*x^(3/2) + 2610*d^4*...
 

Rubi [A] (verified)

Time = 2.17 (sec) , antiderivative size = 913, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2904, 2848, 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.

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

\(\Big \downarrow \) 2904

\(\displaystyle -2 \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^{5/2}}d\frac {1}{\sqrt {x}}\)

\(\Big \downarrow \) 2848

\(\displaystyle -2 \int \left (-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3 d^5}{e^5}+\frac {5 \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3 d^4}{e^5}-\frac {10 \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3 d^3}{e^5}+\frac {10 \left (d+\frac {e}{\sqrt {x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3 d^2}{e^5}-\frac {5 \left (d+\frac {e}{\sqrt {x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3 d}{e^5}+\frac {\left (d+\frac {e}{\sqrt {x}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^5}\right )d\frac {1}{\sqrt {x}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -2 \left (-\frac {b^3 n^3 \left (d+\frac {e}{\sqrt {x}}\right )^6}{216 e^6}+\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3 \left (d+\frac {e}{\sqrt {x}}\right )^6}{6 e^6}-\frac {b n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \left (d+\frac {e}{\sqrt {x}}\right )^6}{12 e^6}+\frac {b^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \left (d+\frac {e}{\sqrt {x}}\right )^6}{36 e^6}+\frac {6 b^3 d n^3 \left (d+\frac {e}{\sqrt {x}}\right )^5}{125 e^6}-\frac {d \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3 \left (d+\frac {e}{\sqrt {x}}\right )^5}{e^6}+\frac {3 b d n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \left (d+\frac {e}{\sqrt {x}}\right )^5}{5 e^6}-\frac {6 b^2 d n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \left (d+\frac {e}{\sqrt {x}}\right )^5}{25 e^6}-\frac {15 b^3 d^2 n^3 \left (d+\frac {e}{\sqrt {x}}\right )^4}{64 e^6}+\frac {5 d^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3 \left (d+\frac {e}{\sqrt {x}}\right )^4}{2 e^6}-\frac {15 b d^2 n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \left (d+\frac {e}{\sqrt {x}}\right )^4}{8 e^6}+\frac {15 b^2 d^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \left (d+\frac {e}{\sqrt {x}}\right )^4}{16 e^6}+\frac {20 b^3 d^3 n^3 \left (d+\frac {e}{\sqrt {x}}\right )^3}{27 e^6}-\frac {10 d^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3 \left (d+\frac {e}{\sqrt {x}}\right )^3}{3 e^6}+\frac {10 b d^3 n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \left (d+\frac {e}{\sqrt {x}}\right )^3}{3 e^6}-\frac {20 b^2 d^3 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \left (d+\frac {e}{\sqrt {x}}\right )^3}{9 e^6}-\frac {15 b^3 d^4 n^3 \left (d+\frac {e}{\sqrt {x}}\right )^2}{8 e^6}+\frac {5 d^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3 \left (d+\frac {e}{\sqrt {x}}\right )^2}{2 e^6}-\frac {15 b d^4 n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \left (d+\frac {e}{\sqrt {x}}\right )^2}{4 e^6}+\frac {15 b^2 d^4 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \left (d+\frac {e}{\sqrt {x}}\right )^2}{4 e^6}-\frac {d^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3 \left (d+\frac {e}{\sqrt {x}}\right )}{e^6}+\frac {3 b d^5 n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \left (d+\frac {e}{\sqrt {x}}\right )}{e^6}-\frac {6 b^3 d^5 n^2 \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right ) \left (d+\frac {e}{\sqrt {x}}\right )}{e^6}+\frac {6 b^3 d^5 n^3}{e^5 \sqrt {x}}-\frac {6 a b^2 d^5 n^2}{e^5 \sqrt {x}}\right )\)

Input:

Int[(a + b*Log[c*(d + e/Sqrt[x])^n])^3/x^4,x]
 

Output:

-2*((-15*b^3*d^4*n^3*(d + e/Sqrt[x])^2)/(8*e^6) + (20*b^3*d^3*n^3*(d + e/S 
qrt[x])^3)/(27*e^6) - (15*b^3*d^2*n^3*(d + e/Sqrt[x])^4)/(64*e^6) + (6*b^3 
*d*n^3*(d + e/Sqrt[x])^5)/(125*e^6) - (b^3*n^3*(d + e/Sqrt[x])^6)/(216*e^6 
) - (6*a*b^2*d^5*n^2)/(e^5*Sqrt[x]) + (6*b^3*d^5*n^3)/(e^5*Sqrt[x]) - (6*b 
^3*d^5*n^2*(d + e/Sqrt[x])*Log[c*(d + e/Sqrt[x])^n])/e^6 + (15*b^2*d^4*n^2 
*(d + e/Sqrt[x])^2*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(4*e^6) - (20*b^2*d^3 
*n^2*(d + e/Sqrt[x])^3*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(9*e^6) + (15*b^2 
*d^2*n^2*(d + e/Sqrt[x])^4*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(16*e^6) - (6 
*b^2*d*n^2*(d + e/Sqrt[x])^5*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(25*e^6) + 
(b^2*n^2*(d + e/Sqrt[x])^6*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(36*e^6) + (3 
*b*d^5*n*(d + e/Sqrt[x])*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/e^6 - (15*b*d 
^4*n*(d + e/Sqrt[x])^2*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/(4*e^6) + (10*b 
*d^3*n*(d + e/Sqrt[x])^3*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/(3*e^6) - (15 
*b*d^2*n*(d + e/Sqrt[x])^4*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/(8*e^6) + ( 
3*b*d*n*(d + e/Sqrt[x])^5*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/(5*e^6) - (b 
*n*(d + e/Sqrt[x])^6*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/(12*e^6) - (d^5*( 
d + e/Sqrt[x])*(a + b*Log[c*(d + e/Sqrt[x])^n])^3)/e^6 + (5*d^4*(d + e/Sqr 
t[x])^2*(a + b*Log[c*(d + e/Sqrt[x])^n])^3)/(2*e^6) - (10*d^3*(d + e/Sqrt[ 
x])^3*(a + b*Log[c*(d + e/Sqrt[x])^n])^3)/(3*e^6) + (5*d^2*(d + e/Sqrt[x]) 
^4*(a + b*Log[c*(d + e/Sqrt[x])^n])^3)/(2*e^6) - (d*(d + e/Sqrt[x])^5*(...
 

Defintions of rubi rules used

rule 2009

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

rule 2848

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

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]

\[\int \frac {{\left (a +b \ln \left (c \left (d +\frac {e}{\sqrt {x}}\right )^{n}\right )\right )}^{3}}{x^{4}}d x\]

Input:

int((a+b*ln(c*(d+e/x^(1/2))^n))^3/x^4,x)
 

Output:

int((a+b*ln(c*(d+e/x^(1/2))^n))^3/x^4,x)
 

Fricas [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 1203, normalized size of antiderivative = 1.33 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^4} \, dx=\text {Too large to display} \] Input:

integrate((a+b*log(c*(d+e/x^(1/2))^n))^3/x^4,x, algorithm="fricas")
 

Output:

1/108000*(1000*b^3*e^6*n^3 - 36000*b^3*e^6*log(c)^3 - 6000*a*b^2*e^6*n^2 + 
 18000*a^2*b*e^6*n - 36000*a^3*e^6 + 36000*(b^3*d^6*n^3*x^3 - b^3*e^6*n^3) 
*log((d*x + e*sqrt(x))/x)^3 + 30*(4669*b^3*d^4*e^2*n^3 - 5220*a*b^2*d^4*e^ 
2*n^2 + 1800*a^2*b*d^4*e^2*n)*x^2 + 9000*(6*b^3*d^4*e^2*n*x^2 + 3*b^3*d^2* 
e^4*n*x + 2*b^3*e^6*n - 12*a*b^2*e^6)*log(c)^2 + 1800*(30*b^3*d^4*e^2*n^3* 
x^2 + 15*b^3*d^2*e^4*n^3*x + 10*b^3*e^6*n^3 - 60*a*b^2*e^6*n^2 - 3*(49*b^3 
*d^6*n^3 - 20*a*b^2*d^6*n^2)*x^3 + 60*(b^3*d^6*n^2*x^3 - b^3*e^6*n^2)*log( 
c) - 4*(15*b^3*d^5*e*n^3*x^2 + 5*b^3*d^3*e^3*n^3*x + 3*b^3*d*e^5*n^3)*sqrt 
(x))*log((d*x + e*sqrt(x))/x)^2 + 15*(919*b^3*d^2*e^4*n^3 - 2220*a*b^2*d^2 
*e^4*n^2 + 1800*a^2*b*d^2*e^4*n)*x - 300*(20*b^3*e^6*n^2 - 120*a*b^2*e^6*n 
 + 360*a^2*b*e^6 + 18*(29*b^3*d^4*e^2*n^2 - 20*a*b^2*d^4*e^2*n)*x^2 + 3*(3 
7*b^3*d^2*e^4*n^2 - 60*a*b^2*d^2*e^4*n)*x)*log(c) - 60*(100*b^3*e^6*n^3 - 
600*a*b^2*e^6*n^2 + 1800*a^2*b*e^6*n - (13489*b^3*d^6*n^3 - 8820*a*b^2*d^6 
*n^2 + 1800*a^2*b*d^6*n)*x^3 + 90*(29*b^3*d^4*e^2*n^3 - 20*a*b^2*d^4*e^2*n 
^2)*x^2 - 1800*(b^3*d^6*n*x^3 - b^3*e^6*n)*log(c)^2 + 15*(37*b^3*d^2*e^4*n 
^3 - 60*a*b^2*d^2*e^4*n^2)*x - 60*(30*b^3*d^4*e^2*n^2*x^2 + 15*b^3*d^2*e^4 
*n^2*x + 10*b^3*e^6*n^2 - 60*a*b^2*e^6*n - 3*(49*b^3*d^6*n^2 - 20*a*b^2*d^ 
6*n)*x^3)*log(c) - 12*(22*b^3*d*e^5*n^3 - 60*a*b^2*d*e^5*n^2 + 15*(49*b^3* 
d^5*e*n^3 - 20*a*b^2*d^5*e*n^2)*x^2 + 5*(19*b^3*d^3*e^3*n^3 - 20*a*b^2*d^3 
*e^3*n^2)*x - 20*(15*b^3*d^5*e*n^2*x^2 + 5*b^3*d^3*e^3*n^2*x + 3*b^3*d*...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^4} \, dx=\text {Timed out} \] Input:

integrate((a+b*ln(c*(d+e/x**(1/2))**n))**3/x**4,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 864, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^4} \, dx=\text {Too large to display} \] Input:

integrate((a+b*log(c*(d+e/x^(1/2))^n))^3/x^4,x, algorithm="maxima")
 

Output:

1/60*a^2*b*e*n*(60*d^6*log(d*sqrt(x) + e)/e^7 - 30*d^6*log(x)/e^7 - (60*d^ 
5*x^(5/2) - 30*d^4*e*x^2 + 20*d^3*e^2*x^(3/2) - 15*d^2*e^3*x + 12*d*e^4*sq 
rt(x) - 10*e^5)/(e^6*x^3)) + 1/1800*(60*e*n*(60*d^6*log(d*sqrt(x) + e)/e^7 
 - 30*d^6*log(x)/e^7 - (60*d^5*x^(5/2) - 30*d^4*e*x^2 + 20*d^3*e^2*x^(3/2) 
 - 15*d^2*e^3*x + 12*d*e^4*sqrt(x) - 10*e^5)/(e^6*x^3))*log(c*(d + e/sqrt( 
x))^n) - (1800*d^6*x^3*log(d*sqrt(x) + e)^2 + 450*d^6*x^3*log(x)^2 - 4410* 
d^6*x^3*log(x) - 8820*d^5*e*x^(5/2) + 2610*d^4*e^2*x^2 - 1140*d^3*e^3*x^(3 
/2) + 555*d^2*e^4*x - 264*d*e^5*sqrt(x) + 100*e^6 - 180*(10*d^6*x^3*log(x) 
 - 49*d^6*x^3)*log(d*sqrt(x) + e))*n^2/(e^6*x^3))*a*b^2 + 1/108000*(1800*e 
*n*(60*d^6*log(d*sqrt(x) + e)/e^7 - 30*d^6*log(x)/e^7 - (60*d^5*x^(5/2) - 
30*d^4*e*x^2 + 20*d^3*e^2*x^(3/2) - 15*d^2*e^3*x + 12*d*e^4*sqrt(x) - 10*e 
^5)/(e^6*x^3))*log(c*(d + e/sqrt(x))^n)^2 + e*n*((36000*d^6*x^3*log(d*sqrt 
(x) + e)^3 - 4500*d^6*x^3*log(x)^3 + 66150*d^6*x^3*log(x)^2 - 404670*d^6*x 
^3*log(x) - 809340*d^5*e*x^(5/2) + 140070*d^4*e^2*x^2 - 41180*d^3*e^3*x^(3 
/2) + 13785*d^2*e^4*x - 4368*d*e^5*sqrt(x) + 1000*e^6 - 5400*(10*d^6*x^3*l 
og(x) - 49*d^6*x^3)*log(d*sqrt(x) + e)^2 + 60*(450*d^6*x^3*log(x)^2 - 4410 
*d^6*x^3*log(x) + 13489*d^6*x^3)*log(d*sqrt(x) + e))*n^2/(e^7*x^3) - 60*(1 
800*d^6*x^3*log(d*sqrt(x) + e)^2 + 450*d^6*x^3*log(x)^2 - 4410*d^6*x^3*log 
(x) - 8820*d^5*e*x^(5/2) + 2610*d^4*e^2*x^2 - 1140*d^3*e^3*x^(3/2) + 555*d 
^2*e^4*x - 264*d*e^5*sqrt(x) + 100*e^6 - 180*(10*d^6*x^3*log(x) - 49*d^...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1747 vs. \(2 (787) = 1574\).

Time = 0.23 (sec) , antiderivative size = 1747, normalized size of antiderivative = 1.93 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^4} \, dx=\text {Too large to display} \] Input:

integrate((a+b*log(c*(d+e/x^(1/2))^n))^3/x^4,x, algorithm="giac")
 

Output:

1/108000*(36000*(6*(d*sqrt(x) + e)*b^3*d^5*n^3/(e^5*sqrt(x)) - 15*(d*sqrt( 
x) + e)^2*b^3*d^4*n^3/(e^5*x) + 20*(d*sqrt(x) + e)^3*b^3*d^3*n^3/(e^5*x^(3 
/2)) - 15*(d*sqrt(x) + e)^4*b^3*d^2*n^3/(e^5*x^2) + 6*(d*sqrt(x) + e)^5*b^ 
3*d*n^3/(e^5*x^(5/2)) - (d*sqrt(x) + e)^6*b^3*n^3/(e^5*x^3))*log((d*sqrt(x 
) + e)/sqrt(x))^3 + 1800*(10*(b^3*n^3 - 6*b^3*n^2*log(c) - 6*a*b^2*n^2)*(d 
*sqrt(x) + e)^6/(e^5*x^3) - 72*(b^3*d*n^3 - 5*b^3*d*n^2*log(c) - 5*a*b^2*d 
*n^2)*(d*sqrt(x) + e)^5/(e^5*x^(5/2)) + 225*(b^3*d^2*n^3 - 4*b^3*d^2*n^2*l 
og(c) - 4*a*b^2*d^2*n^2)*(d*sqrt(x) + e)^4/(e^5*x^2) - 400*(b^3*d^3*n^3 - 
3*b^3*d^3*n^2*log(c) - 3*a*b^2*d^3*n^2)*(d*sqrt(x) + e)^3/(e^5*x^(3/2)) + 
450*(b^3*d^4*n^3 - 2*b^3*d^4*n^2*log(c) - 2*a*b^2*d^4*n^2)*(d*sqrt(x) + e) 
^2/(e^5*x) - 360*(b^3*d^5*n^3 - b^3*d^5*n^2*log(c) - a*b^2*d^5*n^2)*(d*sqr 
t(x) + e)/(e^5*sqrt(x)))*log((d*sqrt(x) + e)/sqrt(x))^2 - 60*(100*(b^3*n^3 
 - 6*b^3*n^2*log(c) + 18*b^3*n*log(c)^2 - 6*a*b^2*n^2 + 36*a*b^2*n*log(c) 
+ 18*a^2*b*n)*(d*sqrt(x) + e)^6/(e^5*x^3) - 432*(2*b^3*d*n^3 - 10*b^3*d*n^ 
2*log(c) + 25*b^3*d*n*log(c)^2 - 10*a*b^2*d*n^2 + 50*a*b^2*d*n*log(c) + 25 
*a^2*b*d*n)*(d*sqrt(x) + e)^5/(e^5*x^(5/2)) + 3375*(b^3*d^2*n^3 - 4*b^3*d^ 
2*n^2*log(c) + 8*b^3*d^2*n*log(c)^2 - 4*a*b^2*d^2*n^2 + 16*a*b^2*d^2*n*log 
(c) + 8*a^2*b*d^2*n)*(d*sqrt(x) + e)^4/(e^5*x^2) - 4000*(2*b^3*d^3*n^3 - 6 
*b^3*d^3*n^2*log(c) + 9*b^3*d^3*n*log(c)^2 - 6*a*b^2*d^3*n^2 + 18*a*b^2*d^ 
3*n*log(c) + 9*a^2*b*d^3*n)*(d*sqrt(x) + e)^3/(e^5*x^(3/2)) + 13500*(b^...
 

Mupad [B] (verification not implemented)

Time = 21.01 (sec) , antiderivative size = 989, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^4} \, dx=\text {Too large to display} \] Input:

int((a + b*log(c*(d + e/x^(1/2))^n))^3/x^4,x)
 

Output:

(b^3*n^3)/(108*x^3) - (b^3*log(c*(d + e/x^(1/2))^n)^3)/(3*x^3) - a^3/(3*x^ 
3) - (a*b^2*log(c*(d + e/x^(1/2))^n)^2)/x^3 + (b^3*n*log(c*(d + e/x^(1/2)) 
^n)^2)/(6*x^3) - (b^3*n^2*log(c*(d + e/x^(1/2))^n))/(18*x^3) - (a*b^2*n^2) 
/(18*x^3) + (b^3*d^6*log(c*(d + e/x^(1/2))^n)^3)/(3*e^6) - (a^2*b*log(c*(d 
 + e/x^(1/2))^n))/x^3 + (a^2*b*n)/(6*x^3) + (a*b^2*n*log(c*(d + e/x^(1/2)) 
^n))/(3*x^3) + (13489*b^3*d^6*n^3*log(d + e/x^(1/2)))/(1800*e^6) + (919*b^ 
3*d^2*n^3)/(7200*e^2*x^2) + (4669*b^3*d^4*n^3)/(3600*e^4*x) - (2059*b^3*d^ 
3*n^3)/(5400*e^3*x^(3/2)) - (13489*b^3*d^5*n^3)/(1800*e^5*x^(1/2)) + (a*b^ 
2*d^6*log(c*(d + e/x^(1/2))^n)^2)/e^6 - (49*b^3*d^6*n*log(c*(d + e/x^(1/2) 
)^n)^2)/(20*e^6) - (91*b^3*d*n^3)/(2250*e*x^(5/2)) + (a^2*b*d^6*n*log(d + 
e/x^(1/2)))/e^6 - (b^3*d*n*log(c*(d + e/x^(1/2))^n)^2)/(5*e*x^(5/2)) + (11 
*b^3*d*n^2*log(c*(d + e/x^(1/2))^n))/(75*e*x^(5/2)) + (a^2*b*d^2*n)/(4*e^2 
*x^2) + (a^2*b*d^4*n)/(2*e^4*x) + (11*a*b^2*d*n^2)/(75*e*x^(5/2)) - (a^2*b 
*d^3*n)/(3*e^3*x^(3/2)) - (a^2*b*d^5*n)/(e^5*x^(1/2)) - (49*a*b^2*d^6*n^2* 
log(d + e/x^(1/2)))/(10*e^6) + (b^3*d^2*n*log(c*(d + e/x^(1/2))^n)^2)/(4*e 
^2*x^2) - (37*b^3*d^2*n^2*log(c*(d + e/x^(1/2))^n))/(120*e^2*x^2) + (b^3*d 
^4*n*log(c*(d + e/x^(1/2))^n)^2)/(2*e^4*x) - (29*b^3*d^4*n^2*log(c*(d + e/ 
x^(1/2))^n))/(20*e^4*x) - (b^3*d^3*n*log(c*(d + e/x^(1/2))^n)^2)/(3*e^3*x^ 
(3/2)) + (19*b^3*d^3*n^2*log(c*(d + e/x^(1/2))^n))/(30*e^3*x^(3/2)) - (b^3 
*d^5*n*log(c*(d + e/x^(1/2))^n)^2)/(e^5*x^(1/2)) + (49*b^3*d^5*n^2*log(...
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 1153, normalized size of antiderivative = 1.27 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^4} \, dx =\text {Too large to display} \] Input:

int((a+b*log(c*(d+e/x^(1/2))^n))^3/x^4,x)
 

Output:

( - 108000*sqrt(x)*log(((sqrt(x)*d + e)**n*c)/x**(n/2))**2*b**3*d**5*e*n*x 
**2 - 36000*sqrt(x)*log(((sqrt(x)*d + e)**n*c)/x**(n/2))**2*b**3*d**3*e**3 
*n*x - 21600*sqrt(x)*log(((sqrt(x)*d + e)**n*c)/x**(n/2))**2*b**3*d*e**5*n 
 - 216000*sqrt(x)*log(((sqrt(x)*d + e)**n*c)/x**(n/2))*a*b**2*d**5*e*n*x** 
2 - 72000*sqrt(x)*log(((sqrt(x)*d + e)**n*c)/x**(n/2))*a*b**2*d**3*e**3*n* 
x - 43200*sqrt(x)*log(((sqrt(x)*d + e)**n*c)/x**(n/2))*a*b**2*d*e**5*n + 5 
29200*sqrt(x)*log(((sqrt(x)*d + e)**n*c)/x**(n/2))*b**3*d**5*e*n**2*x**2 + 
 68400*sqrt(x)*log(((sqrt(x)*d + e)**n*c)/x**(n/2))*b**3*d**3*e**3*n**2*x 
+ 15840*sqrt(x)*log(((sqrt(x)*d + e)**n*c)/x**(n/2))*b**3*d*e**5*n**2 - 10 
8000*sqrt(x)*a**2*b*d**5*e*n*x**2 - 36000*sqrt(x)*a**2*b*d**3*e**3*n*x - 2 
1600*sqrt(x)*a**2*b*d*e**5*n + 529200*sqrt(x)*a*b**2*d**5*e*n**2*x**2 + 68 
400*sqrt(x)*a*b**2*d**3*e**3*n**2*x + 15840*sqrt(x)*a*b**2*d*e**5*n**2 - 8 
09340*sqrt(x)*b**3*d**5*e*n**3*x**2 - 41180*sqrt(x)*b**3*d**3*e**3*n**3*x 
- 4368*sqrt(x)*b**3*d*e**5*n**3 + 36000*log(((sqrt(x)*d + e)**n*c)/x**(n/2 
))**3*b**3*d**6*x**3 - 36000*log(((sqrt(x)*d + e)**n*c)/x**(n/2))**3*b**3* 
e**6 + 108000*log(((sqrt(x)*d + e)**n*c)/x**(n/2))**2*a*b**2*d**6*x**3 - 1 
08000*log(((sqrt(x)*d + e)**n*c)/x**(n/2))**2*a*b**2*e**6 - 264600*log(((s 
qrt(x)*d + e)**n*c)/x**(n/2))**2*b**3*d**6*n*x**3 + 54000*log(((sqrt(x)*d 
+ e)**n*c)/x**(n/2))**2*b**3*d**4*e**2*n*x**2 + 27000*log(((sqrt(x)*d + e) 
**n*c)/x**(n/2))**2*b**3*d**2*e**4*n*x + 18000*log(((sqrt(x)*d + e)**n*...