\(\int x^2 (a+b \log (c (d+e \sqrt {x})^n))^3 \, dx\) [415]

Optimal result

Integrand size = 24, antiderivative size = 907 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx =\text {Too large to display} \] 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^6-1/108*b^3*n^3*(d+e*x^(1/2))^6 
/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^6-20/3*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^6-15/4*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*x^(1/2)/e^5+1/3*(d+e*x^(1/2))^6*(a+b*ln(c*(d+e*x^(1/2))^n))^3/e^6-12* 
a*b^2*d^5*n^2*x^(1/2)/e^5-12*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*l 
n(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
 

Mathematica [A] (verified)

Time = 0.51 (sec) , antiderivative size = 577, normalized size of antiderivative = 0.64 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx=\frac {b^3 e n^3 \sqrt {x} \left (809340 d^5-140070 d^4 e \sqrt {x}+41180 d^3 e^2 x-13785 d^2 e^3 x^{3/2}+4368 d e^4 x^2-1000 e^5 x^{5/2}\right )+1800 a^2 b n \left (147 d^6+60 d^5 e \sqrt {x}-30 d^4 e^2 x+20 d^3 e^3 x^{3/2}-15 d^2 e^4 x^2+12 d e^5 x^{5/2}-10 e^6 x^3\right )-36000 a^3 \left (d^6-e^6 x^3\right )+60 a b^2 n^2 \left (8111 d^6-8820 d^5 e \sqrt {x}+2610 d^4 e^2 x-1140 d^3 e^3 x^{3/2}+555 d^2 e^4 x^2-264 d e^5 x^{5/2}+100 e^6 x^3\right )-60 b \left (b^2 n^2 \left (13489 d^6+8820 d^5 e \sqrt {x}-2610 d^4 e^2 x+1140 d^3 e^3 x^{3/2}-555 d^2 e^4 x^2+264 d e^5 x^{5/2}-100 e^6 x^3\right )-60 a b n \left (147 d^6+60 d^5 e \sqrt {x}-30 d^4 e^2 x+20 d^3 e^3 x^{3/2}-15 d^2 e^4 x^2+12 d e^5 x^{5/2}-10 e^6 x^3\right )+1800 a^2 \left (d^6-e^6 x^3\right )\right ) \log \left (c \left (d+e \sqrt {x}\right )^n\right )-1800 b^2 \left (60 a \left (d^6-e^6 x^3\right )+b n \left (-147 d^6-60 d^5 e \sqrt {x}+30 d^4 e^2 x-20 d^3 e^3 x^{3/2}+15 d^2 e^4 x^2-12 d e^5 x^{5/2}+10 e^6 x^3\right )\right ) \log ^2\left (c \left (d+e \sqrt {x}\right )^n\right )-36000 b^3 \left (d^6-e^6 x^3\right ) \log ^3\left (c \left (d+e \sqrt {x}\right )^n\right )}{108000 e^6} \] Input:

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

Output:

(b^3*e*n^3*Sqrt[x]*(809340*d^5 - 140070*d^4*e*Sqrt[x] + 41180*d^3*e^2*x - 
13785*d^2*e^3*x^(3/2) + 4368*d*e^4*x^2 - 1000*e^5*x^(5/2)) + 1800*a^2*b*n* 
(147*d^6 + 60*d^5*e*Sqrt[x] - 30*d^4*e^2*x + 20*d^3*e^3*x^(3/2) - 15*d^2*e 
^4*x^2 + 12*d*e^5*x^(5/2) - 10*e^6*x^3) - 36000*a^3*(d^6 - e^6*x^3) + 60*a 
*b^2*n^2*(8111*d^6 - 8820*d^5*e*Sqrt[x] + 2610*d^4*e^2*x - 1140*d^3*e^3*x^ 
(3/2) + 555*d^2*e^4*x^2 - 264*d*e^5*x^(5/2) + 100*e^6*x^3) - 60*b*(b^2*n^2 
*(13489*d^6 + 8820*d^5*e*Sqrt[x] - 2610*d^4*e^2*x + 1140*d^3*e^3*x^(3/2) - 
 555*d^2*e^4*x^2 + 264*d*e^5*x^(5/2) - 100*e^6*x^3) - 60*a*b*n*(147*d^6 + 
60*d^5*e*Sqrt[x] - 30*d^4*e^2*x + 20*d^3*e^3*x^(3/2) - 15*d^2*e^4*x^2 + 12 
*d*e^5*x^(5/2) - 10*e^6*x^3) + 1800*a^2*(d^6 - e^6*x^3))*Log[c*(d + e*Sqrt 
[x])^n] - 1800*b^2*(60*a*(d^6 - e^6*x^3) + b*n*(-147*d^6 - 60*d^5*e*Sqrt[x 
] + 30*d^4*e^2*x - 20*d^3*e^3*x^(3/2) + 15*d^2*e^4*x^2 - 12*d*e^5*x^(5/2) 
+ 10*e^6*x^3))*Log[c*(d + e*Sqrt[x])^n]^2 - 36000*b^3*(d^6 - e^6*x^3)*Log[ 
c*(d + e*Sqrt[x])^n]^3)/(108000*e^6)
 

Rubi [A] (verified)

Time = 2.28 (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.

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

\(\Big \downarrow \) 2904

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

\(\Big \downarrow \) 2848

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

\(\Big \downarrow \) 2009

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

Input:

Int[x^2*(a + b*Log[c*(d + e*Sqrt[x])^n])^3,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*Sq 
rt[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*Sqrt[x])/e^5 + (6*b^3*d^5*n^3*Sqrt[x])/e^5 - (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*Sqrt[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*(a + b...
 

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]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 1197, normalized size of antiderivative = 1.32 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

1/108000*(36000*b^3*e^6*x^3*log(c)^3 - 1000*(b^3*e^6*n^3 - 6*a*b^2*e^6*n^2 
 + 18*a^2*b*e^6*n - 36*a^3*e^6)*x^3 + 36000*(b^3*e^6*n^3*x^3 - b^3*d^6*n^3 
)*log(e*sqrt(x) + d)^3 - 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^2 - 1800*(15*b^3*d^2*e^4*n^3*x^2 + 30*b^3*d^4*e^ 
2*n^3*x - 147*b^3*d^6*n^3 + 60*a*b^2*d^6*n^2 + 10*(b^3*e^6*n^3 - 6*a*b^2*e 
^6*n^2)*x^3 - 60*(b^3*e^6*n^2*x^3 - b^3*d^6*n^2)*log(c) - 4*(3*b^3*d*e^5*n 
^3*x^2 + 5*b^3*d^3*e^3*n^3*x + 15*b^3*d^5*e*n^3)*sqrt(x))*log(e*sqrt(x) + 
d)^2 - 9000*(3*b^3*d^2*e^4*n*x^2 + 6*b^3*d^4*e^2*n*x + 2*(b^3*e^6*n - 6*a* 
b^2*e^6)*x^3)*log(c)^2 - 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 - 60*(13489*b^3*d^6*n^3 - 8820*a*b^2*d^6*n^2 + 
1800*a^2*b*d^6*n - 100*(b^3*e^6*n^3 - 6*a*b^2*e^6*n^2 + 18*a^2*b*e^6*n)*x^ 
3 - 15*(37*b^3*d^2*e^4*n^3 - 60*a*b^2*d^2*e^4*n^2)*x^2 - 1800*(b^3*e^6*n*x 
^3 - b^3*d^6*n)*log(c)^2 - 90*(29*b^3*d^4*e^2*n^3 - 20*a*b^2*d^4*e^2*n^2)* 
x + 60*(15*b^3*d^2*e^4*n^2*x^2 + 30*b^3*d^4*e^2*n^2*x - 147*b^3*d^6*n^2 + 
60*a*b^2*d^6*n + 10*(b^3*e^6*n^2 - 6*a*b^2*e^6*n)*x^3)*log(c) + 12*(735*b^ 
3*d^5*e*n^3 - 300*a*b^2*d^5*e*n^2 + 2*(11*b^3*d*e^5*n^3 - 30*a*b^2*d*e^5*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*(3*b^3*d*e^ 
5*n^2*x^2 + 5*b^3*d^3*e^3*n^2*x + 15*b^3*d^5*e*n^2)*log(c))*sqrt(x))*log(e 
*sqrt(x) + d) + 300*(20*(b^3*e^6*n^2 - 6*a*b^2*e^6*n + 18*a^2*b*e^6)*x^3 + 
 3*(37*b^3*d^2*e^4*n^2 - 60*a*b^2*d^2*e^4*n)*x^2 + 18*(29*b^3*d^4*e^2*n...
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 666, normalized size of antiderivative = 0.73 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx =\text {Too large to display} \] Input:

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

Output:

1/3*b^3*x^3*log((e*sqrt(x) + d)^n*c)^3 + a*b^2*x^3*log((e*sqrt(x) + d)^n*c 
)^2 + a^2*b*x^3*log((e*sqrt(x) + d)^n*c) + 1/3*a^3*x^3 - 1/60*a^2*b*e*n*(6 
0*d^6*log(e*sqrt(x) + d)/e^7 + (10*e^5*x^3 - 12*d*e^4*x^(5/2) + 15*d^2*e^3 
*x^2 - 20*d^3*e^2*x^(3/2) + 30*d^4*e*x - 60*d^5*sqrt(x))/e^6) - 1/1800*(60 
*e*n*(60*d^6*log(e*sqrt(x) + d)/e^7 + (10*e^5*x^3 - 12*d*e^4*x^(5/2) + 15* 
d^2*e^3*x^2 - 20*d^3*e^2*x^(3/2) + 30*d^4*e*x - 60*d^5*sqrt(x))/e^6)*log(( 
e*sqrt(x) + d)^n*c) - (100*e^6*x^3 - 264*d*e^5*x^(5/2) + 555*d^2*e^4*x^2 + 
 1800*d^6*log(e*sqrt(x) + d)^2 - 1140*d^3*e^3*x^(3/2) + 2610*d^4*e^2*x + 8 
820*d^6*log(e*sqrt(x) + d) - 8820*d^5*e*sqrt(x))*n^2/e^6)*a*b^2 - 1/108000 
*(1800*e*n*(60*d^6*log(e*sqrt(x) + d)/e^7 + (10*e^5*x^3 - 12*d*e^4*x^(5/2) 
 + 15*d^2*e^3*x^2 - 20*d^3*e^2*x^(3/2) + 30*d^4*e*x - 60*d^5*sqrt(x))/e^6) 
*log((e*sqrt(x) + d)^n*c)^2 + e*n*((1000*e^6*x^3 + 36000*d^6*log(e*sqrt(x) 
 + d)^3 - 4368*d*e^5*x^(5/2) + 13785*d^2*e^4*x^2 + 264600*d^6*log(e*sqrt(x 
) + d)^2 - 41180*d^3*e^3*x^(3/2) + 140070*d^4*e^2*x + 809340*d^6*log(e*sqr 
t(x) + d) - 809340*d^5*e*sqrt(x))*n^2/e^7 - 60*(100*e^6*x^3 - 264*d*e^5*x^ 
(5/2) + 555*d^2*e^4*x^2 + 1800*d^6*log(e*sqrt(x) + d)^2 - 1140*d^3*e^3*x^( 
3/2) + 2610*d^4*e^2*x + 8820*d^6*log(e*sqrt(x) + d) - 8820*d^5*e*sqrt(x))* 
n*log((e*sqrt(x) + d)^n*c)/e^7))*b^3
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2160 vs. \(2 (787) = 1574\).

Time = 0.15 (sec) , antiderivative size = 2160, normalized size of antiderivative = 2.38 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

1/108000*(36000*b^3*e*x^3*log(c)^3 + 108000*a*b^2*e*x^3*log(c)^2 + 108000* 
a^2*b*e*x^3*log(c) + (36000*(e*sqrt(x) + d)^6*log(e*sqrt(x) + d)^3/e^5 - 2 
16000*(e*sqrt(x) + d)^5*d*log(e*sqrt(x) + d)^3/e^5 + 540000*(e*sqrt(x) + d 
)^4*d^2*log(e*sqrt(x) + d)^3/e^5 - 720000*(e*sqrt(x) + d)^3*d^3*log(e*sqrt 
(x) + d)^3/e^5 + 540000*(e*sqrt(x) + d)^2*d^4*log(e*sqrt(x) + d)^3/e^5 - 2 
16000*(e*sqrt(x) + d)*d^5*log(e*sqrt(x) + d)^3/e^5 - 18000*(e*sqrt(x) + d) 
^6*log(e*sqrt(x) + d)^2/e^5 + 129600*(e*sqrt(x) + d)^5*d*log(e*sqrt(x) + d 
)^2/e^5 - 405000*(e*sqrt(x) + d)^4*d^2*log(e*sqrt(x) + d)^2/e^5 + 720000*( 
e*sqrt(x) + d)^3*d^3*log(e*sqrt(x) + d)^2/e^5 - 810000*(e*sqrt(x) + d)^2*d 
^4*log(e*sqrt(x) + d)^2/e^5 + 648000*(e*sqrt(x) + d)*d^5*log(e*sqrt(x) + d 
)^2/e^5 + 6000*(e*sqrt(x) + d)^6*log(e*sqrt(x) + d)/e^5 - 51840*(e*sqrt(x) 
 + d)^5*d*log(e*sqrt(x) + d)/e^5 + 202500*(e*sqrt(x) + d)^4*d^2*log(e*sqrt 
(x) + d)/e^5 - 480000*(e*sqrt(x) + d)^3*d^3*log(e*sqrt(x) + d)/e^5 + 81000 
0*(e*sqrt(x) + d)^2*d^4*log(e*sqrt(x) + d)/e^5 - 1296000*(e*sqrt(x) + d)*d 
^5*log(e*sqrt(x) + d)/e^5 - 1000*(e*sqrt(x) + d)^6/e^5 + 10368*(e*sqrt(x) 
+ d)^5*d/e^5 - 50625*(e*sqrt(x) + d)^4*d^2/e^5 + 160000*(e*sqrt(x) + d)^3* 
d^3/e^5 - 405000*(e*sqrt(x) + d)^2*d^4/e^5 + 1296000*(e*sqrt(x) + d)*d^5/e 
^5)*b^3*n^3 + 36000*a^3*e*x^3 + 60*(1800*(e*sqrt(x) + d)^6*log(e*sqrt(x) + 
 d)^2/e^5 - 10800*(e*sqrt(x) + d)^5*d*log(e*sqrt(x) + d)^2/e^5 + 27000*(e* 
sqrt(x) + d)^4*d^2*log(e*sqrt(x) + d)^2/e^5 - 36000*(e*sqrt(x) + d)^3*d...
 

Mupad [B] (verification not implemented)

Time = 21.30 (sec) , antiderivative size = 976, normalized size of antiderivative = 1.08 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx =\text {Too large to display} \] Input:

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

Output:

(a^3*x^3)/3 + (b^3*x^3*log(c*(d + e*x^(1/2))^n)^3)/3 - (b^3*n^3*x^3)/108 + 
 a*b^2*x^3*log(c*(d + e*x^(1/2))^n)^2 - (b^3*n*x^3*log(c*(d + e*x^(1/2))^n 
)^2)/6 + (b^3*n^2*x^3*log(c*(d + e*x^(1/2))^n))/18 + (a*b^2*n^2*x^3)/18 - 
(b^3*d^6*log(c*(d + e*x^(1/2))^n)^3)/(3*e^6) + a^2*b*x^3*log(c*(d + e*x^(1 
/2))^n) - (a^2*b*n*x^3)/6 - (a*b^2*n*x^3*log(c*(d + e*x^(1/2))^n))/3 - (13 
489*b^3*d^6*n^3*log(d + e*x^(1/2)))/(1800*e^6) - (919*b^3*d^2*n^3*x^2)/(72 
00*e^2) + (2059*b^3*d^3*n^3*x^(3/2))/(5400*e^3) + (13489*b^3*d^5*n^3*x^(1/ 
2))/(1800*e^5) - (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*x^(5/2))/(2250*e) - 
 (4669*b^3*d^4*n^3*x)/(3600*e^4) - (a^2*b*d^6*n*log(d + e*x^(1/2)))/e^6 + 
(b^3*d*n*x^(5/2)*log(c*(d + e*x^(1/2))^n)^2)/(5*e) - (11*b^3*d*n^2*x^(5/2) 
*log(c*(d + e*x^(1/2))^n))/(75*e) - (b^3*d^4*n*x*log(c*(d + e*x^(1/2))^n)^ 
2)/(2*e^4) + (29*b^3*d^4*n^2*x*log(c*(d + e*x^(1/2))^n))/(20*e^4) - (a^2*b 
*d^2*n*x^2)/(4*e^2) - (11*a*b^2*d*n^2*x^(5/2))/(75*e) + (29*a*b^2*d^4*n^2* 
x)/(20*e^4) + (a^2*b*d^3*n*x^(3/2))/(3*e^3) + (a^2*b*d^5*n*x^(1/2))/e^5 + 
(49*a*b^2*d^6*n^2*log(d + e*x^(1/2)))/(10*e^6) - (b^3*d^2*n*x^2*log(c*(d + 
 e*x^(1/2))^n)^2)/(4*e^2) + (37*b^3*d^2*n^2*x^2*log(c*(d + e*x^(1/2))^n))/ 
(120*e^2) + (b^3*d^3*n*x^(3/2)*log(c*(d + e*x^(1/2))^n)^2)/(3*e^3) - (19*b 
^3*d^3*n^2*x^(3/2)*log(c*(d + e*x^(1/2))^n))/(30*e^3) + (b^3*d^5*n*x^(1/2) 
*log(c*(d + e*x^(1/2))^n)^2)/e^5 - (49*b^3*d^5*n^2*x^(1/2)*log(c*(d + e...
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 973, normalized size of antiderivative = 1.07 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx =\text {Too large to display} \] Input:

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

Output:

(108000*sqrt(x)*log((sqrt(x)*e + d)**n*c)**2*b**3*d**5*e*n + 36000*sqrt(x) 
*log((sqrt(x)*e + d)**n*c)**2*b**3*d**3*e**3*n*x + 21600*sqrt(x)*log((sqrt 
(x)*e + d)**n*c)**2*b**3*d*e**5*n*x**2 + 216000*sqrt(x)*log((sqrt(x)*e + d 
)**n*c)*a*b**2*d**5*e*n + 72000*sqrt(x)*log((sqrt(x)*e + d)**n*c)*a*b**2*d 
**3*e**3*n*x + 43200*sqrt(x)*log((sqrt(x)*e + d)**n*c)*a*b**2*d*e**5*n*x** 
2 - 529200*sqrt(x)*log((sqrt(x)*e + d)**n*c)*b**3*d**5*e*n**2 - 68400*sqrt 
(x)*log((sqrt(x)*e + d)**n*c)*b**3*d**3*e**3*n**2*x - 15840*sqrt(x)*log((s 
qrt(x)*e + d)**n*c)*b**3*d*e**5*n**2*x**2 + 108000*sqrt(x)*a**2*b*d**5*e*n 
 + 36000*sqrt(x)*a**2*b*d**3*e**3*n*x + 21600*sqrt(x)*a**2*b*d*e**5*n*x**2 
 - 529200*sqrt(x)*a*b**2*d**5*e*n**2 - 68400*sqrt(x)*a*b**2*d**3*e**3*n**2 
*x - 15840*sqrt(x)*a*b**2*d*e**5*n**2*x**2 + 809340*sqrt(x)*b**3*d**5*e*n* 
*3 + 41180*sqrt(x)*b**3*d**3*e**3*n**3*x + 4368*sqrt(x)*b**3*d*e**5*n**3*x 
**2 - 36000*log((sqrt(x)*e + d)**n*c)**3*b**3*d**6 + 36000*log((sqrt(x)*e 
+ d)**n*c)**3*b**3*e**6*x**3 - 108000*log((sqrt(x)*e + d)**n*c)**2*a*b**2* 
d**6 + 108000*log((sqrt(x)*e + d)**n*c)**2*a*b**2*e**6*x**3 + 264600*log(( 
sqrt(x)*e + d)**n*c)**2*b**3*d**6*n - 54000*log((sqrt(x)*e + d)**n*c)**2*b 
**3*d**4*e**2*n*x - 27000*log((sqrt(x)*e + d)**n*c)**2*b**3*d**2*e**4*n*x* 
*2 - 18000*log((sqrt(x)*e + d)**n*c)**2*b**3*e**6*n*x**3 - 108000*log((sqr 
t(x)*e + d)**n*c)*a**2*b*d**6 + 108000*log((sqrt(x)*e + d)**n*c)*a**2*b*e* 
*6*x**3 + 529200*log((sqrt(x)*e + d)**n*c)*a*b**2*d**6*n - 108000*log((...