\(\int x (a+b \log (c (d+e \sqrt {x})^n))^3 \, dx\) [416]

Optimal result

Integrand size = 22, antiderivative size = 595 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx=-\frac {9 b^3 d^2 n^3 \left (d+e \sqrt {x}\right )^2}{4 e^4}+\frac {4 b^3 d n^3 \left (d+e \sqrt {x}\right )^3}{9 e^4}-\frac {3 b^3 n^3 \left (d+e \sqrt {x}\right )^4}{64 e^4}-\frac {12 a b^2 d^3 n^2 \sqrt {x}}{e^3}+\frac {12 b^3 d^3 n^3 \sqrt {x}}{e^3}-\frac {12 b^3 d^3 n^2 \left (d+e \sqrt {x}\right ) \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{e^4}+\frac {9 b^2 d^2 n^2 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 e^4}-\frac {4 b^2 d n^2 \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{3 e^4}+\frac {3 b^2 n^2 \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{16 e^4}+\frac {6 b d^3 n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^4}-\frac {9 b d^2 n \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 e^4}+\frac {2 b d n \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^4}-\frac {3 b n \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{8 e^4}-\frac {2 d^3 \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^4}+\frac {3 d^2 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^4}-\frac {2 d \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^4}+\frac {\left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{2 e^4} \] Output:

-9/4*b^3*d^2*n^3*(d+e*x^(1/2))^2/e^4+4/9*b^3*d*n^3*(d+e*x^(1/2))^3/e^4-3/6 
4*b^3*n^3*(d+e*x^(1/2))^4/e^4-12*a*b^2*d^3*n^2*x^(1/2)/e^3+12*b^3*d^3*n^3* 
x^(1/2)/e^3-12*b^3*d^3*n^2*(d+e*x^(1/2))*ln(c*(d+e*x^(1/2))^n)/e^4+9/2*b^2 
*d^2*n^2*(d+e*x^(1/2))^2*(a+b*ln(c*(d+e*x^(1/2))^n))/e^4-4/3*b^2*d*n^2*(d+ 
e*x^(1/2))^3*(a+b*ln(c*(d+e*x^(1/2))^n))/e^4+3/16*b^2*n^2*(d+e*x^(1/2))^4* 
(a+b*ln(c*(d+e*x^(1/2))^n))/e^4+6*b*d^3*n*(d+e*x^(1/2))*(a+b*ln(c*(d+e*x^( 
1/2))^n))^2/e^4-9/2*b*d^2*n*(d+e*x^(1/2))^2*(a+b*ln(c*(d+e*x^(1/2))^n))^2/ 
e^4+2*b*d*n*(d+e*x^(1/2))^3*(a+b*ln(c*(d+e*x^(1/2))^n))^2/e^4-3/8*b*n*(d+e 
*x^(1/2))^4*(a+b*ln(c*(d+e*x^(1/2))^n))^2/e^4-2*d^3*(d+e*x^(1/2))*(a+b*ln( 
c*(d+e*x^(1/2))^n))^3/e^4+3*d^2*(d+e*x^(1/2))^2*(a+b*ln(c*(d+e*x^(1/2))^n) 
)^3/e^4-2*d*(d+e*x^(1/2))^3*(a+b*ln(c*(d+e*x^(1/2))^n))^3/e^4+1/2*(d+e*x^( 
1/2))^4*(a+b*ln(c*(d+e*x^(1/2))^n))^3/e^4
 

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 433, normalized size of antiderivative = 0.73 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx=\frac {b^3 e n^3 \sqrt {x} \left (4980 d^3-690 d^2 e \sqrt {x}+148 d e^2 x-27 e^3 x^{3/2}\right )+72 a^2 b n \left (25 d^4+12 d^3 e \sqrt {x}-6 d^2 e^2 x+4 d e^3 x^{3/2}-3 e^4 x^2\right )-288 a^3 \left (d^4-e^4 x^2\right )+12 a b^2 n^2 \left (161 d^4-300 d^3 e \sqrt {x}+78 d^2 e^2 x-28 d e^3 x^{3/2}+9 e^4 x^2\right )-12 b \left (b^2 n^2 \left (415 d^4+300 d^3 e \sqrt {x}-78 d^2 e^2 x+28 d e^3 x^{3/2}-9 e^4 x^2\right )-12 a b n \left (25 d^4+12 d^3 e \sqrt {x}-6 d^2 e^2 x+4 d e^3 x^{3/2}-3 e^4 x^2\right )+72 a^2 \left (d^4-e^4 x^2\right )\right ) \log \left (c \left (d+e \sqrt {x}\right )^n\right )-72 b^2 \left (12 a \left (d^4-e^4 x^2\right )+b n \left (-25 d^4-12 d^3 e \sqrt {x}+6 d^2 e^2 x-4 d e^3 x^{3/2}+3 e^4 x^2\right )\right ) \log ^2\left (c \left (d+e \sqrt {x}\right )^n\right )-288 b^3 \left (d^4-e^4 x^2\right ) \log ^3\left (c \left (d+e \sqrt {x}\right )^n\right )}{576 e^4} \] Input:

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

Output:

(b^3*e*n^3*Sqrt[x]*(4980*d^3 - 690*d^2*e*Sqrt[x] + 148*d*e^2*x - 27*e^3*x^ 
(3/2)) + 72*a^2*b*n*(25*d^4 + 12*d^3*e*Sqrt[x] - 6*d^2*e^2*x + 4*d*e^3*x^( 
3/2) - 3*e^4*x^2) - 288*a^3*(d^4 - e^4*x^2) + 12*a*b^2*n^2*(161*d^4 - 300* 
d^3*e*Sqrt[x] + 78*d^2*e^2*x - 28*d*e^3*x^(3/2) + 9*e^4*x^2) - 12*b*(b^2*n 
^2*(415*d^4 + 300*d^3*e*Sqrt[x] - 78*d^2*e^2*x + 28*d*e^3*x^(3/2) - 9*e^4* 
x^2) - 12*a*b*n*(25*d^4 + 12*d^3*e*Sqrt[x] - 6*d^2*e^2*x + 4*d*e^3*x^(3/2) 
 - 3*e^4*x^2) + 72*a^2*(d^4 - e^4*x^2))*Log[c*(d + e*Sqrt[x])^n] - 72*b^2* 
(12*a*(d^4 - e^4*x^2) + b*n*(-25*d^4 - 12*d^3*e*Sqrt[x] + 6*d^2*e^2*x - 4* 
d*e^3*x^(3/2) + 3*e^4*x^2))*Log[c*(d + e*Sqrt[x])^n]^2 - 288*b^3*(d^4 - e^ 
4*x^2)*Log[c*(d + e*Sqrt[x])^n]^3)/(576*e^4)
 

Rubi [A] (verified)

Time = 1.55 (sec) , antiderivative size = 598, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, 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.

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

\(\Big \downarrow \) 2904

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

\(\Big \downarrow \) 2848

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

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (\frac {9 b^2 d^2 n^2 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{4 e^4}+\frac {3 b^2 n^2 \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{32 e^4}-\frac {2 b^2 d n^2 \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{3 e^4}-\frac {6 a b^2 d^3 n^2 \sqrt {x}}{e^3}-\frac {d^3 \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^4}+\frac {3 b d^3 n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^4}+\frac {3 d^2 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{2 e^4}-\frac {9 b d^2 n \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{4 e^4}+\frac {\left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{4 e^4}-\frac {3 b n \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{16 e^4}-\frac {d \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^4}+\frac {b d n \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^4}-\frac {6 b^3 d^3 n^2 \left (d+e \sqrt {x}\right ) \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{e^4}+\frac {6 b^3 d^3 n^3 \sqrt {x}}{e^3}-\frac {9 b^3 d^2 n^3 \left (d+e \sqrt {x}\right )^2}{8 e^4}-\frac {3 b^3 n^3 \left (d+e \sqrt {x}\right )^4}{128 e^4}+\frac {2 b^3 d n^3 \left (d+e \sqrt {x}\right )^3}{9 e^4}\right )\)

Input:

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

Output:

2*((-9*b^3*d^2*n^3*(d + e*Sqrt[x])^2)/(8*e^4) + (2*b^3*d*n^3*(d + e*Sqrt[x 
])^3)/(9*e^4) - (3*b^3*n^3*(d + e*Sqrt[x])^4)/(128*e^4) - (6*a*b^2*d^3*n^2 
*Sqrt[x])/e^3 + (6*b^3*d^3*n^3*Sqrt[x])/e^3 - (6*b^3*d^3*n^2*(d + e*Sqrt[x 
])*Log[c*(d + e*Sqrt[x])^n])/e^4 + (9*b^2*d^2*n^2*(d + e*Sqrt[x])^2*(a + b 
*Log[c*(d + e*Sqrt[x])^n]))/(4*e^4) - (2*b^2*d*n^2*(d + e*Sqrt[x])^3*(a + 
b*Log[c*(d + e*Sqrt[x])^n]))/(3*e^4) + (3*b^2*n^2*(d + e*Sqrt[x])^4*(a + b 
*Log[c*(d + e*Sqrt[x])^n]))/(32*e^4) + (3*b*d^3*n*(d + e*Sqrt[x])*(a + b*L 
og[c*(d + e*Sqrt[x])^n])^2)/e^4 - (9*b*d^2*n*(d + e*Sqrt[x])^2*(a + b*Log[ 
c*(d + e*Sqrt[x])^n])^2)/(4*e^4) + (b*d*n*(d + e*Sqrt[x])^3*(a + b*Log[c*( 
d + e*Sqrt[x])^n])^2)/e^4 - (3*b*n*(d + e*Sqrt[x])^4*(a + b*Log[c*(d + e*S 
qrt[x])^n])^2)/(16*e^4) - (d^3*(d + e*Sqrt[x])*(a + b*Log[c*(d + e*Sqrt[x] 
)^n])^3)/e^4 + (3*d^2*(d + e*Sqrt[x])^2*(a + b*Log[c*(d + e*Sqrt[x])^n])^3 
)/(2*e^4) - (d*(d + e*Sqrt[x])^3*(a + b*Log[c*(d + e*Sqrt[x])^n])^3)/e^4 + 
 ((d + e*Sqrt[x])^4*(a + b*Log[c*(d + e*Sqrt[x])^n])^3)/(4*e^4))
 

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]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 861, normalized size of antiderivative = 1.45 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx =\text {Too large to display} \] Input:

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

Output:

1/576*(288*b^3*e^4*x^2*log(c)^3 + 288*(b^3*e^4*n^3*x^2 - b^3*d^4*n^3)*log( 
e*sqrt(x) + d)^3 - 9*(3*b^3*e^4*n^3 - 12*a*b^2*e^4*n^2 + 24*a^2*b*e^4*n - 
32*a^3*e^4)*x^2 - 72*(6*b^3*d^2*e^2*n^3*x - 25*b^3*d^4*n^3 + 12*a*b^2*d^4* 
n^2 + 3*(b^3*e^4*n^3 - 4*a*b^2*e^4*n^2)*x^2 - 12*(b^3*e^4*n^2*x^2 - b^3*d^ 
4*n^2)*log(c) - 4*(b^3*d*e^3*n^3*x + 3*b^3*d^3*e*n^3)*sqrt(x))*log(e*sqrt( 
x) + d)^2 - 216*(2*b^3*d^2*e^2*n*x + (b^3*e^4*n - 4*a*b^2*e^4)*x^2)*log(c) 
^2 - 6*(115*b^3*d^2*e^2*n^3 - 156*a*b^2*d^2*e^2*n^2 + 72*a^2*b*d^2*e^2*n)* 
x - 12*(415*b^3*d^4*n^3 - 300*a*b^2*d^4*n^2 + 72*a^2*b*d^4*n - 9*(b^3*e^4* 
n^3 - 4*a*b^2*e^4*n^2 + 8*a^2*b*e^4*n)*x^2 - 72*(b^3*e^4*n*x^2 - b^3*d^4*n 
)*log(c)^2 - 6*(13*b^3*d^2*e^2*n^3 - 12*a*b^2*d^2*e^2*n^2)*x + 12*(6*b^3*d 
^2*e^2*n^2*x - 25*b^3*d^4*n^2 + 12*a*b^2*d^4*n + 3*(b^3*e^4*n^2 - 4*a*b^2* 
e^4*n)*x^2)*log(c) + 4*(75*b^3*d^3*e*n^3 - 36*a*b^2*d^3*e*n^2 + (7*b^3*d*e 
^3*n^3 - 12*a*b^2*d*e^3*n^2)*x - 12*(b^3*d*e^3*n^2*x + 3*b^3*d^3*e*n^2)*lo 
g(c))*sqrt(x))*log(e*sqrt(x) + d) + 36*(3*(b^3*e^4*n^2 - 4*a*b^2*e^4*n + 8 
*a^2*b*e^4)*x^2 + 2*(13*b^3*d^2*e^2*n^2 - 12*a*b^2*d^2*e^2*n)*x)*log(c) + 
4*(1245*b^3*d^3*e*n^3 - 900*a*b^2*d^3*e*n^2 + 216*a^2*b*d^3*e*n + 72*(b^3* 
d*e^3*n*x + 3*b^3*d^3*e*n)*log(c)^2 + (37*b^3*d*e^3*n^3 - 84*a*b^2*d*e^3*n 
^2 + 72*a^2*b*d*e^3*n)*x - 12*(75*b^3*d^3*e*n^2 - 36*a*b^2*d^3*e*n + (7*b^ 
3*d*e^3*n^2 - 12*a*b^2*d*e^3*n)*x)*log(c))*sqrt(x))/e^4
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 536, normalized size of antiderivative = 0.90 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx=\frac {1}{2} \, b^{3} x^{2} \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )^{3} + \frac {3}{2} \, a b^{2} x^{2} \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )^{2} - \frac {1}{8} \, a^{2} b e n {\left (\frac {12 \, d^{4} \log \left (e \sqrt {x} + d\right )}{e^{5}} + \frac {3 \, e^{3} x^{2} - 4 \, d e^{2} x^{\frac {3}{2}} + 6 \, d^{2} e x - 12 \, d^{3} \sqrt {x}}{e^{4}}\right )} + \frac {3}{2} \, a^{2} b x^{2} \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + \frac {1}{2} \, a^{3} x^{2} - \frac {1}{48} \, {\left (12 \, e n {\left (\frac {12 \, d^{4} \log \left (e \sqrt {x} + d\right )}{e^{5}} + \frac {3 \, e^{3} x^{2} - 4 \, d e^{2} x^{\frac {3}{2}} + 6 \, d^{2} e x - 12 \, d^{3} \sqrt {x}}{e^{4}}\right )} \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) - \frac {{\left (9 \, e^{4} x^{2} + 72 \, d^{4} \log \left (e \sqrt {x} + d\right )^{2} - 28 \, d e^{3} x^{\frac {3}{2}} + 78 \, d^{2} e^{2} x + 300 \, d^{4} \log \left (e \sqrt {x} + d\right ) - 300 \, d^{3} e \sqrt {x}\right )} n^{2}}{e^{4}}\right )} a b^{2} - \frac {1}{576} \, {\left (72 \, e n {\left (\frac {12 \, d^{4} \log \left (e \sqrt {x} + d\right )}{e^{5}} + \frac {3 \, e^{3} x^{2} - 4 \, d e^{2} x^{\frac {3}{2}} + 6 \, d^{2} e x - 12 \, d^{3} \sqrt {x}}{e^{4}}\right )} \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )^{2} + e n {\left (\frac {{\left (288 \, d^{4} \log \left (e \sqrt {x} + d\right )^{3} + 27 \, e^{4} x^{2} + 1800 \, d^{4} \log \left (e \sqrt {x} + d\right )^{2} - 148 \, d e^{3} x^{\frac {3}{2}} + 690 \, d^{2} e^{2} x + 4980 \, d^{4} \log \left (e \sqrt {x} + d\right ) - 4980 \, d^{3} e \sqrt {x}\right )} n^{2}}{e^{5}} - \frac {12 \, {\left (9 \, e^{4} x^{2} + 72 \, d^{4} \log \left (e \sqrt {x} + d\right )^{2} - 28 \, d e^{3} x^{\frac {3}{2}} + 78 \, d^{2} e^{2} x + 300 \, d^{4} \log \left (e \sqrt {x} + d\right ) - 300 \, d^{3} e \sqrt {x}\right )} n \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )}{e^{5}}\right )}\right )} b^{3} \] Input:

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

Output:

1/2*b^3*x^2*log((e*sqrt(x) + d)^n*c)^3 + 3/2*a*b^2*x^2*log((e*sqrt(x) + d) 
^n*c)^2 - 1/8*a^2*b*e*n*(12*d^4*log(e*sqrt(x) + d)/e^5 + (3*e^3*x^2 - 4*d* 
e^2*x^(3/2) + 6*d^2*e*x - 12*d^3*sqrt(x))/e^4) + 3/2*a^2*b*x^2*log((e*sqrt 
(x) + d)^n*c) + 1/2*a^3*x^2 - 1/48*(12*e*n*(12*d^4*log(e*sqrt(x) + d)/e^5 
+ (3*e^3*x^2 - 4*d*e^2*x^(3/2) + 6*d^2*e*x - 12*d^3*sqrt(x))/e^4)*log((e*s 
qrt(x) + d)^n*c) - (9*e^4*x^2 + 72*d^4*log(e*sqrt(x) + d)^2 - 28*d*e^3*x^( 
3/2) + 78*d^2*e^2*x + 300*d^4*log(e*sqrt(x) + d) - 300*d^3*e*sqrt(x))*n^2/ 
e^4)*a*b^2 - 1/576*(72*e*n*(12*d^4*log(e*sqrt(x) + d)/e^5 + (3*e^3*x^2 - 4 
*d*e^2*x^(3/2) + 6*d^2*e*x - 12*d^3*sqrt(x))/e^4)*log((e*sqrt(x) + d)^n*c) 
^2 + e*n*((288*d^4*log(e*sqrt(x) + d)^3 + 27*e^4*x^2 + 1800*d^4*log(e*sqrt 
(x) + d)^2 - 148*d*e^3*x^(3/2) + 690*d^2*e^2*x + 4980*d^4*log(e*sqrt(x) + 
d) - 4980*d^3*e*sqrt(x))*n^2/e^5 - 12*(9*e^4*x^2 + 72*d^4*log(e*sqrt(x) + 
d)^2 - 28*d*e^3*x^(3/2) + 78*d^2*e^2*x + 300*d^4*log(e*sqrt(x) + d) - 300* 
d^3*e*sqrt(x))*n*log((e*sqrt(x) + d)^n*c)/e^5))*b^3
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1440 vs. \(2 (519) = 1038\).

Time = 0.15 (sec) , antiderivative size = 1440, normalized size of antiderivative = 2.42 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

1/576*(288*b^3*e*x^2*log(c)^3 + 864*a*b^2*e*x^2*log(c)^2 + (288*(e*sqrt(x) 
 + d)^4*log(e*sqrt(x) + d)^3/e^3 - 1152*(e*sqrt(x) + d)^3*d*log(e*sqrt(x) 
+ d)^3/e^3 + 1728*(e*sqrt(x) + d)^2*d^2*log(e*sqrt(x) + d)^3/e^3 - 1152*(e 
*sqrt(x) + d)*d^3*log(e*sqrt(x) + d)^3/e^3 - 216*(e*sqrt(x) + d)^4*log(e*s 
qrt(x) + d)^2/e^3 + 1152*(e*sqrt(x) + d)^3*d*log(e*sqrt(x) + d)^2/e^3 - 25 
92*(e*sqrt(x) + d)^2*d^2*log(e*sqrt(x) + d)^2/e^3 + 3456*(e*sqrt(x) + d)*d 
^3*log(e*sqrt(x) + d)^2/e^3 + 108*(e*sqrt(x) + d)^4*log(e*sqrt(x) + d)/e^3 
 - 768*(e*sqrt(x) + d)^3*d*log(e*sqrt(x) + d)/e^3 + 2592*(e*sqrt(x) + d)^2 
*d^2*log(e*sqrt(x) + d)/e^3 - 6912*(e*sqrt(x) + d)*d^3*log(e*sqrt(x) + d)/ 
e^3 - 27*(e*sqrt(x) + d)^4/e^3 + 256*(e*sqrt(x) + d)^3*d/e^3 - 1296*(e*sqr 
t(x) + d)^2*d^2/e^3 + 6912*(e*sqrt(x) + d)*d^3/e^3)*b^3*n^3 + 12*(72*(e*sq 
rt(x) + d)^4*log(e*sqrt(x) + d)^2/e^3 - 288*(e*sqrt(x) + d)^3*d*log(e*sqrt 
(x) + d)^2/e^3 + 432*(e*sqrt(x) + d)^2*d^2*log(e*sqrt(x) + d)^2/e^3 - 288* 
(e*sqrt(x) + d)*d^3*log(e*sqrt(x) + d)^2/e^3 - 36*(e*sqrt(x) + d)^4*log(e* 
sqrt(x) + d)/e^3 + 192*(e*sqrt(x) + d)^3*d*log(e*sqrt(x) + d)/e^3 - 432*(e 
*sqrt(x) + d)^2*d^2*log(e*sqrt(x) + d)/e^3 + 576*(e*sqrt(x) + d)*d^3*log(e 
*sqrt(x) + d)/e^3 + 9*(e*sqrt(x) + d)^4/e^3 - 64*(e*sqrt(x) + d)^3*d/e^3 + 
 216*(e*sqrt(x) + d)^2*d^2/e^3 - 576*(e*sqrt(x) + d)*d^3/e^3)*b^3*n^2*log( 
c) + 864*a^2*b*e*x^2*log(c) + 72*(12*(e*sqrt(x) + d)^4*log(e*sqrt(x) + d)/ 
e^3 - 48*(e*sqrt(x) + d)^3*d*log(e*sqrt(x) + d)/e^3 + 72*(e*sqrt(x) + d...
 

Mupad [B] (verification not implemented)

Time = 15.28 (sec) , antiderivative size = 840, normalized size of antiderivative = 1.41 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx =\text {Too large to display} \] Input:

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

Output:

log(c*(d + e*x^(1/2))^n)^3*((b^3*x^2)/2 - (b^3*d^4)/(2*e^4)) - x^(3/2)*((d 
*(2*a^3 - (3*b^3*n^3)/16 + (3*a*b^2*n^2)/4 - (3*a^2*b*n)/2))/(3*e) - (d*(2 
4*a^3 + 7*b^3*n^3 - 12*a*b^2*n^2))/(36*e)) - log(c*(d + e*x^(1/2))^n)^2*(( 
x^(3/2)*((b^2*d*(4*a - b*n))/e - (4*a*b^2*d)/e))/2 - (3*b^2*x^2*(4*a - b*n 
))/8 + (d*(12*a*b^2*d^3 - 25*b^3*d^3*n))/(8*e^4) + (d^2*x^(1/2)*((6*b^2*d* 
(4*a - b*n))/e - (24*a*b^2*d)/e))/(4*e^2) - (d*x*((6*b^2*d*(4*a - b*n))/e 
- (24*a*b^2*d)/e))/(8*e)) + x*((d*((d*(2*a^3 - (3*b^3*n^3)/16 + (3*a*b^2*n 
^2)/4 - (3*a^2*b*n)/2))/e - (d*(24*a^3 + 7*b^3*n^3 - 12*a*b^2*n^2))/(12*e) 
))/(2*e) + (b^2*d^2*n^2*(12*a - 13*b*n))/(16*e^2)) - x^(1/2)*((d*((d*((d*( 
2*a^3 - (3*b^3*n^3)/16 + (3*a*b^2*n^2)/4 - (3*a^2*b*n)/2))/e - (d*(24*a^3 
+ 7*b^3*n^3 - 12*a*b^2*n^2))/(12*e)))/e + (b^2*d^2*n^2*(12*a - 13*b*n))/(8 
*e^2)))/e + (b^2*d^3*n^2*(12*a - 25*b*n))/(4*e^3)) + x^2*(a^3/2 - (3*b^3*n 
^3)/64 + (3*a*b^2*n^2)/16 - (3*a^2*b*n)/8) + (log(c*(d + e*x^(1/2))^n)*((x 
^(3/2)*(16*b*d*e^3*(6*a^2 - b^2*n^2) - 12*b*d*e^3*(8*a^2 + b^2*n^2 - 4*a*b 
*n)))/(12*e^2) - (x*((d*(16*b*d*e^3*(6*a^2 - b^2*n^2) - 12*b*d*e^3*(8*a^2 
+ b^2*n^2 - 4*a*b*n)))/e - 24*b^3*d^2*e^2*n^2))/(8*e^2) + (x^(1/2)*((d*((d 
*(16*b*d*e^3*(6*a^2 - b^2*n^2) - 12*b*d*e^3*(8*a^2 + b^2*n^2 - 4*a*b*n)))/ 
e - 24*b^3*d^2*e^2*n^2))/e - 48*b^3*d^3*e*n^2))/(4*e^2) + (3*b*e^2*x^2*(8* 
a^2 + b^2*n^2 - 4*a*b*n))/4))/(4*e^2) - (log(d + e*x^(1/2))*(415*b^3*d^4*n 
^3 - 300*a*b^2*d^4*n^2 + 72*a^2*b*d^4*n))/(48*e^4)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 693, normalized size of antiderivative = 1.16 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx =\text {Too large to display} \] Input:

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

Output:

(864*sqrt(x)*log((sqrt(x)*e + d)**n*c)**2*b**3*d**3*e*n + 288*sqrt(x)*log( 
(sqrt(x)*e + d)**n*c)**2*b**3*d*e**3*n*x + 1728*sqrt(x)*log((sqrt(x)*e + d 
)**n*c)*a*b**2*d**3*e*n + 576*sqrt(x)*log((sqrt(x)*e + d)**n*c)*a*b**2*d*e 
**3*n*x - 3600*sqrt(x)*log((sqrt(x)*e + d)**n*c)*b**3*d**3*e*n**2 - 336*sq 
rt(x)*log((sqrt(x)*e + d)**n*c)*b**3*d*e**3*n**2*x + 864*sqrt(x)*a**2*b*d* 
*3*e*n + 288*sqrt(x)*a**2*b*d*e**3*n*x - 3600*sqrt(x)*a*b**2*d**3*e*n**2 - 
 336*sqrt(x)*a*b**2*d*e**3*n**2*x + 4980*sqrt(x)*b**3*d**3*e*n**3 + 148*sq 
rt(x)*b**3*d*e**3*n**3*x - 288*log((sqrt(x)*e + d)**n*c)**3*b**3*d**4 + 28 
8*log((sqrt(x)*e + d)**n*c)**3*b**3*e**4*x**2 - 864*log((sqrt(x)*e + d)**n 
*c)**2*a*b**2*d**4 + 864*log((sqrt(x)*e + d)**n*c)**2*a*b**2*e**4*x**2 + 1 
800*log((sqrt(x)*e + d)**n*c)**2*b**3*d**4*n - 432*log((sqrt(x)*e + d)**n* 
c)**2*b**3*d**2*e**2*n*x - 216*log((sqrt(x)*e + d)**n*c)**2*b**3*e**4*n*x* 
*2 - 864*log((sqrt(x)*e + d)**n*c)*a**2*b*d**4 + 864*log((sqrt(x)*e + d)** 
n*c)*a**2*b*e**4*x**2 + 3600*log((sqrt(x)*e + d)**n*c)*a*b**2*d**4*n - 864 
*log((sqrt(x)*e + d)**n*c)*a*b**2*d**2*e**2*n*x - 432*log((sqrt(x)*e + d)* 
*n*c)*a*b**2*e**4*n*x**2 - 4980*log((sqrt(x)*e + d)**n*c)*b**3*d**4*n**2 + 
 936*log((sqrt(x)*e + d)**n*c)*b**3*d**2*e**2*n**2*x + 108*log((sqrt(x)*e 
+ d)**n*c)*b**3*e**4*n**2*x**2 + 288*a**3*e**4*x**2 - 432*a**2*b*d**2*e**2 
*n*x - 216*a**2*b*e**4*n*x**2 + 936*a*b**2*d**2*e**2*n**2*x + 108*a*b**2*e 
**4*n**2*x**2 - 690*b**3*d**2*e**2*n**3*x - 27*b**3*e**4*n**3*x**2)/(57...