\(\int (a+b \log (c (d+e \sqrt {x})^n))^3 \, dx\) [417]

Optimal result

Integrand size = 20, antiderivative size = 284 \[ \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx=-\frac {3 b^3 n^3 \left (d+e \sqrt {x}\right )^2}{4 e^2}-\frac {12 a b^2 d n^2 \sqrt {x}}{e}+\frac {12 b^3 d n^3 \sqrt {x}}{e}-\frac {12 b^3 d n^2 \left (d+e \sqrt {x}\right ) \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{e^2}+\frac {3 b^2 n^2 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 e^2}+\frac {6 b d n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^2}-\frac {3 b n \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 e^2}-\frac {2 d \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^2}+\frac {\left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.85 \[ \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx=\frac {-8 d \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3+4 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3+24 b d n \left (\left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2-2 b n \left (e (a-b n) \sqrt {x}+b \left (d+e \sqrt {x}\right ) \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )\right )-3 b n \left (2 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2+b n \left (b e n \left (2 d \sqrt {x}+e x\right )-2 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )\right )\right )}{4 e^2} \] Input:

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

Output:

(-8*d*(d + e*Sqrt[x])*(a + b*Log[c*(d + e*Sqrt[x])^n])^3 + 4*(d + e*Sqrt[x 
])^2*(a + b*Log[c*(d + e*Sqrt[x])^n])^3 + 24*b*d*n*((d + e*Sqrt[x])*(a + b 
*Log[c*(d + e*Sqrt[x])^n])^2 - 2*b*n*(e*(a - b*n)*Sqrt[x] + b*(d + e*Sqrt[ 
x])*Log[c*(d + e*Sqrt[x])^n])) - 3*b*n*(2*(d + e*Sqrt[x])^2*(a + b*Log[c*( 
d + e*Sqrt[x])^n])^2 + b*n*(b*e*n*(2*d*Sqrt[x] + e*x) - 2*(d + e*Sqrt[x])^ 
2*(a + b*Log[c*(d + e*Sqrt[x])^n]))))/(4*e^2)
 

Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2901, 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.

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

\(\Big \downarrow \) 2901

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

\(\Big \downarrow \) 2848

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

2*((-3*b^3*n^3*(d + e*Sqrt[x])^2)/(8*e^2) - (6*a*b^2*d*n^2*Sqrt[x])/e + (6 
*b^3*d*n^3*Sqrt[x])/e - (6*b^3*d*n^2*(d + e*Sqrt[x])*Log[c*(d + e*Sqrt[x]) 
^n])/e^2 + (3*b^2*n^2*(d + e*Sqrt[x])^2*(a + b*Log[c*(d + e*Sqrt[x])^n]))/ 
(4*e^2) + (3*b*d*n*(d + e*Sqrt[x])*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/e^2 
 - (3*b*n*(d + e*Sqrt[x])^2*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/(4*e^2) - 
(d*(d + e*Sqrt[x])*(a + b*Log[c*(d + e*Sqrt[x])^n])^3)/e^2 + ((d + e*Sqrt[ 
x])^2*(a + b*Log[c*(d + e*Sqrt[x])^n])^3)/(2*e^2))
 

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 2901

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_), x_Symbo 
l] :> With[{k = Denominator[n]}, Simp[k   Subst[Int[x^(k - 1)*(a + b*Log[c* 
(d + e*x^(k*n))^p])^q, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, 
 x] && FractionQ[n]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 527 vs. \(2 (248) = 496\).

Time = 0.10 (sec) , antiderivative size = 527, normalized size of antiderivative = 1.86 \[ \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx=\frac {4 \, b^{3} e^{2} x \log \left (c\right )^{3} + 4 \, {\left (b^{3} e^{2} n^{3} x - b^{3} d^{2} n^{3}\right )} \log \left (e \sqrt {x} + d\right )^{3} - 6 \, {\left (b^{3} e^{2} n - 2 \, a b^{2} e^{2}\right )} x \log \left (c\right )^{2} + 6 \, {\left (2 \, b^{3} d e n^{3} \sqrt {x} + 3 \, b^{3} d^{2} n^{3} - 2 \, a b^{2} d^{2} n^{2} - {\left (b^{3} e^{2} n^{3} - 2 \, a b^{2} e^{2} n^{2}\right )} x + 2 \, {\left (b^{3} e^{2} n^{2} x - b^{3} d^{2} n^{2}\right )} \log \left (c\right )\right )} \log \left (e \sqrt {x} + d\right )^{2} + 6 \, {\left (b^{3} e^{2} n^{2} - 2 \, a b^{2} e^{2} n + 2 \, a^{2} b e^{2}\right )} x \log \left (c\right ) - {\left (3 \, b^{3} e^{2} n^{3} - 6 \, a b^{2} e^{2} n^{2} + 6 \, a^{2} b e^{2} n - 4 \, a^{3} e^{2}\right )} x - 6 \, {\left (7 \, b^{3} d^{2} n^{3} - 6 \, a b^{2} d^{2} n^{2} + 2 \, a^{2} b d^{2} n - 2 \, {\left (b^{3} e^{2} n x - b^{3} d^{2} n\right )} \log \left (c\right )^{2} - {\left (b^{3} e^{2} n^{3} - 2 \, a b^{2} e^{2} n^{2} + 2 \, a^{2} b e^{2} n\right )} x - 2 \, {\left (3 \, b^{3} d^{2} n^{2} - 2 \, a b^{2} d^{2} n - {\left (b^{3} e^{2} n^{2} - 2 \, a b^{2} e^{2} n\right )} x\right )} \log \left (c\right ) + 2 \, {\left (3 \, b^{3} d e n^{3} - 2 \, b^{3} d e n^{2} \log \left (c\right ) - 2 \, a b^{2} d e n^{2}\right )} \sqrt {x}\right )} \log \left (e \sqrt {x} + d\right ) + 6 \, {\left (7 \, b^{3} d e n^{3} + 2 \, b^{3} d e n \log \left (c\right )^{2} - 6 \, a b^{2} d e n^{2} + 2 \, a^{2} b d e n - 2 \, {\left (3 \, b^{3} d e n^{2} - 2 \, a b^{2} d e n\right )} \log \left (c\right )\right )} \sqrt {x}}{4 \, e^{2}} \] Input:

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

Output:

1/4*(4*b^3*e^2*x*log(c)^3 + 4*(b^3*e^2*n^3*x - b^3*d^2*n^3)*log(e*sqrt(x) 
+ d)^3 - 6*(b^3*e^2*n - 2*a*b^2*e^2)*x*log(c)^2 + 6*(2*b^3*d*e*n^3*sqrt(x) 
 + 3*b^3*d^2*n^3 - 2*a*b^2*d^2*n^2 - (b^3*e^2*n^3 - 2*a*b^2*e^2*n^2)*x + 2 
*(b^3*e^2*n^2*x - b^3*d^2*n^2)*log(c))*log(e*sqrt(x) + d)^2 + 6*(b^3*e^2*n 
^2 - 2*a*b^2*e^2*n + 2*a^2*b*e^2)*x*log(c) - (3*b^3*e^2*n^3 - 6*a*b^2*e^2* 
n^2 + 6*a^2*b*e^2*n - 4*a^3*e^2)*x - 6*(7*b^3*d^2*n^3 - 6*a*b^2*d^2*n^2 + 
2*a^2*b*d^2*n - 2*(b^3*e^2*n*x - b^3*d^2*n)*log(c)^2 - (b^3*e^2*n^3 - 2*a* 
b^2*e^2*n^2 + 2*a^2*b*e^2*n)*x - 2*(3*b^3*d^2*n^2 - 2*a*b^2*d^2*n - (b^3*e 
^2*n^2 - 2*a*b^2*e^2*n)*x)*log(c) + 2*(3*b^3*d*e*n^3 - 2*b^3*d*e*n^2*log(c 
) - 2*a*b^2*d*e*n^2)*sqrt(x))*log(e*sqrt(x) + d) + 6*(7*b^3*d*e*n^3 + 2*b^ 
3*d*e*n*log(c)^2 - 6*a*b^2*d*e*n^2 + 2*a^2*b*d*e*n - 2*(3*b^3*d*e*n^2 - 2* 
a*b^2*d*e*n)*log(c))*sqrt(x))/e^2
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.34 \[ \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx=-\frac {3}{2} \, {\left (e n {\left (\frac {2 \, d^{2} \log \left (e \sqrt {x} + d\right )}{e^{3}} + \frac {e x - 2 \, d \sqrt {x}}{e^{2}}\right )} - 2 \, x \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )\right )} a^{2} b - \frac {3}{2} \, {\left (2 \, e n {\left (\frac {2 \, d^{2} \log \left (e \sqrt {x} + d\right )}{e^{3}} + \frac {e x - 2 \, d \sqrt {x}}{e^{2}}\right )} \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) - 2 \, x \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )^{2} - \frac {{\left (2 \, d^{2} \log \left (e \sqrt {x} + d\right )^{2} + e^{2} x + 6 \, d^{2} \log \left (e \sqrt {x} + d\right ) - 6 \, d e \sqrt {x}\right )} n^{2}}{e^{2}}\right )} a b^{2} - \frac {1}{4} \, {\left (6 \, e n {\left (\frac {2 \, d^{2} \log \left (e \sqrt {x} + d\right )}{e^{3}} + \frac {e x - 2 \, d \sqrt {x}}{e^{2}}\right )} \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )^{2} - 4 \, x \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )^{3} + e n {\left (\frac {{\left (4 \, d^{2} \log \left (e \sqrt {x} + d\right )^{3} + 18 \, d^{2} \log \left (e \sqrt {x} + d\right )^{2} + 3 \, e^{2} x + 42 \, d^{2} \log \left (e \sqrt {x} + d\right ) - 42 \, d e \sqrt {x}\right )} n^{2}}{e^{3}} - \frac {6 \, {\left (2 \, d^{2} \log \left (e \sqrt {x} + d\right )^{2} + e^{2} x + 6 \, d^{2} \log \left (e \sqrt {x} + d\right ) - 6 \, d e \sqrt {x}\right )} n \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )}{e^{3}}\right )}\right )} b^{3} + a^{3} x \] Input:

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

Output:

-3/2*(e*n*(2*d^2*log(e*sqrt(x) + d)/e^3 + (e*x - 2*d*sqrt(x))/e^2) - 2*x*l 
og((e*sqrt(x) + d)^n*c))*a^2*b - 3/2*(2*e*n*(2*d^2*log(e*sqrt(x) + d)/e^3 
+ (e*x - 2*d*sqrt(x))/e^2)*log((e*sqrt(x) + d)^n*c) - 2*x*log((e*sqrt(x) + 
 d)^n*c)^2 - (2*d^2*log(e*sqrt(x) + d)^2 + e^2*x + 6*d^2*log(e*sqrt(x) + d 
) - 6*d*e*sqrt(x))*n^2/e^2)*a*b^2 - 1/4*(6*e*n*(2*d^2*log(e*sqrt(x) + d)/e 
^3 + (e*x - 2*d*sqrt(x))/e^2)*log((e*sqrt(x) + d)^n*c)^2 - 4*x*log((e*sqrt 
(x) + d)^n*c)^3 + e*n*((4*d^2*log(e*sqrt(x) + d)^3 + 18*d^2*log(e*sqrt(x) 
+ d)^2 + 3*e^2*x + 42*d^2*log(e*sqrt(x) + d) - 42*d*e*sqrt(x))*n^2/e^3 - 6 
*(2*d^2*log(e*sqrt(x) + d)^2 + e^2*x + 6*d^2*log(e*sqrt(x) + d) - 6*d*e*sq 
rt(x))*n*log((e*sqrt(x) + d)^n*c)/e^3))*b^3 + a^3*x
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 714 vs. \(2 (248) = 496\).

Time = 0.13 (sec) , antiderivative size = 714, normalized size of antiderivative = 2.51 \[ \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx =\text {Too large to display} \] Input:

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

Output:

1/4*((4*(e*sqrt(x) + d)^2*log(e*sqrt(x) + d)^3 - 8*(e*sqrt(x) + d)*d*log(e 
*sqrt(x) + d)^3 - 6*(e*sqrt(x) + d)^2*log(e*sqrt(x) + d)^2 + 24*(e*sqrt(x) 
 + d)*d*log(e*sqrt(x) + d)^2 + 6*(e*sqrt(x) + d)^2*log(e*sqrt(x) + d) - 48 
*(e*sqrt(x) + d)*d*log(e*sqrt(x) + d) - 3*(e*sqrt(x) + d)^2 + 48*(e*sqrt(x 
) + d)*d)*b^3*n^3/e + 6*(2*(e*sqrt(x) + d)^2*log(e*sqrt(x) + d)^2 - 4*(e*s 
qrt(x) + d)*d*log(e*sqrt(x) + d)^2 - 2*(e*sqrt(x) + d)^2*log(e*sqrt(x) + d 
) + 8*(e*sqrt(x) + d)*d*log(e*sqrt(x) + d) + (e*sqrt(x) + d)^2 - 8*(e*sqrt 
(x) + d)*d)*b^3*n^2*log(c)/e + 6*(2*(e*sqrt(x) + d)^2*log(e*sqrt(x) + d) - 
 4*(e*sqrt(x) + d)*d*log(e*sqrt(x) + d) - (e*sqrt(x) + d)^2 + 4*(e*sqrt(x) 
 + d)*d)*b^3*n*log(c)^2/e + 4*((e*sqrt(x) + d)^2 - 2*(e*sqrt(x) + d)*d)*b^ 
3*log(c)^3/e + 6*(2*(e*sqrt(x) + d)^2*log(e*sqrt(x) + d)^2 - 4*(e*sqrt(x) 
+ d)*d*log(e*sqrt(x) + d)^2 - 2*(e*sqrt(x) + d)^2*log(e*sqrt(x) + d) + 8*( 
e*sqrt(x) + d)*d*log(e*sqrt(x) + d) + (e*sqrt(x) + d)^2 - 8*(e*sqrt(x) + d 
)*d)*a*b^2*n^2/e + 12*(2*(e*sqrt(x) + d)^2*log(e*sqrt(x) + d) - 4*(e*sqrt( 
x) + d)*d*log(e*sqrt(x) + d) - (e*sqrt(x) + d)^2 + 4*(e*sqrt(x) + d)*d)*a* 
b^2*n*log(c)/e + 12*((e*sqrt(x) + d)^2 - 2*(e*sqrt(x) + d)*d)*a*b^2*log(c) 
^2/e + 6*(2*(e*sqrt(x) + d)^2*log(e*sqrt(x) + d) - 4*(e*sqrt(x) + d)*d*log 
(e*sqrt(x) + d) - (e*sqrt(x) + d)^2 + 4*(e*sqrt(x) + d)*d)*a^2*b*n/e + 12* 
((e*sqrt(x) + d)^2 - 2*(e*sqrt(x) + d)*d)*a^2*b*log(c)/e + 4*((e*sqrt(x) + 
 d)^2 - 2*(e*sqrt(x) + d)*d)*a^3/e)/e
 

Mupad [B] (verification not implemented)

Time = 15.05 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.23 \[ \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx=x\,\left (a^3-\frac {3\,a^2\,b\,n}{2}+\frac {3\,a\,b^2\,n^2}{2}-\frac {3\,b^3\,n^3}{4}\right )-\sqrt {x}\,\left (\frac {d\,\left (2\,a^3-3\,a^2\,b\,n+3\,a\,b^2\,n^2-\frac {3\,b^3\,n^3}{2}\right )}{e}-\frac {d\,\left (2\,a^3-6\,a\,b^2\,n^2+9\,b^3\,n^3\right )}{e}\right )+{\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}^3\,\left (b^3\,x-\frac {b^3\,d^2}{e^2}\right )-\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )\,\left (\sqrt {x}\,\left (\frac {3\,b\,d\,\left (2\,a^2-2\,a\,b\,n+b^2\,n^2\right )}{e}-\frac {6\,b\,d\,\left (a^2-b^2\,n^2\right )}{e}\right )-\frac {3\,b\,x\,\left (2\,a^2-2\,a\,b\,n+b^2\,n^2\right )}{2}\right )-{\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}^2\,\left (\sqrt {x}\,\left (\frac {3\,b^2\,d\,\left (2\,a-b\,n\right )}{e}-\frac {6\,a\,b^2\,d}{e}\right )+\frac {3\,d\,\left (2\,a\,b^2\,d-3\,b^3\,d\,n\right )}{2\,e^2}-\frac {3\,b^2\,x\,\left (2\,a-b\,n\right )}{2}\right )-\frac {\ln \left (d+e\,\sqrt {x}\right )\,\left (6\,a^2\,b\,d^2\,n-18\,a\,b^2\,d^2\,n^2+21\,b^3\,d^2\,n^3\right )}{2\,e^2} \] Input:

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

Output:

x*(a^3 - (3*b^3*n^3)/4 + (3*a*b^2*n^2)/2 - (3*a^2*b*n)/2) - x^(1/2)*((d*(2 
*a^3 - (3*b^3*n^3)/2 + 3*a*b^2*n^2 - 3*a^2*b*n))/e - (d*(2*a^3 + 9*b^3*n^3 
 - 6*a*b^2*n^2))/e) + log(c*(d + e*x^(1/2))^n)^3*(b^3*x - (b^3*d^2)/e^2) - 
 log(c*(d + e*x^(1/2))^n)*(x^(1/2)*((3*b*d*(2*a^2 + b^2*n^2 - 2*a*b*n))/e 
- (6*b*d*(a^2 - b^2*n^2))/e) - (3*b*x*(2*a^2 + b^2*n^2 - 2*a*b*n))/2) - lo 
g(c*(d + e*x^(1/2))^n)^2*(x^(1/2)*((3*b^2*d*(2*a - b*n))/e - (6*a*b^2*d)/e 
) + (3*d*(2*a*b^2*d - 3*b^3*d*n))/(2*e^2) - (3*b^2*x*(2*a - b*n))/2) - (lo 
g(d + e*x^(1/2))*(21*b^3*d^2*n^3 - 18*a*b^2*d^2*n^2 + 6*a^2*b*d^2*n))/(2*e 
^2)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.47 \[ \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx=\frac {12 \sqrt {x}\, \mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right )^{2} b^{3} d e n +24 \sqrt {x}\, \mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right ) a \,b^{2} d e n -36 \sqrt {x}\, \mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right ) b^{3} d e \,n^{2}+12 \sqrt {x}\, a^{2} b d e n -36 \sqrt {x}\, a \,b^{2} d e \,n^{2}+42 \sqrt {x}\, b^{3} d e \,n^{3}-4 \mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right )^{3} b^{3} d^{2}+4 \mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right )^{3} b^{3} e^{2} x -12 \mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right )^{2} a \,b^{2} d^{2}+12 \mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right )^{2} a \,b^{2} e^{2} x +18 \mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right )^{2} b^{3} d^{2} n -6 \mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right )^{2} b^{3} e^{2} n x -12 \,\mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right ) a^{2} b \,d^{2}+12 \,\mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right ) a^{2} b \,e^{2} x +36 \,\mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right ) a \,b^{2} d^{2} n -12 \,\mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right ) a \,b^{2} e^{2} n x -42 \,\mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right ) b^{3} d^{2} n^{2}+6 \,\mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right ) b^{3} e^{2} n^{2} x +4 a^{3} e^{2} x -6 a^{2} b \,e^{2} n x +6 a \,b^{2} e^{2} n^{2} x -3 b^{3} e^{2} n^{3} x}{4 e^{2}} \] Input:

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

Output:

(12*sqrt(x)*log((sqrt(x)*e + d)**n*c)**2*b**3*d*e*n + 24*sqrt(x)*log((sqrt 
(x)*e + d)**n*c)*a*b**2*d*e*n - 36*sqrt(x)*log((sqrt(x)*e + d)**n*c)*b**3* 
d*e*n**2 + 12*sqrt(x)*a**2*b*d*e*n - 36*sqrt(x)*a*b**2*d*e*n**2 + 42*sqrt( 
x)*b**3*d*e*n**3 - 4*log((sqrt(x)*e + d)**n*c)**3*b**3*d**2 + 4*log((sqrt( 
x)*e + d)**n*c)**3*b**3*e**2*x - 12*log((sqrt(x)*e + d)**n*c)**2*a*b**2*d* 
*2 + 12*log((sqrt(x)*e + d)**n*c)**2*a*b**2*e**2*x + 18*log((sqrt(x)*e + d 
)**n*c)**2*b**3*d**2*n - 6*log((sqrt(x)*e + d)**n*c)**2*b**3*e**2*n*x - 12 
*log((sqrt(x)*e + d)**n*c)*a**2*b*d**2 + 12*log((sqrt(x)*e + d)**n*c)*a**2 
*b*e**2*x + 36*log((sqrt(x)*e + d)**n*c)*a*b**2*d**2*n - 12*log((sqrt(x)*e 
 + d)**n*c)*a*b**2*e**2*n*x - 42*log((sqrt(x)*e + d)**n*c)*b**3*d**2*n**2 
+ 6*log((sqrt(x)*e + d)**n*c)*b**3*e**2*n**2*x + 4*a**3*e**2*x - 6*a**2*b* 
e**2*n*x + 6*a*b**2*e**2*n**2*x - 3*b**3*e**2*n**3*x)/(4*e**2)