\(\int x (a+b \log (c (d+e \sqrt {x})^n))^2 \, dx\) [409]

Optimal result

Integrand size = 22, antiderivative size = 342 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \, dx=\frac {3 b^2 d^2 n^2 \left (d+e \sqrt {x}\right )^2}{2 e^4}-\frac {4 b^2 d n^2 \left (d+e \sqrt {x}\right )^3}{9 e^4}+\frac {b^2 n^2 \left (d+e \sqrt {x}\right )^4}{16 e^4}-\frac {4 b^2 d^3 n^2 \sqrt {x}}{e^3}+\frac {b^2 d^4 n^2 \log ^2\left (d+e \sqrt {x}\right )}{2 e^4}+\frac {4 b d^3 n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{e^4}-\frac {3 b d^2 n \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{e^4}+\frac {4 b d n \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{3 e^4}-\frac {b n \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{4 e^4}-\frac {b d^4 n \log \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{e^4}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \] 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+1/16 
*b^2*n^2*(d+e*x^(1/2))^4/e^4-4*b^2*d^3*n^2*x^(1/2)/e^3+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*ln 
(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*x^2*(a+b*ln(c*(d+e*x^(1/2))^n))^2
 

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.65 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \, dx=\frac {e \sqrt {x} \left (72 a^2 e^3 x^{3/2}+12 a b n \left (12 d^3-6 d^2 e \sqrt {x}+4 d e^2 x-3 e^3 x^{3/2}\right )+b^2 n^2 \left (-300 d^3+78 d^2 e \sqrt {x}-28 d e^2 x+9 e^3 x^{3/2}\right )\right )-12 b \left (12 a \left (d^4-e^4 x^2\right )+b n \left (-25 d^4-12 d^3 e \sqrt {x}+6 d^2 e^2 x-4 d e^3 x^{3/2}+3 e^4 x^2\right )\right ) \log \left (c \left (d+e \sqrt {x}\right )^n\right )-72 b^2 \left (d^4-e^4 x^2\right ) \log ^2\left (c \left (d+e \sqrt {x}\right )^n\right )}{144 e^4} \] Input:

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

Output:

(e*Sqrt[x]*(72*a^2*e^3*x^(3/2) + 12*a*b*n*(12*d^3 - 6*d^2*e*Sqrt[x] + 4*d* 
e^2*x - 3*e^3*x^(3/2)) + b^2*n^2*(-300*d^3 + 78*d^2*e*Sqrt[x] - 28*d*e^2*x 
 + 9*e^3*x^(3/2))) - 12*b*(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] - 72*b^2*(d^4 - e^4*x^2)*Log[c*(d + e*Sqrt[x])^n]^2)/(144*e^4)
 

Rubi [A] (warning: unable to verify)

Time = 0.88 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.65, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, 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.

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

\(\Big \downarrow \) 2904

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

\(\Big \downarrow \) 2845

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

\(\Big \downarrow \) 2858

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2772

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.04 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \, dx=\frac {72 \, b^{2} e^{4} x^{2} \log \left (c\right )^{2} + 9 \, {\left (b^{2} e^{4} n^{2} - 4 \, a b e^{4} n + 8 \, a^{2} e^{4}\right )} x^{2} + 72 \, {\left (b^{2} e^{4} n^{2} x^{2} - b^{2} d^{4} n^{2}\right )} \log \left (e \sqrt {x} + d\right )^{2} + 6 \, {\left (13 \, b^{2} d^{2} e^{2} n^{2} - 12 \, a b d^{2} e^{2} n\right )} x - 12 \, {\left (6 \, b^{2} d^{2} e^{2} n^{2} x - 25 \, b^{2} d^{4} n^{2} + 12 \, a b d^{4} n + 3 \, {\left (b^{2} e^{4} n^{2} - 4 \, a b e^{4} n\right )} x^{2} - 12 \, {\left (b^{2} e^{4} n x^{2} - b^{2} d^{4} n\right )} \log \left (c\right ) - 4 \, {\left (b^{2} d e^{3} n^{2} x + 3 \, b^{2} d^{3} e n^{2}\right )} \sqrt {x}\right )} \log \left (e \sqrt {x} + d\right ) - 36 \, {\left (2 \, b^{2} d^{2} e^{2} n x + {\left (b^{2} e^{4} n - 4 \, a b e^{4}\right )} x^{2}\right )} \log \left (c\right ) - 4 \, {\left (75 \, b^{2} d^{3} e n^{2} - 36 \, a b d^{3} e n + {\left (7 \, b^{2} d e^{3} n^{2} - 12 \, a b d e^{3} n\right )} x - 12 \, {\left (b^{2} d e^{3} n x + 3 \, b^{2} d^{3} e n\right )} \log \left (c\right )\right )} \sqrt {x}}{144 \, e^{4}} \] Input:

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

Output:

1/144*(72*b^2*e^4*x^2*log(c)^2 + 9*(b^2*e^4*n^2 - 4*a*b*e^4*n + 8*a^2*e^4) 
*x^2 + 72*(b^2*e^4*n^2*x^2 - b^2*d^4*n^2)*log(e*sqrt(x) + d)^2 + 6*(13*b^2 
*d^2*e^2*n^2 - 12*a*b*d^2*e^2*n)*x - 12*(6*b^2*d^2*e^2*n^2*x - 25*b^2*d^4* 
n^2 + 12*a*b*d^4*n + 3*(b^2*e^4*n^2 - 4*a*b*e^4*n)*x^2 - 12*(b^2*e^4*n*x^2 
 - b^2*d^4*n)*log(c) - 4*(b^2*d*e^3*n^2*x + 3*b^2*d^3*e*n^2)*sqrt(x))*log( 
e*sqrt(x) + d) - 36*(2*b^2*d^2*e^2*n*x + (b^2*e^4*n - 4*a*b*e^4)*x^2)*log( 
c) - 4*(75*b^2*d^3*e*n^2 - 36*a*b*d^3*e*n + (7*b^2*d*e^3*n^2 - 12*a*b*d*e^ 
3*n)*x - 12*(b^2*d*e^3*n*x + 3*b^2*d^3*e*n)*log(c))*sqrt(x))/e^4
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.75 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \, dx=\frac {1}{2} \, b^{2} x^{2} \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )^{2} - \frac {1}{12} \, a b e n {\left (\frac {12 \, d^{4} \log \left (e \sqrt {x} + d\right )}{e^{5}} + \frac {3 \, e^{3} x^{2} - 4 \, d e^{2} x^{\frac {3}{2}} + 6 \, d^{2} e x - 12 \, d^{3} \sqrt {x}}{e^{4}}\right )} + a b x^{2} \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + \frac {1}{2} \, a^{2} x^{2} - \frac {1}{144} \, {\left (12 \, e n {\left (\frac {12 \, d^{4} \log \left (e \sqrt {x} + d\right )}{e^{5}} + \frac {3 \, e^{3} x^{2} - 4 \, d e^{2} x^{\frac {3}{2}} + 6 \, d^{2} e x - 12 \, d^{3} \sqrt {x}}{e^{4}}\right )} \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) - \frac {{\left (9 \, e^{4} x^{2} + 72 \, d^{4} \log \left (e \sqrt {x} + d\right )^{2} - 28 \, d e^{3} x^{\frac {3}{2}} + 78 \, d^{2} e^{2} x + 300 \, d^{4} \log \left (e \sqrt {x} + d\right ) - 300 \, d^{3} e \sqrt {x}\right )} n^{2}}{e^{4}}\right )} b^{2} \] Input:

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

Output:

1/2*b^2*x^2*log((e*sqrt(x) + d)^n*c)^2 - 1/12*a*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) + a*b*x^2*log((e*sqrt(x) + d)^n*c) + 1/2*a^2*x^2 - 1/144*(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*sqrt(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)*b^2
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 624 vs. \(2 (296) = 592\).

Time = 0.13 (sec) , antiderivative size = 624, normalized size of antiderivative = 1.82 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \, dx =\text {Too large to display} \] Input:

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

Output:

1/144*(72*b^2*e*x^2*log(c)^2 + 144*a*b*e*x^2*log(c) + (72*(e*sqrt(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*sqr 
t(x) + d)^2*d^2/e^3 - 576*(e*sqrt(x) + d)*d^3/e^3)*b^2*n^2 + 72*a^2*e*x^2 
+ 12*(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)^2*d^2*log(e*sqrt(x) + d)/e^3 
- 48*(e*sqrt(x) + d)*d^3*log(e*sqrt(x) + d)/e^3 - 3*(e*sqrt(x) + d)^4/e^3 
+ 16*(e*sqrt(x) + d)^3*d/e^3 - 36*(e*sqrt(x) + d)^2*d^2/e^3 + 48*(e*sqrt(x 
) + d)*d^3/e^3)*b^2*n*log(c) + 12*(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)^ 
2*d^2*log(e*sqrt(x) + d)/e^3 - 48*(e*sqrt(x) + d)*d^3*log(e*sqrt(x) + d)/e 
^3 - 3*(e*sqrt(x) + d)^4/e^3 + 16*(e*sqrt(x) + d)^3*d/e^3 - 36*(e*sqrt(x) 
+ d)^2*d^2/e^3 + 48*(e*sqrt(x) + d)*d^3/e^3)*a*b*n)/e
 

Mupad [B] (verification not implemented)

Time = 15.24 (sec) , antiderivative size = 420, normalized size of antiderivative = 1.23 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \, dx=x\,\left (\frac {d\,\left (\frac {d\,\left (2\,a^2-a\,b\,n+\frac {b^2\,n^2}{4}\right )}{e}-\frac {d\,\left (6\,a^2-b^2\,n^2\right )}{3\,e}\right )}{2\,e}+\frac {b^2\,d^2\,n^2}{4\,e^2}\right )-x^{3/2}\,\left (\frac {d\,\left (2\,a^2-a\,b\,n+\frac {b^2\,n^2}{4}\right )}{3\,e}-\frac {d\,\left (6\,a^2-b^2\,n^2\right )}{9\,e}\right )+{\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}^2\,\left (\frac {b^2\,x^2}{2}-\frac {b^2\,d^4}{2\,e^4}\right )+x^2\,\left (\frac {a^2}{2}-\frac {a\,b\,n}{4}+\frac {b^2\,n^2}{16}\right )-\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )\,\left (x^{3/2}\,\left (\frac {b\,d\,\left (4\,a-b\,n\right )}{3\,e}-\frac {4\,a\,b\,d}{3\,e}\right )-\frac {b\,x^2\,\left (4\,a-b\,n\right )}{4}+\frac {d^2\,\sqrt {x}\,\left (\frac {b\,d\,\left (4\,a-b\,n\right )}{e}-\frac {4\,a\,b\,d}{e}\right )}{e^2}-\frac {d\,x\,\left (\frac {b\,d\,\left (4\,a-b\,n\right )}{e}-\frac {4\,a\,b\,d}{e}\right )}{2\,e}\right )-\sqrt {x}\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (2\,a^2-a\,b\,n+\frac {b^2\,n^2}{4}\right )}{e}-\frac {d\,\left (6\,a^2-b^2\,n^2\right )}{3\,e}\right )}{e}+\frac {b^2\,d^2\,n^2}{2\,e^2}\right )}{e}+\frac {b^2\,d^3\,n^2}{e^3}\right )+\frac {\ln \left (d+e\,\sqrt {x}\right )\,\left (25\,b^2\,d^4\,n^2-12\,a\,b\,d^4\,n\right )}{12\,e^4} \] Input:

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

Output:

x*((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^(3/2)*((d*(2*a^2 + (b^2*n^2)/4 - a*b*n 
))/(3*e) - (d*(6*a^2 - b^2*n^2))/(9*e)) + log(c*(d + e*x^(1/2))^n)^2*((b^2 
*x^2)/2 - (b^2*d^4)/(2*e^4)) + x^2*(a^2/2 + (b^2*n^2)/16 - (a*b*n)/4) - lo 
g(c*(d + e*x^(1/2))^n)*(x^(3/2)*((b*d*(4*a - b*n))/(3*e) - (4*a*b*d)/(3*e) 
) - (b*x^2*(4*a - b*n))/4 + (d^2*x^(1/2)*((b*d*(4*a - b*n))/e - (4*a*b*d)/ 
e))/e^2 - (d*x*((b*d*(4*a - b*n))/e - (4*a*b*d)/e))/(2*e)) - x^(1/2)*((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) + (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 = 0.18 (sec) , antiderivative size = 319, normalized size of antiderivative = 0.93 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \, dx=\frac {144 \sqrt {x}\, \mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right ) b^{2} d^{3} e n +48 \sqrt {x}\, \mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right ) b^{2} d \,e^{3} n x +144 \sqrt {x}\, a b \,d^{3} e n +48 \sqrt {x}\, a b d \,e^{3} n x -300 \sqrt {x}\, b^{2} d^{3} e \,n^{2}-28 \sqrt {x}\, b^{2} d \,e^{3} n^{2} x -72 \mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right )^{2} b^{2} d^{4}+72 \mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right )^{2} b^{2} e^{4} x^{2}-144 \,\mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right ) a b \,d^{4}+144 \,\mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right ) a b \,e^{4} x^{2}+300 \,\mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right ) b^{2} d^{4} n -72 \,\mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right ) b^{2} d^{2} e^{2} n x -36 \,\mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right ) b^{2} e^{4} n \,x^{2}+72 a^{2} e^{4} x^{2}-72 a b \,d^{2} e^{2} n x -36 a b \,e^{4} n \,x^{2}+78 b^{2} d^{2} e^{2} n^{2} x +9 b^{2} e^{4} n^{2} x^{2}}{144 e^{4}} \] Input:

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

Output:

(144*sqrt(x)*log((sqrt(x)*e + d)**n*c)*b**2*d**3*e*n + 48*sqrt(x)*log((sqr 
t(x)*e + d)**n*c)*b**2*d*e**3*n*x + 144*sqrt(x)*a*b*d**3*e*n + 48*sqrt(x)* 
a*b*d*e**3*n*x - 300*sqrt(x)*b**2*d**3*e*n**2 - 28*sqrt(x)*b**2*d*e**3*n** 
2*x - 72*log((sqrt(x)*e + d)**n*c)**2*b**2*d**4 + 72*log((sqrt(x)*e + d)** 
n*c)**2*b**2*e**4*x**2 - 144*log((sqrt(x)*e + d)**n*c)*a*b*d**4 + 144*log( 
(sqrt(x)*e + d)**n*c)*a*b*e**4*x**2 + 300*log((sqrt(x)*e + d)**n*c)*b**2*d 
**4*n - 72*log((sqrt(x)*e + d)**n*c)*b**2*d**2*e**2*n*x - 36*log((sqrt(x)* 
e + d)**n*c)*b**2*e**4*n*x**2 + 72*a**2*e**4*x**2 - 72*a*b*d**2*e**2*n*x - 
 36*a*b*e**4*n*x**2 + 78*b**2*d**2*e**2*n**2*x + 9*b**2*e**4*n**2*x**2)/(1 
44*e**4)