\(\int (a+b \log (c (d+e \sqrt {x})^n))^2 \, dx\) [410]

Optimal result

Integrand size = 20, antiderivative size = 195 \[ \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \, dx=\frac {b^2 n^2 \left (d+e \sqrt {x}\right )^2}{2 e^2}+\frac {4 a b d n \sqrt {x}}{e}-\frac {4 b^2 d n^2 \sqrt {x}}{e}+\frac {4 b^2 d n \left (d+e \sqrt {x}\right ) \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{e^2}-\frac {b n \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{e^2}-\frac {2 d \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^2}+\frac {\left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.77 \[ \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \, dx=\frac {-2 a b n \left (d-e \sqrt {x}\right )^2+b^2 e n^2 \left (-6 d+e \sqrt {x}\right ) \sqrt {x}-2 a^2 \left (d^2-e^2 x\right )+2 b \left (d+e \sqrt {x}\right ) \left (-2 a d+3 b d n+2 a e \sqrt {x}-b e n \sqrt {x}\right ) \log \left (c \left (d+e \sqrt {x}\right )^n\right )-2 b^2 \left (d^2-e^2 x\right ) \log ^2\left (c \left (d+e \sqrt {x}\right )^n\right )}{2 e^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.04, 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.

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

\(\Big \downarrow \) 2901

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

\(\Big \downarrow \) 2848

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.15 \[ \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \, dx=\frac {2 \, b^{2} e^{2} x \log \left (c\right )^{2} + 2 \, {\left (b^{2} e^{2} n^{2} x - b^{2} d^{2} n^{2}\right )} \log \left (e \sqrt {x} + d\right )^{2} - 2 \, {\left (b^{2} e^{2} n - 2 \, a b e^{2}\right )} x \log \left (c\right ) + {\left (b^{2} e^{2} n^{2} - 2 \, a b e^{2} n + 2 \, a^{2} e^{2}\right )} x + 2 \, {\left (2 \, b^{2} d e n^{2} \sqrt {x} + 3 \, b^{2} d^{2} n^{2} - 2 \, a b d^{2} n - {\left (b^{2} e^{2} n^{2} - 2 \, a b e^{2} n\right )} x + 2 \, {\left (b^{2} e^{2} n x - b^{2} d^{2} n\right )} \log \left (c\right )\right )} \log \left (e \sqrt {x} + d\right ) - 2 \, {\left (3 \, b^{2} d e n^{2} - 2 \, b^{2} d e n \log \left (c\right ) - 2 \, a b d e n\right )} \sqrt {x}}{2 \, e^{2}} \] Input:

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

Output:

1/2*(2*b^2*e^2*x*log(c)^2 + 2*(b^2*e^2*n^2*x - b^2*d^2*n^2)*log(e*sqrt(x) 
+ d)^2 - 2*(b^2*e^2*n - 2*a*b*e^2)*x*log(c) + (b^2*e^2*n^2 - 2*a*b*e^2*n + 
 2*a^2*e^2)*x + 2*(2*b^2*d*e*n^2*sqrt(x) + 3*b^2*d^2*n^2 - 2*a*b*d^2*n - ( 
b^2*e^2*n^2 - 2*a*b*e^2*n)*x + 2*(b^2*e^2*n*x - b^2*d^2*n)*log(c))*log(e*s 
qrt(x) + d) - 2*(3*b^2*d*e*n^2 - 2*b^2*d*e*n*log(c) - 2*a*b*d*e*n)*sqrt(x) 
)/e^2
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.92 \[ \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \, dx=-{\left (e n {\left (\frac {2 \, d^{2} \log \left (e \sqrt {x} + d\right )}{e^{3}} + \frac {e x - 2 \, d \sqrt {x}}{e^{2}}\right )} - 2 \, x \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )\right )} a b - \frac {1}{2} \, {\left (2 \, e n {\left (\frac {2 \, d^{2} \log \left (e \sqrt {x} + d\right )}{e^{3}} + \frac {e x - 2 \, d \sqrt {x}}{e^{2}}\right )} \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) - 2 \, x \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )^{2} - \frac {{\left (2 \, d^{2} \log \left (e \sqrt {x} + d\right )^{2} + e^{2} x + 6 \, d^{2} \log \left (e \sqrt {x} + d\right ) - 6 \, d e \sqrt {x}\right )} n^{2}}{e^{2}}\right )} b^{2} + a^{2} x \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.74 \[ \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \, dx=\frac {\frac {{\left (2 \, {\left (e \sqrt {x} + d\right )}^{2} \log \left (e \sqrt {x} + d\right )^{2} - 4 \, {\left (e \sqrt {x} + d\right )} d \log \left (e \sqrt {x} + d\right )^{2} - 2 \, {\left (e \sqrt {x} + d\right )}^{2} \log \left (e \sqrt {x} + d\right ) + 8 \, {\left (e \sqrt {x} + d\right )} d \log \left (e \sqrt {x} + d\right ) + {\left (e \sqrt {x} + d\right )}^{2} - 8 \, {\left (e \sqrt {x} + d\right )} d\right )} b^{2} n^{2}}{e} + \frac {2 \, {\left (2 \, {\left (e \sqrt {x} + d\right )}^{2} \log \left (e \sqrt {x} + d\right ) - 4 \, {\left (e \sqrt {x} + d\right )} d \log \left (e \sqrt {x} + d\right ) - {\left (e \sqrt {x} + d\right )}^{2} + 4 \, {\left (e \sqrt {x} + d\right )} d\right )} b^{2} n \log \left (c\right )}{e} + \frac {2 \, {\left ({\left (e \sqrt {x} + d\right )}^{2} - 2 \, {\left (e \sqrt {x} + d\right )} d\right )} b^{2} \log \left (c\right )^{2}}{e} + \frac {2 \, {\left (2 \, {\left (e \sqrt {x} + d\right )}^{2} \log \left (e \sqrt {x} + d\right ) - 4 \, {\left (e \sqrt {x} + d\right )} d \log \left (e \sqrt {x} + d\right ) - {\left (e \sqrt {x} + d\right )}^{2} + 4 \, {\left (e \sqrt {x} + d\right )} d\right )} a b n}{e} + \frac {4 \, {\left ({\left (e \sqrt {x} + d\right )}^{2} - 2 \, {\left (e \sqrt {x} + d\right )} d\right )} a b \log \left (c\right )}{e} + \frac {2 \, {\left ({\left (e \sqrt {x} + d\right )}^{2} - 2 \, {\left (e \sqrt {x} + d\right )} d\right )} a^{2}}{e}}{2 \, e} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 15.19 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.95 \[ \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \, dx=x\,\left (a^2-a\,b\,n+\frac {b^2\,n^2}{2}\right )-\sqrt {x}\,\left (\frac {d\,\left (2\,a^2-2\,a\,b\,n+b^2\,n^2\right )}{e}-\frac {2\,d\,\left (a^2-b^2\,n^2\right )}{e}\right )+{\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}^2\,\left (b^2\,x-\frac {b^2\,d^2}{e^2}\right )-\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )\,\left (\sqrt {x}\,\left (\frac {2\,b\,d\,\left (2\,a-b\,n\right )}{e}-\frac {4\,a\,b\,d}{e}\right )-b\,x\,\left (2\,a-b\,n\right )\right )+\frac {\ln \left (d+e\,\sqrt {x}\right )\,\left (3\,b^2\,d^2\,n^2-2\,a\,b\,d^2\,n\right )}{e^2} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.02 \[ \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \, dx=\frac {4 \sqrt {x}\, \mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right ) b^{2} d e n +4 \sqrt {x}\, a b d e n -6 \sqrt {x}\, b^{2} d e \,n^{2}-2 \mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right )^{2} b^{2} d^{2}+2 \mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right )^{2} b^{2} e^{2} x -4 \,\mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right ) a b \,d^{2}+4 \,\mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right ) a b \,e^{2} x +6 \,\mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right ) b^{2} d^{2} n -2 \,\mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right ) b^{2} e^{2} n x +2 a^{2} e^{2} x -2 a b \,e^{2} n x +b^{2} e^{2} n^{2} x}{2 e^{2}} \] Input:

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

Output:

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