3.1.94 \(\int x^2 (a+b \text {arcsinh}(c+d x))^n \, dx\) [94]

3.1.94.1 Optimal result

Integrand size = 16, antiderivative size = 545 \[ \int x^2 (a+b \text {arcsinh}(c+d x))^n \, dx=\frac {3^{-1-n} e^{-\frac {3 a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{8 d^3}-\frac {2^{-2-n} c e^{-\frac {2 a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{d^3}-\frac {e^{-\frac {a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{8 d^3}+\frac {c^2 e^{-\frac {a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{2 d^3}+\frac {e^{a/b} (a+b \text {arcsinh}(c+d x))^n \left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{8 d^3}-\frac {c^2 e^{a/b} (a+b \text {arcsinh}(c+d x))^n \left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{2 d^3}-\frac {2^{-2-n} c e^{\frac {2 a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{d^3}-\frac {3^{-1-n} e^{\frac {3 a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{8 d^3} \]

output

1/8*3^(-1-n)*(a+b*arcsinh(d*x+c))^n*GAMMA(1+n,-3*(a+b*arcsinh(d*x+c))/b)/d 
^3/exp(3*a/b)/(((-a-b*arcsinh(d*x+c))/b)^n)-2^(-2-n)*c*(a+b*arcsinh(d*x+c) 
)^n*GAMMA(1+n,-2*(a+b*arcsinh(d*x+c))/b)/d^3/exp(2*a/b)/(((-a-b*arcsinh(d* 
x+c))/b)^n)-1/8*(a+b*arcsinh(d*x+c))^n*GAMMA(1+n,(-a-b*arcsinh(d*x+c))/b)/ 
d^3/exp(a/b)/(((-a-b*arcsinh(d*x+c))/b)^n)+1/2*c^2*(a+b*arcsinh(d*x+c))^n* 
GAMMA(1+n,(-a-b*arcsinh(d*x+c))/b)/d^3/exp(a/b)/(((-a-b*arcsinh(d*x+c))/b) 
^n)+1/8*exp(a/b)*(a+b*arcsinh(d*x+c))^n*GAMMA(1+n,(a+b*arcsinh(d*x+c))/b)/ 
d^3/(((a+b*arcsinh(d*x+c))/b)^n)-1/2*c^2*exp(a/b)*(a+b*arcsinh(d*x+c))^n*G 
AMMA(1+n,(a+b*arcsinh(d*x+c))/b)/d^3/(((a+b*arcsinh(d*x+c))/b)^n)-2^(-2-n) 
*c*exp(2*a/b)*(a+b*arcsinh(d*x+c))^n*GAMMA(1+n,2*(a+b*arcsinh(d*x+c))/b)/d 
^3/(((a+b*arcsinh(d*x+c))/b)^n)-1/8*3^(-1-n)*exp(3*a/b)*(a+b*arcsinh(d*x+c 
))^n*GAMMA(1+n,3*(a+b*arcsinh(d*x+c))/b)/d^3/(((a+b*arcsinh(d*x+c))/b)^n)
 

3.1.94.2 Mathematica [A] (verified)

Time = 0.66 (sec) , antiderivative size = 345, normalized size of antiderivative = 0.63 \[ \int x^2 (a+b \text {arcsinh}(c+d x))^n \, dx=\frac {2^{-3-n} 3^{-1-n} e^{-\frac {3 a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (-\frac {(a+b \text {arcsinh}(c+d x))^2}{b^2}\right )^{-n} \left (-2^n 3^{1+n} \left (-1+4 c^2\right ) e^{\frac {4 a}{b}} \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^n \Gamma \left (1+n,\frac {a}{b}+\text {arcsinh}(c+d x)\right )+2^n \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )^n \Gamma \left (1+n,-\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )-2\ 3^{1+n} c e^{a/b} \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )^n \Gamma \left (1+n,-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )+2^n 3^{1+n} \left (-1+4 c^2\right ) e^{\frac {2 a}{b}} \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )^n \Gamma \left (1+n,-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )-e^{\frac {5 a}{b}} \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^n \left (2\ 3^{1+n} c \Gamma \left (1+n,\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )+2^n e^{a/b} \Gamma \left (1+n,\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )\right )}{d^3} \]

input

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

output

(2^(-3 - n)*3^(-1 - n)*(a + b*ArcSinh[c + d*x])^n*(-(2^n*3^(1 + n)*(-1 + 4 
*c^2)*E^((4*a)/b)*(-((a + b*ArcSinh[c + d*x])/b))^n*Gamma[1 + n, a/b + Arc 
Sinh[c + d*x]]) + 2^n*(a/b + ArcSinh[c + d*x])^n*Gamma[1 + n, (-3*(a + b*A 
rcSinh[c + d*x]))/b] - 2*3^(1 + n)*c*E^(a/b)*(a/b + ArcSinh[c + d*x])^n*Ga 
mma[1 + n, (-2*(a + b*ArcSinh[c + d*x]))/b] + 2^n*3^(1 + n)*(-1 + 4*c^2)*E 
^((2*a)/b)*(a/b + ArcSinh[c + d*x])^n*Gamma[1 + n, -((a + b*ArcSinh[c + d* 
x])/b)] - E^((5*a)/b)*(-((a + b*ArcSinh[c + d*x])/b))^n*(2*3^(1 + n)*c*Gam 
ma[1 + n, (2*(a + b*ArcSinh[c + d*x]))/b] + 2^n*E^(a/b)*Gamma[1 + n, (3*(a 
 + b*ArcSinh[c + d*x]))/b])))/(d^3*E^((3*a)/b)*(-((a + b*ArcSinh[c + d*x]) 
^2/b^2))^n)
 

3.1.94.3 Rubi [A] (verified)

Time = 1.14 (sec) , antiderivative size = 525, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {6274, 27, 6245, 7293, 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.

\(\displaystyle \int x^2 (a+b \text {arcsinh}(c+d x))^n \, dx\)

\(\Big \downarrow \) 6274

\(\displaystyle \frac {\int x^2 (a+b \text {arcsinh}(c+d x))^nd(c+d x)}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int d^2 x^2 (a+b \text {arcsinh}(c+d x))^nd(c+d x)}{d^3}\)

\(\Big \downarrow \) 6245

\(\displaystyle \frac {\int d^2 x^2 \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^nd\text {arcsinh}(c+d x)}{d^3}\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {\int \left (c^2 \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^n+(c+d x)^2 \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^n-2 c (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^n\right )d\text {arcsinh}(c+d x)}{d^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {1}{2} c^2 e^{-\frac {a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )-\frac {1}{2} c^2 e^{a/b} (a+b \text {arcsinh}(c+d x))^n \left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {a+b \text {arcsinh}(c+d x)}{b}\right )+\frac {1}{8} 3^{-n-1} e^{-\frac {3 a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )-c 2^{-n-2} e^{-\frac {2 a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )-\frac {1}{8} e^{-\frac {a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )+\frac {1}{8} e^{a/b} (a+b \text {arcsinh}(c+d x))^n \left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {a+b \text {arcsinh}(c+d x)}{b}\right )-c 2^{-n-2} e^{\frac {2 a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )-\frac {1}{8} 3^{-n-1} e^{\frac {3 a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{d^3}\)

input

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

output

((3^(-1 - n)*(a + b*ArcSinh[c + d*x])^n*Gamma[1 + n, (-3*(a + b*ArcSinh[c 
+ d*x]))/b])/(8*E^((3*a)/b)*(-((a + b*ArcSinh[c + d*x])/b))^n) - (2^(-2 - 
n)*c*(a + b*ArcSinh[c + d*x])^n*Gamma[1 + n, (-2*(a + b*ArcSinh[c + d*x])) 
/b])/(E^((2*a)/b)*(-((a + b*ArcSinh[c + d*x])/b))^n) - ((a + b*ArcSinh[c + 
 d*x])^n*Gamma[1 + n, -((a + b*ArcSinh[c + d*x])/b)])/(8*E^(a/b)*(-((a + b 
*ArcSinh[c + d*x])/b))^n) + (c^2*(a + b*ArcSinh[c + d*x])^n*Gamma[1 + n, - 
((a + b*ArcSinh[c + d*x])/b)])/(2*E^(a/b)*(-((a + b*ArcSinh[c + d*x])/b))^ 
n) + (E^(a/b)*(a + b*ArcSinh[c + d*x])^n*Gamma[1 + n, (a + b*ArcSinh[c + d 
*x])/b])/(8*((a + b*ArcSinh[c + d*x])/b)^n) - (c^2*E^(a/b)*(a + b*ArcSinh[ 
c + d*x])^n*Gamma[1 + n, (a + b*ArcSinh[c + d*x])/b])/(2*((a + b*ArcSinh[c 
 + d*x])/b)^n) - (2^(-2 - n)*c*E^((2*a)/b)*(a + b*ArcSinh[c + d*x])^n*Gamm 
a[1 + n, (2*(a + b*ArcSinh[c + d*x]))/b])/((a + b*ArcSinh[c + d*x])/b)^n - 
 (3^(-1 - n)*E^((3*a)/b)*(a + b*ArcSinh[c + d*x])^n*Gamma[1 + n, (3*(a + b 
*ArcSinh[c + d*x]))/b])/(8*((a + b*ArcSinh[c + d*x])/b)^n))/d^3
 

3.1.94.3.1 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 6245

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x 
_Symbol] :> Simp[1/c^(m + 1)   Subst[Int[(a + b*x)^n*Cosh[x]*(c*d + e*Sinh[ 
x])^m, x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[m, 
0]
 

rule 6274

Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^( 
m_.), x_Symbol] :> Simp[1/d   Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b* 
ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.1.94.4 Maple [F]

\[\int x^{2} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{n}d x\]

input

int(x^2*(a+b*arcsinh(d*x+c))^n,x)
 

output

int(x^2*(a+b*arcsinh(d*x+c))^n,x)
 

3.1.94.5 Fricas [F]

\[ \int x^2 (a+b \text {arcsinh}(c+d x))^n \, dx=\int { {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{n} x^{2} \,d x } \]

input

integrate(x^2*(a+b*arcsinh(d*x+c))^n,x, algorithm="fricas")
 

output

integral((b*arcsinh(d*x + c) + a)^n*x^2, x)
 

3.1.94.6 Sympy [F]

\[ \int x^2 (a+b \text {arcsinh}(c+d x))^n \, dx=\int x^{2} \left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{n}\, dx \]

input

integrate(x**2*(a+b*asinh(d*x+c))**n,x)
 

output

Integral(x**2*(a + b*asinh(c + d*x))**n, x)
 

3.1.94.7 Maxima [F]

\[ \int x^2 (a+b \text {arcsinh}(c+d x))^n \, dx=\int { {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{n} x^{2} \,d x } \]

input

integrate(x^2*(a+b*arcsinh(d*x+c))^n,x, algorithm="maxima")
 

output

integrate((b*arcsinh(d*x + c) + a)^n*x^2, x)
 

3.1.94.8 Giac [F]

\[ \int x^2 (a+b \text {arcsinh}(c+d x))^n \, dx=\int { {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{n} x^{2} \,d x } \]

input

integrate(x^2*(a+b*arcsinh(d*x+c))^n,x, algorithm="giac")
 

output

integrate((b*arcsinh(d*x + c) + a)^n*x^2, x)
 

3.1.94.9 Mupad [F(-1)]

Timed out. \[ \int x^2 (a+b \text {arcsinh}(c+d x))^n \, dx=\int x^2\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^n \,d x \]

input

int(x^2*(a + b*asinh(c + d*x))^n,x)
 

output

int(x^2*(a + b*asinh(c + d*x))^n, x)