3.1.95 \(\int x (a+b \text {arcsinh}(c+d x))^n \, dx\) [95]

3.1.95.1 Optimal result

Integrand size = 14, antiderivative size = 267 \[ \int x (a+b \text {arcsinh}(c+d x))^n \, dx=\frac {2^{-3-n} e^{-\frac {2 a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{d^2}-\frac {c e^{-\frac {a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{2 d^2}+\frac {c e^{a/b} (a+b \text {arcsinh}(c+d x))^n \left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{2 d^2}+\frac {2^{-3-n} e^{\frac {2 a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{d^2} \]

output

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

3.1.95.2 Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.85 \[ \int x (a+b \text {arcsinh}(c+d x))^n \, dx=\frac {2^{-3-n} e^{-\frac {2 a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (-\frac {(a+b \text {arcsinh}(c+d x))^2}{b^2}\right )^{-n} \left (2^{2+n} c e^{\frac {3 a}{b}} \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^n \Gamma \left (1+n,\frac {a}{b}+\text {arcsinh}(c+d x)\right )+\left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )^n \Gamma \left (1+n,-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )-2^{2+n} c e^{a/b} \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )^n \Gamma \left (1+n,-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )+e^{\frac {4 a}{b}} \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^n \Gamma \left (1+n,\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )}{d^2} \]

input

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

output

(2^(-3 - n)*(a + b*ArcSinh[c + d*x])^n*(2^(2 + n)*c*E^((3*a)/b)*(-((a + b* 
ArcSinh[c + d*x])/b))^n*Gamma[1 + n, a/b + ArcSinh[c + d*x]] + (a/b + ArcS 
inh[c + d*x])^n*Gamma[1 + n, (-2*(a + b*ArcSinh[c + d*x]))/b] - 2^(2 + n)* 
c*E^(a/b)*(a/b + ArcSinh[c + d*x])^n*Gamma[1 + n, -((a + b*ArcSinh[c + d*x 
])/b)] + E^((4*a)/b)*(-((a + b*ArcSinh[c + d*x])/b))^n*Gamma[1 + n, (2*(a 
+ b*ArcSinh[c + d*x]))/b]))/(d^2*E^((2*a)/b)*(-((a + b*ArcSinh[c + d*x])^2 
/b^2))^n)
 

3.1.95.3 Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6274, 25, 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.

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

\(\Big \downarrow \) 6274

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6245

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

input

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

output

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

3.1.95.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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] 
]
 

3.1.95.4 Maple [F]

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

input

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

output

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

3.1.95.5 Fricas [F]

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

input

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

output

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

3.1.95.6 Sympy [F]

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

input

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

output

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

3.1.95.7 Maxima [F]

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

input

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

output

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

3.1.95.8 Giac [F]

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

input

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

output

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

3.1.95.9 Mupad [F(-1)]

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

input

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

output

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