\(\int x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \, dx\) [462]

Optimal result

Integrand size = 26, antiderivative size = 355 \[ \int x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \, dx=\frac {3^{-1-n} e^{-\frac {3 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{8 c^2 \sqrt {1+c^2 x^2}}+\frac {e^{-\frac {a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 c^2 \sqrt {1+c^2 x^2}}+\frac {e^{a/b} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 c^2 \sqrt {1+c^2 x^2}}+\frac {3^{-1-n} e^{\frac {3 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{8 c^2 \sqrt {1+c^2 x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.68 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.65 \[ \int x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \, dx=\frac {d e^{-\frac {3 a}{b}} \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^n \left (3 e^{\frac {4 a}{b}} \left (\frac {a}{b}+\text {arcsinh}(c x)\right )^{-n} \Gamma \left (1+n,\frac {a}{b}+\text {arcsinh}(c x)\right )+\left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \left (3^{-n} \Gamma \left (1+n,-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )+3 e^{\frac {2 a}{b}} \Gamma \left (1+n,-\frac {a+b \text {arcsinh}(c x)}{b}\right )+3^{-n} e^{\frac {6 a}{b}} \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{2 n} \left (-\frac {(a+b \text {arcsinh}(c x))^2}{b^2}\right )^{-n} \Gamma \left (1+n,\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )\right )\right )}{24 c^2 \sqrt {d \left (1+c^2 x^2\right )}} \] Input:

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

Output:

(d*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*((3*E^((4*a)/b)*Gamma[1 + n, a 
/b + ArcSinh[c*x]])/(a/b + ArcSinh[c*x])^n + (Gamma[1 + n, (-3*(a + b*ArcS 
inh[c*x]))/b]/3^n + 3*E^((2*a)/b)*Gamma[1 + n, -((a + b*ArcSinh[c*x])/b)] 
+ (E^((6*a)/b)*(-((a + b*ArcSinh[c*x])/b))^(2*n)*Gamma[1 + n, (3*(a + b*Ar 
cSinh[c*x]))/b])/(3^n*(-((a + b*ArcSinh[c*x])^2/b^2))^n))/(-((a + b*ArcSin 
h[c*x])/b))^n))/(24*c^2*E^((3*a)/b)*Sqrt[d*(1 + c^2*x^2)])
 

Rubi [A] (verified)

Time = 0.93 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.77, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {6234, 25, 5971, 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.

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

\(\Big \downarrow \) 6234

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 5971

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(Sqrt[d + c^2*d*x^2]*((3^(-1 - n)*b*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (- 
3*(a + b*ArcSinh[c*x]))/b])/(8*E^((3*a)/b)*(-((a + b*ArcSinh[c*x])/b))^n) 
+ (b*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, -((a + b*ArcSinh[c*x])/b)])/(8*E^ 
(a/b)*(-((a + b*ArcSinh[c*x])/b))^n) + (b*E^(a/b)*(a + b*ArcSinh[c*x])^n*G 
amma[1 + n, (a + b*ArcSinh[c*x])/b])/(8*((a + b*ArcSinh[c*x])/b)^n) + (3^( 
-1 - n)*b*E^((3*a)/b)*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (3*(a + b*ArcSin 
h[c*x]))/b])/(8*((a + b*ArcSinh[c*x])/b)^n)))/(b*c^2*Sqrt[1 + c^2*x^2])
 

Defintions of rubi rules used

rule 25

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

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5971

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + 
(b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + 
b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & 
& IGtQ[p, 0]
 

rule 6234

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_) 
^2)^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))*Simp[(d + e*x^2)^p/(1 + c^2* 
x^2)^p]   Subst[Int[x^n*Sinh[-a/b + x/b]^m*Cosh[-a/b + x/b]^(2*p + 1), x], 
x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] 
 && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

\[ \int x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \, dx=\int x \sqrt {d \left (c^{2} x^{2} + 1\right )} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{n}\, dx \] Input:

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

Output:

Integral(x*sqrt(d*(c**2*x**2 + 1))*(a + b*asinh(c*x))**n, x)
 

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \, dx=\sqrt {d}\, \left (\int \sqrt {c^{2} x^{2}+1}\, \left (\mathit {asinh} \left (c x \right ) b +a \right )^{n} x d x \right ) \] Input:

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

Output:

sqrt(d)*int(sqrt(c**2*x**2 + 1)*(asinh(c*x)*b + a)**n*x,x)