\(\int x (d+c^2 d x^2)^{3/2} (a+b \text {arcsinh}(c x))^n \, dx\) [467]

Optimal result

Integrand size = 26, antiderivative size = 542 \[ \int x \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^n \, dx=\frac {5^{-1-n} d e^{-\frac {5 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{32 c^2 \sqrt {1+c^2 x^2}}+\frac {3^{-n} d e^{-\frac {3 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{32 c^2 \sqrt {1+c^2 x^2}}+\frac {d e^{-\frac {a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{16 c^2 \sqrt {1+c^2 x^2}}+\frac {d e^{a/b} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {a+b \text {arcsinh}(c x)}{b}\right )}{16 c^2 \sqrt {1+c^2 x^2}}+\frac {3^{-n} d e^{\frac {3 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{32 c^2 \sqrt {1+c^2 x^2}}+\frac {5^{-1-n} d e^{\frac {5 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{32 c^2 \sqrt {1+c^2 x^2}} \] Output:

1/32*5^(-1-n)*d*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^n*GAMMA(1+n,(-5*a-5 
*b*arcsinh(c*x))/b)/c^2/exp(5*a/b)/(c^2*x^2+1)^(1/2)/((-(a+b*arcsinh(c*x)) 
/b)^n)+1/32*d*(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)/(3^n)/c^2/exp(3*a/b)/(c^2*x^2+1)^(1/2)/((-(a+b*arcsinh(c 
*x))/b)^n)+1/16*d*(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)+1/16*d*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)+1/32*d* 
exp(3*a/b)*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^n*GAMMA(1+n,3*(a+b*arcsi 
nh(c*x))/b)/(3^n)/c^2/(c^2*x^2+1)^(1/2)/(((a+b*arcsinh(c*x))/b)^n)+1/32*5^ 
(-1-n)*d*exp(5*a/b)*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^n*GAMMA(1+n,5*( 
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 = 1.24 (sec) , antiderivative size = 390, normalized size of antiderivative = 0.72 \[ \int x \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^n \, dx=\frac {15^{-1-n} d^2 e^{-\frac {5 a}{b}} \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {(a+b \text {arcsinh}(c x))^2}{b^2}\right )^{-2 n} \left (2\ 15^{1+n} e^{\frac {6 a}{b}} \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^n \left (-\frac {(a+b \text {arcsinh}(c x))^2}{b^2}\right )^n \Gamma \left (1+n,\frac {a}{b}+\text {arcsinh}(c x)\right )+3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )^n \left (3^n \left (-\frac {(a+b \text {arcsinh}(c x))^2}{b^2}\right )^n \Gamma \left (1+n,-\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )+5^{1+n} e^{\frac {2 a}{b}} \left (-\frac {(a+b \text {arcsinh}(c x))^2}{b^2}\right )^n \Gamma \left (1+n,-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )+2\ 3^n 5^{1+n} e^{\frac {4 a}{b}} \left (-\frac {(a+b \text {arcsinh}(c x))^2}{b^2}\right )^n \Gamma \left (1+n,-\frac {a+b \text {arcsinh}(c x)}{b}\right )+5^{1+n} e^{\frac {8 a}{b}} \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{2 n} \Gamma \left (1+n,\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )+3^n e^{\frac {10 a}{b}} \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{2 n} \Gamma \left (1+n,\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )\right )\right )}{32 c^2 \sqrt {d+c^2 d x^2}} \] Input:

Integrate[x*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^n,x]
 

Output:

(15^(-1 - n)*d^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*(2*15^(1 + n)*E^ 
((6*a)/b)*(-((a + b*ArcSinh[c*x])/b))^n*(-((a + b*ArcSinh[c*x])^2/b^2))^n* 
Gamma[1 + n, a/b + ArcSinh[c*x]] + 3*(a/b + ArcSinh[c*x])^n*(3^n*(-((a + b 
*ArcSinh[c*x])^2/b^2))^n*Gamma[1 + n, (-5*(a + b*ArcSinh[c*x]))/b] + 5^(1 
+ n)*E^((2*a)/b)*(-((a + b*ArcSinh[c*x])^2/b^2))^n*Gamma[1 + n, (-3*(a + b 
*ArcSinh[c*x]))/b] + 2*3^n*5^(1 + n)*E^((4*a)/b)*(-((a + b*ArcSinh[c*x])^2 
/b^2))^n*Gamma[1 + n, -((a + b*ArcSinh[c*x])/b)] + 5^(1 + n)*E^((8*a)/b)*( 
-((a + b*ArcSinh[c*x])/b))^(2*n)*Gamma[1 + n, (3*(a + b*ArcSinh[c*x]))/b] 
+ 3^n*E^((10*a)/b)*(-((a + b*ArcSinh[c*x])/b))^(2*n)*Gamma[1 + n, (5*(a + 
b*ArcSinh[c*x]))/b])))/(32*c^2*E^((5*a)/b)*Sqrt[d + c^2*d*x^2]*(-((a + b*A 
rcSinh[c*x])^2/b^2))^(2*n))
 

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 397, normalized size of antiderivative = 0.73, 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.

\(\displaystyle \int x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^n \, dx\)

\(\Big \downarrow \) 6234

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 5971

\(\displaystyle -\frac {d \sqrt {c^2 d x^2+d} \int \left (\frac {1}{16} \sinh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arcsinh}(c x))}{b}\right ) (a+b \text {arcsinh}(c x))^n+\frac {3}{16} \sinh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right ) (a+b \text {arcsinh}(c x))^n+\frac {1}{8} \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) (a+b \text {arcsinh}(c x))^n\right )d(a+b \text {arcsinh}(c x))}{b c^2 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d \sqrt {c^2 d x^2+d} \left (\frac {1}{32} b 5^{-n-1} e^{-\frac {5 a}{b}} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {1}{32} b 3^{-n} e^{-\frac {3 a}{b}} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {1}{16} b e^{-\frac {a}{b}} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {a+b \text {arcsinh}(c x)}{b}\right )+\frac {1}{16} b e^{a/b} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {a+b \text {arcsinh}(c x)}{b}\right )+\frac {1}{32} b 3^{-n} e^{\frac {3 a}{b}} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {1}{32} b 5^{-n-1} e^{\frac {5 a}{b}} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )\right )}{b c^2 \sqrt {c^2 x^2+1}}\)

Input:

Int[x*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^n,x]
 

Output:

(d*Sqrt[d + c^2*d*x^2]*((5^(-1 - n)*b*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, 
(-5*(a + b*ArcSinh[c*x]))/b])/(32*E^((5*a)/b)*(-((a + b*ArcSinh[c*x])/b))^ 
n) + (b*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (-3*(a + b*ArcSinh[c*x]))/b])/ 
(32*3^n*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)])/(16*E^(a/b)*(-((a + b*ArcSin 
h[c*x])/b))^n) + (b*E^(a/b)*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (a + b*Arc 
Sinh[c*x])/b])/(16*((a + b*ArcSinh[c*x])/b)^n) + (b*E^((3*a)/b)*(a + b*Arc 
Sinh[c*x])^n*Gamma[1 + n, (3*(a + b*ArcSinh[c*x]))/b])/(32*3^n*((a + b*Arc 
Sinh[c*x])/b)^n) + (5^(-1 - n)*b*E^((5*a)/b)*(a + b*ArcSinh[c*x])^n*Gamma[ 
1 + n, (5*(a + b*ArcSinh[c*x]))/b])/(32*((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]

\[\int x \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{n}d x\]

Input:

int(x*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(x*c))^n,x)
 

Output:

int(x*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(x*c))^n,x)
 

Fricas [F]

\[ \int x \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^n \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{n} x \,d x } \] Input:

integrate(x*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))^n,x, algorithm="fricas" 
)
 

Output:

integral((c^2*d*x^3 + d*x)*sqrt(c^2*d*x^2 + d)*(b*arcsinh(c*x) + a)^n, x)
 

Sympy [F(-1)]

Timed out. \[ \int x \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^n \, dx=\text {Timed out} \] Input:

integrate(x*(c**2*d*x**2+d)**(3/2)*(a+b*asinh(c*x))**n,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int x \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^n \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{n} x \,d x } \] Input:

integrate(x*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))^n,x, algorithm="maxima" 
)
 

Output:

integrate((c^2*d*x^2 + d)^(3/2)*(b*arcsinh(c*x) + a)^n*x, x)
 

Giac [F(-2)]

Exception generated. \[ \int x \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^n \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x*(c^2*d*x^2+d)^(3/2)*(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. \[ \int x \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^n \, dx=\int x\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^n\,{\left (d\,c^2\,x^2+d\right )}^{3/2} \,d x \] Input:

int(x*(a + b*asinh(c*x))^n*(d + c^2*d*x^2)^(3/2),x)
 

Output:

int(x*(a + b*asinh(c*x))^n*(d + c^2*d*x^2)^(3/2), x)
 

Reduce [F]

\[ \int x \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^n \, dx=\sqrt {d}\, d \left (\left (\int \sqrt {c^{2} x^{2}+1}\, \left (\mathit {asinh} \left (c x \right ) b +a \right )^{n} x^{3}d x \right ) c^{2}+\int \sqrt {c^{2} x^{2}+1}\, \left (\mathit {asinh} \left (c x \right ) b +a \right )^{n} x d x \right ) \] Input:

int(x*(c^2*d*x^2+d)^(3/2)*(a+b*asinh(c*x))^n,x)
 

Output:

sqrt(d)*d*(int(sqrt(c**2*x**2 + 1)*(asinh(c*x)*b + a)**n*x**3,x)*c**2 + in 
t(sqrt(c**2*x**2 + 1)*(asinh(c*x)*b + a)**n*x,x))