\(\int x^2 (d+c^2 d x^2)^{5/2} (a+b \text {arcsinh}(c x))^n \, dx\) [471]

Optimal result

Integrand size = 28, antiderivative size = 816 \[ \int x^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^n \, dx=-\frac {5 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^{1+n}}{128 b c^3 (1+n) \sqrt {1+c^2 x^2}}+\frac {2^{-11-3 n} d^2 e^{-\frac {8 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 {8 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}+\frac {2^{-7-n} 3^{-1-n} d^2 e^{-\frac {6 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 {6 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}+\frac {2^{-2 (4+n)} d^2 e^{-\frac {4 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 {4 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}-\frac {2^{-7-n} d^2 e^{-\frac {2 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 {2 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}+\frac {2^{-7-n} d^2 e^{\frac {2 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 {2 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}-\frac {2^{-2 (4+n)} d^2 e^{\frac {4 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 {4 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}-\frac {2^{-7-n} 3^{-1-n} d^2 e^{\frac {6 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 {6 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}-\frac {2^{-11-3 n} d^2 e^{\frac {8 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 {8 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}} \] Output:

-5/128*d^2*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^(1+n)/b/c^3/(1+n)/(c^2*x 
^2+1)^(1/2)+2^(-11-3*n)*d^2*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^n*GAMMA 
(1+n,(-8*a-8*b*arcsinh(c*x))/b)/c^3/exp(8*a/b)/(c^2*x^2+1)^(1/2)/((-(a+b*a 
rcsinh(c*x))/b)^n)+2^(-7-n)*3^(-1-n)*d^2*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh( 
c*x))^n*GAMMA(1+n,(-6*a-6*b*arcsinh(c*x))/b)/c^3/exp(6*a/b)/(c^2*x^2+1)^(1 
/2)/((-(a+b*arcsinh(c*x))/b)^n)+d^2*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x)) 
^n*GAMMA(1+n,(-4*a-4*b*arcsinh(c*x))/b)/(2^(8+2*n))/c^3/exp(4*a/b)/(c^2*x^ 
2+1)^(1/2)/((-(a+b*arcsinh(c*x))/b)^n)-2^(-7-n)*d^2*(c^2*d*x^2+d)^(1/2)*(a 
+b*arcsinh(c*x))^n*GAMMA(1+n,(-2*a-2*b*arcsinh(c*x))/b)/c^3/exp(2*a/b)/(c^ 
2*x^2+1)^(1/2)/((-(a+b*arcsinh(c*x))/b)^n)+2^(-7-n)*d^2*exp(2*a/b)*(c^2*d* 
x^2+d)^(1/2)*(a+b*arcsinh(c*x))^n*GAMMA(1+n,2*(a+b*arcsinh(c*x))/b)/c^3/(c 
^2*x^2+1)^(1/2)/(((a+b*arcsinh(c*x))/b)^n)-d^2*exp(4*a/b)*(c^2*d*x^2+d)^(1 
/2)*(a+b*arcsinh(c*x))^n*GAMMA(1+n,4*(a+b*arcsinh(c*x))/b)/(2^(8+2*n))/c^3 
/(c^2*x^2+1)^(1/2)/(((a+b*arcsinh(c*x))/b)^n)-2^(-7-n)*3^(-1-n)*d^2*exp(6* 
a/b)*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^n*GAMMA(1+n,6*(a+b*arcsinh(c*x 
))/b)/c^3/(c^2*x^2+1)^(1/2)/(((a+b*arcsinh(c*x))/b)^n)-2^(-11-3*n)*d^2*exp 
(8*a/b)*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^n*GAMMA(1+n,8*(a+b*arcsinh( 
c*x))/b)/c^3/(c^2*x^2+1)^(1/2)/(((a+b*arcsinh(c*x))/b)^n)
 

Mathematica [A] (verified)

Time = 4.89 (sec) , antiderivative size = 667, normalized size of antiderivative = 0.82 \[ \int x^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^n \, dx=-\frac {2^{-11-3 n} 3^{-1-n} d^3 e^{-\frac {8 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 )^{-n} \left (-3^{1+n} b (1+n) \left (\frac {a}{b}+\text {arcsinh}(c x)\right )^n \Gamma \left (1+n,-\frac {8 (a+b \text {arcsinh}(c x))}{b}\right )-4^{2+n} b e^{\frac {2 a}{b}} (1+n) \left (\frac {a}{b}+\text {arcsinh}(c x)\right )^n \Gamma \left (1+n,-\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )-2^{3+n} 3^{1+n} b e^{\frac {4 a}{b}} (1+n) \left (\frac {a}{b}+\text {arcsinh}(c x)\right )^n \Gamma \left (1+n,-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )+3^{1+n} 4^{2+n} b e^{\frac {6 a}{b}} (1+n) \left (\frac {a}{b}+\text {arcsinh}(c x)\right )^n \Gamma \left (1+n,-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )+e^{\frac {8 a}{b}} \left (5\ 2^{4+3 n} 3^{1+n} a \left (-\frac {(a+b \text {arcsinh}(c x))^2}{b^2}\right )^n+5\ 2^{4+3 n} 3^{1+n} b \text {arcsinh}(c x) \left (-\frac {(a+b \text {arcsinh}(c x))^2}{b^2}\right )^n-3^{1+n} 4^{2+n} b e^{\frac {2 a}{b}} (1+n) \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^n \Gamma \left (1+n,\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )+2^{3+n} 3^{1+n} b e^{\frac {4 a}{b}} (1+n) \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^n \Gamma \left (1+n,\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )+4^{2+n} b e^{\frac {6 a}{b}} \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^n \Gamma \left (1+n,\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )+4^{2+n} b e^{\frac {6 a}{b}} n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^n \Gamma \left (1+n,\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )+3^{1+n} b e^{\frac {8 a}{b}} \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^n \Gamma \left (1+n,\frac {8 (a+b \text {arcsinh}(c x))}{b}\right )+3^{1+n} b e^{\frac {8 a}{b}} n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^n \Gamma \left (1+n,\frac {8 (a+b \text {arcsinh}(c x))}{b}\right )\right )\right )}{b c^3 (1+n) \sqrt {d+c^2 d x^2}} \] Input:

Integrate[x^2*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^n,x]
 

Output:

-((2^(-11 - 3*n)*3^(-1 - n)*d^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*( 
-(3^(1 + n)*b*(1 + n)*(a/b + ArcSinh[c*x])^n*Gamma[1 + n, (-8*(a + b*ArcSi 
nh[c*x]))/b]) - 4^(2 + n)*b*E^((2*a)/b)*(1 + n)*(a/b + ArcSinh[c*x])^n*Gam 
ma[1 + n, (-6*(a + b*ArcSinh[c*x]))/b] - 2^(3 + n)*3^(1 + n)*b*E^((4*a)/b) 
*(1 + n)*(a/b + ArcSinh[c*x])^n*Gamma[1 + n, (-4*(a + b*ArcSinh[c*x]))/b] 
+ 3^(1 + n)*4^(2 + n)*b*E^((6*a)/b)*(1 + n)*(a/b + ArcSinh[c*x])^n*Gamma[1 
 + n, (-2*(a + b*ArcSinh[c*x]))/b] + E^((8*a)/b)*(5*2^(4 + 3*n)*3^(1 + n)* 
a*(-((a + b*ArcSinh[c*x])^2/b^2))^n + 5*2^(4 + 3*n)*3^(1 + n)*b*ArcSinh[c* 
x]*(-((a + b*ArcSinh[c*x])^2/b^2))^n - 3^(1 + n)*4^(2 + n)*b*E^((2*a)/b)*( 
1 + n)*(-((a + b*ArcSinh[c*x])/b))^n*Gamma[1 + n, (2*(a + b*ArcSinh[c*x])) 
/b] + 2^(3 + n)*3^(1 + n)*b*E^((4*a)/b)*(1 + n)*(-((a + b*ArcSinh[c*x])/b) 
)^n*Gamma[1 + n, (4*(a + b*ArcSinh[c*x]))/b] + 4^(2 + n)*b*E^((6*a)/b)*(-( 
(a + b*ArcSinh[c*x])/b))^n*Gamma[1 + n, (6*(a + b*ArcSinh[c*x]))/b] + 4^(2 
 + n)*b*E^((6*a)/b)*n*(-((a + b*ArcSinh[c*x])/b))^n*Gamma[1 + n, (6*(a + b 
*ArcSinh[c*x]))/b] + 3^(1 + n)*b*E^((8*a)/b)*(-((a + b*ArcSinh[c*x])/b))^n 
*Gamma[1 + n, (8*(a + b*ArcSinh[c*x]))/b] + 3^(1 + n)*b*E^((8*a)/b)*n*(-(( 
a + b*ArcSinh[c*x])/b))^n*Gamma[1 + n, (8*(a + b*ArcSinh[c*x]))/b])))/(b*c 
^3*E^((8*a)/b)*(1 + n)*Sqrt[d + c^2*d*x^2]*(-((a + b*ArcSinh[c*x])^2/b^2)) 
^n))
 

Rubi [A] (verified)

Time = 1.64 (sec) , antiderivative size = 561, normalized size of antiderivative = 0.69, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {6234, 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.

\(\displaystyle \int x^2 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^n \, dx\)

\(\Big \downarrow \) 6234

\(\displaystyle \frac {d^2 \sqrt {c^2 d x^2+d} \int (a+b \text {arcsinh}(c x))^n \cosh ^6\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh ^2\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )d(a+b \text {arcsinh}(c x))}{b c^3 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 5971

\(\displaystyle \frac {d^2 \sqrt {c^2 d x^2+d} \int \left (\frac {1}{128} \cosh \left (\frac {8 a}{b}-\frac {8 (a+b \text {arcsinh}(c x))}{b}\right ) (a+b \text {arcsinh}(c x))^n+\frac {1}{32} \cosh \left (\frac {6 a}{b}-\frac {6 (a+b \text {arcsinh}(c x))}{b}\right ) (a+b \text {arcsinh}(c x))^n+\frac {1}{32} \cosh \left (\frac {4 a}{b}-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right ) (a+b \text {arcsinh}(c x))^n-\frac {1}{32} \cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right ) (a+b \text {arcsinh}(c x))^n-\frac {5}{128} (a+b \text {arcsinh}(c x))^n\right )d(a+b \text {arcsinh}(c x))}{b c^3 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d^2 \sqrt {c^2 d x^2+d} \left (-\frac {5 (a+b \text {arcsinh}(c x))^{n+1}}{128 (n+1)}+b 2^{-3 n-11} e^{-\frac {8 a}{b}} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {8 (a+b \text {arcsinh}(c x))}{b}\right )+b 2^{-n-7} 3^{-n-1} e^{-\frac {6 a}{b}} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )+b 2^{-2 (n+4)} e^{-\frac {4 a}{b}} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )-b 2^{-n-7} e^{-\frac {2 a}{b}} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )+b 2^{-n-7} e^{\frac {2 a}{b}} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )-b 2^{-2 (n+4)} e^{\frac {4 a}{b}} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )-b 2^{-n-7} 3^{-n-1} e^{\frac {6 a}{b}} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )-b 2^{-3 n-11} e^{\frac {8 a}{b}} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {8 (a+b \text {arcsinh}(c x))}{b}\right )\right )}{b c^3 \sqrt {c^2 x^2+1}}\)

Input:

Int[x^2*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^n,x]
 

Output:

(d^2*Sqrt[d + c^2*d*x^2]*((-5*(a + b*ArcSinh[c*x])^(1 + n))/(128*(1 + n)) 
+ (2^(-11 - 3*n)*b*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (-8*(a + b*ArcSinh[ 
c*x]))/b])/(E^((8*a)/b)*(-((a + b*ArcSinh[c*x])/b))^n) + (2^(-7 - n)*3^(-1 
 - n)*b*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (-6*(a + b*ArcSinh[c*x]))/b])/ 
(E^((6*a)/b)*(-((a + b*ArcSinh[c*x])/b))^n) + (b*(a + b*ArcSinh[c*x])^n*Ga 
mma[1 + n, (-4*(a + b*ArcSinh[c*x]))/b])/(2^(2*(4 + n))*E^((4*a)/b)*(-((a 
+ b*ArcSinh[c*x])/b))^n) - (2^(-7 - n)*b*(a + b*ArcSinh[c*x])^n*Gamma[1 + 
n, (-2*(a + b*ArcSinh[c*x]))/b])/(E^((2*a)/b)*(-((a + b*ArcSinh[c*x])/b))^ 
n) + (2^(-7 - n)*b*E^((2*a)/b)*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (2*(a + 
 b*ArcSinh[c*x]))/b])/((a + b*ArcSinh[c*x])/b)^n - (b*E^((4*a)/b)*(a + b*A 
rcSinh[c*x])^n*Gamma[1 + n, (4*(a + b*ArcSinh[c*x]))/b])/(2^(2*(4 + n))*(( 
a + b*ArcSinh[c*x])/b)^n) - (2^(-7 - n)*3^(-1 - n)*b*E^((6*a)/b)*(a + b*Ar 
cSinh[c*x])^n*Gamma[1 + n, (6*(a + b*ArcSinh[c*x]))/b])/((a + b*ArcSinh[c* 
x])/b)^n - (2^(-11 - 3*n)*b*E^((8*a)/b)*(a + b*ArcSinh[c*x])^n*Gamma[1 + n 
, (8*(a + b*ArcSinh[c*x]))/b])/((a + b*ArcSinh[c*x])/b)^n))/(b*c^3*Sqrt[1 
+ c^2*x^2])
 

Defintions of rubi rules used

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]

\[\int x^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{n}d x\]

Input:

int(x^2*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(x*c))^n,x)
 

Output:

int(x^2*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(x*c))^n,x)
 

Fricas [F]

\[ \int x^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^n \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{n} x^{2} \,d x } \] Input:

integrate(x^2*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^n,x, algorithm="frica 
s")
 

Output:

integral((c^4*d^2*x^6 + 2*c^2*d^2*x^4 + d^2*x^2)*sqrt(c^2*d*x^2 + d)*(b*ar 
csinh(c*x) + a)^n, x)
 

Sympy [F(-1)]

Timed out. \[ \int x^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^n \, dx=\text {Timed out} \] Input:

integrate(x**2*(c**2*d*x**2+d)**(5/2)*(a+b*asinh(c*x))**n,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int x^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^n \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{n} x^{2} \,d x } \] Input:

integrate(x^2*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^n,x, algorithm="maxim 
a")
 

Output:

integrate((c^2*d*x^2 + d)^(5/2)*(b*arcsinh(c*x) + a)^n*x^2, x)
 

Giac [F]

\[ \int x^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^n \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{n} x^{2} \,d x } \] Input:

integrate(x^2*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^n,x, algorithm="giac" 
)
 

Output:

integrate((c^2*d*x^2 + d)^(5/2)*(b*arcsinh(c*x) + a)^n*x^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int x^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^n \, dx=\int x^2\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^n\,{\left (d\,c^2\,x^2+d\right )}^{5/2} \,d x \] Input:

int(x^2*(a + b*asinh(c*x))^n*(d + c^2*d*x^2)^(5/2),x)
 

Output:

int(x^2*(a + b*asinh(c*x))^n*(d + c^2*d*x^2)^(5/2), x)
 

Reduce [F]

\[ \int x^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^n \, dx=\sqrt {d}\, d^{2} \left (\left (\int \sqrt {c^{2} x^{2}+1}\, \left (\mathit {asinh} \left (c x \right ) b +a \right )^{n} x^{6}d x \right ) c^{4}+2 \left (\int \sqrt {c^{2} x^{2}+1}\, \left (\mathit {asinh} \left (c x \right ) b +a \right )^{n} x^{4}d x \right ) c^{2}+\int \sqrt {c^{2} x^{2}+1}\, \left (\mathit {asinh} \left (c x \right ) b +a \right )^{n} x^{2}d x \right ) \] Input:

int(x^2*(c^2*d*x^2+d)^(5/2)*(a+b*asinh(c*x))^n,x)
 

Output:

sqrt(d)*d**2*(int(sqrt(c**2*x**2 + 1)*(asinh(c*x)*b + a)**n*x**6,x)*c**4 + 
 2*int(sqrt(c**2*x**2 + 1)*(asinh(c*x)*b + a)**n*x**4,x)*c**2 + int(sqrt(c 
**2*x**2 + 1)*(asinh(c*x)*b + a)**n*x**2,x))