\(\int x^2 (d+c^2 d x^2)^{3/2} (a+b \text {arcsinh}(c x))^n \, dx\) [466]

Optimal result

Integrand size = 28, antiderivative size = 616 \[ \int x^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^n \, dx=-\frac {d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^{1+n}}{16 b c^3 (1+n) \sqrt {1+c^2 x^2}}+\frac {2^{-7-n} 3^{-1-n} d e^{-\frac {6 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}+\frac {2^{-7-2 n} d e^{-\frac {4 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}-\frac {2^{-7-n} d e^{-\frac {2 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}+\frac {2^{-7-n} d e^{\frac {2 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}-\frac {2^{-7-2 n} d e^{\frac {4 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}-\frac {2^{-7-n} 3^{-1-n} d e^{\frac {6 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}} \] Output:

-1/16*d*(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^(-7-n)*3^(-1-n)*d*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^n*GAMM 
A(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)+2^(-7-2*n)*d*(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)/c^3/exp(4*a/b)/(c^2*x^2+1)^(1/2)/((-( 
a+b*arcsinh(c*x))/b)^n)-2^(-7-n)*d*(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*exp(2*a/b)*(c^2*d*x^2+d)^(1/2)*(a+b*a 
rcsinh(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)-2^(-7-2*n)*d*exp(4*a/b)*(c^2*d*x^2+d)^(1/2)*(a+b*a 
rcsinh(c*x))^n*GAMMA(1+n,4*(a+b*arcsinh(c*x))/b)/c^3/(c^2*x^2+1)^(1/2)/((( 
a+b*arcsinh(c*x))/b)^n)-2^(-7-n)*3^(-1-n)*d*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)
 

Mathematica [A] (verified)

Time = 2.04 (sec) , antiderivative size = 429, normalized size of antiderivative = 0.70 \[ \int x^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^n \, dx=-\frac {2^{-7-2 n} 3^{-1-n} d^2 e^{-\frac {6 a}{b}} \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {(a+b \text {arcsinh}(c x))^2}{b^2}\right )^{-n} \left (-2^n b (1+n) \left (\frac {a}{b}+\text {arcsinh}(c x)\right )^n \Gamma \left (1+n,-\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )-3^{1+n} b e^{\frac {2 a}{b}} (1+n) \left (\frac {a}{b}+\text {arcsinh}(c x)\right )^n \Gamma \left (1+n,-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )+2^n 3^{1+n} b e^{\frac {4 a}{b}} (1+n) \left (\frac {a}{b}+\text {arcsinh}(c x)\right )^n \Gamma \left (1+n,-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )-2^n 3^{1+n} b e^{\frac {8 a}{b}} (1+n) \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^n \Gamma \left (1+n,\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )+3^{1+n} b e^{\frac {10 a}{b}} (1+n) \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^n \Gamma \left (1+n,\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )+2^n e^{\frac {6 a}{b}} \left (2^{3+n} 3^{1+n} (a+b \text {arcsinh}(c x)) \left (-\frac {(a+b \text {arcsinh}(c x))^2}{b^2}\right )^n+b e^{\frac {6 a}{b}} (1+n) \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^n \Gamma \left (1+n,\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )\right )\right )}{b c^3 (1+n) \sqrt {d+c^2 d x^2}} \] Input:

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

Output:

-((2^(-7 - 2*n)*3^(-1 - n)*d^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*(- 
(2^n*b*(1 + n)*(a/b + ArcSinh[c*x])^n*Gamma[1 + n, (-6*(a + b*ArcSinh[c*x] 
))/b]) - 3^(1 + n)*b*E^((2*a)/b)*(1 + n)*(a/b + ArcSinh[c*x])^n*Gamma[1 + 
n, (-4*(a + b*ArcSinh[c*x]))/b] + 2^n*3^(1 + n)*b*E^((4*a)/b)*(1 + n)*(a/b 
 + ArcSinh[c*x])^n*Gamma[1 + n, (-2*(a + b*ArcSinh[c*x]))/b] - 2^n*3^(1 + 
n)*b*E^((8*a)/b)*(1 + n)*(-((a + b*ArcSinh[c*x])/b))^n*Gamma[1 + n, (2*(a 
+ b*ArcSinh[c*x]))/b] + 3^(1 + n)*b*E^((10*a)/b)*(1 + n)*(-((a + b*ArcSinh 
[c*x])/b))^n*Gamma[1 + n, (4*(a + b*ArcSinh[c*x]))/b] + 2^n*E^((6*a)/b)*(2 
^(3 + n)*3^(1 + n)*(a + b*ArcSinh[c*x])*(-((a + b*ArcSinh[c*x])^2/b^2))^n 
+ b*E^((6*a)/b)*(1 + n)*(-((a + b*ArcSinh[c*x])/b))^n*Gamma[1 + n, (6*(a + 
 b*ArcSinh[c*x]))/b])))/(b*c^3*E^((6*a)/b)*(1 + n)*Sqrt[d + c^2*d*x^2]*(-( 
(a + b*ArcSinh[c*x])^2/b^2))^n))
 

Rubi [A] (verified)

Time = 0.90 (sec) , antiderivative size = 437, normalized size of antiderivative = 0.71, 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.

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

\(\Big \downarrow \) 6234

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

\(\Big \downarrow \) 5971

\(\displaystyle \frac {d \sqrt {c^2 d x^2+d} \int \left (\frac {1}{32} \cosh \left (\frac {6 a}{b}-\frac {6 (a+b \text {arcsinh}(c x))}{b}\right ) (a+b \text {arcsinh}(c x))^n+\frac {1}{16} \cosh \left (\frac {4 a}{b}-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right ) (a+b \text {arcsinh}(c x))^n-\frac {1}{32} \cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right ) (a+b \text {arcsinh}(c x))^n-\frac {1}{16} (a+b \text {arcsinh}(c x))^n\right )d(a+b \text {arcsinh}(c x))}{b c^3 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d \sqrt {c^2 d x^2+d} \left (-\frac {(a+b \text {arcsinh}(c x))^{n+1}}{16 (n+1)}+b 2^{-n-7} 3^{-n-1} e^{-\frac {6 a}{b}} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )+b 2^{-2 n-7} e^{-\frac {4 a}{b}} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )-b 2^{-n-7} e^{-\frac {2 a}{b}} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )+b 2^{-n-7} e^{\frac {2 a}{b}} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )-b 2^{-2 n-7} e^{\frac {4 a}{b}} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )-b 2^{-n-7} 3^{-n-1} e^{\frac {6 a}{b}} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )\right )}{b c^3 \sqrt {c^2 x^2+1}}\)

Input:

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

Output:

(d*Sqrt[d + c^2*d*x^2]*(-1/16*(a + b*ArcSinh[c*x])^(1 + n)/(1 + 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) + (2^(-7 - 2*n)*b* 
(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (-4*(a + b*ArcSinh[c*x]))/b])/(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 - (2^(-7 - 2*n 
)*b*E^((4*a)/b)*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (4*(a + b*ArcSinh[c*x] 
))/b])/((a + b*ArcSinh[c*x])/b)^n - (2^(-7 - n)*3^(-1 - n)*b*E^((6*a)/b)*( 
a + b*ArcSinh[c*x])^n*Gamma[1 + n, (6*(a + b*ArcSinh[c*x]))/b])/((a + b*Ar 
cSinh[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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^n \, dx=\sqrt {d}\, d \left (\left (\int \sqrt {c^{2} x^{2}+1}\, \left (\mathit {asinh} \left (c x \right ) b +a \right )^{n} x^{4}d x \right ) c^{2}+\int \sqrt {c^{2} x^{2}+1}\, \left (\mathit {asinh} \left (c x \right ) b +a \right )^{n} x^{2}d x \right ) \] Input:

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