\(\int x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \, dx\) [461]

Optimal result

Integrand size = 28, antiderivative size = 235 \[ \int x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \, dx=-\frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^{1+n}}{8 b c^3 (1+n) \sqrt {1+c^2 x^2}}+\frac {2^{-2 (3+n)} e^{-\frac {4 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}-\frac {2^{-2 (3+n)} e^{\frac {4 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{c^3 \sqrt {1+c^2 x^2}} \] Output:

-1/8*(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)+(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^(6+2*n))/c^3/exp(4*a/b)/(c^2*x^2+1)^(1/2)/((-(a+b*arcsinh(c*x 
))/b)^n)-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^(6+2*n))/c^3/(c^2*x^2+1)^(1/2)/(((a+b*arcsinh(c*x) 
)/b)^n)
 

Mathematica [A] (verified)

Time = 0.64 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.73 \[ \int x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \, dx=\frac {d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {8 (a+b \text {arcsinh}(c x))}{b (1+n)}+4^{-n} e^{-\frac {4 a}{b}} \left (-\frac {(a+b \text {arcsinh}(c x))^2}{b^2}\right )^{-n} \left (\left (\frac {a}{b}+\text {arcsinh}(c x)\right )^n \Gamma \left (1+n,-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )-e^{\frac {8 a}{b}} \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^n \Gamma \left (1+n,\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )\right )\right )}{64 c^3 \sqrt {d \left (1+c^2 x^2\right )}} \] Input:

Integrate[x^2*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*((-8*(a + b*ArcSinh[c*x]))/(b* 
(1 + n)) + ((a/b + ArcSinh[c*x])^n*Gamma[1 + n, (-4*(a + b*ArcSinh[c*x]))/ 
b] - E^((8*a)/b)*(-((a + b*ArcSinh[c*x])/b))^n*Gamma[1 + n, (4*(a + b*ArcS 
inh[c*x]))/b])/(4^n*E^((4*a)/b)*(-((a + b*ArcSinh[c*x])^2/b^2))^n)))/(64*c 
^3*Sqrt[d*(1 + c^2*x^2)])
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.76, 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.

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

\(\Big \downarrow \) 6234

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

\(\Big \downarrow \) 5971

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

int(x^2*(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**2,x)