\(\int (d+c^2 d x^2)^{3/2} (a+b \text {arcsinh}(c x))^n \, dx\) [468]

Optimal result

Integrand size = 25, antiderivative size = 420 \[ \int \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^n \, dx=\frac {3 d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^{1+n}}{8 b c (1+n) \sqrt {1+c^2 x^2}}+\frac {4^{-3-n} d e^{-\frac {4 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{c \sqrt {1+c^2 x^2}}+\frac {2^{-3-n} d e^{-\frac {2 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{c \sqrt {1+c^2 x^2}}-\frac {2^{-3-n} d e^{\frac {2 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{c \sqrt {1+c^2 x^2}}-\frac {4^{-3-n} d e^{\frac {4 a}{b}} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{c \sqrt {1+c^2 x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.99 (sec) , antiderivative size = 288, normalized size of antiderivative = 0.69 \[ \int \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^n \, dx=\frac {d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {8 (a+b \text {arcsinh}(c x))}{b (1+n)}+8 \left (\frac {4 a+4 b \text {arcsinh}(c x)}{b+b n}+2^{-n} e^{-\frac {2 a}{b}} \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )-2^{-n} e^{\frac {2 a}{b}} \left (\frac {a}{b}+\text {arcsinh}(c x)\right )^{-n} \Gamma \left (1+n,\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )\right )+4^{-n} e^{-\frac {4 a}{b}} \left (-\frac {(a+b \text {arcsinh}(c x))^2}{b^2}\right )^{-n} \left (\left (\frac {a}{b}+\text {arcsinh}(c x)\right )^n \Gamma \left (1+n,-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )-e^{\frac {8 a}{b}} \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^n \Gamma \left (1+n,\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )\right )\right )}{64 c \sqrt {d+c^2 d x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 301, normalized size of antiderivative = 0.72, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6206, 3042, 3793, 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.

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

\(\Big \downarrow \) 6206

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(d*Sqrt[d + c^2*d*x^2]*((3*(a + b*ArcSinh[c*x])^(1 + n))/(8*(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) + (2^(-3 - n)*b*(a + b*A 
rcSinh[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^(-3 - 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*ArcSinh[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*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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
 

rule 6206

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), 
x_Symbol] :> Simp[(1/(b*c))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]   Subst[Int 
[x^n*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, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

integrate((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)
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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