\(\int x^m \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx\) [205]

Optimal result

Integrand size = 26, antiderivative size = 240 \[ \int x^m \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {b c x^{2+m} \sqrt {d+c^2 d x^2}}{(2+m)^2 \sqrt {1+c^2 x^2}}+\frac {x^{1+m} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2+m}+\frac {x^{1+m} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},-c^2 x^2\right )}{\left (2+3 m+m^2\right ) \sqrt {1+c^2 x^2}}-\frac {b c x^{2+m} \sqrt {d+c^2 d x^2} \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};-c^2 x^2\right )}{(1+m) (2+m)^2 \sqrt {1+c^2 x^2}} \] Output:

-b*c*x^(2+m)*(c^2*d*x^2+d)^(1/2)/(2+m)^2/(c^2*x^2+1)^(1/2)+x^(1+m)*(c^2*d* 
x^2+d)^(1/2)*(a+b*arcsinh(c*x))/(2+m)+x^(1+m)*(c^2*d*x^2+d)^(1/2)*(a+b*arc 
sinh(c*x))*hypergeom([1/2, 1/2+1/2*m],[3/2+1/2*m],-c^2*x^2)/(m^2+3*m+2)/(c 
^2*x^2+1)^(1/2)-b*c*x^(2+m)*(c^2*d*x^2+d)^(1/2)*hypergeom([1, 1+1/2*m, 1+1 
/2*m],[2+1/2*m, 3/2+1/2*m],-c^2*x^2)/(1+m)/(2+m)^2/(c^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.75 \[ \int x^m \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {x^{1+m} \sqrt {d+c^2 d x^2} \left ((1+m) \left (-b c x+a (2+m) \sqrt {1+c^2 x^2}+b (2+m) \sqrt {1+c^2 x^2} \text {arcsinh}(c x)\right )+(2+m) (a+b \text {arcsinh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},-c^2 x^2\right )-b c x \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};-c^2 x^2\right )\right )}{(1+m) (2+m)^2 \sqrt {1+c^2 x^2}} \] Input:

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

Output:

(x^(1 + m)*Sqrt[d + c^2*d*x^2]*((1 + m)*(-(b*c*x) + a*(2 + m)*Sqrt[1 + c^2 
*x^2] + b*(2 + m)*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]) + (2 + m)*(a + b*ArcSinh 
[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, -(c^2*x^2)] - b*c*x*Hy 
pergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, -(c^2*x^2)])) 
/((1 + m)*(2 + m)^2*Sqrt[1 + c^2*x^2])
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.90, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {6221, 15, 6232}

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.

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

\(\Big \downarrow \) 6221

\(\displaystyle \frac {\sqrt {c^2 d x^2+d} \int \frac {x^m (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{(m+2) \sqrt {c^2 x^2+1}}-\frac {b c \sqrt {c^2 d x^2+d} \int x^{m+1}dx}{(m+2) \sqrt {c^2 x^2+1}}+\frac {x^{m+1} \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{m+2}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {\sqrt {c^2 d x^2+d} \int \frac {x^m (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{(m+2) \sqrt {c^2 x^2+1}}+\frac {x^{m+1} \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{m+2}-\frac {b c x^{m+2} \sqrt {c^2 d x^2+d}}{(m+2)^2 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 6232

\(\displaystyle \frac {\sqrt {c^2 d x^2+d} \left (\frac {x^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},-c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{m+1}-\frac {b c x^{m+2} \, _3F_2\left (1,\frac {m}{2}+1,\frac {m}{2}+1;\frac {m}{2}+\frac {3}{2},\frac {m}{2}+2;-c^2 x^2\right )}{m^2+3 m+2}\right )}{(m+2) \sqrt {c^2 x^2+1}}+\frac {x^{m+1} \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{m+2}-\frac {b c x^{m+2} \sqrt {c^2 d x^2+d}}{(m+2)^2 \sqrt {c^2 x^2+1}}\)

Input:

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

Output:

-((b*c*x^(2 + m)*Sqrt[d + c^2*d*x^2])/((2 + m)^2*Sqrt[1 + c^2*x^2])) + (x^ 
(1 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(2 + m) + (Sqrt[d + c^2* 
d*x^2]*((x^(1 + m)*(a + b*ArcSinh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, 
(3 + m)/2, -(c^2*x^2)])/(1 + m) - (b*c*x^(2 + m)*HypergeometricPFQ[{1, 1 + 
 m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, -(c^2*x^2)])/(2 + 3*m + m^2)))/((2 + 
 m)*Sqrt[1 + c^2*x^2])
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 6221

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + 
 (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*Arc 
Sinh[c*x])^n/(f*(m + 2))), x] + (Simp[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/Sqrt 
[1 + c^2*x^2]]   Int[(f*x)^m*((a + b*ArcSinh[c*x])^n/Sqrt[1 + c^2*x^2]), x] 
, x] - Simp[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]]   I 
nt[(f*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d 
, e, f, m}, x] && EqQ[e, c^2*d] && IGtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])
 

rule 6232

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_ 
.)*(x_)^2], x_Symbol] :> Simp[((f*x)^(m + 1)/(f*(m + 1)))*Simp[Sqrt[1 + c^2 
*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSinh[c*x])*Hypergeometric2F1[1/2, (1 + m)/ 
2, (3 + m)/2, (-c^2)*x^2], x] - Simp[b*c*((f*x)^(m + 2)/(f^2*(m + 1)*(m + 2 
)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*HypergeometricPFQ[{1, 1 + m/2, 
1 + m/2}, {3/2 + m/2, 2 + m/2}, (-c^2)*x^2], x] /; FreeQ[{a, b, c, d, e, f, 
 m}, x] && EqQ[e, c^2*d] &&  !IntegerQ[m]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

integrate(x^m*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x)),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. \[ \int x^m \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\int x^m\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\sqrt {d\,c^2\,x^2+d} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

sqrt(d)*(int(x**m*sqrt(c**2*x**2 + 1)*asinh(c*x),x)*b + int(x**m*sqrt(c**2 
*x**2 + 1),x)*a)