\(\int x (d+c^2 d x^2)^{3/2} (a+b \text {arcsinh}(c x)) \, dx\) [139]

Optimal result

Integrand size = 24, antiderivative size = 146 \[ \int x \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {b d x \sqrt {d+c^2 d x^2}}{5 c \sqrt {1+c^2 x^2}}-\frac {2 b c d x^3 \sqrt {d+c^2 d x^2}}{15 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^5 \sqrt {d+c^2 d x^2}}{25 \sqrt {1+c^2 x^2}}+\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^2 d} \] Output:

-1/5*b*d*x*(c^2*d*x^2+d)^(1/2)/c/(c^2*x^2+1)^(1/2)-2/15*b*c*d*x^3*(c^2*d*x 
^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-1/25*b*c^3*d*x^5*(c^2*d*x^2+d)^(1/2)/(c^2*x^ 
2+1)^(1/2)+1/5*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))/c^2/d
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.70 \[ \int x \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {d \sqrt {d+c^2 d x^2} \left (15 a \left (1+c^2 x^2\right )^3-b c x \sqrt {1+c^2 x^2} \left (15+10 c^2 x^2+3 c^4 x^4\right )+15 b \left (1+c^2 x^2\right )^3 \text {arcsinh}(c x)\right )}{75 c^2 \left (1+c^2 x^2\right )} \] Input:

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

Output:

(d*Sqrt[d + c^2*d*x^2]*(15*a*(1 + c^2*x^2)^3 - b*c*x*Sqrt[1 + c^2*x^2]*(15 
 + 10*c^2*x^2 + 3*c^4*x^4) + 15*b*(1 + c^2*x^2)^3*ArcSinh[c*x]))/(75*c^2*( 
1 + c^2*x^2))
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.62, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6213, 210, 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.

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

\(\Big \downarrow \) 6213

\(\displaystyle \frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^2 d}-\frac {b d \sqrt {c^2 d x^2+d} \int \left (c^2 x^2+1\right )^2dx}{5 c \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 210

\(\displaystyle \frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^2 d}-\frac {b d \sqrt {c^2 d x^2+d} \int \left (c^4 x^4+2 c^2 x^2+1\right )dx}{5 c \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^2 d}-\frac {b d \left (\frac {c^4 x^5}{5}+\frac {2 c^2 x^3}{3}+x\right ) \sqrt {c^2 d x^2+d}}{5 c \sqrt {c^2 x^2+1}}\)

Input:

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

Output:

-1/5*(b*d*Sqrt[d + c^2*d*x^2]*(x + (2*c^2*x^3)/3 + (c^4*x^5)/5))/(c*Sqrt[1 
 + c^2*x^2]) + ((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(5*c^2*d)
 

Defintions of rubi rules used

rule 210

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^2 
)^p, x], x] /; FreeQ[{a, b}, x] && IGtQ[p, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6213

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p 
+ 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] 
 Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[ 
{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && NeQ[p, -1]
 

Maple [A] (verified)

Time = 1.49 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.21

method	result	size
orering	\(\frac {\left (27 c^{6} x^{6}+88 c^{4} x^{4}+115 c^{2} x^{2}+30\right ) \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}{75 c^{2} \left (c^{2} x^{2}+1\right )^{2}}-\frac {\left (3 c^{4} x^{4}+10 c^{2} x^{2}+15\right ) \left (\left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )+3 x^{2} \sqrt {c^{2} d \,x^{2}+d}\, \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) c^{2} d +\frac {x \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} b c}{\sqrt {c^{2} x^{2}+1}}\right )}{75 c^{2} \left (c^{2} x^{2}+1\right )}\)	\(176\)
default	\(\frac {a \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{5 c^{2} d}+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (16 c^{6} x^{6}+16 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+28 c^{4} x^{4}+20 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+13 c^{2} x^{2}+5 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (-1+5 \,\operatorname {arcsinh}\left (x c \right )\right ) d}{800 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}+3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (-1+3 \,\operatorname {arcsinh}\left (x c \right )\right ) d}{96 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )-1\right ) d}{16 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )+1\right ) d}{16 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}-3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (1+3 \,\operatorname {arcsinh}\left (x c \right )\right ) d}{96 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (16 c^{6} x^{6}-16 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+28 c^{4} x^{4}-20 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+13 c^{2} x^{2}-5 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (1+5 \,\operatorname {arcsinh}\left (x c \right )\right ) d}{800 c^{2} \left (c^{2} x^{2}+1\right )}\right )\)	\(559\)
parts	\(\frac {a \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{5 c^{2} d}+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (16 c^{6} x^{6}+16 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+28 c^{4} x^{4}+20 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+13 c^{2} x^{2}+5 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (-1+5 \,\operatorname {arcsinh}\left (x c \right )\right ) d}{800 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}+3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (-1+3 \,\operatorname {arcsinh}\left (x c \right )\right ) d}{96 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )-1\right ) d}{16 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )+1\right ) d}{16 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}-3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (1+3 \,\operatorname {arcsinh}\left (x c \right )\right ) d}{96 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (16 c^{6} x^{6}-16 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+28 c^{4} x^{4}-20 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+13 c^{2} x^{2}-5 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (1+5 \,\operatorname {arcsinh}\left (x c \right )\right ) d}{800 c^{2} \left (c^{2} x^{2}+1\right )}\right )\)	\(559\)

Input:

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

Output:

1/75*(27*c^6*x^6+88*c^4*x^4+115*c^2*x^2+30)/c^2/(c^2*x^2+1)^2*(c^2*d*x^2+d 
)^(3/2)*(a+b*arcsinh(x*c))-1/75*(3*c^4*x^4+10*c^2*x^2+15)/c^2/(c^2*x^2+1)* 
((c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(x*c))+3*x^2*(c^2*d*x^2+d)^(1/2)*(a+b*arc 
sinh(x*c))*c^2*d+x*(c^2*d*x^2+d)^(3/2)*b*c/(c^2*x^2+1)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.14 \[ \int x \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {15 \, {\left (b c^{6} d x^{6} + 3 \, b c^{4} d x^{4} + 3 \, b c^{2} d x^{2} + b d\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (15 \, a c^{6} d x^{6} + 45 \, a c^{4} d x^{4} + 45 \, a c^{2} d x^{2} + 15 \, a d - {\left (3 \, b c^{5} d x^{5} + 10 \, b c^{3} d x^{3} + 15 \, b c d x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \sqrt {c^{2} d x^{2} + d}}{75 \, {\left (c^{4} x^{2} + c^{2}\right )}} \] Input:

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

Output:

1/75*(15*(b*c^6*d*x^6 + 3*b*c^4*d*x^4 + 3*b*c^2*d*x^2 + b*d)*sqrt(c^2*d*x^ 
2 + d)*log(c*x + sqrt(c^2*x^2 + 1)) + (15*a*c^6*d*x^6 + 45*a*c^4*d*x^4 + 4 
5*a*c^2*d*x^2 + 15*a*d - (3*b*c^5*d*x^5 + 10*b*c^3*d*x^3 + 15*b*c*d*x)*sqr 
t(c^2*x^2 + 1))*sqrt(c^2*d*x^2 + d))/(c^4*x^2 + c^2)
 

Sympy [F]

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

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

Output:

Integral(x*(d*(c**2*x**2 + 1))**(3/2)*(a + b*asinh(c*x)), x)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.58 \[ \int x \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} b \operatorname {arsinh}\left (c x\right )}{5 \, c^{2} d} + \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} a}{5 \, c^{2} d} - \frac {{\left (3 \, c^{4} d^{\frac {5}{2}} x^{5} + 10 \, c^{2} d^{\frac {5}{2}} x^{3} + 15 \, d^{\frac {5}{2}} x\right )} b}{75 \, c d} \] Input:

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

Output:

1/5*(c^2*d*x^2 + d)^(5/2)*b*arcsinh(c*x)/(c^2*d) + 1/5*(c^2*d*x^2 + d)^(5/ 
2)*a/(c^2*d) - 1/75*(3*c^4*d^(5/2)*x^5 + 10*c^2*d^(5/2)*x^3 + 15*d^(5/2)*x 
)*b/(c*d)
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int x \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\sqrt {d}\, d \left (\sqrt {c^{2} x^{2}+1}\, a \,c^{4} x^{4}+2 \sqrt {c^{2} x^{2}+1}\, a \,c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, a +5 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{3}d x \right ) b \,c^{4}+5 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x d x \right ) b \,c^{2}\right )}{5 c^{2}} \] Input:

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

Output:

(sqrt(d)*d*(sqrt(c**2*x**2 + 1)*a*c**4*x**4 + 2*sqrt(c**2*x**2 + 1)*a*c**2 
*x**2 + sqrt(c**2*x**2 + 1)*a + 5*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x**3, 
x)*b*c**4 + 5*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x,x)*b*c**2))/(5*c**2)