\(\int x (d-c^2 d x^2)^{3/2} (a+b \text {arccosh}(c x)) \, dx\) [80]

Optimal result

Integrand size = 25, antiderivative size = 165 \[ \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\frac {b d x \sqrt {d-c^2 d x^2}}{5 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c d x^3 \sqrt {d-c^2 d x^2}}{15 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{5 c^2 d} \] Output:

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

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.65 \[ \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=-\frac {d \sqrt {d-c^2 d x^2} \left (15 a \left (-1+c^2 x^2\right )^3+b c x \sqrt {-1+c x} \sqrt {1+c x} \left (-15+10 c^2 x^2-3 c^4 x^4\right )+15 b \left (-1+c^2 x^2\right )^3 \text {arccosh}(c x)\right )}{75 c^2 \left (-1+c^2 x^2\right )} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.59, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6329, 39, 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.

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

\(\Big \downarrow \) 6329

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

\(\Big \downarrow \) 39

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

\(\Big \downarrow \) 210

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 39

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[( 
a*c + b*d*x^2)^m, x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b*c + a*d, 0] && ( 
IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))
 

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 6329

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

Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.21

method	result	size
orering	\(\frac {\left (27 c^{6} x^{6}-88 c^{4} x^{4}+115 c^{2} x^{2}-30\right ) \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}{75 c^{2} \left (c x -1\right ) \left (c x +1\right ) \left (c^{2} x^{2}-1\right )}-\frac {\left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) \left (\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )-3 x^{2} \sqrt {-c^{2} d \,x^{2}+d}\, \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) c^{2} d +\frac {x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} b c}{\sqrt {c x -1}\, \sqrt {c x +1}}\right )}{75 c^{2} \left (c x -1\right ) \left (c x +1\right )}\)	\(200\)
default	\(-\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{5 c^{2} d}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (16 c^{6} x^{6}-28 c^{4} x^{4}+16 c^{5} x^{5} \sqrt {c x -1}\, \sqrt {c x +1}+13 c^{2} x^{2}-20 c^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}+5 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -1\right ) \left (-1+5 \,\operatorname {arccosh}\left (c x \right )\right ) d}{800 \left (c x +1\right ) c^{2} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}+4 c^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +1\right ) \left (-1+3 \,\operatorname {arccosh}\left (c x \right )\right ) d}{96 \left (c x +1\right ) c^{2} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (-1+\operatorname {arccosh}\left (c x \right )\right ) d}{16 \left (c x +1\right ) c^{2} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (1+\operatorname {arccosh}\left (c x \right )\right ) d}{16 \left (c x +1\right ) c^{2} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 c^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}+4 c^{4} x^{4}+3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -5 c^{2} x^{2}+1\right ) \left (1+3 \,\operatorname {arccosh}\left (c x \right )\right ) d}{96 \left (c x +1\right ) c^{2} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-16 c^{5} x^{5} \sqrt {c x -1}\, \sqrt {c x +1}+16 c^{6} x^{6}+20 c^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}-28 c^{4} x^{4}-5 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +13 c^{2} x^{2}-1\right ) \left (1+5 \,\operatorname {arccosh}\left (c x \right )\right ) d}{800 \left (c x +1\right ) c^{2} \left (c x -1\right )}\right )\)	\(620\)
parts	\(-\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{5 c^{2} d}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (16 c^{6} x^{6}-28 c^{4} x^{4}+16 c^{5} x^{5} \sqrt {c x -1}\, \sqrt {c x +1}+13 c^{2} x^{2}-20 c^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}+5 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -1\right ) \left (-1+5 \,\operatorname {arccosh}\left (c x \right )\right ) d}{800 \left (c x +1\right ) c^{2} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}+4 c^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +1\right ) \left (-1+3 \,\operatorname {arccosh}\left (c x \right )\right ) d}{96 \left (c x +1\right ) c^{2} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (-1+\operatorname {arccosh}\left (c x \right )\right ) d}{16 \left (c x +1\right ) c^{2} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (1+\operatorname {arccosh}\left (c x \right )\right ) d}{16 \left (c x +1\right ) c^{2} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 c^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}+4 c^{4} x^{4}+3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -5 c^{2} x^{2}+1\right ) \left (1+3 \,\operatorname {arccosh}\left (c x \right )\right ) d}{96 \left (c x +1\right ) c^{2} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-16 c^{5} x^{5} \sqrt {c x -1}\, \sqrt {c x +1}+16 c^{6} x^{6}+20 c^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}-28 c^{4} x^{4}-5 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +13 c^{2} x^{2}-1\right ) \left (1+5 \,\operatorname {arccosh}\left (c x \right )\right ) d}{800 \left (c x +1\right ) c^{2} \left (c x -1\right )}\right )\)	\(620\)

Input:

int(x*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x)),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*x-1)/(c*x+1)/(c^2*x^2-1 
)*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))-1/75*(3*c^4*x^4-10*c^2*x^2+15)/c 
^2/(c*x-1)/(c*x+1)*((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))-3*x^2*(-c^2*d* 
x^2+d)^(1/2)*(a+b*arccosh(c*x))*c^2*d+x*(-c^2*d*x^2+d)^(3/2)*b*c/(c*x-1)^( 
1/2)/(c*x+1)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.12 \[ \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=-\frac {15 \, {\left (b c^{6} d x^{6} - 3 \, b c^{4} d x^{4} + 3 \, b c^{2} d x^{2} - b d\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (3 \, b c^{5} d x^{5} - 10 \, b c^{3} d x^{3} + 15 \, b c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} + 15 \, {\left (a c^{6} d x^{6} - 3 \, a c^{4} d x^{4} + 3 \, a c^{2} d x^{2} - a d\right )} \sqrt {-c^{2} d x^{2} + d}}{75 \, {\left (c^{4} x^{2} - c^{2}\right )}} \] Input:

integrate(x*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(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)) - (3*b*c^5*d*x^5 - 10*b*c^3*d*x^3 + 
15*b*c*d*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) + 15*(a*c^6*d*x^6 - 3*a 
*c^4*d*x^4 + 3*a*c^2*d*x^2 - a*d)*sqrt(-c^2*d*x^2 + d))/(c^4*x^2 - c^2)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.62 \[ \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=-\frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} b \operatorname {arcosh}\left (c x\right )}{5 \, c^{2} d} - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} a}{5 \, c^{2} d} + \frac {{\left (3 \, c^{4} \sqrt {-d} d^{2} x^{5} - 10 \, c^{2} \sqrt {-d} d^{2} x^{3} + 15 \, \sqrt {-d} d^{2} x\right )} b}{75 \, c d} \] Input:

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

int(x*(-c^2*d*x^2+d)^(3/2)*(a+b*acosh(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)*a 
cosh(c*x)*x**3,x)*b*c**4 + 5*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)*x,x)*b* 
c**2))/(5*c**2)