\(\int x^3 (d-c^2 d x^2)^{5/2} (a+b \text {arccosh}(c x)) \, dx\) [95]

Optimal result

Integrand size = 27, antiderivative size = 298 \[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\frac {2 b d^2 x \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 x^3 \sqrt {d-c^2 d x^2}}{189 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 x^5 \sqrt {d-c^2 d x^2}}{21 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {19 b c^3 d^2 x^7 \sqrt {d-c^2 d x^2}}{441 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^9 \sqrt {d-c^2 d x^2}}{81 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^4 d}+\frac {\left (d-c^2 d x^2\right )^{9/2} (a+b \text {arccosh}(c x))}{9 c^4 d^2} \] Output:

2/63*b*d^2*x*(-c^2*d*x^2+d)^(1/2)/c^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/189*b* 
d^2*x^3*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/21*b*c*d^2*x^ 
5*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+19/441*b*c^3*d^2*x^7*(- 
c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/81*b*c^5*d^2*x^9*(-c^2*d* 
x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/7*(-c^2*d*x^2+d)^(7/2)*(a+b*arc 
cosh(c*x))/c^4/d+1/9*(-c^2*d*x^2+d)^(9/2)*(a+b*arccosh(c*x))/c^4/d^2
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.49 \[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\frac {d^2 \sqrt {d-c^2 d x^2} \left (b c \left (126 x+21 c^2 x^3-189 c^4 x^5+171 c^6 x^7-49 c^8 x^9\right )+126 (-1+c x)^{7/2} (1+c x)^{7/2} (a+b \text {arccosh}(c x))+441 c^2 x^2 (-1+c x)^{7/2} (1+c x)^{7/2} (a+b \text {arccosh}(c x))\right )}{3969 c^4 \sqrt {-1+c x} \sqrt {1+c x}} \] Input:

Integrate[x^3*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]),x]
 

Output:

(d^2*Sqrt[d - c^2*d*x^2]*(b*c*(126*x + 21*c^2*x^3 - 189*c^4*x^5 + 171*c^6* 
x^7 - 49*c^8*x^9) + 126*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2)*(a + b*ArcCosh[c* 
x]) + 441*c^2*x^2*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2)*(a + b*ArcCosh[c*x])))/ 
(3969*c^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
 

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.51, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {6337, 27, 290, 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.

\(\displaystyle \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx\)

\(\Big \downarrow \) 6337

\(\displaystyle -\frac {b c \sqrt {d-c^2 d x^2} \int -\frac {d^2 \left (1-c^2 x^2\right )^3 \left (7 c^2 x^2+2\right )}{63 c^4}dx}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {\left (d-c^2 d x^2\right )^{9/2} (a+b \text {arccosh}(c x))}{9 c^4 d^2}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^4 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b d^2 \sqrt {d-c^2 d x^2} \int \left (1-c^2 x^2\right )^3 \left (7 c^2 x^2+2\right )dx}{63 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {\left (d-c^2 d x^2\right )^{9/2} (a+b \text {arccosh}(c x))}{9 c^4 d^2}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^4 d}\)

\(\Big \downarrow \) 290

\(\displaystyle \frac {b d^2 \sqrt {d-c^2 d x^2} \int \left (-7 c^8 x^8+19 c^6 x^6-15 c^4 x^4+c^2 x^2+2\right )dx}{63 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {\left (d-c^2 d x^2\right )^{9/2} (a+b \text {arccosh}(c x))}{9 c^4 d^2}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^4 d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\left (d-c^2 d x^2\right )^{9/2} (a+b \text {arccosh}(c x))}{9 c^4 d^2}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^4 d}+\frac {b d^2 \left (-\frac {7}{9} c^8 x^9+\frac {19 c^6 x^7}{7}-3 c^4 x^5+\frac {c^2 x^3}{3}+2 x\right ) \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {c x-1} \sqrt {c x+1}}\)

Input:

Int[x^3*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]),x]
 

Output:

(b*d^2*Sqrt[d - c^2*d*x^2]*(2*x + (c^2*x^3)/3 - 3*c^4*x^5 + (19*c^6*x^7)/7 
 - (7*c^8*x^9)/9))/(63*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2 
)^(7/2)*(a + b*ArcCosh[c*x]))/(7*c^4*d) + ((d - c^2*d*x^2)^(9/2)*(a + b*Ar 
cCosh[c*x]))/(9*c^4*d^2)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 290

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> I 
nt[ExpandIntegrand[(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d 
}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && IGtQ[q, 0]
 

rule 2009

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

rule 6337

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_ 
), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^2)^p, x]}, Simp[(a + b*ArcCo 
sh[c*x])   u, x] - Simp[b*c*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c 
*x])]   Int[SimplifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b 
, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] 
 && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
 

Maple [A] (verified)

Time = 0.47 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.81

method	result	size
orering	\(\frac {\left (833 c^{10} x^{10}-3153 c^{8} x^{8}+4167 c^{6} x^{6}-1743 c^{4} x^{4}-1008 c^{2} x^{2}+504\right ) \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}{3969 c^{4} \left (c x -1\right )^{2} \left (c x +1\right )^{2} \left (c^{2} x^{2}-1\right )}-\frac {\left (49 c^{8} x^{8}-171 c^{6} x^{6}+189 c^{4} x^{4}-21 c^{2} x^{2}-126\right ) \left (3 x^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )-5 x^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) c^{2} d +\frac {x^{3} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} b c}{\sqrt {c x -1}\, \sqrt {c x +1}}\right )}{3969 x^{2} c^{4} \left (c x -1\right )^{2} \left (c x +1\right )^{2}}\)	\(241\)
default	\(\text {Expression too large to display}\)	\(1102\)
parts	\(\text {Expression too large to display}\)	\(1102\)

Input:

int(x^3*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x)),x,method=_RETURNVERBOSE)
 

Output:

1/3969*(833*c^10*x^10-3153*c^8*x^8+4167*c^6*x^6-1743*c^4*x^4-1008*c^2*x^2+ 
504)/c^4/(c*x-1)^2/(c*x+1)^2/(c^2*x^2-1)*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh 
(c*x))-1/3969/x^2*(49*c^8*x^8-171*c^6*x^6+189*c^4*x^4-21*c^2*x^2-126)/c^4/ 
(c*x-1)^2/(c*x+1)^2*(3*x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))-5*x^4*( 
-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))*c^2*d+x^3*(-c^2*d*x^2+d)^(5/2)*b*c/ 
(c*x-1)^(1/2)/(c*x+1)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 281, normalized size of antiderivative = 0.94 \[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\frac {63 \, {\left (7 \, b c^{10} d^{2} x^{10} - 26 \, b c^{8} d^{2} x^{8} + 34 \, b c^{6} d^{2} x^{6} - 16 \, b c^{4} d^{2} x^{4} - b c^{2} d^{2} x^{2} + 2 \, b d^{2}\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (49 \, b c^{9} d^{2} x^{9} - 171 \, b c^{7} d^{2} x^{7} + 189 \, b c^{5} d^{2} x^{5} - 21 \, b c^{3} d^{2} x^{3} - 126 \, b c d^{2} x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} + 63 \, {\left (7 \, a c^{10} d^{2} x^{10} - 26 \, a c^{8} d^{2} x^{8} + 34 \, a c^{6} d^{2} x^{6} - 16 \, a c^{4} d^{2} x^{4} - a c^{2} d^{2} x^{2} + 2 \, a d^{2}\right )} \sqrt {-c^{2} d x^{2} + d}}{3969 \, {\left (c^{6} x^{2} - c^{4}\right )}} \] Input:

integrate(x^3*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x)),x, algorithm="fricas 
")
 

Output:

1/3969*(63*(7*b*c^10*d^2*x^10 - 26*b*c^8*d^2*x^8 + 34*b*c^6*d^2*x^6 - 16*b 
*c^4*d^2*x^4 - b*c^2*d^2*x^2 + 2*b*d^2)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqr 
t(c^2*x^2 - 1)) - (49*b*c^9*d^2*x^9 - 171*b*c^7*d^2*x^7 + 189*b*c^5*d^2*x^ 
5 - 21*b*c^3*d^2*x^3 - 126*b*c*d^2*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 
1) + 63*(7*a*c^10*d^2*x^10 - 26*a*c^8*d^2*x^8 + 34*a*c^6*d^2*x^6 - 16*a*c^ 
4*d^2*x^4 - a*c^2*d^2*x^2 + 2*a*d^2)*sqrt(-c^2*d*x^2 + d))/(c^6*x^2 - c^4)
 

Sympy [F(-1)]

Timed out. \[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\text {Timed out} \] Input:

integrate(x**3*(-c**2*d*x**2+d)**(5/2)*(a+b*acosh(c*x)),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.62 \[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=-\frac {1}{63} \, {\left (\frac {7 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{4} d}\right )} b \operatorname {arcosh}\left (c x\right ) - \frac {1}{63} \, {\left (\frac {7 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{4} d}\right )} a - \frac {{\left (49 \, c^{8} \sqrt {-d} d^{2} x^{9} - 171 \, c^{6} \sqrt {-d} d^{2} x^{7} + 189 \, c^{4} \sqrt {-d} d^{2} x^{5} - 21 \, c^{2} \sqrt {-d} d^{2} x^{3} - 126 \, \sqrt {-d} d^{2} x\right )} b}{3969 \, c^{3}} \] Input:

integrate(x^3*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x)),x, algorithm="maxima 
")
 

Output:

-1/63*(7*(-c^2*d*x^2 + d)^(7/2)*x^2/(c^2*d) + 2*(-c^2*d*x^2 + d)^(7/2)/(c^ 
4*d))*b*arccosh(c*x) - 1/63*(7*(-c^2*d*x^2 + d)^(7/2)*x^2/(c^2*d) + 2*(-c^ 
2*d*x^2 + d)^(7/2)/(c^4*d))*a - 1/3969*(49*c^8*sqrt(-d)*d^2*x^9 - 171*c^6* 
sqrt(-d)*d^2*x^7 + 189*c^4*sqrt(-d)*d^2*x^5 - 21*c^2*sqrt(-d)*d^2*x^3 - 12 
6*sqrt(-d)*d^2*x)*b/c^3
 

Giac [F(-2)]

Exception generated. \[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^3*(-c^2*d*x^2+d)^(5/2)*(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. \[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\int x^3\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\frac {\sqrt {d}\, d^{2} \left (7 \sqrt {-c^{2} x^{2}+1}\, a \,c^{8} x^{8}-19 \sqrt {-c^{2} x^{2}+1}\, a \,c^{6} x^{6}+15 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} x^{4}-\sqrt {-c^{2} x^{2}+1}\, a \,c^{2} x^{2}-2 \sqrt {-c^{2} x^{2}+1}\, a +63 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{7}d x \right ) b \,c^{8}-126 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{5}d x \right ) b \,c^{6}+63 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{3}d x \right ) b \,c^{4}\right )}{63 c^{4}} \] Input:

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

Output:

(sqrt(d)*d**2*(7*sqrt( - c**2*x**2 + 1)*a*c**8*x**8 - 19*sqrt( - c**2*x**2 
 + 1)*a*c**6*x**6 + 15*sqrt( - c**2*x**2 + 1)*a*c**4*x**4 - sqrt( - c**2*x 
**2 + 1)*a*c**2*x**2 - 2*sqrt( - c**2*x**2 + 1)*a + 63*int(sqrt( - c**2*x* 
*2 + 1)*acosh(c*x)*x**7,x)*b*c**8 - 126*int(sqrt( - c**2*x**2 + 1)*acosh(c 
*x)*x**5,x)*b*c**6 + 63*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)*x**3,x)*b*c* 
*4))/(63*c**4)