\(\int x^3 (d-c^2 d x^2)^{3/2} (a+b \text {arccosh}(c x)) \, dx\) [79]

Optimal result

Integrand size = 27, antiderivative size = 243 \[ \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\frac {2 b d x \sqrt {d-c^2 d x^2}}{35 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d x^3 \sqrt {d-c^2 d x^2}}{105 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {8 b c d x^5 \sqrt {d-c^2 d x^2}}{175 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{5 c^4 d}+\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^4 d^2} \] Output:

2/35*b*d*x*(-c^2*d*x^2+d)^(1/2)/c^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/105*b*d* 
x^3*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-8/175*b*c*d*x^5*(-c 
^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/49*b*c^3*d*x^7*(-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*arccos 
h(c*x))/c^4/d+1/7*(-c^2*d*x^2+d)^(7/2)*(a+b*arccosh(c*x))/c^4/d^2
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.56 \[ \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=-\frac {d \sqrt {d-c^2 d x^2} \left (-b c x \left (210+35 c^2 x^2-168 c^4 x^4+75 c^6 x^6\right )+210 (-1+c x)^{5/2} (1+c x)^{5/2} (a+b \text {arccosh}(c x))+525 c^2 x^2 (-1+c x)^{5/2} (1+c x)^{5/2} (a+b \text {arccosh}(c x))\right )}{3675 c^4 \sqrt {-1+c x} \sqrt {1+c x}} \] Input:

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

Output:

-1/3675*(d*Sqrt[d - c^2*d*x^2]*(-(b*c*x*(210 + 35*c^2*x^2 - 168*c^4*x^4 + 
75*c^6*x^6)) + 210*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2)*(a + b*ArcCosh[c*x]) + 
 525*c^2*x^2*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2)*(a + b*ArcCosh[c*x])))/(c^4* 
Sqrt[-1 + c*x]*Sqrt[1 + c*x])
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.59, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, 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.

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

\(\Big \downarrow \) 6337

\(\displaystyle -\frac {b c \sqrt {d-c^2 d x^2} \int -\frac {d \left (1-c^2 x^2\right )^2 \left (5 c^2 x^2+2\right )}{35 c^4}dx}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^4 d^2}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{5 c^4 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b d \sqrt {d-c^2 d x^2} \int \left (1-c^2 x^2\right )^2 \left (5 c^2 x^2+2\right )dx}{35 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^4 d^2}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{5 c^4 d}\)

\(\Big \downarrow \) 290

\(\displaystyle \frac {b d \sqrt {d-c^2 d x^2} \int \left (5 c^6 x^6-8 c^4 x^4+c^2 x^2+2\right )dx}{35 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^4 d^2}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{5 c^4 d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^4 d^2}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{5 c^4 d}+\frac {b d \left (\frac {5 c^6 x^7}{7}-\frac {8 c^4 x^5}{5}+\frac {c^2 x^3}{3}+2 x\right ) \sqrt {d-c^2 d x^2}}{35 c^3 \sqrt {c x-1} \sqrt {c x+1}}\)

Input:

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

Output:

(b*d*Sqrt[d - c^2*d*x^2]*(2*x + (c^2*x^3)/3 - (8*c^4*x^5)/5 + (5*c^6*x^7)/ 
7))/(35*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2)^(5/2)*(a + b* 
ArcCosh[c*x]))/(5*c^4*d) + ((d - c^2*d*x^2)^(7/2)*(a + b*ArcCosh[c*x]))/(7 
*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 = 0.44 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.93

method	result	size
orering	\(\frac {\left (325 c^{8} x^{8}-866 c^{6} x^{6}+553 c^{4} x^{4}+420 c^{2} x^{2}-280\right ) \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}{1225 c^{4} \left (c x -1\right ) \left (c x +1\right ) \left (c^{2} x^{2}-1\right )}-\frac {\left (75 c^{6} x^{6}-168 c^{4} x^{4}+35 c^{2} x^{2}+210\right ) \left (3 x^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )-3 x^{4} \sqrt {-c^{2} d \,x^{2}+d}\, \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) c^{2} d +\frac {x^{3} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} b c}{\sqrt {c x -1}\, \sqrt {c x +1}}\right )}{3675 x^{2} c^{4} \left (c x -1\right ) \left (c x +1\right )}\)	\(225\)
default	\(a \left (-\frac {x^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{7 c^{2} d}-\frac {2 \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{35 d \,c^{4}}\right )+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (64 c^{8} x^{8}-144 c^{6} x^{6}+64 c^{7} x^{7} \sqrt {c x -1}\, \sqrt {c x +1}+104 c^{4} x^{4}-112 c^{5} x^{5} \sqrt {c x -1}\, \sqrt {c x +1}-25 c^{2} x^{2}+56 c^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}-7 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +1\right ) \left (-1+7 \,\operatorname {arccosh}\left (c x \right )\right ) d}{6272 \left (c x +1\right ) c^{4} \left (c x -1\right )}+\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}{3200 \left (c x +1\right ) c^{4} \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}{384 \left (c x +1\right ) c^{4} \left (c x -1\right )}-\frac {3 \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}{128 \left (c x +1\right ) c^{4} \left (c x -1\right )}-\frac {3 \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}{128 \left (c x +1\right ) c^{4} \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}{384 \left (c x +1\right ) c^{4} \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}{3200 \left (c x +1\right ) c^{4} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-64 c^{7} x^{7} \sqrt {c x -1}\, \sqrt {c x +1}+64 c^{8} x^{8}+112 c^{5} x^{5} \sqrt {c x -1}\, \sqrt {c x +1}-144 c^{6} x^{6}-56 c^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}+104 c^{4} x^{4}+7 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -25 c^{2} x^{2}+1\right ) \left (1+7 \,\operatorname {arccosh}\left (c x \right )\right ) d}{6272 \left (c x +1\right ) c^{4} \left (c x -1\right )}\right )\)	\(966\)
parts	\(a \left (-\frac {x^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{7 c^{2} d}-\frac {2 \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{35 d \,c^{4}}\right )+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (64 c^{8} x^{8}-144 c^{6} x^{6}+64 c^{7} x^{7} \sqrt {c x -1}\, \sqrt {c x +1}+104 c^{4} x^{4}-112 c^{5} x^{5} \sqrt {c x -1}\, \sqrt {c x +1}-25 c^{2} x^{2}+56 c^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}-7 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +1\right ) \left (-1+7 \,\operatorname {arccosh}\left (c x \right )\right ) d}{6272 \left (c x +1\right ) c^{4} \left (c x -1\right )}+\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}{3200 \left (c x +1\right ) c^{4} \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}{384 \left (c x +1\right ) c^{4} \left (c x -1\right )}-\frac {3 \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}{128 \left (c x +1\right ) c^{4} \left (c x -1\right )}-\frac {3 \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}{128 \left (c x +1\right ) c^{4} \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}{384 \left (c x +1\right ) c^{4} \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}{3200 \left (c x +1\right ) c^{4} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-64 c^{7} x^{7} \sqrt {c x -1}\, \sqrt {c x +1}+64 c^{8} x^{8}+112 c^{5} x^{5} \sqrt {c x -1}\, \sqrt {c x +1}-144 c^{6} x^{6}-56 c^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}+104 c^{4} x^{4}+7 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -25 c^{2} x^{2}+1\right ) \left (1+7 \,\operatorname {arccosh}\left (c x \right )\right ) d}{6272 \left (c x +1\right ) c^{4} \left (c x -1\right )}\right )\)	\(966\)

Input:

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

Output:

1/1225*(325*c^8*x^8-866*c^6*x^6+553*c^4*x^4+420*c^2*x^2-280)/c^4/(c*x-1)/( 
c*x+1)/(c^2*x^2-1)*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))-1/3675/x^2*(75* 
c^6*x^6-168*c^4*x^4+35*c^2*x^2+210)/c^4/(c*x-1)/(c*x+1)*(3*x^2*(-c^2*d*x^2 
+d)^(3/2)*(a+b*arccosh(c*x))-3*x^4*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x)) 
*c^2*d+x^3*(-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 = 0.10 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.88 \[ \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=-\frac {105 \, {\left (5 \, b c^{8} d x^{8} - 13 \, b c^{6} d x^{6} + 9 \, b c^{4} d x^{4} + b c^{2} d x^{2} - 2 \, b d\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (75 \, b c^{7} d x^{7} - 168 \, b c^{5} d x^{5} + 35 \, b c^{3} d x^{3} + 210 \, b c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} + 105 \, {\left (5 \, a c^{8} d x^{8} - 13 \, a c^{6} d x^{6} + 9 \, a c^{4} d x^{4} + a c^{2} d x^{2} - 2 \, a d\right )} \sqrt {-c^{2} d x^{2} + d}}{3675 \, {\left (c^{6} x^{2} - c^{4}\right )}} \] Input:

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

Output:

-1/3675*(105*(5*b*c^8*d*x^8 - 13*b*c^6*d*x^6 + 9*b*c^4*d*x^4 + b*c^2*d*x^2 
 - 2*b*d)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)) - (75*b*c^7*d* 
x^7 - 168*b*c^5*d*x^5 + 35*b*c^3*d*x^3 + 210*b*c*d*x)*sqrt(-c^2*d*x^2 + d) 
*sqrt(c^2*x^2 - 1) + 105*(5*a*c^8*d*x^8 - 13*a*c^6*d*x^6 + 9*a*c^4*d*x^4 + 
 a*c^2*d*x^2 - 2*a*d)*sqrt(-c^2*d*x^2 + d))/(c^6*x^2 - c^4)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.66 \[ \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=-\frac {1}{35} \, {\left (\frac {5 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{c^{4} d}\right )} b \operatorname {arcosh}\left (c x\right ) - \frac {1}{35} \, {\left (\frac {5 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{c^{4} d}\right )} a + \frac {{\left (75 \, c^{6} \sqrt {-d} d x^{7} - 168 \, c^{4} \sqrt {-d} d x^{5} + 35 \, c^{2} \sqrt {-d} d x^{3} + 210 \, \sqrt {-d} d x\right )} b}{3675 \, c^{3}} \] Input:

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

Output:

-1/35*(5*(-c^2*d*x^2 + d)^(5/2)*x^2/(c^2*d) + 2*(-c^2*d*x^2 + d)^(5/2)/(c^ 
4*d))*b*arccosh(c*x) - 1/35*(5*(-c^2*d*x^2 + d)^(5/2)*x^2/(c^2*d) + 2*(-c^ 
2*d*x^2 + d)^(5/2)/(c^4*d))*a + 1/3675*(75*c^6*sqrt(-d)*d*x^7 - 168*c^4*sq 
rt(-d)*d*x^5 + 35*c^2*sqrt(-d)*d*x^3 + 210*sqrt(-d)*d*x)*b/c^3
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\frac {\sqrt {d}\, d \left (-5 \sqrt {-c^{2} x^{2}+1}\, a \,c^{6} x^{6}+8 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} x^{4}-\sqrt {-c^{2} x^{2}+1}\, a \,c^{2} x^{2}-2 \sqrt {-c^{2} x^{2}+1}\, a -35 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{5}d x \right ) b \,c^{6}+35 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{3}d x \right ) b \,c^{4}\right )}{35 c^{4}} \] Input:

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

Output:

(sqrt(d)*d*( - 5*sqrt( - c**2*x**2 + 1)*a*c**6*x**6 + 8*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 - 35*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)*x**5,x)*b*c**6 + 35*in 
t(sqrt( - c**2*x**2 + 1)*acosh(c*x)*x**3,x)*b*c**4))/(35*c**4)