\(\int x (d-c^2 d x^2)^{5/2} (a+b \text {arccosh}(c x))^2 \, dx\) [177]

Optimal result

Integrand size = 27, antiderivative size = 405 \[ \int x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=-\frac {32 b^2 d^2 \sqrt {d-c^2 d x^2}}{245 c^2}-\frac {16 b^2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}{735 c^2}-\frac {12 b^2 d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2}}{1225 c^2}-\frac {2 b^2 d^2 (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2}}{343 c^2}+\frac {2 b d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{7 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{7 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {6 b c^3 d^2 x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{35 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 x^7 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{49 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))^2}{7 c^2 d} \] Output:

-32/245*b^2*d^2*(-c^2*d*x^2+d)^(1/2)/c^2-16/735*b^2*d^2*(-c*x+1)*(c*x+1)*( 
-c^2*d*x^2+d)^(1/2)/c^2-12/1225*b^2*d^2*(-c*x+1)^2*(c*x+1)^2*(-c^2*d*x^2+d 
)^(1/2)/c^2-2/343*b^2*d^2*(-c*x+1)^3*(c*x+1)^3*(-c^2*d*x^2+d)^(1/2)/c^2+2/ 
7*b*d^2*x*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/c/(c*x-1)^(1/2)/(c*x+1)^ 
(1/2)-2/7*b*c*d^2*x^3*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/(c*x-1)^(1/2 
)/(c*x+1)^(1/2)+6/35*b*c^3*d^2*x^5*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x)) 
/(c*x-1)^(1/2)/(c*x+1)^(1/2)-2/49*b*c^5*d^2*x^7*(-c^2*d*x^2+d)^(1/2)*(a+b* 
arccosh(c*x))/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/7*(-c^2*d*x^2+d)^(7/2)*(a+b*ar 
ccosh(c*x))^2/c^2/d
 

Mathematica [A] (verified)

Time = 1.48 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.58 \[ \int x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\frac {d^2 \sqrt {d-c^2 d x^2} \left (3675 a^2 \left (-1+c^2 x^2\right )^4-210 a b c x \sqrt {-1+c x} \sqrt {1+c x} \left (-35+35 c^2 x^2-21 c^4 x^4+5 c^6 x^6\right )+2 b^2 \left (2161-2918 c^2 x^2+1108 c^4 x^4-426 c^6 x^6+75 c^8 x^8\right )+210 b \left (35 a \left (-1+c^2 x^2\right )^4+b c x \sqrt {-1+c x} \sqrt {1+c x} \left (35-35 c^2 x^2+21 c^4 x^4-5 c^6 x^6\right )\right ) \text {arccosh}(c x)+3675 b^2 \left (-1+c^2 x^2\right )^4 \text {arccosh}(c x)^2\right )}{25725 c^2 \left (-1+c^2 x^2\right )} \] Input:

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

Output:

(d^2*Sqrt[d - c^2*d*x^2]*(3675*a^2*(-1 + c^2*x^2)^4 - 210*a*b*c*x*Sqrt[-1 
+ c*x]*Sqrt[1 + c*x]*(-35 + 35*c^2*x^2 - 21*c^4*x^4 + 5*c^6*x^6) + 2*b^2*( 
2161 - 2918*c^2*x^2 + 1108*c^4*x^4 - 426*c^6*x^6 + 75*c^8*x^8) + 210*b*(35 
*a*(-1 + c^2*x^2)^4 + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(35 - 35*c^2*x^2 
+ 21*c^4*x^4 - 5*c^6*x^6))*ArcCosh[c*x] + 3675*b^2*(-1 + c^2*x^2)^4*ArcCos 
h[c*x]^2))/(25725*c^2*(-1 + c^2*x^2))
 

Rubi [A] (verified)

Time = 1.53 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.64, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6329, 25, 6304, 6309, 27, 2113, 2331, 2389, 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.

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

\(\Big \downarrow \) 6329

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 6304

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

\(\Big \downarrow \) 6309

\(\displaystyle \frac {2 b d^2 \sqrt {d-c^2 d x^2} \left (-b c \int \frac {x \left (-5 c^6 x^6+21 c^4 x^4-35 c^2 x^2+35\right )}{35 \sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{7} c^6 x^7 (a+b \text {arccosh}(c x))+\frac {3}{5} c^4 x^5 (a+b \text {arccosh}(c x))-c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))\right )}{7 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))^2}{7 c^2 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 b d^2 \sqrt {d-c^2 d x^2} \left (-\frac {1}{35} b c \int \frac {x \left (-5 c^6 x^6+21 c^4 x^4-35 c^2 x^2+35\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{7} c^6 x^7 (a+b \text {arccosh}(c x))+\frac {3}{5} c^4 x^5 (a+b \text {arccosh}(c x))-c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))\right )}{7 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))^2}{7 c^2 d}\)

\(\Big \downarrow \) 2113

\(\displaystyle \frac {2 b d^2 \sqrt {d-c^2 d x^2} \left (-\frac {b c \sqrt {c^2 x^2-1} \int \frac {x \left (-5 c^6 x^6+21 c^4 x^4-35 c^2 x^2+35\right )}{\sqrt {c^2 x^2-1}}dx}{35 \sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{7} c^6 x^7 (a+b \text {arccosh}(c x))+\frac {3}{5} c^4 x^5 (a+b \text {arccosh}(c x))-c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))\right )}{7 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))^2}{7 c^2 d}\)

\(\Big \downarrow \) 2331

\(\displaystyle \frac {2 b d^2 \sqrt {d-c^2 d x^2} \left (-\frac {b c \sqrt {c^2 x^2-1} \int \frac {-5 c^6 x^6+21 c^4 x^4-35 c^2 x^2+35}{\sqrt {c^2 x^2-1}}dx^2}{70 \sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{7} c^6 x^7 (a+b \text {arccosh}(c x))+\frac {3}{5} c^4 x^5 (a+b \text {arccosh}(c x))-c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))\right )}{7 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))^2}{7 c^2 d}\)

\(\Big \downarrow \) 2389

\(\displaystyle \frac {2 b d^2 \sqrt {d-c^2 d x^2} \left (-\frac {b c \sqrt {c^2 x^2-1} \int \left (-5 \left (c^2 x^2-1\right )^{5/2}+6 \left (c^2 x^2-1\right )^{3/2}-8 \sqrt {c^2 x^2-1}+\frac {16}{\sqrt {c^2 x^2-1}}\right )dx^2}{70 \sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{7} c^6 x^7 (a+b \text {arccosh}(c x))+\frac {3}{5} c^4 x^5 (a+b \text {arccosh}(c x))-c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))\right )}{7 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))^2}{7 c^2 d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 b d^2 \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^6 x^7 (a+b \text {arccosh}(c x))+\frac {3}{5} c^4 x^5 (a+b \text {arccosh}(c x))-c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (-\frac {10 \left (c^2 x^2-1\right )^{7/2}}{7 c^2}+\frac {12 \left (c^2 x^2-1\right )^{5/2}}{5 c^2}-\frac {16 \left (c^2 x^2-1\right )^{3/2}}{3 c^2}+\frac {32 \sqrt {c^2 x^2-1}}{c^2}\right )}{70 \sqrt {c x-1} \sqrt {c x+1}}\right )}{7 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))^2}{7 c^2 d}\)

Input:

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

Output:

-1/7*((d - c^2*d*x^2)^(7/2)*(a + b*ArcCosh[c*x])^2)/(c^2*d) + (2*b*d^2*Sqr 
t[d - c^2*d*x^2]*(-1/70*(b*c*Sqrt[-1 + c^2*x^2]*((32*Sqrt[-1 + c^2*x^2])/c 
^2 - (16*(-1 + c^2*x^2)^(3/2))/(3*c^2) + (12*(-1 + c^2*x^2)^(5/2))/(5*c^2) 
 - (10*(-1 + c^2*x^2)^(7/2))/(7*c^2)))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + x* 
(a + b*ArcCosh[c*x]) - c^2*x^3*(a + b*ArcCosh[c*x]) + (3*c^4*x^5*(a + b*Ar 
cCosh[c*x]))/5 - (c^6*x^7*(a + b*ArcCosh[c*x]))/7))/(7*c*Sqrt[-1 + c*x]*Sq 
rt[1 + c*x])
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 2009

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

rule 2113

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_. 
)*(x_))^(p_.), x_Symbol] :> Simp[(a + b*x)^FracPart[m]*((c + d*x)^FracPart[ 
m]/(a*c + b*d*x^2)^FracPart[m])   Int[Px*(a*c + b*d*x^2)^m*(e + f*x)^p, x], 
 x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[b*c + a 
*d, 0] && EqQ[m, n] &&  !IntegerQ[m]
 

rule 2331

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/2   S 
ubst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x)^p, x], x, x^2], x] /; 
 FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]
 

rule 2389

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand 
[Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p 
, 0] || EqQ[n, 1])
 

rule 6304

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_.)*( 
(d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Int[(d1*d2 + e1*e2*x^2)^p*(a + b*A 
rcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[d2*e1 + 
 d1*e2, 0] && IntegerQ[p]
 

rule 6309

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symb 
ol] :> With[{u = IntHide[(d + e*x^2)^p, x]}, Simp[(a + b*ArcCosh[c*x])   u, 
 x] - Simp[b*c   Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x] 
, x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
 

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 = 0.77 (sec) , antiderivative size = 581, normalized size of antiderivative = 1.43

method	result	size
orering	\(\frac {\left (9525 c^{10} x^{10}-41691 c^{8} x^{8}+76515 c^{6} x^{6}-124979 c^{4} x^{4}+26152 c^{2} x^{2}-4322\right ) \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}}{25725 c^{4} \left (c x -1\right ) \left (c x +1\right ) x^{2} \left (c^{2} x^{2}-1\right )^{2}}-\frac {2 \left (675 c^{8} x^{8}-3108 c^{6} x^{6}+6352 c^{4} x^{4}-14480 c^{2} x^{2}+2161\right ) \left (\left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}-5 x^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2} c^{2} d +\frac {2 x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b c}{\sqrt {c x -1}\, \sqrt {c x +1}}\right )}{25725 c^{4} \left (c x -1\right ) \left (c x +1\right ) x^{2} \left (c^{2} x^{2}-1\right )}+\frac {\left (75 c^{6} x^{6}-351 c^{4} x^{4}+757 c^{2} x^{2}-2161\right ) \left (-15 c^{2} d x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}+\frac {4 \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b c}{\sqrt {c x -1}\, \sqrt {c x +1}}+15 x^{3} \sqrt {-c^{2} d \,x^{2}+d}\, \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2} c^{4} d^{2}-\frac {20 x^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) c^{3} d b}{\sqrt {c x -1}\, \sqrt {c x +1}}+\frac {2 x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} b^{2} c^{2}}{\left (c x -1\right ) \left (c x +1\right )}-\frac {x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b \,c^{2}}{\left (c x -1\right )^{\frac {3}{2}} \sqrt {c x +1}}-\frac {x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b \,c^{2}}{\sqrt {c x -1}\, \left (c x +1\right )^{\frac {3}{2}}}\right )}{25725 c^{4} \left (c x -1\right ) \left (c x +1\right ) x}\)	\(581\)
default	\(\text {Expression too large to display}\)	\(1958\)
parts	\(\text {Expression too large to display}\)	\(1958\)

Input:

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

Output:

1/25725*(9525*c^10*x^10-41691*c^8*x^8+76515*c^6*x^6-124979*c^4*x^4+26152*c 
^2*x^2-4322)/c^4/(c*x-1)/(c*x+1)/x^2/(c^2*x^2-1)^2*(-c^2*d*x^2+d)^(5/2)*(a 
+b*arccosh(c*x))^2-2/25725*(675*c^8*x^8-3108*c^6*x^6+6352*c^4*x^4-14480*c^ 
2*x^2+2161)/c^4/(c*x-1)/(c*x+1)/x^2/(c^2*x^2-1)*((-c^2*d*x^2+d)^(5/2)*(a+b 
*arccosh(c*x))^2-5*x^2*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2*c^2*d+2*x 
*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))*b*c/(c*x-1)^(1/2)/(c*x+1)^(1/2))+ 
1/25725*(75*c^6*x^6-351*c^4*x^4+757*c^2*x^2-2161)/c^4/(c*x-1)/(c*x+1)/x*(- 
15*c^2*d*x*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2+4*(-c^2*d*x^2+d)^(5/2 
)*(a+b*arccosh(c*x))*b*c/(c*x-1)^(1/2)/(c*x+1)^(1/2)+15*x^3*(-c^2*d*x^2+d) 
^(1/2)*(a+b*arccosh(c*x))^2*c^4*d^2-20*x^2*(-c^2*d*x^2+d)^(3/2)*(a+b*arcco 
sh(c*x))*c^3*d*b/(c*x-1)^(1/2)/(c*x+1)^(1/2)+2*x*(-c^2*d*x^2+d)^(5/2)*b^2* 
c^2/(c*x-1)/(c*x+1)-x*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))*b*c^2/(c*x-1 
)^(3/2)/(c*x+1)^(1/2)-x*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))*b*c^2/(c*x 
-1)^(1/2)/(c*x+1)^(3/2))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 477, normalized size of antiderivative = 1.18 \[ \int x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\frac {3675 \, {\left (b^{2} c^{8} d^{2} x^{8} - 4 \, b^{2} c^{6} d^{2} x^{6} + 6 \, b^{2} c^{4} d^{2} x^{4} - 4 \, b^{2} c^{2} d^{2} x^{2} + b^{2} d^{2}\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} - 210 \, {\left (5 \, a b c^{7} d^{2} x^{7} - 21 \, a b c^{5} d^{2} x^{5} + 35 \, a b c^{3} d^{2} x^{3} - 35 \, a b c d^{2} x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 210 \, {\left ({\left (5 \, b^{2} c^{7} d^{2} x^{7} - 21 \, b^{2} c^{5} d^{2} x^{5} + 35 \, b^{2} c^{3} d^{2} x^{3} - 35 \, b^{2} c d^{2} x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 35 \, {\left (a b c^{8} d^{2} x^{8} - 4 \, a b c^{6} d^{2} x^{6} + 6 \, a b c^{4} d^{2} x^{4} - 4 \, a b c^{2} d^{2} x^{2} + a b d^{2}\right )} \sqrt {-c^{2} d x^{2} + d}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (75 \, {\left (49 \, a^{2} + 2 \, b^{2}\right )} c^{8} d^{2} x^{8} - 12 \, {\left (1225 \, a^{2} + 71 \, b^{2}\right )} c^{6} d^{2} x^{6} + 2 \, {\left (11025 \, a^{2} + 1108 \, b^{2}\right )} c^{4} d^{2} x^{4} - 4 \, {\left (3675 \, a^{2} + 1459 \, b^{2}\right )} c^{2} d^{2} x^{2} + {\left (3675 \, a^{2} + 4322 \, b^{2}\right )} d^{2}\right )} \sqrt {-c^{2} d x^{2} + d}}{25725 \, {\left (c^{4} x^{2} - c^{2}\right )}} \] Input:

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

Output:

1/25725*(3675*(b^2*c^8*d^2*x^8 - 4*b^2*c^6*d^2*x^6 + 6*b^2*c^4*d^2*x^4 - 4 
*b^2*c^2*d^2*x^2 + b^2*d^2)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 
1))^2 - 210*(5*a*b*c^7*d^2*x^7 - 21*a*b*c^5*d^2*x^5 + 35*a*b*c^3*d^2*x^3 - 
 35*a*b*c*d^2*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) - 210*((5*b^2*c^7* 
d^2*x^7 - 21*b^2*c^5*d^2*x^5 + 35*b^2*c^3*d^2*x^3 - 35*b^2*c*d^2*x)*sqrt(- 
c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) - 35*(a*b*c^8*d^2*x^8 - 4*a*b*c^6*d^2*x^6 
 + 6*a*b*c^4*d^2*x^4 - 4*a*b*c^2*d^2*x^2 + a*b*d^2)*sqrt(-c^2*d*x^2 + d))* 
log(c*x + sqrt(c^2*x^2 - 1)) + (75*(49*a^2 + 2*b^2)*c^8*d^2*x^8 - 12*(1225 
*a^2 + 71*b^2)*c^6*d^2*x^6 + 2*(11025*a^2 + 1108*b^2)*c^4*d^2*x^4 - 4*(367 
5*a^2 + 1459*b^2)*c^2*d^2*x^2 + (3675*a^2 + 4322*b^2)*d^2)*sqrt(-c^2*d*x^2 
 + d))/(c^4*x^2 - c^2)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 337, normalized size of antiderivative = 0.83 \[ \int x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=-\frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} b^{2} \operatorname {arcosh}\left (c x\right )^{2}}{7 \, c^{2} d} - \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} a b \operatorname {arcosh}\left (c x\right )}{7 \, c^{2} d} + \frac {2}{25725} \, b^{2} {\left (\frac {75 \, \sqrt {c^{2} x^{2} - 1} c^{4} \sqrt {-d} d^{3} x^{6} - 351 \, \sqrt {c^{2} x^{2} - 1} c^{2} \sqrt {-d} d^{3} x^{4} + 757 \, \sqrt {c^{2} x^{2} - 1} \sqrt {-d} d^{3} x^{2} - \frac {2161 \, \sqrt {c^{2} x^{2} - 1} \sqrt {-d} d^{3}}{c^{2}}}{d} - \frac {105 \, {\left (5 \, c^{6} \sqrt {-d} d^{3} x^{7} - 21 \, c^{4} \sqrt {-d} d^{3} x^{5} + 35 \, c^{2} \sqrt {-d} d^{3} x^{3} - 35 \, \sqrt {-d} d^{3} x\right )} \operatorname {arcosh}\left (c x\right )}{c d}\right )} - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} a^{2}}{7 \, c^{2} d} - \frac {2 \, {\left (5 \, c^{6} \sqrt {-d} d^{3} x^{7} - 21 \, c^{4} \sqrt {-d} d^{3} x^{5} + 35 \, c^{2} \sqrt {-d} d^{3} x^{3} - 35 \, \sqrt {-d} d^{3} x\right )} a b}{245 \, c d} \] Input:

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

Output:

-1/7*(-c^2*d*x^2 + d)^(7/2)*b^2*arccosh(c*x)^2/(c^2*d) - 2/7*(-c^2*d*x^2 + 
 d)^(7/2)*a*b*arccosh(c*x)/(c^2*d) + 2/25725*b^2*((75*sqrt(c^2*x^2 - 1)*c^ 
4*sqrt(-d)*d^3*x^6 - 351*sqrt(c^2*x^2 - 1)*c^2*sqrt(-d)*d^3*x^4 + 757*sqrt 
(c^2*x^2 - 1)*sqrt(-d)*d^3*x^2 - 2161*sqrt(c^2*x^2 - 1)*sqrt(-d)*d^3/c^2)/ 
d - 105*(5*c^6*sqrt(-d)*d^3*x^7 - 21*c^4*sqrt(-d)*d^3*x^5 + 35*c^2*sqrt(-d 
)*d^3*x^3 - 35*sqrt(-d)*d^3*x)*arccosh(c*x)/(c*d)) - 1/7*(-c^2*d*x^2 + d)^ 
(7/2)*a^2/(c^2*d) - 2/245*(5*c^6*sqrt(-d)*d^3*x^7 - 21*c^4*sqrt(-d)*d^3*x^ 
5 + 35*c^2*sqrt(-d)*d^3*x^3 - 35*sqrt(-d)*d^3*x)*a*b/(c*d)
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\frac {\sqrt {d}\, d^{2} \left (\sqrt {-c^{2} x^{2}+1}\, a^{2} c^{6} x^{6}-3 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{4} x^{4}+3 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{2} x^{2}-\sqrt {-c^{2} x^{2}+1}\, a^{2}+14 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{5}d x \right ) a b \,c^{6}-28 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{3}d x \right ) a b \,c^{4}+14 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x d x \right ) a b \,c^{2}+7 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )^{2} x^{5}d x \right ) b^{2} c^{6}-14 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )^{2} x^{3}d x \right ) b^{2} c^{4}+7 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )^{2} x d x \right ) b^{2} c^{2}\right )}{7 c^{2}} \] Input:

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

Output:

(sqrt(d)*d**2*(sqrt( - c**2*x**2 + 1)*a**2*c**6*x**6 - 3*sqrt( - c**2*x**2 
 + 1)*a**2*c**4*x**4 + 3*sqrt( - c**2*x**2 + 1)*a**2*c**2*x**2 - sqrt( - c 
**2*x**2 + 1)*a**2 + 14*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)*x**5,x)*a*b* 
c**6 - 28*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)*x**3,x)*a*b*c**4 + 14*int( 
sqrt( - c**2*x**2 + 1)*acosh(c*x)*x,x)*a*b*c**2 + 7*int(sqrt( - c**2*x**2 
+ 1)*acosh(c*x)**2*x**5,x)*b**2*c**6 - 14*int(sqrt( - c**2*x**2 + 1)*acosh 
(c*x)**2*x**3,x)*b**2*c**4 + 7*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)**2*x, 
x)*b**2*c**2))/(7*c**2)