\(\int (f+g x)^2 (d-c^2 d x^2)^{3/2} (a+b \text {arccosh}(c x)) \, dx\) [47]

Optimal result

Integrand size = 31, antiderivative size = 722 \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\frac {2 b d f g x \sqrt {d-c^2 d x^2}}{5 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 b c d f^2 x^2 \sqrt {d-c^2 d x^2}}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d g^2 x^2 \sqrt {d-c^2 d x^2}}{32 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {4 b c d f g x^3 \sqrt {d-c^2 d x^2}}{15 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {7 b c d g^2 x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^3 d f g x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d g^2 x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d f^2 (-1+c x)^{3/2} (1+c x)^{3/2} \sqrt {d-c^2 d x^2}}{16 c}+\frac {3}{8} d f^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {d g^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{16 c^2}+\frac {1}{8} d g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {1}{4} d f^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {1}{6} d g^2 x^3 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {2 d f g (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{5 c^2}-\frac {3 d f^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{16 b c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d g^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{32 b c^3 \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

2/5*b*d*f*g*x*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-3/16*b*c* 
d*f^2*x^2*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/32*b*d*g^2*x^ 
2*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-4/15*b*c*d*f*g*x^3*(- 
c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-7/96*b*c*d*g^2*x^4*(-c^2*d* 
x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+2/25*b*c^3*d*f*g*x^5*(-c^2*d*x^2+ 
d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/36*b*c^3*d*g^2*x^6*(-c^2*d*x^2+d)^( 
1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/16*b*d*f^2*(c*x-1)^(3/2)*(c*x+1)^(3/2)* 
(-c^2*d*x^2+d)^(1/2)/c+3/8*d*f^2*x*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x)) 
-1/16*d*g^2*x*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/c^2+1/8*d*g^2*x^3*(- 
c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))+1/4*d*f^2*x*(-c*x+1)*(c*x+1)*(-c^2*d 
*x^2+d)^(1/2)*(a+b*arccosh(c*x))+1/6*d*g^2*x^3*(-c*x+1)*(c*x+1)*(-c^2*d*x^ 
2+d)^(1/2)*(a+b*arccosh(c*x))-2/5*d*f*g*(-c*x+1)^2*(c*x+1)^2*(-c^2*d*x^2+d 
)^(1/2)*(a+b*arccosh(c*x))/c^2-3/16*d*f^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos 
h(c*x))^2/b/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/32*d*g^2*(-c^2*d*x^2+d)^(1/2)* 
(a+b*arccosh(c*x))^2/b/c^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 2.05 (sec) , antiderivative size = 623, normalized size of antiderivative = 0.86 \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\frac {-240 a c d \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2} \left (96 f g \left (-1+c^2 x^2\right )^2+30 c^2 f^2 x \left (-5+2 c^2 x^2\right )+5 g^2 x \left (3-14 c^2 x^2+8 c^4 x^4\right )\right )-3600 a d^{3/2} \left (6 c^2 f^2+g^2\right ) \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+3200 b c d f g \sqrt {d-c^2 d x^2} \left (9 c x+12 \left (\frac {-1+c x}{1+c x}\right )^{3/2} (1+c x)^3 \text {arccosh}(c x)-\cosh (3 \text {arccosh}(c x))\right )-7200 b c^2 d f^2 \sqrt {d-c^2 d x^2} (\cosh (2 \text {arccosh}(c x))+2 \text {arccosh}(c x) (\text {arccosh}(c x)-\sinh (2 \text {arccosh}(c x))))+450 b c^2 d f^2 \sqrt {d-c^2 d x^2} \left (8 \text {arccosh}(c x)^2+\cosh (4 \text {arccosh}(c x))-4 \text {arccosh}(c x) \sinh (4 \text {arccosh}(c x))\right )-450 b d g^2 \sqrt {d-c^2 d x^2} \left (8 \text {arccosh}(c x)^2+\cosh (4 \text {arccosh}(c x))-4 \text {arccosh}(c x) \sinh (4 \text {arccosh}(c x))\right )-32 b c d f g \sqrt {d-c^2 d x^2} \left (450 c x-450 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x)-25 \cosh (3 \text {arccosh}(c x))-9 \cosh (5 \text {arccosh}(c x))+75 \text {arccosh}(c x) \sinh (3 \text {arccosh}(c x))+45 \text {arccosh}(c x) \sinh (5 \text {arccosh}(c x))\right )-25 b d g^2 \sqrt {d-c^2 d x^2} \left (-72 \text {arccosh}(c x)^2+18 \cosh (2 \text {arccosh}(c x))-9 \cosh (4 \text {arccosh}(c x))-2 \cosh (6 \text {arccosh}(c x))+12 \text {arccosh}(c x) (-3 \sinh (2 \text {arccosh}(c x))+3 \sinh (4 \text {arccosh}(c x))+\sinh (6 \text {arccosh}(c x)))\right )}{57600 c^3 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \] Input:

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

Output:

(-240*a*c*d*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(96*f 
*g*(-1 + c^2*x^2)^2 + 30*c^2*f^2*x*(-5 + 2*c^2*x^2) + 5*g^2*x*(3 - 14*c^2* 
x^2 + 8*c^4*x^4)) - 3600*a*d^(3/2)*(6*c^2*f^2 + g^2)*Sqrt[(-1 + c*x)/(1 + 
c*x)]*(1 + c*x)*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] 
 + 3200*b*c*d*f*g*Sqrt[d - c^2*d*x^2]*(9*c*x + 12*((-1 + c*x)/(1 + c*x))^( 
3/2)*(1 + c*x)^3*ArcCosh[c*x] - Cosh[3*ArcCosh[c*x]]) - 7200*b*c^2*d*f^2*S 
qrt[d - c^2*d*x^2]*(Cosh[2*ArcCosh[c*x]] + 2*ArcCosh[c*x]*(ArcCosh[c*x] - 
Sinh[2*ArcCosh[c*x]])) + 450*b*c^2*d*f^2*Sqrt[d - c^2*d*x^2]*(8*ArcCosh[c* 
x]^2 + Cosh[4*ArcCosh[c*x]] - 4*ArcCosh[c*x]*Sinh[4*ArcCosh[c*x]]) - 450*b 
*d*g^2*Sqrt[d - c^2*d*x^2]*(8*ArcCosh[c*x]^2 + Cosh[4*ArcCosh[c*x]] - 4*Ar 
cCosh[c*x]*Sinh[4*ArcCosh[c*x]]) - 32*b*c*d*f*g*Sqrt[d - c^2*d*x^2]*(450*c 
*x - 450*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x] - 25*Cosh[3*Arc 
Cosh[c*x]] - 9*Cosh[5*ArcCosh[c*x]] + 75*ArcCosh[c*x]*Sinh[3*ArcCosh[c*x]] 
 + 45*ArcCosh[c*x]*Sinh[5*ArcCosh[c*x]]) - 25*b*d*g^2*Sqrt[d - c^2*d*x^2]* 
(-72*ArcCosh[c*x]^2 + 18*Cosh[2*ArcCosh[c*x]] - 9*Cosh[4*ArcCosh[c*x]] - 2 
*Cosh[6*ArcCosh[c*x]] + 12*ArcCosh[c*x]*(-3*Sinh[2*ArcCosh[c*x]] + 3*Sinh[ 
4*ArcCosh[c*x]] + Sinh[6*ArcCosh[c*x]])))/(57600*c^3*Sqrt[(-1 + c*x)/(1 + 
c*x)]*(1 + c*x))
 

Rubi [A] (verified)

Time = 1.87 (sec) , antiderivative size = 396, normalized size of antiderivative = 0.55, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {6387, 6390, 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.

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

\(\Big \downarrow \) 6387

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

\(\Big \downarrow \) 6390

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {d \sqrt {d-c^2 d x^2} \left (\frac {g^2 (a+b \text {arccosh}(c x))^2}{32 b c^3}+\frac {2 f g (c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x))}{5 c^2}+\frac {g^2 x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{16 c^2}+\frac {1}{4} f^2 x (c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))-\frac {3}{8} f^2 x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))+\frac {3 f^2 (a+b \text {arccosh}(c x))^2}{16 b c}+\frac {1}{6} g^2 x^3 (c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))-\frac {1}{8} g^2 x^3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))-\frac {1}{16} b c^3 f^2 x^4-\frac {2}{25} b c^3 f g x^5-\frac {1}{36} b c^3 g^2 x^6+\frac {5}{16} b c f^2 x^2+\frac {4}{15} b c f g x^3-\frac {2 b f g x}{5 c}+\frac {7}{96} b c g^2 x^4-\frac {b g^2 x^2}{32 c}\right )}{\sqrt {c x-1} \sqrt {c x+1}}\)

Input:

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

Output:

-((d*Sqrt[d - c^2*d*x^2]*((-2*b*f*g*x)/(5*c) + (5*b*c*f^2*x^2)/16 - (b*g^2 
*x^2)/(32*c) + (4*b*c*f*g*x^3)/15 - (b*c^3*f^2*x^4)/16 + (7*b*c*g^2*x^4)/9 
6 - (2*b*c^3*f*g*x^5)/25 - (b*c^3*g^2*x^6)/36 - (3*f^2*x*Sqrt[-1 + c*x]*Sq 
rt[1 + c*x]*(a + b*ArcCosh[c*x]))/8 + (g^2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]* 
(a + b*ArcCosh[c*x]))/(16*c^2) - (g^2*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a 
+ b*ArcCosh[c*x]))/8 + (f^2*x*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)*(a + b*ArcC 
osh[c*x]))/4 + (g^2*x^3*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)*(a + b*ArcCosh[c* 
x]))/6 + (2*f*g*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2)*(a + b*ArcCosh[c*x]))/(5* 
c^2) + (3*f^2*(a + b*ArcCosh[c*x])^2)/(16*b*c) + (g^2*(a + b*ArcCosh[c*x]) 
^2)/(32*b*c^3)))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]))
 

Defintions of rubi rules used

rule 2009

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

rule 6387

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d 
_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-d)^IntPart[p]*((d + e*x^2)^Fra 
cPart[p]/((-1 + c*x)^FracPart[p]*(1 + c*x)^FracPart[p]))   Int[(f + g*x)^m* 
(-1 + c*x)^p*(1 + c*x)^p*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, 
d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && IntegerQ[m]
 

rule 6390

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_)*(( 
d2_) + (e2_.)*(x_))^(p_)*((f_) + (g_.)*(x_))^(m_.), x_Symbol] :> Int[Expand 
Integrand[(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, (f + g*x)^m, 
x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g}, x] && EqQ[e1 - c*d1, 0] && 
 EqQ[e2 + c*d2, 0] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d1, 0] && LtQ[ 
d2, 0] && IGtQ[n, 0] && ((EqQ[n, 1] && GtQ[p, -1]) || GtQ[p, 0] || EqQ[m, 1 
] || (EqQ[m, 2] && LtQ[p, -2]))
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1708\) vs. \(2(618)=1236\).

Time = 0.67 (sec) , antiderivative size = 1709, normalized size of antiderivative = 2.37

method	result	size
default	\(\text {Expression too large to display}\)	\(1709\)
parts	\(\text {Expression too large to display}\)	\(1709\)

Input:

int((g*x+f)^2*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x)),x,method=_RETURNVERB 
OSE)
 

Output:

a*(f^2*(1/4*x*(-c^2*d*x^2+d)^(3/2)+3/4*d*(1/2*x*(-c^2*d*x^2+d)^(1/2)+1/2*d 
/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))))+g^2*(-1/6*x* 
(-c^2*d*x^2+d)^(5/2)/c^2/d+1/6/c^2*(1/4*x*(-c^2*d*x^2+d)^(3/2)+3/4*d*(1/2* 
x*(-c^2*d*x^2+d)^(1/2)+1/2*d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d* 
x^2+d)^(1/2)))))-2/5*f*g*(-c^2*d*x^2+d)^(5/2)/c^2/d)+b*(-1/32*(-d*(c^2*x^2 
-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/c^3*arccosh(c*x)^2*(6*c^2*f^2+g^2)* 
d-1/2304*(-d*(c^2*x^2-1))^(1/2)*(32*c^7*x^7-64*c^5*x^5+32*c^6*x^6*(c*x-1)^ 
(1/2)*(c*x+1)^(1/2)+38*c^3*x^3-48*c^4*x^4*(c*x-1)^(1/2)*(c*x+1)^(1/2)-6*c* 
x+18*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2-(c*x-1)^(1/2)*(c*x+1)^(1/2))*g^2* 
(-1+6*arccosh(c*x))*d/(c*x+1)/c^3/(c*x-1)-1/400*(-d*(c^2*x^2-1))^(1/2)*(16 
*c^6*x^6-28*c^4*x^4+16*c^5*x^5*(c*x-1)^(1/2)*(c*x+1)^(1/2)+13*c^2*x^2-20*c 
^3*x^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)+5*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x-1)*f* 
g*(-1+5*arccosh(c*x))*d/(c*x+1)/c^2/(c*x-1)-1/512*(-d*(c^2*x^2-1))^(1/2)*( 
8*c^5*x^5-12*c^3*x^3+8*c^4*x^4*(c*x-1)^(1/2)*(c*x+1)^(1/2)+4*c*x-8*(c*x-1) 
^(1/2)*(c*x+1)^(1/2)*c^2*x^2+(c*x-1)^(1/2)*(c*x+1)^(1/2))*(8*arccosh(c*x)* 
c^2*f^2-2*c^2*f^2-4*arccosh(c*x)*g^2+g^2)*d/(c*x+1)/c^3/(c*x-1)+1/48*(-d*( 
c^2*x^2-1))^(1/2)*(4*c^4*x^4-5*c^2*x^2+4*c^3*x^3*(c*x-1)^(1/2)*(c*x+1)^(1/ 
2)-3*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+1)*f*g*(-1+3*arccosh(c*x))*d/(c*x+1)/ 
c^2/(c*x-1)+1/256*(-d*(c^2*x^2-1))^(1/2)*(2*c^3*x^3-2*c*x+2*(c*x-1)^(1/2)* 
(c*x+1)^(1/2)*c^2*x^2-(c*x-1)^(1/2)*(c*x+1)^(1/2))*(32*arccosh(c*x)*c^2...
 

Fricas [F]

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

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

Output:

integral(-(a*c^2*d*g^2*x^4 + 2*a*c^2*d*f*g*x^3 - 2*a*d*f*g*x - a*d*f^2 + ( 
a*c^2*d*f^2 - a*d*g^2)*x^2 + (b*c^2*d*g^2*x^4 + 2*b*c^2*d*f*g*x^3 - 2*b*d* 
f*g*x - b*d*f^2 + (b*c^2*d*f^2 - b*d*g^2)*x^2)*arccosh(c*x))*sqrt(-c^2*d*x 
^2 + d), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/8*(2*(-c^2*d*x^2 + d)^(3/2)*x + 3*sqrt(-c^2*d*x^2 + d)*d*x + 3*d^(3/2)*a 
rcsin(c*x)/c)*a*f^2 + 1/48*a*g^2*(2*(-c^2*d*x^2 + d)^(3/2)*x/c^2 - 8*(-c^2 
*d*x^2 + d)^(5/2)*x/(c^2*d) + 3*sqrt(-c^2*d*x^2 + d)*d*x/c^2 + 3*d^(3/2)*a 
rcsin(c*x)/c^3) - 2/5*(-c^2*d*x^2 + d)^(5/2)*a*f*g/(c^2*d) + integrate((-c 
^2*d*x^2 + d)^(3/2)*b*g^2*x^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + 2*( 
-c^2*d*x^2 + d)^(3/2)*b*f*g*x*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + (-c 
^2*d*x^2 + d)^(3/2)*b*f^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)), x)
 

Giac [F(-2)]

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

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

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\frac {\sqrt {d}\, d \left (90 \mathit {asin} \left (c x \right ) a \,c^{2} f^{2}+15 \mathit {asin} \left (c x \right ) a \,g^{2}-60 \sqrt {-c^{2} x^{2}+1}\, a \,c^{5} f^{2} x^{3}-96 \sqrt {-c^{2} x^{2}+1}\, a \,c^{5} f g \,x^{4}-40 \sqrt {-c^{2} x^{2}+1}\, a \,c^{5} g^{2} x^{5}+150 \sqrt {-c^{2} x^{2}+1}\, a \,c^{3} f^{2} x +192 \sqrt {-c^{2} x^{2}+1}\, a \,c^{3} f g \,x^{2}+70 \sqrt {-c^{2} x^{2}+1}\, a \,c^{3} g^{2} x^{3}-96 \sqrt {-c^{2} x^{2}+1}\, a c f g -15 \sqrt {-c^{2} x^{2}+1}\, a c \,g^{2} x -240 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{4}d x \right ) b \,c^{5} g^{2}-480 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{3}d x \right ) b \,c^{5} f g -240 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{2}d x \right ) b \,c^{5} f^{2}+240 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{2}d x \right ) b \,c^{3} g^{2}+480 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x d x \right ) b \,c^{3} f g +240 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )d x \right ) b \,c^{3} f^{2}+96 a c f g \right )}{240 c^{3}} \] Input:

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

Output:

(sqrt(d)*d*(90*asin(c*x)*a*c**2*f**2 + 15*asin(c*x)*a*g**2 - 60*sqrt( - c* 
*2*x**2 + 1)*a*c**5*f**2*x**3 - 96*sqrt( - c**2*x**2 + 1)*a*c**5*f*g*x**4 
- 40*sqrt( - c**2*x**2 + 1)*a*c**5*g**2*x**5 + 150*sqrt( - c**2*x**2 + 1)* 
a*c**3*f**2*x + 192*sqrt( - c**2*x**2 + 1)*a*c**3*f*g*x**2 + 70*sqrt( - c* 
*2*x**2 + 1)*a*c**3*g**2*x**3 - 96*sqrt( - c**2*x**2 + 1)*a*c*f*g - 15*sqr 
t( - c**2*x**2 + 1)*a*c*g**2*x - 240*int(sqrt( - c**2*x**2 + 1)*acosh(c*x) 
*x**4,x)*b*c**5*g**2 - 480*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)*x**3,x)*b 
*c**5*f*g - 240*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)*x**2,x)*b*c**5*f**2 
+ 240*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)*x**2,x)*b*c**3*g**2 + 480*int( 
sqrt( - c**2*x**2 + 1)*acosh(c*x)*x,x)*b*c**3*f*g + 240*int(sqrt( - c**2*x 
**2 + 1)*acosh(c*x),x)*b*c**3*f**2 + 96*a*c*f*g))/(240*c**3)