\(\int \frac {(f+g x)^2 (a+b \text {arccosh}(c x))}{(d-c^2 d x^2)^{3/2}} \, dx\) [60]

Optimal result

Integrand size = 31, antiderivative size = 300 \[ \int \frac {(f+g x)^2 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {(c f-g)^2 (1-c x) (a+b \text {arccosh}(c x))}{2 c^3 d \sqrt {d-c^2 d x^2}}+\frac {(c f+g)^2 (1+c x) (a+b \text {arccosh}(c x))}{2 c^3 d \sqrt {d-c^2 d x^2}}-\frac {g^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{2 b c^3 d \sqrt {d-c^2 d x^2}}-\frac {b (c f+g)^2 \sqrt {-1+c x} \sqrt {1-c^2 x^2} \log (1-c x)}{2 c^3 d \sqrt {1-c x} \sqrt {d-c^2 d x^2}}-\frac {b (c f-g)^2 \sqrt {-1+c x} \sqrt {1-c^2 x^2} \log (1+c x)}{2 c^3 d \sqrt {1-c x} \sqrt {d-c^2 d x^2}} \] Output:

-1/2*(c*f-g)^2*(-c*x+1)*(a+b*arccosh(c*x))/c^3/d/(-c^2*d*x^2+d)^(1/2)+1/2* 
(c*f+g)^2*(c*x+1)*(a+b*arccosh(c*x))/c^3/d/(-c^2*d*x^2+d)^(1/2)-1/2*g^2*(c 
*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh(c*x))^2/b/c^3/d/(-c^2*d*x^2+d)^(1/2 
)-1/2*b*(c*f+g)^2*(c*x-1)^(1/2)*(-c^2*x^2+1)^(1/2)*ln(-c*x+1)/c^3/d/(-c*x+ 
1)^(1/2)/(-c^2*d*x^2+d)^(1/2)-1/2*b*(c*f-g)^2*(c*x-1)^(1/2)*(-c^2*x^2+1)^( 
1/2)*ln(c*x+1)/c^3/d/(-c*x+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 1.24 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.06 \[ \int \frac {(f+g x)^2 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {2 b c \sqrt {d} \left (2 f g+c^2 f^2 x+g^2 x\right ) \text {arccosh}(c x)-b \sqrt {d} g^2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x)^2+2 a g^2 \sqrt {d-c^2 d x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+2 \sqrt {d} \left (-b \left (c^2 f^2+g^2\right ) \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \log \left (\sqrt {\frac {-1+c x}{1+c x}} (1+c x)\right )+c \left (a \left (2 f g+c^2 f^2 x+g^2 x\right )+2 b f g \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \log \left (\cosh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )-2 b f g \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \log \left (\sinh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )\right )\right )}{2 c^3 d^{3/2} \sqrt {d-c^2 d x^2}} \] Input:

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

Output:

(2*b*c*Sqrt[d]*(2*f*g + c^2*f^2*x + g^2*x)*ArcCosh[c*x] - b*Sqrt[d]*g^2*Sq 
rt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]^2 + 2*a*g^2*Sqrt[d - c^2*d 
*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + 2*Sqrt[ 
d]*(-(b*(c^2*f^2 + g^2)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Log[Sqrt[(-1 
+ c*x)/(1 + c*x)]*(1 + c*x)]) + c*(a*(2*f*g + c^2*f^2*x + g^2*x) + 2*b*f*g 
*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Log[Cosh[ArcCosh[c*x]/2]] - 2*b*f*g* 
Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Log[Sinh[ArcCosh[c*x]/2]])))/(2*c^3*d 
^(3/2)*Sqrt[d - c^2*d*x^2])
 

Rubi [A] (warning: unable to verify)

Time = 1.65 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.43, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {6387, 6396, 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.

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

\(\Big \downarrow \) 6387

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

\(\Big \downarrow \) 6396

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-((Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-1/2*((c*f - g)^2*Sqrt[-1 + c*x]*(a + b*A 
rcCosh[c*x]))/(c^3*Sqrt[1 + c*x]) - ((c*f + g)^2*Sqrt[1 + c*x]*(a + b*ArcC 
osh[c*x]))/(2*c^3*Sqrt[-1 + c*x]) + (g^2*(a + b*ArcCosh[c*x])^2)/(2*b*c^3) 
 - (b*(c*f + g)^2*Sqrt[(1 - c*x)*(1 + c*x)]*Sqrt[1 - c^2*x^2]*Log[Sqrt[-(( 
1 - c*x)/(1 + c*x))]])/(c^3*Sqrt[-1 + c*x]*Sqrt[-((1 - c*x)/(1 + c*x))]*(1 
 + c*x)^(3/2)) + (b*(c*f - g)^2*Sqrt[(1 - c*x)*(1 + c*x)]*Sqrt[1 - c^2*x^2 
]*Log[2/(1 + c*x)])/(2*c^3*Sqrt[-1 + c*x]*Sqrt[-((1 - c*x)/(1 + c*x))]*(1 
+ c*x)^(3/2)) + (b*(c*f + g)^2*Sqrt[(1 - c*x)*(1 + c*x)]*Sqrt[1 - c^2*x^2] 
*Log[2/(1 + c*x)])/(2*c^3*Sqrt[-1 + c*x]*Sqrt[-((1 - c*x)/(1 + c*x))]*(1 + 
 c*x)^(3/2))))/(d*Sqrt[d - c^2*d*x^2]))
 

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 6396

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(876\) vs. \(2(264)=528\).

Time = 0.64 (sec) , antiderivative size = 877, normalized size of antiderivative = 2.92

method	result	size
default	\(a \left (\frac {f^{2} x}{d \sqrt {-c^{2} d \,x^{2}+d}}+g^{2} \left (\frac {x}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {\arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{c^{2} d \sqrt {c^{2} d}}\right )+\frac {2 f g}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}\right )+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, g^{2} \operatorname {arccosh}\left (c x \right )^{2}}{2 d^{2} c^{3} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) f^{2}}{d^{2} c \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) g^{2}}{d^{2} c^{3} \left (c^{2} x^{2}-1\right )}+\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) \left (c x -1\right ) \left (c x +1\right ) f g}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}-\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) x^{2} f g}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) x \,f^{2}}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) x \,g^{2}}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) f^{2}}{d^{2} c \left (c^{2} x^{2}-1\right )}-\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) f g}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) g^{2}}{d^{2} c^{3} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) f^{2}}{d^{2} c \left (c^{2} x^{2}-1\right )}+\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) f g}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) g^{2}}{d^{2} c^{3} \left (c^{2} x^{2}-1\right )}\)	\(877\)
parts	\(a \left (\frac {f^{2} x}{d \sqrt {-c^{2} d \,x^{2}+d}}+g^{2} \left (\frac {x}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {\arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{c^{2} d \sqrt {c^{2} d}}\right )+\frac {2 f g}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}\right )+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, g^{2} \operatorname {arccosh}\left (c x \right )^{2}}{2 d^{2} c^{3} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) f^{2}}{d^{2} c \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) g^{2}}{d^{2} c^{3} \left (c^{2} x^{2}-1\right )}+\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) \left (c x -1\right ) \left (c x +1\right ) f g}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}-\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) x^{2} f g}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) x \,f^{2}}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) x \,g^{2}}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) f^{2}}{d^{2} c \left (c^{2} x^{2}-1\right )}-\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) f g}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) g^{2}}{d^{2} c^{3} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) f^{2}}{d^{2} c \left (c^{2} x^{2}-1\right )}+\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) f g}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) g^{2}}{d^{2} c^{3} \left (c^{2} x^{2}-1\right )}\)	\(877\)

Input:

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

Output:

a*(f^2*x/d/(-c^2*d*x^2+d)^(1/2)+g^2*(x/c^2/d/(-c^2*d*x^2+d)^(1/2)-1/c^2/d/ 
(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2)))+2*f*g/c^2/d/(- 
c^2*d*x^2+d)^(1/2))+1/2*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/ 
2)/d^2/c^3/(c^2*x^2-1)*g^2*arccosh(c*x)^2-b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1) 
^(1/2)*(c*x+1)^(1/2)/d^2/c/(c^2*x^2-1)*arccosh(c*x)*f^2-b*(-d*(c^2*x^2-1)) 
^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/c^3/(c^2*x^2-1)*arccosh(c*x)*g^2+2* 
b*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)/d^2/c^2/(c^2*x^2-1)*(c*x-1)*(c*x+1)* 
f*g-2*b*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)/d^2/(c^2*x^2-1)*x^2*f*g-b*(-d* 
(c^2*x^2-1))^(1/2)*arccosh(c*x)/d^2/(c^2*x^2-1)*x*f^2-b*(-d*(c^2*x^2-1))^( 
1/2)*arccosh(c*x)/d^2/c^2/(c^2*x^2-1)*x*g^2+b*(-d*(c^2*x^2-1))^(1/2)*(c*x- 
1)^(1/2)*(c*x+1)^(1/2)/d^2/c/(c^2*x^2-1)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1 
/2))*f^2-2*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/c^2/(c 
^2*x^2-1)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*f*g+b*(-d*(c^2*x^2-1))^(1/ 
2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/c^3/(c^2*x^2-1)*ln(1+c*x+(c*x-1)^(1/2)* 
(c*x+1)^(1/2))*g^2+b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^ 
2/c/(c^2*x^2-1)*ln((c*x-1)^(1/2)*(c*x+1)^(1/2)+c*x-1)*f^2+2*b*(-d*(c^2*x^2 
-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/c^2/(c^2*x^2-1)*ln((c*x-1)^(1/2 
)*(c*x+1)^(1/2)+c*x-1)*f*g+b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^ 
(1/2)/d^2/c^3/(c^2*x^2-1)*ln((c*x-1)^(1/2)*(c*x+1)^(1/2)+c*x-1)*g^2
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-1/2*b*c*f^2*sqrt(-1/(c^4*d))*log(x^2 - 1/c^2)/d + a*g^2*(x/(sqrt(-c^2*d*x 
^2 + d)*c^2*d) - arcsin(c*x)/(c^3*d^(3/2))) + b*f^2*x*arccosh(c*x)/(sqrt(- 
c^2*d*x^2 + d)*d) + a*f^2*x/(sqrt(-c^2*d*x^2 + d)*d) + 2*a*f*g/(sqrt(-c^2* 
d*x^2 + d)*c^2*d) + integrate(b*g^2*x^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 
 1))/(-c^2*d*x^2 + d)^(3/2) + 2*b*f*g*x*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 
 1))/(-c^2*d*x^2 + d)^(3/2), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

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

Output:

( - sqrt( - c**2*x**2 + 1)*asin(c*x)*a*g**2 - sqrt( - c**2*x**2 + 1)*int(a 
cosh(c*x)/(sqrt( - c**2*x**2 + 1)*c**2*x**2 - sqrt( - c**2*x**2 + 1)),x)*b 
*c**3*f**2 - sqrt( - c**2*x**2 + 1)*int((acosh(c*x)*x**2)/(sqrt( - c**2*x* 
*2 + 1)*c**2*x**2 - sqrt( - c**2*x**2 + 1)),x)*b*c**3*g**2 - 2*sqrt( - c** 
2*x**2 + 1)*int((acosh(c*x)*x)/(sqrt( - c**2*x**2 + 1)*c**2*x**2 - sqrt( - 
 c**2*x**2 + 1)),x)*b*c**3*f*g + a*c**3*f**2*x + 2*a*c*f*g + a*c*g**2*x)/( 
sqrt(d)*sqrt( - c**2*x**2 + 1)*c**3*d)