\(\int (f+g x)^2 (d-c^2 d x^2)^{5/2} (a+b \text {arccosh}(c x)) \, dx\) [51]

Optimal result

Integrand size = 31, antiderivative size = 1000 \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\frac {2 b d^2 f g x \sqrt {d-c^2 d x^2}}{7 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 b c d^2 f^2 x^2 \sqrt {d-c^2 d x^2}}{32 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b d^2 g^2 x^2 \sqrt {d-c^2 d x^2}}{256 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c d^2 f g x^3 \sqrt {d-c^2 d x^2}}{7 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {59 b c d^2 g^2 x^4 \sqrt {d-c^2 d x^2}}{768 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {6 b c^3 d^2 f g x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {17 b c^3 d^2 g^2 x^6 \sqrt {d-c^2 d x^2}}{288 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 f g x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 g^2 x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b d^2 f^2 (-1+c x)^{3/2} (1+c x)^{3/2} \sqrt {d-c^2 d x^2}}{96 c}-\frac {b d^2 f^2 (-1+c x)^{5/2} (1+c x)^{5/2} \sqrt {d-c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {5 d^2 g^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{128 c^2}+\frac {5}{64} d^2 g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {5}{24} d^2 f^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {5}{48} d^2 g^2 x^3 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {1}{6} d^2 f^2 x (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {1}{8} d^2 g^2 x^3 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {2 d^2 f g (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{7 c^2}-\frac {5 d^2 f^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{32 b c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 d^2 g^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{256 b c^3 \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

2/7*b*d^2*f*g*x*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-5/32*b* 
c*d^2*f^2*x^2*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+5/256*b*d^2 
*g^2*x^2*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-2/7*b*c*d^2*f* 
g*x^3*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-59/768*b*c*d^2*g^2* 
x^4*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+6/35*b*c^3*d^2*f*g*x^ 
5*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+17/288*b*c^3*d^2*g^2*x^ 
6*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-2/49*b*c^5*d^2*f*g*x^7* 
(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/64*b*c^5*d^2*g^2*x^8*(- 
c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+5/96*b*d^2*f^2*(c*x-1)^(3/2 
)*(c*x+1)^(3/2)*(-c^2*d*x^2+d)^(1/2)/c-1/36*b*d^2*f^2*(c*x-1)^(5/2)*(c*x+1 
)^(5/2)*(-c^2*d*x^2+d)^(1/2)/c+5/16*d^2*f^2*x*(-c^2*d*x^2+d)^(1/2)*(a+b*ar 
ccosh(c*x))-5/128*d^2*g^2*x*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/c^2+5/ 
64*d^2*g^2*x^3*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))+5/24*d^2*f^2*x*(-c* 
x+1)*(c*x+1)*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))+5/48*d^2*g^2*x^3*(-c* 
x+1)*(c*x+1)*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))+1/6*d^2*f^2*x*(-c*x+1 
)^2*(c*x+1)^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))+1/8*d^2*g^2*x^3*(-c* 
x+1)^2*(c*x+1)^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))-2/7*d^2*f*g*(-c*x 
+1)^3*(c*x+1)^3*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/c^2-5/32*d^2*f^2*( 
-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^2/b/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-5 
/256*d^2*g^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^2/b/c^3/(c*x-1)^(1...
 

Mathematica [A] (warning: unable to verify)

Time = 7.21 (sec) , antiderivative size = 1282, normalized size of antiderivative = 1.28 \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx =\text {Too large to display} \] Input:

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

Output:

Sqrt[-(d*(-1 + c^2*x^2))]*((-2*a*d^2*f*g)/(7*c^2) + (a*d^2*(88*c^2*f^2 - 5 
*g^2)*x)/(128*c^2) + (6*a*d^2*f*g*x^2)/7 + (a*d^2*(-104*c^2*f^2 + 59*g^2)* 
x^3)/192 - (6*a*c^2*d^2*f*g*x^4)/7 + (a*c^2*d^2*(8*c^2*f^2 - 17*g^2)*x^5)/ 
48 + (2*a*c^4*d^2*f*g*x^6)/7 + (a*c^4*d^2*g^2*x^7)/8) - (5*a*d^(5/2)*(8*c^ 
2*f^2 + g^2)*ArcTan[(c*x*Sqrt[-(d*(-1 + c^2*x^2))])/(Sqrt[d]*(-1 + c^2*x^2 
))])/(128*c^3) - (b*d^2*f*g*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]*(-9*c*x - 12*( 
(-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3*ArcCosh[c*x] + Cosh[3*ArcCosh[c*x] 
]))/(18*c^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) - (b*d^2*f^2*Sqrt[-(d*(- 
1 + c*x)*(1 + c*x))]*(Cosh[2*ArcCosh[c*x]] + 2*ArcCosh[c*x]*(ArcCosh[c*x] 
- Sinh[2*ArcCosh[c*x]])))/(8*c*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + (b* 
d^2*f^2*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]*(8*ArcCosh[c*x]^2 + Cosh[4*ArcCosh 
[c*x]] - 4*ArcCosh[c*x]*Sinh[4*ArcCosh[c*x]]))/(64*c*Sqrt[(-1 + c*x)/(1 + 
c*x)]*(1 + c*x)) - (b*d^2*g^2*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]*(8*ArcCosh[c 
*x]^2 + Cosh[4*ArcCosh[c*x]] - 4*ArcCosh[c*x]*Sinh[4*ArcCosh[c*x]]))/(128* 
c^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + (b*d^2*f*g*Sqrt[-(d*(-1 + c*x) 
*(1 + c*x))]*(-450*c*x + 450*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[ 
c*x] + 25*Cosh[3*ArcCosh[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]]))/(900*c^2*Sqr 
t[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + (b*d^2*f^2*Sqrt[-(d*(-1 + c*x)*(1 + c 
*x))]*(18*Cosh[2*ArcCosh[c*x]] - 9*Cosh[4*ArcCosh[c*x]] - 2*(36*ArcCosh...
 

Rubi [A] (verified)

Time = 2.45 (sec) , antiderivative size = 517, normalized size of antiderivative = 0.52, 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.

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

\(\Big \downarrow \) 6387

\(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \int (c x-1)^{5/2} (c x+1)^{5/2} (f+g x)^2 (a+b \text {arccosh}(c x))dx}{\sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 6390

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \left (-\frac {5 g^2 (a+b \text {arccosh}(c x))^2}{256 b c^3}+\frac {2 f g (c x-1)^{7/2} (c x+1)^{7/2} (a+b \text {arccosh}(c x))}{7 c^2}-\frac {5 g^2 x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{128 c^2}+\frac {1}{6} f^2 x (c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x))-\frac {5}{24} f^2 x (c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))+\frac {5}{16} f^2 x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))-\frac {5 f^2 (a+b \text {arccosh}(c x))^2}{32 b c}+\frac {1}{8} g^2 x^3 (c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x))-\frac {5}{48} g^2 x^3 (c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))+\frac {5}{64} g^2 x^3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))-\frac {2}{49} b c^5 f g x^7-\frac {1}{64} b c^5 g^2 x^8+\frac {5}{96} b c^3 f^2 x^4+\frac {6}{35} b c^3 f g x^5+\frac {17}{288} b c^3 g^2 x^6+\frac {b f^2 \left (1-c^2 x^2\right )^3}{36 c}-\frac {25}{96} b c f^2 x^2-\frac {2}{7} b c f g x^3+\frac {2 b f g x}{7 c}-\frac {59}{768} b c g^2 x^4+\frac {5 b g^2 x^2}{256 c}\right )}{\sqrt {c x-1} \sqrt {c x+1}}\)

Input:

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

Output:

(d^2*Sqrt[d - c^2*d*x^2]*((2*b*f*g*x)/(7*c) - (25*b*c*f^2*x^2)/96 + (5*b*g 
^2*x^2)/(256*c) - (2*b*c*f*g*x^3)/7 + (5*b*c^3*f^2*x^4)/96 - (59*b*c*g^2*x 
^4)/768 + (6*b*c^3*f*g*x^5)/35 + (17*b*c^3*g^2*x^6)/288 - (2*b*c^5*f*g*x^7 
)/49 - (b*c^5*g^2*x^8)/64 + (b*f^2*(1 - c^2*x^2)^3)/(36*c) + (5*f^2*x*Sqrt 
[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/16 - (5*g^2*x*Sqrt[-1 + c*x 
]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(128*c^2) + (5*g^2*x^3*Sqrt[-1 + c*x 
]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/64 - (5*f^2*x*(-1 + c*x)^(3/2)*(1 + 
c*x)^(3/2)*(a + b*ArcCosh[c*x]))/24 - (5*g^2*x^3*(-1 + c*x)^(3/2)*(1 + c*x 
)^(3/2)*(a + b*ArcCosh[c*x]))/48 + (f^2*x*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2) 
*(a + b*ArcCosh[c*x]))/6 + (g^2*x^3*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2)*(a + 
b*ArcCosh[c*x]))/8 + (2*f*g*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2)*(a + b*ArcCos 
h[c*x]))/(7*c^2) - (5*f^2*(a + b*ArcCosh[c*x])^2)/(32*b*c) - (5*g^2*(a + b 
*ArcCosh[c*x])^2)/(256*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. \(2522\) vs. \(2(864)=1728\).

Time = 0.76 (sec) , antiderivative size = 2523, normalized size of antiderivative = 2.52

method	result	size
default	\(\text {Expression too large to display}\)	\(2523\)
parts	\(\text {Expression too large to display}\)	\(2523\)

Input:

int((g*x+f)^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x)),x,method=_RETURNVERB 
OSE)
 

Output:

a*(f^2*(1/6*x*(-c^2*d*x^2+d)^(5/2)+5/6*d*(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*(-1/8*x*(-c^2*d*x^2+d)^(7/2)/c^2/d+1/8/c^2*(1/6 
*x*(-c^2*d*x^2+d)^(5/2)+5/6*d*(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))))))-2/7*f*g*(-c^2*d*x^2+d)^(7/2)/c^2/d)+b*(-5/256*(-d*(c^2*x^2-1) 
)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/c^3*arccosh(c*x)^2*(8*c^2*f^2+g^2)*d^2 
+1/16384*(-d*(c^2*x^2-1))^(1/2)*(128*c^9*x^9-320*c^7*x^7+128*(c*x+1)^(1/2) 
*(c*x-1)^(1/2)*x^8*c^8+272*c^5*x^5-256*c^6*x^6*(c*x-1)^(1/2)*(c*x+1)^(1/2) 
-88*c^3*x^3+160*c^4*x^4*(c*x-1)^(1/2)*(c*x+1)^(1/2)+8*c*x-32*(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+8*arccosh(c*x) 
)*d^2/(c*x+1)/c^3/(c*x-1)+1/3136*(-d*(c^2*x^2-1))^(1/2)*(64*c^8*x^8-144*c^ 
6*x^6+64*c^7*x^7*(c*x-1)^(1/2)*(c*x+1)^(1/2)+104*c^4*x^4-112*c^5*x^5*(c*x- 
1)^(1/2)*(c*x+1)^(1/2)-25*c^2*x^2+56*c^3*x^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)-7 
*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+1)*f*g*(-1+7*arccosh(c*x))*d^2/(c*x+1)/c^ 
2/(c*x-1)+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))*(6*arccosh(c*x)*c^2*f^2-c^2*f^2-6*arccosh(c*x)*g^2+g^2)*d^2/(c*x+1)/c^ 
3/(c*x-1)-1/320*(-d*(c^2*x^2-1))^(1/2)*(16*c^6*x^6-28*c^4*x^4+16*c^5*x^...
 

Fricas [F]

\[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{2} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \,d x } \] Input:

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

Output:

integral((a*c^4*d^2*g^2*x^6 + 2*a*c^4*d^2*f*g*x^5 - 4*a*c^2*d^2*f*g*x^3 + 
2*a*d^2*f*g*x + a*d^2*f^2 + (a*c^4*d^2*f^2 - 2*a*c^2*d^2*g^2)*x^4 - (2*a*c 
^2*d^2*f^2 - a*d^2*g^2)*x^2 + (b*c^4*d^2*g^2*x^6 + 2*b*c^4*d^2*f*g*x^5 - 4 
*b*c^2*d^2*f*g*x^3 + 2*b*d^2*f*g*x + b*d^2*f^2 + (b*c^4*d^2*f^2 - 2*b*c^2* 
d^2*g^2)*x^4 - (2*b*c^2*d^2*f^2 - b*d^2*g^2)*x^2)*arccosh(c*x))*sqrt(-c^2* 
d*x^2 + d), x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

\[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{2} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \,d x } \] Input:

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

Output:

1/48*(8*(-c^2*d*x^2 + d)^(5/2)*x + 10*(-c^2*d*x^2 + d)^(3/2)*d*x + 15*sqrt 
(-c^2*d*x^2 + d)*d^2*x + 15*d^(5/2)*arcsin(c*x)/c)*a*f^2 + 1/384*(8*(-c^2* 
d*x^2 + d)^(5/2)*x/c^2 - 48*(-c^2*d*x^2 + d)^(7/2)*x/(c^2*d) + 10*(-c^2*d* 
x^2 + d)^(3/2)*d*x/c^2 + 15*sqrt(-c^2*d*x^2 + d)*d^2*x/c^2 + 15*d^(5/2)*ar 
csin(c*x)/c^3)*a*g^2 - 2/7*(-c^2*d*x^2 + d)^(7/2)*a*f*g/(c^2*d) + integrat 
e((-c^2*d*x^2 + d)^(5/2)*b*g^2*x^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) 
+ 2*(-c^2*d*x^2 + d)^(5/2)*b*f*g*x*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) 
+ (-c^2*d*x^2 + d)^(5/2)*b*f^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)), x)
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\frac {\sqrt {d}\, d^{2} \left (840 \mathit {asin} \left (c x \right ) a \,c^{2} f^{2}+105 \mathit {asin} \left (c x \right ) a \,g^{2}+448 \sqrt {-c^{2} x^{2}+1}\, a \,c^{7} f^{2} x^{5}+768 \sqrt {-c^{2} x^{2}+1}\, a \,c^{7} f g \,x^{6}+336 \sqrt {-c^{2} x^{2}+1}\, a \,c^{7} g^{2} x^{7}-1456 \sqrt {-c^{2} x^{2}+1}\, a \,c^{5} f^{2} x^{3}-2304 \sqrt {-c^{2} x^{2}+1}\, a \,c^{5} f g \,x^{4}-952 \sqrt {-c^{2} x^{2}+1}\, a \,c^{5} g^{2} x^{5}+1848 \sqrt {-c^{2} x^{2}+1}\, a \,c^{3} f^{2} x +2304 \sqrt {-c^{2} x^{2}+1}\, a \,c^{3} f g \,x^{2}+826 \sqrt {-c^{2} x^{2}+1}\, a \,c^{3} g^{2} x^{3}-768 \sqrt {-c^{2} x^{2}+1}\, a c f g -105 \sqrt {-c^{2} x^{2}+1}\, a c \,g^{2} x +2688 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{6}d x \right ) b \,c^{7} g^{2}+5376 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{5}d x \right ) b \,c^{7} f g +2688 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{4}d x \right ) b \,c^{7} f^{2}-5376 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{4}d x \right ) b \,c^{5} g^{2}-10752 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{3}d x \right ) b \,c^{5} f g -5376 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{2}d x \right ) b \,c^{5} f^{2}+2688 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{2}d x \right ) b \,c^{3} g^{2}+5376 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x d x \right ) b \,c^{3} f g +2688 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )d x \right ) b \,c^{3} f^{2}+768 a c f g \right )}{2688 c^{3}} \] Input:

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

Output:

(sqrt(d)*d**2*(840*asin(c*x)*a*c**2*f**2 + 105*asin(c*x)*a*g**2 + 448*sqrt 
( - c**2*x**2 + 1)*a*c**7*f**2*x**5 + 768*sqrt( - c**2*x**2 + 1)*a*c**7*f* 
g*x**6 + 336*sqrt( - c**2*x**2 + 1)*a*c**7*g**2*x**7 - 1456*sqrt( - c**2*x 
**2 + 1)*a*c**5*f**2*x**3 - 2304*sqrt( - c**2*x**2 + 1)*a*c**5*f*g*x**4 - 
952*sqrt( - c**2*x**2 + 1)*a*c**5*g**2*x**5 + 1848*sqrt( - c**2*x**2 + 1)* 
a*c**3*f**2*x + 2304*sqrt( - c**2*x**2 + 1)*a*c**3*f*g*x**2 + 826*sqrt( - 
c**2*x**2 + 1)*a*c**3*g**2*x**3 - 768*sqrt( - c**2*x**2 + 1)*a*c*f*g - 105 
*sqrt( - c**2*x**2 + 1)*a*c*g**2*x + 2688*int(sqrt( - c**2*x**2 + 1)*acosh 
(c*x)*x**6,x)*b*c**7*g**2 + 5376*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)*x** 
5,x)*b*c**7*f*g + 2688*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)*x**4,x)*b*c** 
7*f**2 - 5376*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)*x**4,x)*b*c**5*g**2 - 
10752*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)*x**3,x)*b*c**5*f*g - 5376*int( 
sqrt( - c**2*x**2 + 1)*acosh(c*x)*x**2,x)*b*c**5*f**2 + 2688*int(sqrt( - c 
**2*x**2 + 1)*acosh(c*x)*x**2,x)*b*c**3*g**2 + 5376*int(sqrt( - c**2*x**2 
+ 1)*acosh(c*x)*x,x)*b*c**3*f*g + 2688*int(sqrt( - c**2*x**2 + 1)*acosh(c* 
x),x)*b*c**3*f**2 + 768*a*c*f*g))/(2688*c**3)