\(\int (f+g x)^3 (d-c^2 d x^2)^{5/2} (a+b \text {arccosh}(c x)) \, dx\) [50]

Optimal result

Integrand size = 31, antiderivative size = 1370 \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx =\text {Too large to display} \] Output:

15/64*d^2*f*g^2*x^3*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))+5/96*b*d^2*f^3 
*(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^3*(c*x-1) 
^(5/2)*(c*x+1)^(5/2)*(-c^2*d*x^2+d)^(1/2)/c-15/128*d^2*f*g^2*x*(-c^2*d*x^2 
+d)^(1/2)*(a+b*arccosh(c*x))/c^2+5/24*d^2*f^3*x*(-c*x+1)*(c*x+1)*(-c^2*d*x 
^2+d)^(1/2)*(a+b*arccosh(c*x))+1/6*d^2*f^3*x*(-c*x+1)^2*(c*x+1)^2*(-c^2*d* 
x^2+d)^(1/2)*(a+b*arccosh(c*x))-2/63*d^2*g^3*(-c*x+1)^3*(c*x+1)^3*(-c^2*d* 
x^2+d)^(1/2)*(a+b*arccosh(c*x))/c^4+2/63*b*d^2*g^3*x*(-c^2*d*x^2+d)^(1/2)/ 
c^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)-5/32*b*c*d^2*f^3*x^2*(-c^2*d*x^2+d)^(1/2)/ 
(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/189*b*d^2*g^3*x^3*(-c^2*d*x^2+d)^(1/2)/c/(c* 
x-1)^(1/2)/(c*x+1)^(1/2)-1/21*b*c*d^2*g^3*x^5*(-c^2*d*x^2+d)^(1/2)/(c*x-1) 
^(1/2)/(c*x+1)^(1/2)+19/441*b*c^3*d^2*g^3*x^7*(-c^2*d*x^2+d)^(1/2)/(c*x-1) 
^(1/2)/(c*x+1)^(1/2)-1/81*b*c^5*d^2*g^3*x^9*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^( 
1/2)/(c*x+1)^(1/2)+5/16*d^2*f*g^2*x^3*(-c*x+1)*(c*x+1)*(-c^2*d*x^2+d)^(1/2 
)*(a+b*arccosh(c*x))+3/8*d^2*f*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))-3/7*d^2*f^2*g*(-c*x+1)^3*(c*x+1)^3*(-c^2*d*x^2+d 
)^(1/2)*(a+b*arccosh(c*x))/c^2-1/9*d^2*g^3*x^2*(-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^3*(-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)+5/16*d^2*f^3*x*(-c^2*d* 
x^2+d)^(1/2)*(a+b*arccosh(c*x))+3/7*b*d^2*f^2*g*x*(-c^2*d*x^2+d)^(1/2)/c/( 
c*x-1)^(1/2)/(c*x+1)^(1/2)+15/256*b*d^2*f*g^2*x^2*(-c^2*d*x^2+d)^(1/2)/...
 

Mathematica [A] (warning: unable to verify)

Time = 7.73 (sec) , antiderivative size = 1802, normalized size of antiderivative = 1.32 \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx =\text {Too large to display} \] Input:

Integrate[(f + g*x)^3*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]),x]
 

Output:

Sqrt[-(d*(-1 + c^2*x^2))]*(-1/63*(a*d^2*g*(27*c^2*f^2 + 2*g^2))/c^4 + (a*d 
^2*f*(88*c^2*f^2 - 15*g^2)*x)/(128*c^2) - (a*d^2*g*(-81*c^2*f^2 + g^2)*x^2 
)/(63*c^2) - (a*d^2*f*(104*c^2*f^2 - 177*g^2)*x^3)/192 + (a*d^2*g*(-27*c^2 
*f^2 + 5*g^2)*x^4)/21 + (a*c^2*d^2*f*(8*c^2*f^2 - 51*g^2)*x^5)/48 - (a*c^2 
*d^2*g*(-27*c^2*f^2 + 19*g^2)*x^6)/63 + (3*a*c^4*d^2*f*g^2*x^7)/8 + (a*c^4 
*d^2*g^3*x^8)/9) - (5*a*d^(5/2)*f*(8*c^2*f^2 + 3*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^2*g*Sqr 
t[-(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]]))/(12*c^2*Sqrt[(-1 + c*x)/(1 
+ c*x)]*(1 + c*x)) - (b*d^2*f^3*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]*(Cosh[2*Ar 
cCosh[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^3*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)) - (3*b*d^2*f*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^2*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]]))/(600*c^2*Sqrt[(-1 + c*x)/(1 + c*x)...
 

Rubi [A] (verified)

Time = 3.76 (sec) , antiderivative size = 677, normalized size of antiderivative = 0.49, 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.

\(\displaystyle \int \left (d-c^2 d x^2\right )^{5/2} (f+g x)^3 (a+b \text {arccosh}(c x)) \, dx\)

\(\Big \downarrow \) 6387

\(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \int (c x-1)^{5/2} (c x+1)^{5/2} (f+g x)^3 (a+b \text {arccosh}(c x))dx}{\sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 6390

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \left (\frac {2 g^3 (c x-1)^{7/2} (c x+1)^{7/2} (a+b \text {arccosh}(c x))}{63 c^4}-\frac {15 f g^2 (a+b \text {arccosh}(c x))^2}{256 b c^3}+\frac {3 f^2 g (c x-1)^{7/2} (c x+1)^{7/2} (a+b \text {arccosh}(c x))}{7 c^2}-\frac {15 f g^2 x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{128 c^2}+\frac {g^3 x^2 (c x-1)^{7/2} (c x+1)^{7/2} (a+b \text {arccosh}(c x))}{9 c^2}+\frac {1}{6} f^3 x (c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x))-\frac {5}{24} f^3 x (c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))+\frac {5}{16} f^3 x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))-\frac {5 f^3 (a+b \text {arccosh}(c x))^2}{32 b c}+\frac {3}{8} f g^2 x^3 (c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x))-\frac {5}{16} f g^2 x^3 (c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))+\frac {15}{64} f g^2 x^3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))-\frac {3}{49} b c^5 f^2 g x^7-\frac {3}{64} b c^5 f g^2 x^8-\frac {1}{81} b c^5 g^3 x^9+\frac {5}{96} b c^3 f^3 x^4+\frac {9}{35} b c^3 f^2 g x^5+\frac {17}{96} b c^3 f g^2 x^6+\frac {19}{441} b c^3 g^3 x^7+\frac {2 b g^3 x}{63 c^3}+\frac {b f^3 \left (1-c^2 x^2\right )^3}{36 c}-\frac {25}{96} b c f^3 x^2-\frac {3}{7} b c f^2 g x^3+\frac {3 b f^2 g x}{7 c}-\frac {59}{256} b c f g^2 x^4+\frac {15 b f g^2 x^2}{256 c}-\frac {1}{21} b c g^3 x^5+\frac {b g^3 x^3}{189 c}\right )}{\sqrt {c x-1} \sqrt {c x+1}}\)

Input:

Int[(f + g*x)^3*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]),x]
 

Output:

(d^2*Sqrt[d - c^2*d*x^2]*((3*b*f^2*g*x)/(7*c) + (2*b*g^3*x)/(63*c^3) - (25 
*b*c*f^3*x^2)/96 + (15*b*f*g^2*x^2)/(256*c) - (3*b*c*f^2*g*x^3)/7 + (b*g^3 
*x^3)/(189*c) + (5*b*c^3*f^3*x^4)/96 - (59*b*c*f*g^2*x^4)/256 + (9*b*c^3*f 
^2*g*x^5)/35 - (b*c*g^3*x^5)/21 + (17*b*c^3*f*g^2*x^6)/96 - (3*b*c^5*f^2*g 
*x^7)/49 + (19*b*c^3*g^3*x^7)/441 - (3*b*c^5*f*g^2*x^8)/64 - (b*c^5*g^3*x^ 
9)/81 + (b*f^3*(1 - c^2*x^2)^3)/(36*c) + (5*f^3*x*Sqrt[-1 + c*x]*Sqrt[1 + 
c*x]*(a + b*ArcCosh[c*x]))/16 - (15*f*g^2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*( 
a + b*ArcCosh[c*x]))/(128*c^2) + (15*f*g^2*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x 
]*(a + b*ArcCosh[c*x]))/64 - (5*f^3*x*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)*(a 
+ b*ArcCosh[c*x]))/24 - (5*f*g^2*x^3*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)*(a + 
 b*ArcCosh[c*x]))/16 + (f^3*x*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2)*(a + b*ArcC 
osh[c*x]))/6 + (3*f*g^2*x^3*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2)*(a + b*ArcCos 
h[c*x]))/8 + (3*f^2*g*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2)*(a + b*ArcCosh[c*x] 
))/(7*c^2) + (2*g^3*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2)*(a + b*ArcCosh[c*x])) 
/(63*c^4) + (g^3*x^2*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2)*(a + b*ArcCosh[c*x]) 
)/(9*c^2) - (5*f^3*(a + b*ArcCosh[c*x])^2)/(32*b*c) - (15*f*g^2*(a + b*Arc 
Cosh[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. \(3137\) vs. \(2(1186)=2372\).

Time = 0.85 (sec) , antiderivative size = 3138, normalized size of antiderivative = 2.29

method	result	size
default	\(\text {Expression too large to display}\)	\(3138\)
parts	\(\text {Expression too large to display}\)	\(3138\)

Input:

int((g*x+f)^3*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x)),x,method=_RETURNVERB 
OSE)
 

Output:

a*(f^3*(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^3*(-1/9*x^2*(-c^2*d*x^2+d)^(7/2)/c^2/d-2/63/d/c^4 
*(-c^2*d*x^2+d)^(7/2))+3*f*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))))))-3/7*f^2*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*f*(8*c^2*f^2+ 
3*g^2)*d^2+1/41472*(-d*(c^2*x^2-1))^(1/2)*(256*c^10*x^10-704*c^8*x^8+256*c 
^9*x^9*(c*x-1)^(1/2)*(c*x+1)^(1/2)+688*c^6*x^6-576*c^7*x^7*(c*x-1)^(1/2)*( 
c*x+1)^(1/2)-280*c^4*x^4+432*c^5*x^5*(c*x-1)^(1/2)*(c*x+1)^(1/2)+41*c^2*x^ 
2-120*c^3*x^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)+9*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c* 
x-1)*g^3*(-1+9*arccosh(c*x))*d^2/(c*x+1)/c^4/(c*x-1)+3/16384*(-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))*f*g^2*(-1+8*arccosh(c*x))*d^2/(c*x+1)/c^3/(c 
*x-1)+3/25088*(-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)*(...
 

Fricas [F]

\[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{3} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \,d x } \] Input:

integrate((g*x+f)^3*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x)),x, algorithm=" 
fricas")
 

Output:

integral((a*c^4*d^2*g^3*x^7 + 3*a*c^4*d^2*f*g^2*x^6 + 3*a*d^2*f^2*g*x + a* 
d^2*f^3 + (3*a*c^4*d^2*f^2*g - 2*a*c^2*d^2*g^3)*x^5 + (a*c^4*d^2*f^3 - 6*a 
*c^2*d^2*f*g^2)*x^4 - (6*a*c^2*d^2*f^2*g - a*d^2*g^3)*x^3 - (2*a*c^2*d^2*f 
^3 - 3*a*d^2*f*g^2)*x^2 + (b*c^4*d^2*g^3*x^7 + 3*b*c^4*d^2*f*g^2*x^6 + 3*b 
*d^2*f^2*g*x + b*d^2*f^3 + (3*b*c^4*d^2*f^2*g - 2*b*c^2*d^2*g^3)*x^5 + (b* 
c^4*d^2*f^3 - 6*b*c^2*d^2*f*g^2)*x^4 - (6*b*c^2*d^2*f^2*g - b*d^2*g^3)*x^3 
 - (2*b*c^2*d^2*f^3 - 3*b*d^2*f*g^2)*x^2)*arccosh(c*x))*sqrt(-c^2*d*x^2 + 
d), x)
 

Sympy [F(-1)]

Timed out. \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\text {Timed out} \] Input:

integrate((g*x+f)**3*(-c**2*d*x**2+d)**(5/2)*(a+b*acosh(c*x)),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{3} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \,d x } \] Input:

integrate((g*x+f)^3*(-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^3 + 1/128*(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*f*g^2 - 1/63*(7*(-c^2*d*x^2 + d)^(7/2)*x^2/(c^2*d) + 2*(- 
c^2*d*x^2 + d)^(7/2)/(c^4*d))*a*g^3 - 3/7*(-c^2*d*x^2 + d)^(7/2)*a*f^2*g/( 
c^2*d) + integrate((-c^2*d*x^2 + d)^(5/2)*b*g^3*x^3*log(c*x + sqrt(c*x + 1 
)*sqrt(c*x - 1)) + 3*(-c^2*d*x^2 + d)^(5/2)*b*f*g^2*x^2*log(c*x + sqrt(c*x 
 + 1)*sqrt(c*x - 1)) + 3*(-c^2*d*x^2 + d)^(5/2)*b*f^2*g*x*log(c*x + sqrt(c 
*x + 1)*sqrt(c*x - 1)) + (-c^2*d*x^2 + d)^(5/2)*b*f^3*log(c*x + sqrt(c*x + 
 1)*sqrt(c*x - 1)), x)
 

Giac [F(-2)]

Exception generated. \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((g*x+f)^3*(-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. \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\int {\left (f+g\,x\right )}^3\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \] Input:

int((f + g*x)^3*(a + b*acosh(c*x))*(d - c^2*d*x^2)^(5/2),x)
 

Output:

int((f + g*x)^3*(a + b*acosh(c*x))*(d - c^2*d*x^2)^(5/2), x)
 

Reduce [F]

\[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx =\text {Too large to display} \] Input:

int((g*x+f)^3*(-c^2*d*x^2+d)^(5/2)*(a+b*acosh(c*x)),x)
 

Output:

(sqrt(d)*d**2*(2520*asin(c*x)*a*c**3*f**3 + 945*asin(c*x)*a*c*f*g**2 + 134 
4*sqrt( - c**2*x**2 + 1)*a*c**8*f**3*x**5 + 3456*sqrt( - c**2*x**2 + 1)*a* 
c**8*f**2*g*x**6 + 3024*sqrt( - c**2*x**2 + 1)*a*c**8*f*g**2*x**7 + 896*sq 
rt( - c**2*x**2 + 1)*a*c**8*g**3*x**8 - 4368*sqrt( - c**2*x**2 + 1)*a*c**6 
*f**3*x**3 - 10368*sqrt( - c**2*x**2 + 1)*a*c**6*f**2*g*x**4 - 8568*sqrt( 
- c**2*x**2 + 1)*a*c**6*f*g**2*x**5 - 2432*sqrt( - c**2*x**2 + 1)*a*c**6*g 
**3*x**6 + 5544*sqrt( - c**2*x**2 + 1)*a*c**4*f**3*x + 10368*sqrt( - c**2* 
x**2 + 1)*a*c**4*f**2*g*x**2 + 7434*sqrt( - c**2*x**2 + 1)*a*c**4*f*g**2*x 
**3 + 1920*sqrt( - c**2*x**2 + 1)*a*c**4*g**3*x**4 - 3456*sqrt( - c**2*x** 
2 + 1)*a*c**2*f**2*g - 945*sqrt( - c**2*x**2 + 1)*a*c**2*f*g**2*x - 128*sq 
rt( - c**2*x**2 + 1)*a*c**2*g**3*x**2 - 256*sqrt( - c**2*x**2 + 1)*a*g**3 
+ 8064*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)*x**7,x)*b*c**8*g**3 + 24192*i 
nt(sqrt( - c**2*x**2 + 1)*acosh(c*x)*x**6,x)*b*c**8*f*g**2 + 24192*int(sqr 
t( - c**2*x**2 + 1)*acosh(c*x)*x**5,x)*b*c**8*f**2*g - 16128*int(sqrt( - c 
**2*x**2 + 1)*acosh(c*x)*x**5,x)*b*c**6*g**3 + 8064*int(sqrt( - c**2*x**2 
+ 1)*acosh(c*x)*x**4,x)*b*c**8*f**3 - 48384*int(sqrt( - c**2*x**2 + 1)*aco 
sh(c*x)*x**4,x)*b*c**6*f*g**2 - 48384*int(sqrt( - c**2*x**2 + 1)*acosh(c*x 
)*x**3,x)*b*c**6*f**2*g + 8064*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)*x**3, 
x)*b*c**4*g**3 - 16128*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)*x**2,x)*b*c** 
6*f**3 + 24192*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)*x**2,x)*b*c**4*f*g...