\(\int (f+g x)^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx\) [41]

Optimal result

Integrand size = 31, antiderivative size = 713 \[ \int (f+g x)^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\frac {b f^2 g x \sqrt {d-c^2 d x^2}}{c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b g^3 x \sqrt {d-c^2 d x^2}}{15 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c f^3 x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b f g^2 x^2 \sqrt {d-c^2 d x^2}}{16 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c f^2 g x^3 \sqrt {d-c^2 d x^2}}{3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b g^3 x^3 \sqrt {d-c^2 d x^2}}{45 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 b c f g^2 x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c g^3 x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{2} f^3 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {3 f g^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 c^2}+\frac {3}{4} f g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {f^2 g (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{c^2}-\frac {2 g^3 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{15 c^4}-\frac {g^3 x^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{5 c^2}-\frac {f^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 b c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 f g^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{16 b c^3 \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

b*f^2*g*x*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)+2/15*b*g^3*x* 
(-c^2*d*x^2+d)^(1/2)/c^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/4*b*c*f^3*x^2*(-c^2 
*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+3/16*b*f*g^2*x^2*(-c^2*d*x^2+d 
)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/3*b*c*f^2*g*x^3*(-c^2*d*x^2+d)^(1/ 
2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/45*b*g^3*x^3*(-c^2*d*x^2+d)^(1/2)/c/(c*x- 
1)^(1/2)/(c*x+1)^(1/2)-3/16*b*c*f*g^2*x^4*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/ 
2)/(c*x+1)^(1/2)-1/25*b*c*g^3*x^5*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+ 
1)^(1/2)+1/2*f^3*x*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))-3/8*f*g^2*x*(-c 
^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/c^2+3/4*f*g^2*x^3*(-c^2*d*x^2+d)^(1/2 
)*(a+b*arccosh(c*x))-f^2*g*(-c*x+1)*(c*x+1)*(-c^2*d*x^2+d)^(1/2)*(a+b*arcc 
osh(c*x))/c^2-2/15*g^3*(-c*x+1)*(c*x+1)*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh( 
c*x))/c^4-1/5*g^3*x^2*(-c*x+1)*(c*x+1)*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c 
*x))/c^2-1/4*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)-3/16*f*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 = 1.38 (sec) , antiderivative size = 491, normalized size of antiderivative = 0.69 \[ \int (f+g x)^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\frac {240 a \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2} \left (-16 g^3-c^2 g \left (120 f^2+45 f g x+8 g^2 x^2\right )+6 c^4 x \left (10 f^3+20 f^2 g x+15 f g^2 x^2+4 g^3 x^3\right )\right )-3600 a c \sqrt {d} f \left (4 c^2 f^2+3 g^2\right ) \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+2400 b c^2 f^2 g \sqrt {d-c^2 d x^2} \left (9 c x+12 \left (\frac {-1+c x}{1+c x}\right )^{3/2} (1+c x)^3 \text {arccosh}(c x)-\cosh (3 \text {arccosh}(c x))\right )-3600 b c^3 f^3 \sqrt {d-c^2 d x^2} (\cosh (2 \text {arccosh}(c x))+2 \text {arccosh}(c x) (\text {arccosh}(c x)-\sinh (2 \text {arccosh}(c x))))-675 b c f g^2 \sqrt {d-c^2 d x^2} \left (8 \text {arccosh}(c x)^2+\cosh (4 \text {arccosh}(c x))-4 \text {arccosh}(c x) \sinh (4 \text {arccosh}(c x))\right )+8 b g^3 \sqrt {d-c^2 d x^2} \left (450 c x-450 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x)-25 \cosh (3 \text {arccosh}(c x))-9 \cosh (5 \text {arccosh}(c x))+75 \text {arccosh}(c x) \sinh (3 \text {arccosh}(c x))+45 \text {arccosh}(c x) \sinh (5 \text {arccosh}(c x))\right )}{28800 c^4 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \] Input:

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

Output:

(240*a*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(-16*g^3 - 
 c^2*g*(120*f^2 + 45*f*g*x + 8*g^2*x^2) + 6*c^4*x*(10*f^3 + 20*f^2*g*x + 1 
5*f*g^2*x^2 + 4*g^3*x^3)) - 3600*a*c*Sqrt[d]*f*(4*c^2*f^2 + 3*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))] + 2400*b*c^2*f^2*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]]) - 36 
00*b*c^3*f^3*Sqrt[d - c^2*d*x^2]*(Cosh[2*ArcCosh[c*x]] + 2*ArcCosh[c*x]*(A 
rcCosh[c*x] - Sinh[2*ArcCosh[c*x]])) - 675*b*c*f*g^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]]) + 8*b*g^3*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*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]]))/(28800*c^4*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))
 

Rubi [A] (verified)

Time = 2.45 (sec) , antiderivative size = 401, normalized size of antiderivative = 0.56, 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.

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

\(\Big \downarrow \) 6387

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

\(\Big \downarrow \) 6390

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (\frac {2 g^3 (c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))}{15 c^4}-\frac {3 f g^2 (a+b \text {arccosh}(c x))^2}{16 b c^3}+\frac {f^2 g (c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))}{c^2}-\frac {3 f g^2 x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{8 c^2}+\frac {g^3 x^2 (c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))}{5 c^2}+\frac {1}{2} f^3 x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))-\frac {f^3 (a+b \text {arccosh}(c x))^2}{4 b c}+\frac {3}{4} f g^2 x^3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))+\frac {2 b g^3 x}{15 c^3}-\frac {1}{4} b c f^3 x^2-\frac {1}{3} b c f^2 g x^3+\frac {b f^2 g x}{c}-\frac {3}{16} b c f g^2 x^4+\frac {3 b f g^2 x^2}{16 c}-\frac {1}{25} b c g^3 x^5+\frac {b g^3 x^3}{45 c}\right )}{\sqrt {c x-1} \sqrt {c x+1}}\)

Input:

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

Output:

(Sqrt[d - c^2*d*x^2]*((b*f^2*g*x)/c + (2*b*g^3*x)/(15*c^3) - (b*c*f^3*x^2) 
/4 + (3*b*f*g^2*x^2)/(16*c) - (b*c*f^2*g*x^3)/3 + (b*g^3*x^3)/(45*c) - (3* 
b*c*f*g^2*x^4)/16 - (b*c*g^3*x^5)/25 + (f^3*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] 
*(a + b*ArcCosh[c*x]))/2 - (3*f*g^2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b* 
ArcCosh[c*x]))/(8*c^2) + (3*f*g^2*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b* 
ArcCosh[c*x]))/4 + (f^2*g*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)*(a + b*ArcCosh[ 
c*x]))/c^2 + (2*g^3*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)*(a + b*ArcCosh[c*x])) 
/(15*c^4) + (g^3*x^2*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)*(a + b*ArcCosh[c*x]) 
)/(5*c^2) - (f^3*(a + b*ArcCosh[c*x])^2)/(4*b*c) - (3*f*g^2*(a + b*ArcCosh 
[c*x])^2)/(16*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. \(1417\) vs. \(2(613)=1226\).

Time = 0.61 (sec) , antiderivative size = 1418, normalized size of antiderivative = 1.99

method	result	size
default	\(\text {Expression too large to display}\)	\(1418\)
parts	\(\text {Expression too large to display}\)	\(1418\)

Input:

int((g*x+f)^3*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x)),x,method=_RETURNVERB 
OSE)
 

Output:

a*(f^3*(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*(-1/5*x^2*(-c^2*d*x^2+d)^(3/2)/c^2/d-2/15/d 
/c^4*(-c^2*d*x^2+d)^(3/2))+3*f*g^2*(-1/4*x*(-c^2*d*x^2+d)^(3/2)/c^2/d+1/4/ 
c^2*(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))))-f^2*g*(-c^2*d*x^2+d)^(3/2)/c^2/d)+b*(-1/16*(-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*(4*c^2*f 
^2+3*g^2)+1/800*(-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)*g^3*(-1+5*arccosh(c*x))/(c*x+1)/c^ 
4/(c*x-1)+3/256*(-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))*f*g^2*(-1+4*arccosh(c*x))/(c*x+1)/c^3/(c*x-1)+1/28 
8*(-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)*g*(36*arccosh(c*x)*c^2*f^2- 
12*c^2*f^2+3*arccosh(c*x)*g^2-g^2)/(c*x+1)/c^4/(c*x-1)+1/16*(-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))*f^3*(-1+2*arccosh(c*x))/(c*x+1)/c/(c*x-1)-1/16*(-d*(c^2 
*x^2-1))^(1/2)*((c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)*g*(6*arccosh(c* 
x)*c^2*f^2-6*c^2*f^2+arccosh(c*x)*g^2-g^2)/(c*x+1)/c^4/(c*x-1)-1/16*(-d*(c 
^2*x^2-1))^(1/2)*(-(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)*g*(6*arcc...
 

Fricas [F]

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

integrate((g*x+f)^3*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x)),x, algorithm=" 
fricas")
 

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((g*x+f)^3*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x)),x, algorithm=" 
maxima")
 

Output:

1/2*(sqrt(-c^2*d*x^2 + d)*x + sqrt(d)*arcsin(c*x)/c)*a*f^3 - 1/15*a*g^3*(3 
*(-c^2*d*x^2 + d)^(3/2)*x^2/(c^2*d) + 2*(-c^2*d*x^2 + d)^(3/2)/(c^4*d)) + 
3/8*a*f*g^2*(sqrt(-c^2*d*x^2 + d)*x/c^2 - 2*(-c^2*d*x^2 + d)^(3/2)*x/(c^2* 
d) + sqrt(d)*arcsin(c*x)/c^3) - (-c^2*d*x^2 + d)^(3/2)*a*f^2*g/(c^2*d) + i 
ntegrate(sqrt(-c^2*d*x^2 + d)*b*g^3*x^3*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 
 1)) + 3*sqrt(-c^2*d*x^2 + d)*b*f*g^2*x^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x 
 - 1)) + 3*sqrt(-c^2*d*x^2 + d)*b*f^2*g*x*log(c*x + sqrt(c*x + 1)*sqrt(c*x 
 - 1)) + sqrt(-c^2*d*x^2 + d)*b*f^3*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) 
, x)
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int (f+g x)^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\frac {\sqrt {d}\, \left (60 \mathit {asin} \left (c x \right ) a \,c^{3} f^{3}+45 \mathit {asin} \left (c x \right ) a c f \,g^{2}+60 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} f^{3} x +120 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} f^{2} g \,x^{2}+90 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} f \,g^{2} x^{3}+24 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} g^{3} x^{4}-120 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} f^{2} g -45 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} f \,g^{2} x -8 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} g^{3} x^{2}-16 \sqrt {-c^{2} x^{2}+1}\, a \,g^{3}+120 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{3}d x \right ) b \,c^{4} g^{3}+360 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{2}d x \right ) b \,c^{4} f \,g^{2}+360 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x d x \right ) b \,c^{4} f^{2} g +120 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )d x \right ) b \,c^{4} f^{3}+120 a \,c^{2} f^{2} g +16 a \,g^{3}\right )}{120 c^{4}} \] Input:

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

Output:

(sqrt(d)*(60*asin(c*x)*a*c**3*f**3 + 45*asin(c*x)*a*c*f*g**2 + 60*sqrt( - 
c**2*x**2 + 1)*a*c**4*f**3*x + 120*sqrt( - c**2*x**2 + 1)*a*c**4*f**2*g*x* 
*2 + 90*sqrt( - c**2*x**2 + 1)*a*c**4*f*g**2*x**3 + 24*sqrt( - c**2*x**2 + 
 1)*a*c**4*g**3*x**4 - 120*sqrt( - c**2*x**2 + 1)*a*c**2*f**2*g - 45*sqrt( 
 - c**2*x**2 + 1)*a*c**2*f*g**2*x - 8*sqrt( - c**2*x**2 + 1)*a*c**2*g**3*x 
**2 - 16*sqrt( - c**2*x**2 + 1)*a*g**3 + 120*int(sqrt( - c**2*x**2 + 1)*ac 
osh(c*x)*x**3,x)*b*c**4*g**3 + 360*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)*x 
**2,x)*b*c**4*f*g**2 + 360*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)*x,x)*b*c* 
*4*f**2*g + 120*int(sqrt( - c**2*x**2 + 1)*acosh(c*x),x)*b*c**4*f**3 + 120 
*a*c**2*f**2*g + 16*a*g**3))/(120*c**4)