\(\int (f+g x)^3 (d-c^2 d x^2)^{3/2} (a+b \text {arccosh}(c x)) \, dx\) [46]

Optimal result

Integrand size = 31, antiderivative size = 1026 \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\frac {3 b d f^2 g x \sqrt {d-c^2 d x^2}}{5 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b d g^3 x \sqrt {d-c^2 d x^2}}{35 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 b c d f^3 x^2 \sqrt {d-c^2 d x^2}}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b d f g^2 x^2 \sqrt {d-c^2 d x^2}}{32 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c d f^2 g x^3 \sqrt {d-c^2 d x^2}}{5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d g^3 x^3 \sqrt {d-c^2 d x^2}}{105 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {7 b c d f g^2 x^4 \sqrt {d-c^2 d x^2}}{32 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b c^3 d f^2 g x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {8 b c d g^3 x^5 \sqrt {d-c^2 d x^2}}{175 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d f g^2 x^6 \sqrt {d-c^2 d x^2}}{12 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d g^3 x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d f^3 (-1+c x)^{3/2} (1+c x)^{3/2} \sqrt {d-c^2 d x^2}}{16 c}+\frac {3}{8} d f^3 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {3 d f g^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{16 c^2}+\frac {3}{8} d f g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {1}{4} d f^3 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {1}{2} d f g^2 x^3 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {3 d f^2 g (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{5 c^2}-\frac {2 d g^3 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{35 c^4}-\frac {d g^3 x^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{7 c^2}-\frac {3 d f^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{16 b c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 d f g^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{32 b c^3 \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

3/5*b*d*f^2*g*x*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)+2/35*b* 
d*g^3*x*(-c^2*d*x^2+d)^(1/2)/c^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)-3/16*b*c*d*f^ 
3*x^2*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+3/32*b*d*f*g^2*x^2* 
(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-2/5*b*c*d*f^2*g*x^3*(-c 
^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/105*b*d*g^3*x^3*(-c^2*d*x^ 
2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-7/32*b*c*d*f*g^2*x^4*(-c^2*d*x^2+ 
d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+3/25*b*c^3*d*f^2*g*x^5*(-c^2*d*x^2+d) 
^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-8/175*b*c*d*g^3*x^5*(-c^2*d*x^2+d)^(1/2 
)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/12*b*c^3*d*f*g^2*x^6*(-c^2*d*x^2+d)^(1/2)/ 
(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/49*b*c^3*d*g^3*x^7*(-c^2*d*x^2+d)^(1/2)/(c*x 
-1)^(1/2)/(c*x+1)^(1/2)+1/16*b*d*f^3*(c*x-1)^(3/2)*(c*x+1)^(3/2)*(-c^2*d*x 
^2+d)^(1/2)/c+3/8*d*f^3*x*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))-3/16*d*f 
*g^2*x*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/c^2+3/8*d*f*g^2*x^3*(-c^2*d 
*x^2+d)^(1/2)*(a+b*arccosh(c*x))+1/4*d*f^3*x*(-c*x+1)*(c*x+1)*(-c^2*d*x^2+ 
d)^(1/2)*(a+b*arccosh(c*x))+1/2*d*f*g^2*x^3*(-c*x+1)*(c*x+1)*(-c^2*d*x^2+d 
)^(1/2)*(a+b*arccosh(c*x))-3/5*d*f^2*g*(-c*x+1)^2*(c*x+1)^2*(-c^2*d*x^2+d) 
^(1/2)*(a+b*arccosh(c*x))/c^2-2/35*d*g^3*(-c*x+1)^2*(c*x+1)^2*(-c^2*d*x^2+ 
d)^(1/2)*(a+b*arccosh(c*x))/c^4-1/7*d*g^3*x^2*(-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^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)-3/32*d*f*g^2*(-c^2*d*x^...
 

Mathematica [A] (warning: unable to verify)

Time = 3.28 (sec) , antiderivative size = 901, normalized size of antiderivative = 0.88 \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx =\text {Too large to display} \] Input:

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

Output:

(-5040*a*d*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(32*g^ 
3 + c^2*g*(336*f^2 + 105*f*g*x + 16*g^2*x^2) + 4*c^6*x^3*(35*f^3 + 84*f^2* 
g*x + 70*f*g^2*x^2 + 20*g^3*x^3) - 2*c^4*x*(175*f^3 + 336*f^2*g*x + 245*f* 
g^2*x^2 + 64*g^3*x^3)) - 529200*a*c*d^(3/2)*f*(2*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))] + 235200*b*c^2*d*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]]) - 3 
52800*b*c^3*d*f^3*Sqrt[d - c^2*d*x^2]*(Cosh[2*ArcCosh[c*x]] + 2*ArcCosh[c* 
x]*(ArcCosh[c*x] - Sinh[2*ArcCosh[c*x]])) + 22050*b*c^3*d*f^3*Sqrt[d - c^2 
*d*x^2]*(8*ArcCosh[c*x]^2 + Cosh[4*ArcCosh[c*x]] - 4*ArcCosh[c*x]*Sinh[4*A 
rcCosh[c*x]]) - 66150*b*c*d*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]]) - 2352*b*c^2*d 
*f^2*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*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]]) 
+ 784*b*d*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]]) - 3675*b*c*d*f*g^2*Sqrt[d - c^2*d*x^2]*(-72*ArcCosh[c*x]^2 + 18*Cos 
h[2*ArcCosh[c*x]] - 9*Cosh[4*ArcCosh[c*x]] - 2*Cosh[6*ArcCosh[c*x]] + 1...
 

Rubi [A] (verified)

Time = 2.58 (sec) , antiderivative size = 538, 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.

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

\(\Big \downarrow \) 6387

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

\(\Big \downarrow \) 6390

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {d \sqrt {d-c^2 d x^2} \left (\frac {2 g^3 (c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x))}{35 c^4}+\frac {3 f g^2 (a+b \text {arccosh}(c x))^2}{32 b c^3}+\frac {3 f^2 g (c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x))}{5 c^2}+\frac {3 f g^2 x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{16 c^2}+\frac {g^3 x^2 (c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x))}{7 c^2}+\frac {1}{4} f^3 x (c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))-\frac {3}{8} f^3 x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))+\frac {3 f^3 (a+b \text {arccosh}(c x))^2}{16 b c}+\frac {1}{2} f g^2 x^3 (c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))-\frac {3}{8} f g^2 x^3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))-\frac {1}{16} b c^3 f^3 x^4-\frac {3}{25} b c^3 f^2 g x^5-\frac {1}{12} b c^3 f g^2 x^6-\frac {1}{49} b c^3 g^3 x^7-\frac {2 b g^3 x}{35 c^3}+\frac {5}{16} b c f^3 x^2+\frac {2}{5} b c f^2 g x^3-\frac {3 b f^2 g x}{5 c}+\frac {7}{32} b c f g^2 x^4-\frac {3 b f g^2 x^2}{32 c}+\frac {8}{175} b c g^3 x^5-\frac {b g^3 x^3}{105 c}\right )}{\sqrt {c x-1} \sqrt {c x+1}}\)

Input:

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

Output:

-((d*Sqrt[d - c^2*d*x^2]*((-3*b*f^2*g*x)/(5*c) - (2*b*g^3*x)/(35*c^3) + (5 
*b*c*f^3*x^2)/16 - (3*b*f*g^2*x^2)/(32*c) + (2*b*c*f^2*g*x^3)/5 - (b*g^3*x 
^3)/(105*c) - (b*c^3*f^3*x^4)/16 + (7*b*c*f*g^2*x^4)/32 - (3*b*c^3*f^2*g*x 
^5)/25 + (8*b*c*g^3*x^5)/175 - (b*c^3*f*g^2*x^6)/12 - (b*c^3*g^3*x^7)/49 - 
 (3*f^3*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/8 + (3*f*g^2* 
x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(16*c^2) - (3*f*g^2*x 
^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/8 + (f^3*x*(-1 + c*x 
)^(3/2)*(1 + c*x)^(3/2)*(a + b*ArcCosh[c*x]))/4 + (f*g^2*x^3*(-1 + c*x)^(3 
/2)*(1 + c*x)^(3/2)*(a + b*ArcCosh[c*x]))/2 + (3*f^2*g*(-1 + c*x)^(5/2)*(1 
 + c*x)^(5/2)*(a + b*ArcCosh[c*x]))/(5*c^2) + (2*g^3*(-1 + c*x)^(5/2)*(1 + 
 c*x)^(5/2)*(a + b*ArcCosh[c*x]))/(35*c^4) + (g^3*x^2*(-1 + c*x)^(5/2)*(1 
+ c*x)^(5/2)*(a + b*ArcCosh[c*x]))/(7*c^2) + (3*f^3*(a + b*ArcCosh[c*x])^2 
)/(16*b*c) + (3*f*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. \(2251\) vs. \(2(882)=1764\).

Time = 0.77 (sec) , antiderivative size = 2252, normalized size of antiderivative = 2.19

method	result	size
default	\(\text {Expression too large to display}\)	\(2252\)
parts	\(\text {Expression too large to display}\)	\(2252\)

Input:

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

Output:

a*(f^3*(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*(-1/7*x^ 
2*(-c^2*d*x^2+d)^(5/2)/c^2/d-2/35/d/c^4*(-c^2*d*x^2+d)^(5/2))+3*f*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)))))-3/5*f^2*g*(-c^2*d*x^2+d)^(5/2)/c^2/d)+b*(-3/32*(-d*(c 
^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/c^3*arccosh(c*x)^2*f*(2*c^2*f 
^2+g^2)*d-1/6272*(-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)*g^3*(-1+7*arccosh(c*x))*d/(c*x+1)/c^4/(c*x-1)-1/768*(- 
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))*f*g^2*(-1+6*arc 
cosh(c*x))*d/(c*x+1)/c^3/(c*x-1)-1/3200*(-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*(60*arcc 
osh(c*x)*c^2*f^2-12*c^2*f^2-5*arccosh(c*x)*g^2+g^2)*d/(c*x+1)/c^4/(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...
 

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

integrate((g*x+f)^3*(-c^2*d*x^2+d)^(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^3 - 1/35*(5*(-c^2*d*x^2 + d)^(5/2)*x^2/(c^2*d) + 2*(-c^2 
*d*x^2 + d)^(5/2)/(c^4*d))*a*g^3 + 1/16*a*f*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)*arcsin(c*x)/c^3) - 3/5*(-c^2*d*x^2 + d)^(5/2)*a*f^2*g/(c^2*d 
) + integrate((-c^2*d*x^2 + d)^(3/2)*b*g^3*x^3*log(c*x + sqrt(c*x + 1)*sqr 
t(c*x - 1)) + 3*(-c^2*d*x^2 + d)^(3/2)*b*f*g^2*x^2*log(c*x + sqrt(c*x + 1) 
*sqrt(c*x - 1)) + 3*(-c^2*d*x^2 + d)^(3/2)*b*f^2*g*x*log(c*x + sqrt(c*x + 
1)*sqrt(c*x - 1)) + (-c^2*d*x^2 + d)^(3/2)*b*f^3*log(c*x + sqrt(c*x + 1)*s 
qrt(c*x - 1)), x)
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\frac {\sqrt {d}\, d \left (210 \mathit {asin} \left (c x \right ) a \,c^{3} f^{3}+105 \mathit {asin} \left (c x \right ) a c f \,g^{2}-140 \sqrt {-c^{2} x^{2}+1}\, a \,c^{6} f^{3} x^{3}-336 \sqrt {-c^{2} x^{2}+1}\, a \,c^{6} f^{2} g \,x^{4}-280 \sqrt {-c^{2} x^{2}+1}\, a \,c^{6} f \,g^{2} x^{5}-80 \sqrt {-c^{2} x^{2}+1}\, a \,c^{6} g^{3} x^{6}+350 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} f^{3} x +672 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} f^{2} g \,x^{2}+490 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} f \,g^{2} x^{3}+128 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} g^{3} x^{4}-336 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} f^{2} g -105 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} f \,g^{2} x -16 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} g^{3} x^{2}-32 \sqrt {-c^{2} x^{2}+1}\, a \,g^{3}-560 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{5}d x \right ) b \,c^{6} g^{3}-1680 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{4}d x \right ) b \,c^{6} f \,g^{2}-1680 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{3}d x \right ) b \,c^{6} f^{2} g +560 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{3}d x \right ) b \,c^{4} g^{3}-560 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{2}d x \right ) b \,c^{6} f^{3}+1680 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{2}d x \right ) b \,c^{4} f \,g^{2}+1680 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x d x \right ) b \,c^{4} f^{2} g +560 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )d x \right ) b \,c^{4} f^{3}+336 a \,c^{2} f^{2} g +32 a \,g^{3}\right )}{560 c^{4}} \] Input:

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

Output:

(sqrt(d)*d*(210*asin(c*x)*a*c**3*f**3 + 105*asin(c*x)*a*c*f*g**2 - 140*sqr 
t( - c**2*x**2 + 1)*a*c**6*f**3*x**3 - 336*sqrt( - c**2*x**2 + 1)*a*c**6*f 
**2*g*x**4 - 280*sqrt( - c**2*x**2 + 1)*a*c**6*f*g**2*x**5 - 80*sqrt( - c* 
*2*x**2 + 1)*a*c**6*g**3*x**6 + 350*sqrt( - c**2*x**2 + 1)*a*c**4*f**3*x + 
 672*sqrt( - c**2*x**2 + 1)*a*c**4*f**2*g*x**2 + 490*sqrt( - c**2*x**2 + 1 
)*a*c**4*f*g**2*x**3 + 128*sqrt( - c**2*x**2 + 1)*a*c**4*g**3*x**4 - 336*s 
qrt( - c**2*x**2 + 1)*a*c**2*f**2*g - 105*sqrt( - c**2*x**2 + 1)*a*c**2*f* 
g**2*x - 16*sqrt( - c**2*x**2 + 1)*a*c**2*g**3*x**2 - 32*sqrt( - c**2*x**2 
 + 1)*a*g**3 - 560*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)*x**5,x)*b*c**6*g* 
*3 - 1680*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)*x**4,x)*b*c**6*f*g**2 - 16 
80*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)*x**3,x)*b*c**6*f**2*g + 560*int(s 
qrt( - c**2*x**2 + 1)*acosh(c*x)*x**3,x)*b*c**4*g**3 - 560*int(sqrt( - c** 
2*x**2 + 1)*acosh(c*x)*x**2,x)*b*c**6*f**3 + 1680*int(sqrt( - c**2*x**2 + 
1)*acosh(c*x)*x**2,x)*b*c**4*f*g**2 + 1680*int(sqrt( - c**2*x**2 + 1)*acos 
h(c*x)*x,x)*b*c**4*f**2*g + 560*int(sqrt( - c**2*x**2 + 1)*acosh(c*x),x)*b 
*c**4*f**3 + 336*a*c**2*f**2*g + 32*a*g**3))/(560*c**4)