\(\int (f+g x)^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx\) [34]

Optimal result

Integrand size = 30, antiderivative size = 620 \[ \int (f+g x)^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {b f^2 g x \sqrt {d+c^2 d x^2}}{c \sqrt {1+c^2 x^2}}+\frac {2 b g^3 x \sqrt {d+c^2 d x^2}}{15 c^3 \sqrt {1+c^2 x^2}}-\frac {b c f^3 x^2 \sqrt {d+c^2 d x^2}}{4 \sqrt {1+c^2 x^2}}-\frac {3 b f g^2 x^2 \sqrt {d+c^2 d x^2}}{16 c \sqrt {1+c^2 x^2}}-\frac {b c f^2 g x^3 \sqrt {d+c^2 d x^2}}{3 \sqrt {1+c^2 x^2}}-\frac {b g^3 x^3 \sqrt {d+c^2 d x^2}}{45 c \sqrt {1+c^2 x^2}}-\frac {3 b c f g^2 x^4 \sqrt {d+c^2 d x^2}}{16 \sqrt {1+c^2 x^2}}-\frac {b c g^3 x^5 \sqrt {d+c^2 d x^2}}{25 \sqrt {1+c^2 x^2}}+\frac {1}{2} f^3 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {3 f g^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{8 c^2}+\frac {3}{4} f g^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {f^2 g \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{c^2 d}-\frac {g^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^4 d}+\frac {g^3 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4 d^2}+\frac {f^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{4 b c \sqrt {1+c^2 x^2}}-\frac {3 f g^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{16 b c^3 \sqrt {1+c^2 x^2}} \] Output:

-b*f^2*g*x*(c^2*d*x^2+d)^(1/2)/c/(c^2*x^2+1)^(1/2)+2/15*b*g^3*x*(c^2*d*x^2 
+d)^(1/2)/c^3/(c^2*x^2+1)^(1/2)-1/4*b*c*f^3*x^2*(c^2*d*x^2+d)^(1/2)/(c^2*x 
^2+1)^(1/2)-3/16*b*f*g^2*x^2*(c^2*d*x^2+d)^(1/2)/c/(c^2*x^2+1)^(1/2)-1/3*b 
*c*f^2*g*x^3*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-1/45*b*g^3*x^3*(c^2*d*x 
^2+d)^(1/2)/c/(c^2*x^2+1)^(1/2)-3/16*b*c*f*g^2*x^4*(c^2*d*x^2+d)^(1/2)/(c^ 
2*x^2+1)^(1/2)-1/25*b*c*g^3*x^5*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)+1/2* 
f^3*x*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))+3/8*f*g^2*x*(c^2*d*x^2+d)^(1/ 
2)*(a+b*arcsinh(c*x))/c^2+3/4*f*g^2*x^3*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c 
*x))+f^2*g*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))/c^2/d-1/3*g^3*(c^2*d*x^2 
+d)^(3/2)*(a+b*arcsinh(c*x))/c^4/d+1/5*g^3*(c^2*d*x^2+d)^(5/2)*(a+b*arcsin 
h(c*x))/c^4/d^2+1/4*f^3*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2/b/c/(c^2* 
x^2+1)^(1/2)-3/16*f*g^2*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2/b/c^3/(c^ 
2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.72 (sec) , antiderivative size = 410, normalized size of antiderivative = 0.66 \[ \int (f+g x)^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {240 a d \left (1+c^2 x^2\right )^{3/2} \left (-16 g^3+c^2 g \left (120 f^2+45 f g x+8 g^2 x^2\right )+6 c^4 x \left (10 f^3+20 f^2 g x+15 f g^2 x^2+4 g^3 x^3\right )\right )-9600 b c^2 d f^2 g \left (3 c x+4 c^3 x^3+c^5 x^5-3 \left (1+c^2 x^2\right )^{5/2} \text {arcsinh}(c x)\right )+128 b d g^3 \left (1+c^2 x^2\right ) \left (30 c x-5 c^3 x^3-9 c^5 x^5+15 \sqrt {1+c^2 x^2} \left (-2+c^2 x^2+3 c^4 x^4\right ) \text {arcsinh}(c x)\right )+3600 a c \sqrt {d} f \left (4 c^2 f^2-3 g^2\right ) \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )-3600 b c^3 d f^3 \left (1+c^2 x^2\right ) (\cosh (2 \text {arcsinh}(c x))-2 \text {arcsinh}(c x) (\text {arcsinh}(c x)+\sinh (2 \text {arcsinh}(c x))))-675 b c d f g^2 \left (1+c^2 x^2\right ) \left (8 \text {arcsinh}(c x)^2+\cosh (4 \text {arcsinh}(c x))-4 \text {arcsinh}(c x) \sinh (4 \text {arcsinh}(c x))\right )}{28800 c^4 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}} \] Input:

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

Output:

(240*a*d*(1 + c^2*x^2)^(3/2)*(-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 + 15*f*g^2*x^2 + 4*g^3*x^3)) - 9600*b* 
c^2*d*f^2*g*(3*c*x + 4*c^3*x^3 + c^5*x^5 - 3*(1 + c^2*x^2)^(5/2)*ArcSinh[c 
*x]) + 128*b*d*g^3*(1 + c^2*x^2)*(30*c*x - 5*c^3*x^3 - 9*c^5*x^5 + 15*Sqrt 
[1 + c^2*x^2]*(-2 + c^2*x^2 + 3*c^4*x^4)*ArcSinh[c*x]) + 3600*a*c*Sqrt[d]* 
f*(4*c^2*f^2 - 3*g^2)*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]*Log[c*d*x + Sq 
rt[d]*Sqrt[d + c^2*d*x^2]] - 3600*b*c^3*d*f^3*(1 + c^2*x^2)*(Cosh[2*ArcSin 
h[c*x]] - 2*ArcSinh[c*x]*(ArcSinh[c*x] + Sinh[2*ArcSinh[c*x]])) - 675*b*c* 
d*f*g^2*(1 + c^2*x^2)*(8*ArcSinh[c*x]^2 + Cosh[4*ArcSinh[c*x]] - 4*ArcSinh 
[c*x]*Sinh[4*ArcSinh[c*x]]))/(28800*c^4*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x 
^2])
 

Rubi [A] (verified)

Time = 1.19 (sec) , antiderivative size = 363, normalized size of antiderivative = 0.59, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6260, 6253, 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.

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

\(\Big \downarrow \) 6260

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

\(\Big \downarrow \) 6253

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

\(\Big \downarrow \) 2009

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

Input:

Int[(f + g*x)^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[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^2*x^2]*(a + b* 
ArcSinh[c*x]))/2 + (3*f*g^2*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(8*c 
^2) + (3*f*g^2*x^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/4 + (f^2*g*(1 + 
 c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/c^2 - (g^3*(1 + c^2*x^2)^(3/2)*(a + 
b*ArcSinh[c*x]))/(3*c^4) + (g^3*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/ 
(5*c^4) + (f^3*(a + b*ArcSinh[c*x])^2)/(4*b*c) - (3*f*g^2*(a + b*ArcSinh[c 
*x])^2)/(16*b*c^3)))/Sqrt[1 + c^2*x^2]
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6253

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d 
_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^p*(a 
+ b*ArcSinh[c*x])^n, (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] 
 && EqQ[e, c^2*d] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d, 0] && IGtQ[n 
, 0] && ((EqQ[n, 1] && GtQ[p, -1]) || GtQ[p, 0] || EqQ[m, 1] || (EqQ[m, 2] 
&& LtQ[p, -2]))
 

rule 6260

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d 
_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[Simp[(d + e*x^2)^p/(1 + c^2*x^2) 
^p]   Int[(f + g*x)^m*(1 + c^2*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] /; Fre 
eQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e, c^2*d] && IntegerQ[m] && IntegerQ 
[p - 1/2] &&  !GtQ[d, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1308\) vs. \(2(540)=1080\).

Time = 1.32 (sec) , antiderivative size = 1309, normalized size of antiderivative = 2.11

method	result	size
default	\(\text {Expression too large to display}\)	\(1309\)
parts	\(\text {Expression too large to display}\)	\(1309\)

Input:

int((g*x+f)^3*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(x*c)),x,method=_RETURNVERBO 
SE)
 

Output:

a*(f^3*(1/2*x*(c^2*d*x^2+d)^(1/2)+1/2*d*ln(x*c^2*d/(c^2*d)^(1/2)+(c^2*d*x^ 
2+d)^(1/2))/(c^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*ln(x*c^2*d/(c^2*d)^(1/2)+(c^2*d*x^2+d)^(1/ 
2))/(c^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)*f*arcsinh(x*c)^2*(4*c^2*f^2-3*g^2)/(c^2*x^2+1)^(1/2)/c^3+1/800*(d 
*(c^2*x^2+1))^(1/2)*(16*c^6*x^6+16*(c^2*x^2+1)^(1/2)*x^5*c^5+28*c^4*x^4+20 
*(c^2*x^2+1)^(1/2)*c^3*x^3+13*c^2*x^2+5*(c^2*x^2+1)^(1/2)*x*c+1)*g^3*(-1+5 
*arcsinh(x*c))/c^4/(c^2*x^2+1)+3/256*(d*(c^2*x^2+1))^(1/2)*(8*x^5*c^5+8*x^ 
4*c^4*(c^2*x^2+1)^(1/2)+12*x^3*c^3+8*x^2*c^2*(c^2*x^2+1)^(1/2)+4*x*c+(c^2* 
x^2+1)^(1/2))*f*g^2*(-1+4*arcsinh(x*c))/c^3/(c^2*x^2+1)+1/288*(d*(c^2*x^2+ 
1))^(1/2)*(4*c^4*x^4+4*(c^2*x^2+1)^(1/2)*c^3*x^3+5*c^2*x^2+3*(c^2*x^2+1)^( 
1/2)*x*c+1)*g*(36*arcsinh(x*c)*c^2*f^2-12*c^2*f^2-3*arcsinh(x*c)*g^2+g^2)/ 
c^4/(c^2*x^2+1)+1/16*(d*(c^2*x^2+1))^(1/2)*(2*x^3*c^3+2*x^2*c^2*(c^2*x^2+1 
)^(1/2)+2*x*c+(c^2*x^2+1)^(1/2))*f^3*(-1+2*arcsinh(x*c))/c/(c^2*x^2+1)+1/1 
6*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2+(c^2*x^2+1)^(1/2)*x*c+1)*g*(6*arcsinh(x*c 
)*c^2*f^2-6*c^2*f^2-arcsinh(x*c)*g^2+g^2)/c^4/(c^2*x^2+1)+1/16*(d*(c^2*x^2 
+1))^(1/2)*(c^2*x^2-(c^2*x^2+1)^(1/2)*x*c+1)*g*(6*arcsinh(x*c)*c^2*f^2+6*c 
^2*f^2-arcsinh(x*c)*g^2-g^2)/c^4/(c^2*x^2+1)+1/16*(d*(c^2*x^2+1))^(1/2)*(2 
*x^3*c^3-2*x^2*c^2*(c^2*x^2+1)^(1/2)+2*x*c-(c^2*x^2+1)^(1/2))*f^3*(1+2*...
 

Fricas [F]

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

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

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)*arcsinh(c*x))*sqrt(c^2*d*x^2 + d), x)
 

Sympy [F]

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

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

Output:

Integral(sqrt(d*(c**2*x**2 + 1))*(a + b*asinh(c*x))*(f + g*x)**3, x)
 

Maxima [F(-2)]

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

integrate((g*x+f)^3*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x)),x, algorithm="m 
axima")
 

Output:

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati 
ve exponent.
 

Giac [F(-2)]

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

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

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. \[ \int (f+g x)^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\int {\left (f+g\,x\right )}^3\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\sqrt {d\,c^2\,x^2+d} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(d)*(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)*asinh(c* 
x)*x**3,x)*b*c**4*g**3 + 360*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x**2,x)*b* 
c**4*f*g**2 + 360*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x,x)*b*c**4*f**2*g + 
120*int(sqrt(c**2*x**2 + 1)*asinh(c*x),x)*b*c**4*f**3 + 60*log(sqrt(c**2*x 
**2 + 1) + c*x)*a*c**3*f**3 - 45*log(sqrt(c**2*x**2 + 1) + c*x)*a*c*f*g**2 
))/(120*c**4)