\(\int (f+g x)^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx\) [35]

Optimal result

Integrand size = 30, antiderivative size = 425 \[ \int (f+g x)^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {2 b f g x \sqrt {d+c^2 d x^2}}{3 c \sqrt {1+c^2 x^2}}-\frac {b c f^2 x^2 \sqrt {d+c^2 d x^2}}{4 \sqrt {1+c^2 x^2}}-\frac {b g^2 x^2 \sqrt {d+c^2 d x^2}}{16 c \sqrt {1+c^2 x^2}}-\frac {2 b c f g x^3 \sqrt {d+c^2 d x^2}}{9 \sqrt {1+c^2 x^2}}-\frac {b c g^2 x^4 \sqrt {d+c^2 d x^2}}{16 \sqrt {1+c^2 x^2}}+\frac {1}{2} f^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {g^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{8 c^2}+\frac {1}{4} g^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {2 f g \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2 d}+\frac {f^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{4 b c \sqrt {1+c^2 x^2}}-\frac {g^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{16 b c^3 \sqrt {1+c^2 x^2}} \] Output:

-2/3*b*f*g*x*(c^2*d*x^2+d)^(1/2)/c/(c^2*x^2+1)^(1/2)-1/4*b*c*f^2*x^2*(c^2* 
d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-1/16*b*g^2*x^2*(c^2*d*x^2+d)^(1/2)/c/(c^2 
*x^2+1)^(1/2)-2/9*b*c*f*g*x^3*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-1/16*b 
*c*g^2*x^4*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)+1/2*f^2*x*(c^2*d*x^2+d)^( 
1/2)*(a+b*arcsinh(c*x))+1/8*g^2*x*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))/c 
^2+1/4*g^2*x^3*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))+2/3*f*g*(c^2*d*x^2+d 
)^(3/2)*(a+b*arcsinh(c*x))/c^2/d+1/4*f^2*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh( 
c*x))^2/b/c/(c^2*x^2+1)^(1/2)-1/16*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 = 0.68 (sec) , antiderivative size = 301, normalized size of antiderivative = 0.71 \[ \int (f+g x)^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {48 a c \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (12 c^2 f^2 x+3 g^2 x \left (1+2 c^2 x^2\right )+16 f \left (g+c^2 g x^2\right )\right )-256 b c f g \sqrt {d+c^2 d x^2} \left (3 c x+c^3 x^3-3 \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x)\right )+144 a \sqrt {d} (2 c f-g) (2 c f+g) \sqrt {1+c^2 x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )-144 b c^2 f^2 \sqrt {d+c^2 d x^2} (\cosh (2 \text {arcsinh}(c x))-2 \text {arcsinh}(c x) (\text {arcsinh}(c x)+\sinh (2 \text {arcsinh}(c x))))-9 b g^2 \sqrt {d+c^2 d x^2} \left (8 \text {arcsinh}(c x)^2+\cosh (4 \text {arcsinh}(c x))-4 \text {arcsinh}(c x) \sinh (4 \text {arcsinh}(c x))\right )}{1152 c^3 \sqrt {1+c^2 x^2}} \] Input:

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

Output:

(48*a*c*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]*(12*c^2*f^2*x + 3*g^2*x*(1 + 
 2*c^2*x^2) + 16*f*(g + c^2*g*x^2)) - 256*b*c*f*g*Sqrt[d + c^2*d*x^2]*(3*c 
*x + c^3*x^3 - 3*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]) + 144*a*Sqrt[d]*(2*c*f 
- g)*(2*c*f + g)*Sqrt[1 + c^2*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2] 
] - 144*b*c^2*f^2*Sqrt[d + c^2*d*x^2]*(Cosh[2*ArcSinh[c*x]] - 2*ArcSinh[c* 
x]*(ArcSinh[c*x] + Sinh[2*ArcSinh[c*x]])) - 9*b*g^2*Sqrt[d + c^2*d*x^2]*(8 
*ArcSinh[c*x]^2 + Cosh[4*ArcSinh[c*x]] - 4*ArcSinh[c*x]*Sinh[4*ArcSinh[c*x 
]]))/(1152*c^3*Sqrt[1 + c^2*x^2])
 

Rubi [A] (verified)

Time = 1.35 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.60, 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.

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

\(\Big \downarrow \) 6260

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

\(\Big \downarrow \) 6253

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {c^2 d x^2+d} \left (-\frac {g^2 (a+b \text {arcsinh}(c x))^2}{16 b c^3}+\frac {1}{2} f^2 x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {2 f g \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2}+\frac {g^2 x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{8 c^2}+\frac {1}{4} g^2 x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {f^2 (a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c f^2 x^2-\frac {2}{9} b c f g x^3-\frac {2 b f g x}{3 c}-\frac {1}{16} b c g^2 x^4-\frac {b g^2 x^2}{16 c}\right )}{\sqrt {c^2 x^2+1}}\)

Input:

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

Output:

(Sqrt[d + c^2*d*x^2]*((-2*b*f*g*x)/(3*c) - (b*c*f^2*x^2)/4 - (b*g^2*x^2)/( 
16*c) - (2*b*c*f*g*x^3)/9 - (b*c*g^2*x^4)/16 + (f^2*x*Sqrt[1 + c^2*x^2]*(a 
 + b*ArcSinh[c*x]))/2 + (g^2*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(8* 
c^2) + (g^2*x^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/4 + (2*f*g*(1 + c^ 
2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(3*c^2) + (f^2*(a + b*ArcSinh[c*x])^2)/ 
(4*b*c) - (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. \(912\) vs. \(2(367)=734\).

Time = 1.29 (sec) , antiderivative size = 913, normalized size of antiderivative = 2.15

method	result	size
default	\(a \left (f^{2} \left (\frac {x \sqrt {c^{2} d \,x^{2}+d}}{2}+\frac {d \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 \sqrt {c^{2} d}}\right )+g^{2} \left (\frac {x \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4 c^{2} d}-\frac {\frac {x \sqrt {c^{2} d \,x^{2}+d}}{2}+\frac {d \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 \sqrt {c^{2} d}}}{4 c^{2}}\right )+\frac {2 f g \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}\right )+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2} \left (4 c^{2} f^{2}-g^{2}\right )}{16 \sqrt {c^{2} x^{2}+1}\, c^{3}}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 x^{5} c^{5}+8 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+12 x^{3} c^{3}+8 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+4 x c +\sqrt {c^{2} x^{2}+1}\right ) g^{2} \left (-1+4 \,\operatorname {arcsinh}\left (x c \right )\right )}{256 c^{3} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}+3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) f g \left (-1+3 \,\operatorname {arcsinh}\left (x c \right )\right )}{36 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}+2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c +\sqrt {c^{2} x^{2}+1}\right ) f^{2} \left (-1+2 \,\operatorname {arcsinh}\left (x c \right )\right )}{16 c \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +1\right ) f g \left (\operatorname {arcsinh}\left (x c \right )-1\right )}{4 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) f g \left (\operatorname {arcsinh}\left (x c \right )+1\right )}{4 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}-2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c -\sqrt {c^{2} x^{2}+1}\right ) f^{2} \left (1+2 \,\operatorname {arcsinh}\left (x c \right )\right )}{16 c \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}-3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) f g \left (1+3 \,\operatorname {arcsinh}\left (x c \right )\right )}{36 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 x^{5} c^{5}-8 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+12 x^{3} c^{3}-8 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+4 x c -\sqrt {c^{2} x^{2}+1}\right ) g^{2} \left (1+4 \,\operatorname {arcsinh}\left (x c \right )\right )}{256 c^{3} \left (c^{2} x^{2}+1\right )}\right )\)	\(913\)
parts	\(a \left (f^{2} \left (\frac {x \sqrt {c^{2} d \,x^{2}+d}}{2}+\frac {d \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 \sqrt {c^{2} d}}\right )+g^{2} \left (\frac {x \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4 c^{2} d}-\frac {\frac {x \sqrt {c^{2} d \,x^{2}+d}}{2}+\frac {d \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 \sqrt {c^{2} d}}}{4 c^{2}}\right )+\frac {2 f g \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}\right )+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2} \left (4 c^{2} f^{2}-g^{2}\right )}{16 \sqrt {c^{2} x^{2}+1}\, c^{3}}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 x^{5} c^{5}+8 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+12 x^{3} c^{3}+8 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+4 x c +\sqrt {c^{2} x^{2}+1}\right ) g^{2} \left (-1+4 \,\operatorname {arcsinh}\left (x c \right )\right )}{256 c^{3} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}+3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) f g \left (-1+3 \,\operatorname {arcsinh}\left (x c \right )\right )}{36 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}+2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c +\sqrt {c^{2} x^{2}+1}\right ) f^{2} \left (-1+2 \,\operatorname {arcsinh}\left (x c \right )\right )}{16 c \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +1\right ) f g \left (\operatorname {arcsinh}\left (x c \right )-1\right )}{4 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) f g \left (\operatorname {arcsinh}\left (x c \right )+1\right )}{4 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}-2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c -\sqrt {c^{2} x^{2}+1}\right ) f^{2} \left (1+2 \,\operatorname {arcsinh}\left (x c \right )\right )}{16 c \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}-3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) f g \left (1+3 \,\operatorname {arcsinh}\left (x c \right )\right )}{36 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 x^{5} c^{5}-8 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+12 x^{3} c^{3}-8 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+4 x c -\sqrt {c^{2} x^{2}+1}\right ) g^{2} \left (1+4 \,\operatorname {arcsinh}\left (x c \right )\right )}{256 c^{3} \left (c^{2} x^{2}+1\right )}\right )\)	\(913\)

Input:

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

Output:

a*(f^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))+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)))+2/3*f*g*(c^2*d*x^2+d)^(3/2)/c^2/d)+b*(1/16*(d*(c^2*x^2+ 
1))^(1/2)*arcsinh(x*c)^2*(4*c^2*f^2-g^2)/(c^2*x^2+1)^(1/2)/c^3+1/256*(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))*g^2*(-1+4*arcsinh(x*c))/c^3/ 
(c^2*x^2+1)+1/36*(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)*f*g*(-1+3*arcsinh(x*c))/c^2/(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^2*(-1+2*arcsinh(x*c))/c/(c^2*x^2+1)+1/4*(d*(c^2* 
x^2+1))^(1/2)*(c^2*x^2+(c^2*x^2+1)^(1/2)*x*c+1)*f*g*(arcsinh(x*c)-1)/c^2/( 
c^2*x^2+1)+1/4*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2-(c^2*x^2+1)^(1/2)*x*c+1)*f*g 
*(arcsinh(x*c)+1)/c^2/(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^2*(1+2*arcsinh(x*c))/ 
c/(c^2*x^2+1)+1/36*(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)*f*g*(1+3*arcsinh(x*c))/c^2/(c^2 
*x^2+1)+1/256*(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))*g^2*(1+4* 
arcsinh(x*c))/c^3/(c^2*x^2+1))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

integrate((g*x+f)**2*(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)**2, x)
 

Maxima [F(-2)]

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

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

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

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

Output:

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

Reduce [F]

\[ \int (f+g x)^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\sqrt {d}\, \left (12 \sqrt {c^{2} x^{2}+1}\, a \,c^{3} f^{2} x +16 \sqrt {c^{2} x^{2}+1}\, a \,c^{3} f g \,x^{2}+6 \sqrt {c^{2} x^{2}+1}\, a \,c^{3} g^{2} x^{3}+16 \sqrt {c^{2} x^{2}+1}\, a c f g +3 \sqrt {c^{2} x^{2}+1}\, a c \,g^{2} x +24 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{2}d x \right ) b \,c^{3} g^{2}+48 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x d x \right ) b \,c^{3} f g +24 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )d x \right ) b \,c^{3} f^{2}+12 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a \,c^{2} f^{2}-3 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a \,g^{2}\right )}{24 c^{3}} \] Input:

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

Output:

(sqrt(d)*(12*sqrt(c**2*x**2 + 1)*a*c**3*f**2*x + 16*sqrt(c**2*x**2 + 1)*a* 
c**3*f*g*x**2 + 6*sqrt(c**2*x**2 + 1)*a*c**3*g**2*x**3 + 16*sqrt(c**2*x**2 
 + 1)*a*c*f*g + 3*sqrt(c**2*x**2 + 1)*a*c*g**2*x + 24*int(sqrt(c**2*x**2 + 
 1)*asinh(c*x)*x**2,x)*b*c**3*g**2 + 48*int(sqrt(c**2*x**2 + 1)*asinh(c*x) 
*x,x)*b*c**3*f*g + 24*int(sqrt(c**2*x**2 + 1)*asinh(c*x),x)*b*c**3*f**2 + 
12*log(sqrt(c**2*x**2 + 1) + c*x)*a*c**2*f**2 - 3*log(sqrt(c**2*x**2 + 1) 
+ c*x)*a*g**2))/(24*c**3)