\(\int (f+g x) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx\) [36]

Optimal result

Integrand size = 28, antiderivative size = 221 \[ \int (f+g x) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {b g x \sqrt {d+c^2 d x^2}}{3 c \sqrt {1+c^2 x^2}}-\frac {b c f x^2 \sqrt {d+c^2 d x^2}}{4 \sqrt {1+c^2 x^2}}-\frac {b c g x^3 \sqrt {d+c^2 d x^2}}{9 \sqrt {1+c^2 x^2}}+\frac {1}{2} f x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {g \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2 d}+\frac {f \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{4 b c \sqrt {1+c^2 x^2}} \] Output:

-1/3*b*g*x*(c^2*d*x^2+d)^(1/2)/c/(c^2*x^2+1)^(1/2)-1/4*b*c*f*x^2*(c^2*d*x^ 
2+d)^(1/2)/(c^2*x^2+1)^(1/2)-1/9*b*c*g*x^3*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1) 
^(1/2)+1/2*f*x*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))+1/3*g*(c^2*d*x^2+d)^ 
(3/2)*(a+b*arcsinh(c*x))/c^2/d+1/4*f*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x) 
)^2/b/c/(c^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.85 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.94 \[ \int (f+g x) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{6} a \sqrt {d+c^2 d x^2} \left (\frac {2 g}{c^2}+x (3 f+2 g x)\right )-\frac {b g \sqrt {d+c^2 d x^2} \left (3 c x+c^3 x^3-3 \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x)\right )}{9 c^2 \sqrt {1+c^2 x^2}}+\frac {a \sqrt {d} f \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )}{2 c}+\frac {b f \sqrt {d+c^2 d x^2} (-\cosh (2 \text {arcsinh}(c x))+2 \text {arcsinh}(c x) (\text {arcsinh}(c x)+\sinh (2 \text {arcsinh}(c x))))}{8 c \sqrt {1+c^2 x^2}} \] Input:

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

Output:

(a*Sqrt[d + c^2*d*x^2]*((2*g)/c^2 + x*(3*f + 2*g*x)))/6 - (b*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]))/(9*c^2*Sq 
rt[1 + c^2*x^2]) + (a*Sqrt[d]*f*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]])/ 
(2*c) + (b*f*Sqrt[d + c^2*d*x^2]*(-Cosh[2*ArcSinh[c*x]] + 2*ArcSinh[c*x]*( 
ArcSinh[c*x] + Sinh[2*ArcSinh[c*x]])))/(8*c*Sqrt[1 + c^2*x^2])
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.62, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, 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.

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

\(\Big \downarrow \) 6260

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

\(\Big \downarrow \) 6253

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(Sqrt[d + c^2*d*x^2]*(-1/3*(b*g*x)/c - (b*c*f*x^2)/4 - (b*c*g*x^3)/9 + (f* 
x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/2 + (g*(1 + c^2*x^2)^(3/2)*(a + 
b*ArcSinh[c*x]))/(3*c^2) + (f*(a + b*ArcSinh[c*x])^2)/(4*b*c)))/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. \(581\) vs. \(2(189)=378\).

Time = 1.62 (sec) , antiderivative size = 582, normalized size of antiderivative = 2.63

method	result	size
default	\(\frac {a f x \sqrt {c^{2} d \,x^{2}+d}}{2}+\frac {a f d \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 \sqrt {c^{2} d}}+\frac {a g \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, f \operatorname {arcsinh}\left (x c \right )^{2}}{4 \sqrt {c^{2} x^{2}+1}\, c}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}+3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) g \left (-1+3 \,\operatorname {arcsinh}\left (x c \right )\right )}{72 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}+2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c +\sqrt {c^{2} x^{2}+1}\right ) f \left (-1+2 \,\operatorname {arcsinh}\left (x c \right )\right )}{16 \left (c^{2} x^{2}+1\right ) c}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +1\right ) g \left (\operatorname {arcsinh}\left (x c \right )-1\right )}{8 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) g \left (\operatorname {arcsinh}\left (x c \right )+1\right )}{8 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}-2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c -\sqrt {c^{2} x^{2}+1}\right ) f \left (1+2 \,\operatorname {arcsinh}\left (x c \right )\right )}{16 \left (c^{2} x^{2}+1\right ) c}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}-3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) g \left (1+3 \,\operatorname {arcsinh}\left (x c \right )\right )}{72 c^{2} \left (c^{2} x^{2}+1\right )}\right )\)	\(582\)
parts	\(\frac {a f x \sqrt {c^{2} d \,x^{2}+d}}{2}+\frac {a f d \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 \sqrt {c^{2} d}}+\frac {a g \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, f \operatorname {arcsinh}\left (x c \right )^{2}}{4 \sqrt {c^{2} x^{2}+1}\, c}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}+3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) g \left (-1+3 \,\operatorname {arcsinh}\left (x c \right )\right )}{72 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}+2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c +\sqrt {c^{2} x^{2}+1}\right ) f \left (-1+2 \,\operatorname {arcsinh}\left (x c \right )\right )}{16 \left (c^{2} x^{2}+1\right ) c}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +1\right ) g \left (\operatorname {arcsinh}\left (x c \right )-1\right )}{8 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) g \left (\operatorname {arcsinh}\left (x c \right )+1\right )}{8 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}-2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c -\sqrt {c^{2} x^{2}+1}\right ) f \left (1+2 \,\operatorname {arcsinh}\left (x c \right )\right )}{16 \left (c^{2} x^{2}+1\right ) c}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}-3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) g \left (1+3 \,\operatorname {arcsinh}\left (x c \right )\right )}{72 c^{2} \left (c^{2} x^{2}+1\right )}\right )\)	\(582\)

Input:

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

Output:

1/2*a*f*x*(c^2*d*x^2+d)^(1/2)+1/2*a*f*d*ln(x*c^2*d/(c^2*d)^(1/2)+(c^2*d*x^ 
2+d)^(1/2))/(c^2*d)^(1/2)+1/3*a*g*(c^2*d*x^2+d)^(3/2)/c^2/d+b*(1/4*(d*(c^2 
*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c*f*arcsinh(x*c)^2+1/72*(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*(-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*(-1+2*arcs 
inh(x*c))/(c^2*x^2+1)/c+1/8*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2+(c^2*x^2+1)^(1/ 
2)*x*c+1)*g*(arcsinh(x*c)-1)/c^2/(c^2*x^2+1)+1/8*(d*(c^2*x^2+1))^(1/2)*(c^ 
2*x^2-(c^2*x^2+1)^(1/2)*x*c+1)*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*(1+2*arcsinh(x*c))/(c^2*x^2+1)/c+1/72*(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 
*(1+3*arcsinh(x*c))/c^2/(c^2*x^2+1))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Maxima [F(-2)]

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

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

Output:

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

Giac [F(-2)]

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

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

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(d)*(3*sqrt(c**2*x**2 + 1)*a*c**2*f*x + 2*sqrt(c**2*x**2 + 1)*a*c**2* 
g*x**2 + 2*sqrt(c**2*x**2 + 1)*a*g + 6*int(sqrt(c**2*x**2 + 1)*asinh(c*x)* 
x,x)*b*c**2*g + 6*int(sqrt(c**2*x**2 + 1)*asinh(c*x),x)*b*c**2*f + 3*log(s 
qrt(c**2*x**2 + 1) + c*x)*a*c*f))/(6*c**2)