\(\int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{f+g x} \, dx\) [37]

Optimal result

Integrand size = 30, antiderivative size = 664 \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{f+g x} \, dx=\frac {a \sqrt {d+c^2 d x^2}}{g}-\frac {b c x \sqrt {d+c^2 d x^2}}{g \sqrt {1+c^2 x^2}}+\frac {b \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{g}-\frac {c x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b g \sqrt {1+c^2 x^2}}-\frac {\left (1+\frac {c^2 f^2}{g^2}\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x) \sqrt {1+c^2 x^2}}+\frac {\sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x)}-\frac {a \sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2} \text {arctanh}\left (\frac {g-c^2 f x}{\sqrt {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}\right )}{g^2 \sqrt {1+c^2 x^2}}+\frac {b \sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2} \text {arcsinh}(c x) \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{g^2 \sqrt {1+c^2 x^2}}-\frac {b \sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2} \text {arcsinh}(c x) \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{g^2 \sqrt {1+c^2 x^2}}+\frac {b \sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{g^2 \sqrt {1+c^2 x^2}}-\frac {b \sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{g^2 \sqrt {1+c^2 x^2}} \] Output:

a*(c^2*d*x^2+d)^(1/2)/g-b*c*x*(c^2*d*x^2+d)^(1/2)/g/(c^2*x^2+1)^(1/2)+b*(c 
^2*d*x^2+d)^(1/2)*arcsinh(c*x)/g-1/2*c*x*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh( 
c*x))^2/b/g/(c^2*x^2+1)^(1/2)-1/2*(1+c^2*f^2/g^2)*(c^2*d*x^2+d)^(1/2)*(a+b 
*arcsinh(c*x))^2/b/c/(g*x+f)/(c^2*x^2+1)^(1/2)+1/2*(c^2*x^2+1)^(1/2)*(c^2* 
d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2/b/c/(g*x+f)-a*(c^2*f^2+g^2)^(1/2)*(c^2 
*d*x^2+d)^(1/2)*arctanh((-c^2*f*x+g)/(c^2*f^2+g^2)^(1/2)/(c^2*x^2+1)^(1/2) 
)/g^2/(c^2*x^2+1)^(1/2)+b*(c^2*f^2+g^2)^(1/2)*(c^2*d*x^2+d)^(1/2)*arcsinh( 
c*x)*ln(1+(c*x+(c^2*x^2+1)^(1/2))*g/(c*f-(c^2*f^2+g^2)^(1/2)))/g^2/(c^2*x^ 
2+1)^(1/2)-b*(c^2*f^2+g^2)^(1/2)*(c^2*d*x^2+d)^(1/2)*arcsinh(c*x)*ln(1+(c* 
x+(c^2*x^2+1)^(1/2))*g/(c*f+(c^2*f^2+g^2)^(1/2)))/g^2/(c^2*x^2+1)^(1/2)+b* 
(c^2*f^2+g^2)^(1/2)*(c^2*d*x^2+d)^(1/2)*polylog(2,-(c*x+(c^2*x^2+1)^(1/2)) 
*g/(c*f-(c^2*f^2+g^2)^(1/2)))/g^2/(c^2*x^2+1)^(1/2)-b*(c^2*f^2+g^2)^(1/2)* 
(c^2*d*x^2+d)^(1/2)*polylog(2,-(c*x+(c^2*x^2+1)^(1/2))*g/(c*f+(c^2*f^2+g^2 
)^(1/2)))/g^2/(c^2*x^2+1)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 4.50 (sec) , antiderivative size = 1358, normalized size of antiderivative = 2.05 \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{f+g x} \, dx =\text {Too large to display} \] Input:

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

Output:

(2*a*g*Sqrt[d + c^2*d*x^2] + 2*a*Sqrt[d]*Sqrt[c^2*f^2 + g^2]*Log[f + g*x] 
- 2*a*c*Sqrt[d]*f*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] - 2*a*Sqrt[d]*S 
qrt[c^2*f^2 + g^2]*Log[d*(g - c^2*f*x) + Sqrt[d]*Sqrt[c^2*f^2 + g^2]*Sqrt[ 
d + c^2*d*x^2]] + b*Sqrt[d + c^2*d*x^2]*((-2*c*g*x)/Sqrt[1 + c^2*x^2] + 2* 
g*ArcSinh[c*x] - (c*f*ArcSinh[c*x]^2)/Sqrt[1 + c^2*x^2] + (2*((-I)*c*f + g 
)*(I*c*f + g)*(((-I)*Pi*ArcTanh[(-g + c*f*Tanh[ArcSinh[c*x]/2])/Sqrt[c^2*f 
^2 + g^2]])/Sqrt[c^2*f^2 + g^2] - (2*ArcCos[((-I)*c*f)/g]*ArcTanh[((c*f + 
I*g)*Cot[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]] + (Pi - (2* 
I)*ArcSinh[c*x])*ArcTanh[((c*f - I*g)*Tan[(Pi + (2*I)*ArcSinh[c*x])/4])/Sq 
rt[-(c^2*f^2) - g^2]] + (ArcCos[((-I)*c*f)/g] - (2*I)*ArcTanh[((c*f + I*g) 
*Cot[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]] - (2*I)*ArcTanh 
[((c*f - I*g)*Tan[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]])*L 
og[((1/2 - I/2)*Sqrt[-(c^2*f^2) - g^2])/(E^(ArcSinh[c*x]/2)*Sqrt[(-I)*g]*S 
qrt[c*(f + g*x)])] + (ArcCos[((-I)*c*f)/g] + (2*I)*(ArcTanh[((c*f + I*g)*C 
ot[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]] + ArcTanh[((c*f - 
 I*g)*Tan[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]]))*Log[((1/ 
2 + I/2)*E^(ArcSinh[c*x]/2)*Sqrt[-(c^2*f^2) - g^2])/(Sqrt[(-I)*g]*Sqrt[c*( 
f + g*x)])] - (ArcCos[((-I)*c*f)/g] + (2*I)*ArcTanh[((c*f + I*g)*Cot[(Pi + 
 (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]])*Log[((I*c*f + g)*((-I)*c 
*f + g + Sqrt[-(c^2*f^2) - g^2])*(1 + I*Cot[(Pi + (2*I)*ArcSinh[c*x])/4...
 

Rubi [A] (verified)

Time = 2.57 (sec) , antiderivative size = 461, normalized size of antiderivative = 0.69, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {6260, 6254, 25, 6250, 25, 6271, 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.

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

\(\Big \downarrow \) 6260

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

\(\Big \downarrow \) 6254

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 6250

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 6271

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {c^2 d x^2+d} \left (\frac {2 b c \left (-\frac {a \sqrt {c^2 f^2+g^2} \text {arctanh}\left (\frac {g-c^2 f x}{\sqrt {c^2 x^2+1} \sqrt {c^2 f^2+g^2}}\right )}{g^2}+\frac {a \sqrt {c^2 x^2+1}}{g}+\frac {b \sqrt {c^2 f^2+g^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{g^2}-\frac {b \sqrt {c^2 f^2+g^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{g^2}+\frac {b \text {arcsinh}(c x) \sqrt {c^2 f^2+g^2} \log \left (\frac {g e^{\text {arcsinh}(c x)}}{c f-\sqrt {c^2 f^2+g^2}}+1\right )}{g^2}-\frac {b \text {arcsinh}(c x) \sqrt {c^2 f^2+g^2} \log \left (\frac {g e^{\text {arcsinh}(c x)}}{\sqrt {c^2 f^2+g^2}+c f}+1\right )}{g^2}+\frac {b \sqrt {c^2 x^2+1} \text {arcsinh}(c x)}{g}-\frac {b c x}{g}\right )-\frac {\left (\frac {c^2 f^2}{g^2}+1\right ) (a+b \text {arcsinh}(c x))^2}{f+g x}-\frac {c^2 x (a+b \text {arcsinh}(c x))^2}{g}}{2 b c}+\frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x)}\right )}{\sqrt {c^2 x^2+1}}\)

Input:

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

Output:

(Sqrt[d + c^2*d*x^2]*(((1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/(2*b*c*(f + g 
*x)) + (-((c^2*x*(a + b*ArcSinh[c*x])^2)/g) - ((1 + (c^2*f^2)/g^2)*(a + b* 
ArcSinh[c*x])^2)/(f + g*x) + 2*b*c*(-((b*c*x)/g) + (a*Sqrt[1 + c^2*x^2])/g 
 + (b*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/g - (a*Sqrt[c^2*f^2 + g^2]*ArcTanh[( 
g - c^2*f*x)/(Sqrt[c^2*f^2 + g^2]*Sqrt[1 + c^2*x^2])])/g^2 + (b*Sqrt[c^2*f 
^2 + g^2]*ArcSinh[c*x]*Log[1 + (E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^ 
2])])/g^2 - (b*Sqrt[c^2*f^2 + g^2]*ArcSinh[c*x]*Log[1 + (E^ArcSinh[c*x]*g) 
/(c*f + Sqrt[c^2*f^2 + g^2])])/g^2 + (b*Sqrt[c^2*f^2 + g^2]*PolyLog[2, -(( 
E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2]))])/g^2 - (b*Sqrt[c^2*f^2 + g 
^2]*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))])/g^2))/( 
2*b*c)))/Sqrt[1 + c^2*x^2]
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2009

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

rule 6250

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*((f_.) + (g_.)*(x_) + (h_.)*( 
x_)^2)^(p_.))/((d_) + (e_.)*(x_))^2, x_Symbol] :> With[{u = IntHide[(f + g* 
x + h*x^2)^p/(d + e*x)^2, x]}, Simp[(a + b*ArcSinh[c*x])^n   u, x] - Simp[b 
*c*n   Int[SimplifyIntegrand[u*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x 
^2]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && IGtQ[n, 0] && IG 
tQ[p, 0] && EqQ[e*g - 2*d*h, 0]
 

rule 6254

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.) + (g_.)*(x_))^(m_)*Sqr 
t[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(f + g*x)^m*(d + e*x^2)*((a + b*A 
rcSinh[c*x])^(n + 1)/(b*c*Sqrt[d]*(n + 1))), x] - Simp[1/(b*c*Sqrt[d]*(n + 
1))   Int[(d*g*m + 2*e*f*x + e*g*(m + 2)*x^2)*(f + g*x)^(m - 1)*(a + b*ArcS 
inh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[e, c^2* 
d] && ILtQ[m, 0] && GtQ[d, 0] && IGtQ[n, 0]
 

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]
 

rule 6271

Int[(ArcSinh[(c_.)*(x_)]*(b_.) + (a_))^(n_.)*(RFx_)*((d_) + (e_.)*(x_)^2)^( 
p_), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^p, RFx*(a + b*ArcSinh[c*x 
])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && RationalFunctionQ[RFx, x] && I 
GtQ[n, 0] && EqQ[e, c^2*d] && IntegerQ[p - 1/2]
 

Maple [A] (verified)

Time = 1.54 (sec) , antiderivative size = 747, normalized size of antiderivative = 1.12

method	result	size
default	\(\frac {a \left (\sqrt {\left (x +\frac {f}{g}\right )^{2} c^{2} d -\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+\frac {d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}}-\frac {c^{2} d f \ln \left (\frac {-\frac {c^{2} d f}{g}+\left (x +\frac {f}{g}\right ) c^{2} d}{\sqrt {c^{2} d}}+\sqrt {\left (x +\frac {f}{g}\right )^{2} c^{2} d -\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+\frac {d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}}\right )}{g \sqrt {c^{2} d}}-\frac {d \left (c^{2} f^{2}+g^{2}\right ) \ln \left (\frac {\frac {2 d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}-\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+2 \sqrt {\frac {d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}}\, \sqrt {\left (x +\frac {f}{g}\right )^{2} c^{2} d -\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+\frac {d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}}}{x +\frac {f}{g}}\right )}{g^{2} \sqrt {\frac {d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}}}\right )}{g}+b \left (-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, f \operatorname {arcsinh}\left (x c \right )^{2} c}{2 \sqrt {c^{2} x^{2}+1}\, g^{2}}+\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 ) \left (\operatorname {arcsinh}\left (x c \right )-1\right )}{2 \left (c^{2} x^{2}+1\right ) g}+\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 ) \left (\operatorname {arcsinh}\left (x c \right )+1\right )}{2 \left (c^{2} x^{2}+1\right ) g}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \sqrt {c^{2} f^{2}+g^{2}}\, \left (\operatorname {arcsinh}\left (x c \right ) \ln \left (\frac {-\left (x c +\sqrt {c^{2} x^{2}+1}\right ) g -c f +\sqrt {c^{2} f^{2}+g^{2}}}{-c f +\sqrt {c^{2} f^{2}+g^{2}}}\right )-\operatorname {arcsinh}\left (x c \right ) \ln \left (\frac {\left (x c +\sqrt {c^{2} x^{2}+1}\right ) g +c f +\sqrt {c^{2} f^{2}+g^{2}}}{c f +\sqrt {c^{2} f^{2}+g^{2}}}\right )+\operatorname {dilog}\left (\frac {-\left (x c +\sqrt {c^{2} x^{2}+1}\right ) g -c f +\sqrt {c^{2} f^{2}+g^{2}}}{-c f +\sqrt {c^{2} f^{2}+g^{2}}}\right )-\operatorname {dilog}\left (\frac {\left (x c +\sqrt {c^{2} x^{2}+1}\right ) g +c f +\sqrt {c^{2} f^{2}+g^{2}}}{c f +\sqrt {c^{2} f^{2}+g^{2}}}\right )\right )}{\sqrt {c^{2} x^{2}+1}\, g^{2}}\right )\)	\(747\)
parts	\(\frac {a \left (\sqrt {\left (x +\frac {f}{g}\right )^{2} c^{2} d -\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+\frac {d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}}-\frac {c^{2} d f \ln \left (\frac {-\frac {c^{2} d f}{g}+\left (x +\frac {f}{g}\right ) c^{2} d}{\sqrt {c^{2} d}}+\sqrt {\left (x +\frac {f}{g}\right )^{2} c^{2} d -\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+\frac {d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}}\right )}{g \sqrt {c^{2} d}}-\frac {d \left (c^{2} f^{2}+g^{2}\right ) \ln \left (\frac {\frac {2 d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}-\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+2 \sqrt {\frac {d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}}\, \sqrt {\left (x +\frac {f}{g}\right )^{2} c^{2} d -\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+\frac {d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}}}{x +\frac {f}{g}}\right )}{g^{2} \sqrt {\frac {d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}}}\right )}{g}+b \left (-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, f \operatorname {arcsinh}\left (x c \right )^{2} c}{2 \sqrt {c^{2} x^{2}+1}\, g^{2}}+\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 ) \left (\operatorname {arcsinh}\left (x c \right )-1\right )}{2 \left (c^{2} x^{2}+1\right ) g}+\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 ) \left (\operatorname {arcsinh}\left (x c \right )+1\right )}{2 \left (c^{2} x^{2}+1\right ) g}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \sqrt {c^{2} f^{2}+g^{2}}\, \left (\operatorname {arcsinh}\left (x c \right ) \ln \left (\frac {-\left (x c +\sqrt {c^{2} x^{2}+1}\right ) g -c f +\sqrt {c^{2} f^{2}+g^{2}}}{-c f +\sqrt {c^{2} f^{2}+g^{2}}}\right )-\operatorname {arcsinh}\left (x c \right ) \ln \left (\frac {\left (x c +\sqrt {c^{2} x^{2}+1}\right ) g +c f +\sqrt {c^{2} f^{2}+g^{2}}}{c f +\sqrt {c^{2} f^{2}+g^{2}}}\right )+\operatorname {dilog}\left (\frac {-\left (x c +\sqrt {c^{2} x^{2}+1}\right ) g -c f +\sqrt {c^{2} f^{2}+g^{2}}}{-c f +\sqrt {c^{2} f^{2}+g^{2}}}\right )-\operatorname {dilog}\left (\frac {\left (x c +\sqrt {c^{2} x^{2}+1}\right ) g +c f +\sqrt {c^{2} f^{2}+g^{2}}}{c f +\sqrt {c^{2} f^{2}+g^{2}}}\right )\right )}{\sqrt {c^{2} x^{2}+1}\, g^{2}}\right )\)	\(747\)

Input:

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

Output:

a/g*(((x+f/g)^2*c^2*d-2*c^2*d*f/g*(x+f/g)+d*(c^2*f^2+g^2)/g^2)^(1/2)-c^2*d 
*f/g*ln((-c^2*d*f/g+(x+f/g)*c^2*d)/(c^2*d)^(1/2)+((x+f/g)^2*c^2*d-2*c^2*d* 
f/g*(x+f/g)+d*(c^2*f^2+g^2)/g^2)^(1/2))/(c^2*d)^(1/2)-d*(c^2*f^2+g^2)/g^2/ 
(d*(c^2*f^2+g^2)/g^2)^(1/2)*ln((2*d*(c^2*f^2+g^2)/g^2-2*c^2*d*f/g*(x+f/g)+ 
2*(d*(c^2*f^2+g^2)/g^2)^(1/2)*((x+f/g)^2*c^2*d-2*c^2*d*f/g*(x+f/g)+d*(c^2* 
f^2+g^2)/g^2)^(1/2))/(x+f/g)))+b*(-1/2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^( 
1/2)*f*arcsinh(x*c)^2*c/g^2+1/2*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2+(c^2*x^2+1) 
^(1/2)*x*c+1)*(arcsinh(x*c)-1)/(c^2*x^2+1)/g+1/2*(d*(c^2*x^2+1))^(1/2)*(c^ 
2*x^2-(c^2*x^2+1)^(1/2)*x*c+1)*(arcsinh(x*c)+1)/(c^2*x^2+1)/g+(d*(c^2*x^2+ 
1))^(1/2)*(c^2*f^2+g^2)^(1/2)/(c^2*x^2+1)^(1/2)*(arcsinh(x*c)*ln((-(x*c+(c 
^2*x^2+1)^(1/2))*g-c*f+(c^2*f^2+g^2)^(1/2))/(-c*f+(c^2*f^2+g^2)^(1/2)))-ar 
csinh(x*c)*ln(((x*c+(c^2*x^2+1)^(1/2))*g+c*f+(c^2*f^2+g^2)^(1/2))/(c*f+(c^ 
2*f^2+g^2)^(1/2)))+dilog((-(x*c+(c^2*x^2+1)^(1/2))*g-c*f+(c^2*f^2+g^2)^(1/ 
2))/(-c*f+(c^2*f^2+g^2)^(1/2)))-dilog(((x*c+(c^2*x^2+1)^(1/2))*g+c*f+(c^2* 
f^2+g^2)^(1/2))/(c*f+(c^2*f^2+g^2)^(1/2))))/g^2)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-(c*sqrt(d)*f*arcsinh(c*x)/g^2 - sqrt(c^2*d*f^2/g^2 + d)*arcsinh(c*f*x/abs 
(g*x + f) - g/(c*abs(g*x + f)))/g - sqrt(c^2*d*x^2 + d)/g)*a + b*integrate 
(sqrt(c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 + 1))/(g*x + f), x)
 

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(d)*(2*sqrt(c**2*f**2 + g**2)*atan((sqrt(c**2*x**2 + 1)*g*i + c*f*i + 
 c*g*i*x)/sqrt(c**2*f**2 + g**2))*a*i + sqrt(c**2*x**2 + 1)*a*g + int((sqr 
t(c**2*x**2 + 1)*asinh(c*x))/(f + g*x),x)*b*g**2 - log(sqrt(c**2*x**2 + 1) 
 + c*x)*a*c*f))/g**2