\(\int \frac {(d+c^2 d x^2)^{3/2} (a+b \text {arcsinh}(c x))}{f+g x} \, dx\) [42]

Optimal result

Integrand size = 30, antiderivative size = 974 \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{f+g x} \, dx=\frac {a d \left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2}}{g^3}-\frac {b c d x \sqrt {d+c^2 d x^2}}{3 g \sqrt {1+c^2 x^2}}-\frac {b c d \left (c^2 f^2+g^2\right ) x \sqrt {d+c^2 d x^2}}{g^3 \sqrt {1+c^2 x^2}}+\frac {b c^3 d f x^2 \sqrt {d+c^2 d x^2}}{4 g^2 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^3 \sqrt {d+c^2 d x^2}}{9 g \sqrt {1+c^2 x^2}}+\frac {b d \left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{g^3}-\frac {c^2 d f x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2 g^2}+\frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 g}-\frac {c d f \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{4 b g^2 \sqrt {1+c^2 x^2}}-\frac {c d \left (c^2 f^2+g^2\right ) x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b g^3 \sqrt {1+c^2 x^2}}-\frac {d \left (c^2 f^2+g^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c g^4 (f+g x) \sqrt {1+c^2 x^2}}+\frac {d \left (c^2 f^2+g^2\right ) \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c g^2 (f+g x)}-\frac {a d \left (c^2 f^2+g^2\right )^{3/2} \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^4 \sqrt {1+c^2 x^2}}+\frac {b d \left (c^2 f^2+g^2\right )^{3/2} \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^4 \sqrt {1+c^2 x^2}}-\frac {b d \left (c^2 f^2+g^2\right )^{3/2} \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^4 \sqrt {1+c^2 x^2}}+\frac {b d \left (c^2 f^2+g^2\right )^{3/2} \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^4 \sqrt {1+c^2 x^2}}-\frac {b d \left (c^2 f^2+g^2\right )^{3/2} \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^4 \sqrt {1+c^2 x^2}} \] Output:

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

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 11.88 (sec) , antiderivative size = 2869, normalized size of antiderivative = 2.95 \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{f+g x} \, dx=\text {Result too large to show} \] Input:

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

Output:

(a*d*Sqrt[d + c^2*d*x^2]*(8*g^2 + c^2*(6*f^2 - 3*f*g*x + 2*g^2*x^2)))/(6*g 
^3) + (a*d^(3/2)*(c^2*f^2 + g^2)^(3/2)*Log[f + g*x])/g^4 - (a*c*d^(3/2)*f* 
(2*c^2*f^2 + 3*g^2)*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]])/(2*g^4) - (a 
*d^(3/2)*(c^2*f^2 + g^2)^(3/2)*Log[d*(g - c^2*f*x) + Sqrt[d]*Sqrt[c^2*f^2 
+ g^2]*Sqrt[d + c^2*d*x^2]])/g^4 + (b*d*Sqrt[d + c^2*d*x^2]*((-2*c*g*x)/Sq 
rt[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])/Sqrt[-(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]])*Log[((1/2 - I/2)*Sqrt[-(c^2*f^2) - g^2])/(E^(ArcSinh[c* 
x]/2)*Sqrt[(-I)*g]*Sqrt[c*(f + g*x)])] + (ArcCos[((-I)*c*f)/g] + (2*I)*(Ar 
cTanh[((c*f + I*g)*Cot[(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])/(S 
qrt[(-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]])*L...
 

Rubi [A] (verified)

Time = 2.29 (sec) , antiderivative size = 641, normalized size of antiderivative = 0.66, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6260, 6255, 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.

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

\(\Big \downarrow \) 6260

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

\(\Big \downarrow \) 6255

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(d*Sqrt[d + c^2*d*x^2]*(-1/3*(b*c*x)/g - (b*c*(c^2*f^2 + g^2)*x)/g^3 + (b* 
c^3*f*x^2)/(4*g^2) - (b*c^3*x^3)/(9*g) + (a*(c^2*f^2 + g^2)*Sqrt[1 + c^2*x 
^2])/g^3 + (b*(c^2*f^2 + g^2)*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/g^3 - (c^2*f 
*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(2*g^2) + ((1 + c^2*x^2)^(3/2)* 
(a + b*ArcSinh[c*x]))/(3*g) - (c*f*(a + b*ArcSinh[c*x])^2)/(4*b*g^2) - (c* 
(c^2*f^2 + g^2)*x*(a + b*ArcSinh[c*x])^2)/(2*b*g^3) - ((c^2*f^2 + g^2)^2*( 
a + b*ArcSinh[c*x])^2)/(2*b*c*g^4*(f + g*x)) + ((c^2*f^2 + g^2)*(1 + c^2*x 
^2)*(a + b*ArcSinh[c*x])^2)/(2*b*c*g^2*(f + g*x)) - (a*(c^2*f^2 + g^2)^(3/ 
2)*ArcTanh[(g - c^2*f*x)/(Sqrt[c^2*f^2 + g^2]*Sqrt[1 + c^2*x^2])])/g^4 + ( 
b*(c^2*f^2 + g^2)^(3/2)*ArcSinh[c*x]*Log[1 + (E^ArcSinh[c*x]*g)/(c*f - Sqr 
t[c^2*f^2 + g^2])])/g^4 - (b*(c^2*f^2 + g^2)^(3/2)*ArcSinh[c*x]*Log[1 + (E 
^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2])])/g^4 + (b*(c^2*f^2 + g^2)^(3 
/2)*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2]))])/g^4 - ( 
b*(c^2*f^2 + g^2)^(3/2)*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^ 
2 + g^2]))])/g^4))/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 6255

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

Maple [A] (verified)

Time = 1.45 (sec) , antiderivative size = 1557, normalized size of antiderivative = 1.60

method	result	size
default	\(\text {Expression too large to display}\)	\(1557\)
parts	\(\text {Expression too large to display}\)	\(1557\)

Input:

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

Output:

a/g*(1/3*((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)^(3/2)-c 
^2*d*f/g*(1/4*(2*(x+f/g)*c^2*d-2*c^2*d*f/g)/c^2/d*((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)+1/8*(4*c^2*d^2*(c^2*f^2+g^2)/g^2-4 
*c^4*d^2*f^2/g^2)/c^2/d*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*(((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*(d*(c^2*x^2+1))^(1/2)*d/ 
(c^2*x^2+1)/g^3*arcsinh(x*c)*x^2*c^4*f^2-b*(d*(c^2*x^2+1))^(1/2)*d/(c^2*x^ 
2+1)^(1/2)/g^3*x*c^3*f^2-1/2*b*(d*(c^2*x^2+1))^(1/2)*f*c^4*d/(c^2*x^2+1)/g 
^2*arcsinh(x*c)*x^3+1/4*b*(d*(c^2*x^2+1))^(1/2)*f*c^3*d/(c^2*x^2+1)^(1/2)/ 
g^2*x^2-1/2*b*(d*(c^2*x^2+1))^(1/2)*f*c^2*d/(c^2*x^2+1)/g^2*arcsinh(x*c)*x 
+b*(c^2*f^2+g^2)^(3/2)*d*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/g^4*arcsi 
nh(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)))-b*(c^2*f^2+g^2)^(3/2)*d*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1 
)^(1/2)/g^4*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)))+4/3*b*(d*(c^2*x^2+1))^(1/2)*d/(c^2*x^2...
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{f+g x} \, dx=\frac {\sqrt {d}\, d \left (12 \sqrt {c^{2} f^{2}+g^{2}}\, \mathit {atan} \left (\frac {\sqrt {c^{2} x^{2}+1}\, g i +c f i +c g i x}{\sqrt {c^{2} f^{2}+g^{2}}}\right ) a \,c^{2} f^{2} i +12 \sqrt {c^{2} f^{2}+g^{2}}\, \mathit {atan} \left (\frac {\sqrt {c^{2} x^{2}+1}\, g i +c f i +c g i x}{\sqrt {c^{2} f^{2}+g^{2}}}\right ) a \,g^{2} i +6 \sqrt {c^{2} x^{2}+1}\, a \,c^{2} f^{2} g -3 \sqrt {c^{2} x^{2}+1}\, a \,c^{2} f \,g^{2} x +2 \sqrt {c^{2} x^{2}+1}\, a \,c^{2} g^{3} x^{2}+8 \sqrt {c^{2} x^{2}+1}\, a \,g^{3}+6 \left (\int \frac {\sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{2}}{g x +f}d x \right ) b \,c^{2} g^{4}+6 \left (\int \frac {\sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )}{g x +f}d x \right ) b \,g^{4}-6 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a \,c^{3} f^{3}-9 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a c f \,g^{2}\right )}{6 g^{4}} \] Input:

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

Output:

(sqrt(d)*d*(12*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*c**2*f**2*i + 12*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*g**2*i + 6*sqrt(c**2*x**2 + 1)*a*c**2*f**2*g - 3*sqrt(c**2*x**2 + 1) 
*a*c**2*f*g**2*x + 2*sqrt(c**2*x**2 + 1)*a*c**2*g**3*x**2 + 8*sqrt(c**2*x* 
*2 + 1)*a*g**3 + 6*int((sqrt(c**2*x**2 + 1)*asinh(c*x)*x**2)/(f + g*x),x)* 
b*c**2*g**4 + 6*int((sqrt(c**2*x**2 + 1)*asinh(c*x))/(f + g*x),x)*b*g**4 - 
 6*log(sqrt(c**2*x**2 + 1) + c*x)*a*c**3*f**3 - 9*log(sqrt(c**2*x**2 + 1) 
+ c*x)*a*c*f*g**2))/(6*g**4)