\(\int \frac {x^2}{\sqrt {a+b \text {arcsinh}(c+d x)}} \, dx\) [48]

Optimal result

Integrand size = 18, antiderivative size = 411 \[ \int \frac {x^2}{\sqrt {a+b \text {arcsinh}(c+d x)}} \, dx=-\frac {e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d^3}+\frac {c^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}+\frac {c e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}+\frac {e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d^3}-\frac {e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d^3}+\frac {c^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}-\frac {c e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}+\frac {e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d^3} \] Output:

-1/8*exp(a/b)*Pi^(1/2)*erf((a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))/b^(1/2)/d^3 
+1/2*c^2*exp(a/b)*Pi^(1/2)*erf((a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))/b^(1/2) 
/d^3+1/4*c*exp(2*a/b)*2^(1/2)*Pi^(1/2)*erf(2^(1/2)*(a+b*arcsinh(d*x+c))^(1 
/2)/b^(1/2))/b^(1/2)/d^3+1/24*exp(3*a/b)*3^(1/2)*Pi^(1/2)*erf(3^(1/2)*(a+b 
*arcsinh(d*x+c))^(1/2)/b^(1/2))/b^(1/2)/d^3-1/8*Pi^(1/2)*erfi((a+b*arcsinh 
(d*x+c))^(1/2)/b^(1/2))/b^(1/2)/d^3/exp(a/b)+1/2*c^2*Pi^(1/2)*erfi((a+b*ar 
csinh(d*x+c))^(1/2)/b^(1/2))/b^(1/2)/d^3/exp(a/b)-1/4*c*2^(1/2)*Pi^(1/2)*e 
rfi(2^(1/2)*(a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))/b^(1/2)/d^3/exp(2*a/b)+1/2 
4*3^(1/2)*Pi^(1/2)*erfi(3^(1/2)*(a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))/b^(1/2 
)/d^3/exp(3*a/b)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 1.55 (sec) , antiderivative size = 498, normalized size of antiderivative = 1.21 \[ \int \frac {x^2}{\sqrt {a+b \text {arcsinh}(c+d x)}} \, dx=\frac {\sqrt {\frac {\pi }{6}} \left (\sqrt {2} \cosh \left (\frac {3 a}{b}\right ) \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\sqrt {6} \cosh \left (\frac {a}{b}\right ) \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+4 \sqrt {6} c^2 \cosh \left (\frac {a}{b}\right ) \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-4 \sqrt {3} c \cosh \left (\frac {2 a}{b}\right ) \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\sqrt {2} \cosh \left (\frac {3 a}{b}\right ) \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\sqrt {6} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {a}{b}\right )-4 \sqrt {6} c^2 \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {a}{b}\right )+\sqrt {6} \left (-1+4 c^2\right ) \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )+4 \sqrt {3} c \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {2 a}{b}\right )+4 \sqrt {3} c \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )+\sinh \left (\frac {2 a}{b}\right )\right )+\sqrt {2} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {3 a}{b}\right )-\sqrt {2} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {3 a}{b}\right )\right )}{8 \sqrt {b} d^3} \] Input:

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

Output:

(Sqrt[Pi/6]*(Sqrt[2]*Cosh[(3*a)/b]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x 
]])/Sqrt[b]] - Sqrt[6]*Cosh[a/b]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b] 
] + 4*Sqrt[6]*c^2*Cosh[a/b]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]] - 4 
*Sqrt[3]*c*Cosh[(2*a)/b]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[ 
b]] + Sqrt[2]*Cosh[(3*a)/b]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sq 
rt[b]] + Sqrt[6]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*Sinh[a/b] - 4* 
Sqrt[6]*c^2*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*Sinh[a/b] + Sqrt[6] 
*(-1 + 4*c^2)*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[ 
a/b]) + 4*Sqrt[3]*c*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]]*S 
inh[(2*a)/b] + 4*Sqrt[3]*c*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt 
[b]]*(Cosh[(2*a)/b] + Sinh[(2*a)/b]) + Sqrt[2]*Erf[(Sqrt[3]*Sqrt[a + b*Arc 
Sinh[c + d*x]])/Sqrt[b]]*Sinh[(3*a)/b] - Sqrt[2]*Erfi[(Sqrt[3]*Sqrt[a + b* 
ArcSinh[c + d*x]])/Sqrt[b]]*Sinh[(3*a)/b]))/(8*Sqrt[b]*d^3)
 

Rubi [A] (verified)

Time = 1.66 (sec) , antiderivative size = 395, normalized size of antiderivative = 0.96, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {6274, 27, 6245, 7267, 7292, 7293, 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.

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

\(\Big \downarrow \) 6274

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6245

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

\(\Big \downarrow \) 7267

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

\(\Big \downarrow \) 7292

\(\displaystyle \frac {2 \int d^2 x^2 \cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )d\sqrt {a+b \text {arcsinh}(c+d x)}}{b d^3}\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {2 \int \left (\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right ) c^2+\sinh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right ) c+\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right ) \sinh ^2\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )\right )d\sqrt {a+b \text {arcsinh}(c+d x)}}{b d^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \left (\frac {1}{4} \sqrt {\pi } \sqrt {b} c^2 e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{4} \sqrt {\pi } \sqrt {b} c^2 e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {b} c e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{16} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{16} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {b} c e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{16} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{16} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )}{b d^3}\)

Input:

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

Output:

(2*(-1/16*(Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[ 
b]]) + (Sqrt[b]*c^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt 
[b]])/4 + (Sqrt[b]*c*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSin 
h[c + d*x]])/Sqrt[b]])/4 + (Sqrt[b]*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sq 
rt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/16 - (Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + 
 b*ArcSinh[c + d*x]]/Sqrt[b]])/(16*E^(a/b)) + (Sqrt[b]*c^2*Sqrt[Pi]*Erfi[S 
qrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(4*E^(a/b)) - (Sqrt[b]*c*Sqrt[Pi/2]* 
Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(4*E^((2*a)/b)) + (S 
qrt[b]*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(1 
6*E^((3*a)/b))))/(b*d^3)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

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

rule 6245

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x 
_Symbol] :> Simp[1/c^(m + 1)   Subst[Int[(a + b*x)^n*Cosh[x]*(c*d + e*Sinh[ 
x])^m, x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[m, 
0]
 

rule 6274

Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^( 
m_.), x_Symbol] :> Simp[1/d   Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b* 
ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
 

rule 7267

Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si 
mp[lst[[2]]*lst[[4]]   Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x 
] /;  !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
 

rule 7292

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =! 
= u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [F]

\[\int \frac {x^{2}}{\sqrt {a +b \,\operatorname {arcsinh}\left (d x +c \right )}}d x\]

Input:

int(x^2/(a+b*arcsinh(d*x+c))^(1/2),x)
 

Output:

int(x^2/(a+b*arcsinh(d*x+c))^(1/2),x)
 

Fricas [F(-2)]

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

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

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

integrate(x^2/sqrt(b*arcsinh(d*x + c) + a), x)
 

Giac [F]

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

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

Output:

integrate(x^2/sqrt(b*arcsinh(d*x + c) + a), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int((sqrt(asinh(c + d*x)*b + a)*x**2)/(asinh(c + d*x)*b + a),x)