3.1.98 \(\int x^2 \sqrt {a+b \text {arcsinh}(c+d x)} \, dx\) [98]

3.1.98.1 Optimal result

Integrand size = 18, antiderivative size = 496 \[ \int x^2 \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=\frac {c^2 (c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}}{d^3}+\frac {(c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}}{3 d^3}-\frac {c \sqrt {a+b \text {arcsinh}(c+d x)} \cosh (2 \text {arcsinh}(c+d x))}{2 d^3}-\frac {\sqrt {b} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{16 d^3}+\frac {\sqrt {b} c^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{4 d^3}+\frac {\sqrt {b} c e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 d^3}+\frac {\sqrt {b} e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{48 d^3}+\frac {\sqrt {b} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{16 d^3}-\frac {\sqrt {b} c^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{4 d^3}+\frac {\sqrt {b} c e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 d^3}-\frac {\sqrt {b} e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{48 d^3} \]

output

1/144*exp(3*a/b)*erf(3^(1/2)*(a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*b^(1/2)*3 
^(1/2)*Pi^(1/2)/d^3-1/144*erfi(3^(1/2)*(a+b*arcsinh(d*x+c))^(1/2)/b^(1/2)) 
*b^(1/2)*3^(1/2)*Pi^(1/2)/d^3/exp(3*a/b)+1/16*c*exp(2*a/b)*erf(2^(1/2)*(a+ 
b*arcsinh(d*x+c))^(1/2)/b^(1/2))*b^(1/2)*2^(1/2)*Pi^(1/2)/d^3+1/16*c*erfi( 
2^(1/2)*(a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*b^(1/2)*2^(1/2)*Pi^(1/2)/d^3/e 
xp(2*a/b)-1/16*exp(a/b)*erf((a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*b^(1/2)*Pi 
^(1/2)/d^3+1/4*c^2*exp(a/b)*erf((a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*b^(1/2 
)*Pi^(1/2)/d^3+1/16*erfi((a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*b^(1/2)*Pi^(1 
/2)/d^3/exp(a/b)-1/4*c^2*erfi((a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*b^(1/2)* 
Pi^(1/2)/d^3/exp(a/b)+c^2*(d*x+c)*(a+b*arcsinh(d*x+c))^(1/2)/d^3+1/3*(d*x+ 
c)^3*(a+b*arcsinh(d*x+c))^(1/2)/d^3-1/2*c*cosh(2*arcsinh(d*x+c))*(a+b*arcs 
inh(d*x+c))^(1/2)/d^3
 

3.1.98.2 Mathematica [A] (verified)

Time = 2.53 (sec) , antiderivative size = 656, normalized size of antiderivative = 1.32 \[ \int x^2 \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=\frac {-36 (c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}+144 c^2 (c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}-72 c \sqrt {a+b \text {arcsinh}(c+d x)} \cosh (2 \text {arcsinh}(c+d x))+\sqrt {b} \sqrt {3 \pi } \cosh \left (\frac {3 a}{b}\right ) \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+9 \sqrt {b} \sqrt {\pi } \cosh \left (\frac {a}{b}\right ) \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-36 \sqrt {b} c^2 \sqrt {\pi } \cosh \left (\frac {a}{b}\right ) \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+9 \sqrt {b} c \sqrt {2 \pi } \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 {b} \sqrt {3 \pi } \cosh \left (\frac {3 a}{b}\right ) \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-9 \sqrt {b} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {a}{b}\right )+36 \sqrt {b} c^2 \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {a}{b}\right )+9 \sqrt {b} \left (-1+4 c^2\right ) \sqrt {\pi } \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 )-9 \sqrt {b} c \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {2 a}{b}\right )+9 \sqrt {b} c \sqrt {2 \pi } \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 {b} \sqrt {3 \pi } \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 {b} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {3 a}{b}\right )+12 \sqrt {a+b \text {arcsinh}(c+d x)} \sinh (3 \text {arcsinh}(c+d x))}{144 d^3} \]

input

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

output

(-36*(c + d*x)*Sqrt[a + b*ArcSinh[c + d*x]] + 144*c^2*(c + d*x)*Sqrt[a + b 
*ArcSinh[c + d*x]] - 72*c*Sqrt[a + b*ArcSinh[c + d*x]]*Cosh[2*ArcSinh[c + 
d*x]] + Sqrt[b]*Sqrt[3*Pi]*Cosh[(3*a)/b]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c 
 + d*x]])/Sqrt[b]] + 9*Sqrt[b]*Sqrt[Pi]*Cosh[a/b]*Erfi[Sqrt[a + b*ArcSinh[ 
c + d*x]]/Sqrt[b]] - 36*Sqrt[b]*c^2*Sqrt[Pi]*Cosh[a/b]*Erfi[Sqrt[a + b*Arc 
Sinh[c + d*x]]/Sqrt[b]] + 9*Sqrt[b]*c*Sqrt[2*Pi]*Cosh[(2*a)/b]*Erfi[(Sqrt[ 
2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]] - Sqrt[b]*Sqrt[3*Pi]*Cosh[(3*a)/ 
b]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]] - 9*Sqrt[b]*Sqrt[P 
i]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*Sinh[a/b] + 36*Sqrt[b]*c^2*S 
qrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*Sinh[a/b] + 9*Sqrt[b]*( 
-1 + 4*c^2)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] 
+ Sinh[a/b]) - 9*Sqrt[b]*c*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + 
 d*x]])/Sqrt[b]]*Sinh[(2*a)/b] + 9*Sqrt[b]*c*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[ 
a + b*ArcSinh[c + d*x]])/Sqrt[b]]*(Cosh[(2*a)/b] + Sinh[(2*a)/b]) + Sqrt[b 
]*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]]*Sinh[(3*a 
)/b] + Sqrt[b]*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt 
[b]]*Sinh[(3*a)/b] + 12*Sqrt[a + b*ArcSinh[c + d*x]]*Sinh[3*ArcSinh[c + d* 
x]])/(144*d^3)
 

3.1.98.3 Rubi [A] (verified)

Time = 1.86 (sec) , antiderivative size = 525, normalized size of antiderivative = 1.06, 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.

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

\(\Big \downarrow \) 6274

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6245

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

\(\Big \downarrow \) 7267

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \left (\frac {1}{8} \sqrt {\pi } b^{3/2} c^2 e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } b^{3/2} c^2 e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{32} \sqrt {\pi } b^{3/2} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{16} \sqrt {\frac {\pi }{2}} b^{3/2} c e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{96} \sqrt {\frac {\pi }{3}} b^{3/2} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{32} \sqrt {\pi } b^{3/2} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{16} \sqrt {\frac {\pi }{2}} b^{3/2} c e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{96} \sqrt {\frac {\pi }{3}} b^{3/2} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{2} b c^2 \sqrt {a+b \text {arcsinh}(c+d x)} \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )-\frac {1}{6} b \sqrt {a+b \text {arcsinh}(c+d x)} \sinh ^3\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )-\frac {1}{4} b c \sqrt {a+b \text {arcsinh}(c+d x)} \cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )}{b d^3}\)

input

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

output

(2*(-1/4*(b*c*Sqrt[a + b*ArcSinh[c + d*x]]*Cosh[(2*a)/b - (2*(a + b*ArcSin 
h[c + d*x]))/b]) - (b^(3/2)*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d* 
x]]/Sqrt[b]])/32 + (b^(3/2)*c^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c 
+ d*x]]/Sqrt[b]])/8 + (b^(3/2)*c*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[ 
a + b*ArcSinh[c + d*x]])/Sqrt[b]])/16 + (b^(3/2)*E^((3*a)/b)*Sqrt[Pi/3]*Er 
f[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/96 + (b^(3/2)*Sqrt[Pi]* 
Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(32*E^(a/b)) - (b^(3/2)*c^2*Sq 
rt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(8*E^(a/b)) + (b^(3/2)* 
c*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(16*E^( 
(2*a)/b)) - (b^(3/2)*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]] 
)/Sqrt[b]])/(96*E^((3*a)/b)) - (b*c^2*Sqrt[a + b*ArcSinh[c + d*x]]*Sinh[a/ 
b - (a + b*ArcSinh[c + d*x])/b])/2 - (b*Sqrt[a + b*ArcSinh[c + d*x]]*Sinh[ 
a/b - (a + b*ArcSinh[c + d*x])/b]^3)/6))/(b*d^3)
 

3.1.98.3.1 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] 
]
 

3.1.98.4 Maple [F]

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

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)
 

3.1.98.5 Fricas [F(-2)]

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

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)
 

3.1.98.6 Sympy [F]

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

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)
 

3.1.98.7 Maxima [F]

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

input

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

output

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

3.1.98.8 Giac [F]

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

input

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

output

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

3.1.98.9 Mupad [F(-1)]

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

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)