3.2.1 \(\int x (a+b \text {arcsinh}(c+d x))^{3/2} \, dx\) [101]

3.2.1.1 Optimal result

Integrand size = 16, antiderivative size = 326 \[ \int x (a+b \text {arcsinh}(c+d x))^{3/2} \, dx=\frac {3 b c \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{2 d^2}-\frac {c (c+d x) (a+b \text {arcsinh}(c+d x))^{3/2}}{d^2}+\frac {(a+b \text {arcsinh}(c+d x))^{3/2} \cosh (2 \text {arcsinh}(c+d x))}{4 d^2}-\frac {3 b^{3/2} c e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 d^2}-\frac {3 b^{3/2} e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{64 d^2}-\frac {3 b^{3/2} c e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 d^2}+\frac {3 b^{3/2} e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{64 d^2}-\frac {3 b \sqrt {a+b \text {arcsinh}(c+d x)} \sinh (2 \text {arcsinh}(c+d x))}{16 d^2} \]

output

-c*(d*x+c)*(a+b*arcsinh(d*x+c))^(3/2)/d^2+1/4*(a+b*arcsinh(d*x+c))^(3/2)*c 
osh(2*arcsinh(d*x+c))/d^2-3/128*b^(3/2)*exp(2*a/b)*erf(2^(1/2)*(a+b*arcsin 
h(d*x+c))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/d^2+3/128*b^(3/2)*erfi(2^(1/2)*( 
a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/d^2/exp(2*a/b)-3/8*b^( 
3/2)*c*exp(a/b)*erf((a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*Pi^(1/2)/d^2-3/8*b 
^(3/2)*c*erfi((a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*Pi^(1/2)/d^2/exp(a/b)-3/ 
16*b*sinh(2*arcsinh(d*x+c))*(a+b*arcsinh(d*x+c))^(1/2)/d^2+3/2*b*c*(1+(d*x 
+c)^2)^(1/2)*(a+b*arcsinh(d*x+c))^(1/2)/d^2
 

3.2.1.2 Mathematica [A] (verified)

Time = 4.14 (sec) , antiderivative size = 582, normalized size of antiderivative = 1.79 \[ \int x (a+b \text {arcsinh}(c+d x))^{3/2} \, dx=\frac {-64 a c e^{-\frac {a}{b}} \sqrt {a+b \text {arcsinh}(c+d x)} \left (-\frac {e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\text {arcsinh}(c+d x)\right )}{\sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)}}+\frac {\Gamma \left (\frac {3}{2},-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{\sqrt {-\frac {a+b \text {arcsinh}(c+d x)}{b}}}\right )-16 \sqrt {b} c \left (4 \sqrt {b} \sqrt {a+b \text {arcsinh}(c+d x)} \left (-3 \sqrt {1+(c+d x)^2}+2 (c+d x) \text {arcsinh}(c+d x)\right )+(2 a+3 b) \sqrt {\pi } \text {erfi}\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 )+(-2 a+3 b) \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 )\right )+4 a \left (8 \sqrt {a+b \text {arcsinh}(c+d x)} \cosh (2 \text {arcsinh}(c+d x))+\sqrt {b} \sqrt {2 \pi } \text {erfi}\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 {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 )\right )+\sqrt {b} \left ((4 a+3 b) \sqrt {2 \pi } \text {erfi}\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 )+(4 a-3 b) \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 )+8 \sqrt {b} \sqrt {a+b \text {arcsinh}(c+d x)} (4 \text {arcsinh}(c+d x) \cosh (2 \text {arcsinh}(c+d x))-3 \sinh (2 \text {arcsinh}(c+d x)))\right )}{128 d^2} \]

input

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

output

((-64*a*c*Sqrt[a + b*ArcSinh[c + d*x]]*(-((E^((2*a)/b)*Gamma[3/2, a/b + Ar 
cSinh[c + d*x]])/Sqrt[a/b + ArcSinh[c + d*x]]) + Gamma[3/2, -((a + b*ArcSi 
nh[c + d*x])/b)]/Sqrt[-((a + b*ArcSinh[c + d*x])/b)]))/E^(a/b) - 16*Sqrt[b 
]*c*(4*Sqrt[b]*Sqrt[a + b*ArcSinh[c + d*x]]*(-3*Sqrt[1 + (c + d*x)^2] + 2* 
(c + d*x)*ArcSinh[c + d*x]) + (2*a + 3*b)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh 
[c + d*x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]) + (-2*a + 3*b)*Sqrt[Pi]*Erf[Sq 
rt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b])) + 4*a*(8*Sqrt 
[a + b*ArcSinh[c + d*x]]*Cosh[2*ArcSinh[c + d*x]] + Sqrt[b]*Sqrt[2*Pi]*Erf 
i[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]]*(-Cosh[(2*a)/b] + Sinh[( 
2*a)/b]) - Sqrt[b]*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/S 
qrt[b]]*(Cosh[(2*a)/b] + Sinh[(2*a)/b])) + Sqrt[b]*((4*a + 3*b)*Sqrt[2*Pi] 
*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]]*(Cosh[(2*a)/b] - Sin 
h[(2*a)/b]) + (4*a - 3*b)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d 
*x]])/Sqrt[b]]*(Cosh[(2*a)/b] + Sinh[(2*a)/b]) + 8*Sqrt[b]*Sqrt[a + b*ArcS 
inh[c + d*x]]*(4*ArcSinh[c + d*x]*Cosh[2*ArcSinh[c + d*x]] - 3*Sinh[2*ArcS 
inh[c + d*x]])))/(128*d^2)
 

3.2.1.3 Rubi [A] (verified)

Time = 1.25 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.14, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6274, 25, 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.

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

\(\Big \downarrow \) 6274

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6245

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

\(\Big \downarrow \) 7267

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 \left (\frac {3}{16} \sqrt {\pi } b^{5/2} c e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {3}{128} \sqrt {\frac {\pi }{2}} b^{5/2} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {3}{16} \sqrt {\pi } b^{5/2} c e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {3}{128} \sqrt {\frac {\pi }{2}} b^{5/2} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {3}{32} b^2 \sqrt {a+b \text {arcsinh}(c+d x)} \sinh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )-\frac {3}{4} b^2 c \sqrt {a+b \text {arcsinh}(c+d x)} \cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )-\frac {1}{2} b c (a+b \text {arcsinh}(c+d x))^{3/2} \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )-\frac {1}{8} b (a+b \text {arcsinh}(c+d x))^{3/2} \cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )}{b d^2}\)

input

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

output

(-2*(-1/8*(b*(a + b*ArcSinh[c + d*x])^(3/2)*Cosh[(2*a)/b - (2*(a + b*ArcSi 
nh[c + d*x]))/b]) - (3*b^2*c*Sqrt[a + b*ArcSinh[c + d*x]]*Cosh[a/b - (a + 
b*ArcSinh[c + d*x])/b])/4 + (3*b^(5/2)*c*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*A 
rcSinh[c + d*x]]/Sqrt[b]])/16 + (3*b^(5/2)*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqr 
t[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/128 + (3*b^(5/2)*c*Sqrt[Pi]*E 
rfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(16*E^(a/b)) - (3*b^(5/2)*Sqrt[ 
Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(128*E^((2*a)/ 
b)) - (3*b^2*Sqrt[a + b*ArcSinh[c + d*x]]*Sinh[(2*a)/b - (2*(a + b*ArcSinh 
[c + d*x]))/b])/32 - (b*c*(a + b*ArcSinh[c + d*x])^(3/2)*Sinh[a/b - (a + b 
*ArcSinh[c + d*x])/b])/2))/(b*d^2)
 

3.2.1.3.1 Defintions of rubi rules used

rule 25

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

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] 
]
 

3.2.1.4 Maple [F]

\[\int x \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{\frac {3}{2}}d x\]

input

int(x*(a+b*arcsinh(d*x+c))^(3/2),x)
 

output

int(x*(a+b*arcsinh(d*x+c))^(3/2),x)
 

3.2.1.5 Fricas [F(-2)]

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

input

integrate(x*(a+b*arcsinh(d*x+c))^(3/2),x, algorithm="fricas")
 

output

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

3.2.1.6 Sympy [F]

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

input

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

output

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

3.2.1.7 Maxima [F]

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

input

integrate(x*(a+b*arcsinh(d*x+c))^(3/2),x, algorithm="maxima")
 

output

integrate((b*arcsinh(d*x + c) + a)^(3/2)*x, x)
 

3.2.1.8 Giac [F]

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

input

integrate(x*(a+b*arcsinh(d*x+c))^(3/2),x, algorithm="giac")
 

output

integrate((b*arcsinh(d*x + c) + a)^(3/2)*x, x)
 

3.2.1.9 Mupad [F(-1)]

Timed out. \[ \int x (a+b \text {arcsinh}(c+d x))^{3/2} \, dx=\int x\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^{3/2} \,d x \]

input

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

output

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