3.2.2 \(\int (a+b \text {arcsinh}(c+d x))^{3/2} \, dx\) [102]

3.2.2.1 Optimal result

Integrand size = 14, antiderivative size = 150 \[ \int (a+b \text {arcsinh}(c+d x))^{3/2} \, dx=-\frac {3 b \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{2 d}+\frac {(c+d x) (a+b \text {arcsinh}(c+d x))^{3/2}}{d}+\frac {3 b^{3/2} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 d}+\frac {3 b^{3/2} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 d} \]

output

(d*x+c)*(a+b*arcsinh(d*x+c))^(3/2)/d+3/8*b^(3/2)*exp(a/b)*erf((a+b*arcsinh 
(d*x+c))^(1/2)/b^(1/2))*Pi^(1/2)/d+3/8*b^(3/2)*erfi((a+b*arcsinh(d*x+c))^( 
1/2)/b^(1/2))*Pi^(1/2)/d/exp(a/b)-3/2*b*(1+(d*x+c)^2)^(1/2)*(a+b*arcsinh(d 
*x+c))^(1/2)/d
 

3.2.2.2 Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.81 \[ \int (a+b \text {arcsinh}(c+d x))^{3/2} \, dx=\frac {a e^{-\frac {a}{b}} \sqrt {a+b \text {arcsinh}(c+d x)} \left (-\frac {e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\text {arcsinh}(c+d x)\right )}{\sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)}}+\frac {\Gamma \left (\frac {3}{2},-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{\sqrt {-\frac {a+b \text {arcsinh}(c+d x)}{b}}}\right )}{2 d}+\frac {\sqrt {b} \left (4 \sqrt {b} \sqrt {a+b \text {arcsinh}(c+d x)} \left (-3 \sqrt {1+(c+d x)^2}+2 (c+d x) \text {arcsinh}(c+d x)\right )+(2 a+3 b) \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )-\sinh \left (\frac {a}{b}\right )\right )+(-2 a+3 b) \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )\right )}{8 d} \]

input

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

output

(a*Sqrt[a + b*ArcSinh[c + d*x]]*(-((E^((2*a)/b)*Gamma[3/2, a/b + ArcSinh[c 
 + d*x]])/Sqrt[a/b + ArcSinh[c + d*x]]) + Gamma[3/2, -((a + b*ArcSinh[c + 
d*x])/b)]/Sqrt[-((a + b*ArcSinh[c + d*x])/b)]))/(2*d*E^(a/b)) + (Sqrt[b]*( 
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[Sqrt[a 
+ b*ArcSinh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b])))/(8*d)
 

3.2.2.3 Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.99, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {6273, 6187, 6213, 6189, 3042, 3788, 26, 2611, 2633, 2634}

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.

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

\(\Big \downarrow \) 6273

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

\(\Big \downarrow \) 6187

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

\(\Big \downarrow \) 6213

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

\(\Big \downarrow \) 6189

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3788

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 2611

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

\(\Big \downarrow \) 2633

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

\(\Big \downarrow \) 2634

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

input

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

output

((c + d*x)*(a + b*ArcSinh[c + d*x])^(3/2) - (3*b*(Sqrt[1 + (c + d*x)^2]*Sq 
rt[a + b*ArcSinh[c + d*x]] + (-1/2*(Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + 
b*ArcSinh[c + d*x]]/Sqrt[b]]) - (Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[ 
c + d*x]]/Sqrt[b]])/(2*E^(a/b)))/2))/2)/d
 

3.2.2.3.1 Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2611

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : 
> Simp[2/d   Subst[Int[F^(g*(e - c*(f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d 
*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]
 

rule 2633

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt 
[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ 
F, a, b, c, d}, x] && PosQ[b]
 

rule 2634

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt 
[Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F], 2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; Fr 
eeQ[{F, a, b, c, d}, x] && NegQ[b]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3788

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol 
] :> Simp[I/2   Int[(c + d*x)^m/(E^(I*k*Pi)*E^(I*(e + f*x))), x], x] - Simp 
[I/2   Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e 
, f, m}, x] && IntegerQ[2*k]
 

rule 6187

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*A 
rcSinh[c*x])^n, x] - Simp[b*c*n   Int[x*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[ 
1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
 

rule 6189

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[1/(b*c)   S 
ubst[Int[x^n*Cosh[-a/b + x/b], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, 
b, c, n}, x]
 

rule 6213

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

rule 6273

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

3.2.2.4 Maple [F]

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

input

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

output

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

3.2.2.5 Fricas [F(-2)]

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

input

integrate((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)
 

3.2.2.6 Sympy [F]

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

input

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

output

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

3.2.2.7 Maxima [F]

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

input

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

output

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

3.2.2.8 Giac [F]

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

input

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

output

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

3.2.2.9 Mupad [F(-1)]

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

input

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

output

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