3.2.40 \(\int x (a+b \text {arcsinh}(c x))^{3/2} \, dx\) [140]

3.2.40.1 Optimal result

Integrand size = 14, antiderivative size = 179 \[ \int x (a+b \text {arcsinh}(c x))^{3/2} \, dx=-\frac {3 b x \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{8 c}+\frac {(a+b \text {arcsinh}(c x))^{3/2}}{4 c^2}+\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))^{3/2}-\frac {3 b^{3/2} e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 c^2}+\frac {3 b^{3/2} e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 c^2} \]

output

1/4*(a+b*arcsinh(c*x))^(3/2)/c^2+1/2*x^2*(a+b*arcsinh(c*x))^(3/2)-3/128*b^ 
(3/2)*exp(2*a/b)*erf(2^(1/2)*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))*2^(1/2)*Pi^ 
(1/2)/c^2+3/128*b^(3/2)*erfi(2^(1/2)*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))*2^( 
1/2)*Pi^(1/2)/c^2/exp(2*a/b)-3/8*b*x*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))^ 
(1/2)/c
 

3.2.40.2 Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.64 \[ \int x (a+b \text {arcsinh}(c x))^{3/2} \, dx=\frac {e^{-\frac {2 a}{b}} \left (b^2 \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {5}{2},-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )+b^2 e^{\frac {4 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {5}{2},\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )\right )}{16 \sqrt {2} c^2 \sqrt {a+b \text {arcsinh}(c x)}} \]

input

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

output

(b^2*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[5/2, (-2*(a + b*ArcSinh[c*x]))/ 
b] + b^2*E^((4*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*Gamma[5/2, (2*(a + b*ArcSinh 
[c*x]))/b])/(16*Sqrt[2]*c^2*E^((2*a)/b)*Sqrt[a + b*ArcSinh[c*x]])
 

3.2.40.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.42 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.09, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.929, Rules used = {6192, 6227, 6195, 25, 5971, 27, 3042, 26, 3789, 2611, 2633, 2634, 6198}

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.

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

\(\Big \downarrow \) 6192

\(\displaystyle \frac {1}{2} x^2 (a+b \text {arcsinh}(c x))^{3/2}-\frac {3}{4} b c \int \frac {x^2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {c^2 x^2+1}}dx\)

\(\Big \downarrow \) 6227

\(\displaystyle \frac {1}{2} x^2 (a+b \text {arcsinh}(c x))^{3/2}-\frac {3}{4} b c \left (-\frac {\int \frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {c^2 x^2+1}}dx}{2 c^2}-\frac {b \int \frac {x}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{4 c}+\frac {x \sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}{2 c^2}\right )\)

\(\Big \downarrow \) 6195

\(\displaystyle \frac {1}{2} x^2 (a+b \text {arcsinh}(c x))^{3/2}-\frac {3}{4} b c \left (-\frac {\int -\frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{4 c^3}-\frac {\int \frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {c^2 x^2+1}}dx}{2 c^2}+\frac {x \sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}{2 c^2}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{2} x^2 (a+b \text {arcsinh}(c x))^{3/2}-\frac {3}{4} b c \left (\frac {\int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{4 c^3}-\frac {\int \frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {c^2 x^2+1}}dx}{2 c^2}+\frac {x \sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}{2 c^2}\right )\)

\(\Big \downarrow \) 5971

\(\displaystyle \frac {1}{2} x^2 (a+b \text {arcsinh}(c x))^{3/2}-\frac {3}{4} b c \left (\frac {\int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{2 \sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{4 c^3}-\frac {\int \frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {c^2 x^2+1}}dx}{2 c^2}+\frac {x \sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}{2 c^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} x^2 (a+b \text {arcsinh}(c x))^{3/2}-\frac {3}{4} b c \left (\frac {\int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{8 c^3}-\frac {\int \frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {c^2 x^2+1}}dx}{2 c^2}+\frac {x \sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}{2 c^2}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} x^2 (a+b \text {arcsinh}(c x))^{3/2}-\frac {3}{4} b c \left (\frac {\int -\frac {i \sin \left (\frac {2 i a}{b}-\frac {2 i (a+b \text {arcsinh}(c x))}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{8 c^3}-\frac {\int \frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {c^2 x^2+1}}dx}{2 c^2}+\frac {x \sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}{2 c^2}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {1}{2} x^2 (a+b \text {arcsinh}(c x))^{3/2}-\frac {3}{4} b c \left (-\frac {i \int \frac {\sin \left (\frac {2 i a}{b}-\frac {2 i (a+b \text {arcsinh}(c x))}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{8 c^3}-\frac {\int \frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {c^2 x^2+1}}dx}{2 c^2}+\frac {x \sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}{2 c^2}\right )\)

\(\Big \downarrow \) 3789

\(\displaystyle \frac {1}{2} x^2 (a+b \text {arcsinh}(c x))^{3/2}-\frac {3}{4} b c \left (-\frac {i \left (\frac {1}{2} i \int \frac {e^{-2 \text {arcsinh}(c x)}}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))-\frac {1}{2} i \int \frac {e^{2 \text {arcsinh}(c x)}}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))\right )}{8 c^3}-\frac {\int \frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {c^2 x^2+1}}dx}{2 c^2}+\frac {x \sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}{2 c^2}\right )\)

\(\Big \downarrow \) 2611

\(\displaystyle \frac {1}{2} x^2 (a+b \text {arcsinh}(c x))^{3/2}-\frac {3}{4} b c \left (-\frac {i \left (i \int e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}}d\sqrt {a+b \text {arcsinh}(c x)}-i \int e^{\frac {2 (a+b \text {arcsinh}(c x))}{b}-\frac {2 a}{b}}d\sqrt {a+b \text {arcsinh}(c x)}\right )}{8 c^3}-\frac {\int \frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {c^2 x^2+1}}dx}{2 c^2}+\frac {x \sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}{2 c^2}\right )\)

\(\Big \downarrow \) 2633

\(\displaystyle \frac {1}{2} x^2 (a+b \text {arcsinh}(c x))^{3/2}-\frac {3}{4} b c \left (-\frac {i \left (i \int e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}}d\sqrt {a+b \text {arcsinh}(c x)}-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )\right )}{8 c^3}-\frac {\int \frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {c^2 x^2+1}}dx}{2 c^2}+\frac {x \sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}{2 c^2}\right )\)

\(\Big \downarrow \) 2634

\(\displaystyle \frac {1}{2} x^2 (a+b \text {arcsinh}(c x))^{3/2}-\frac {3}{4} b c \left (-\frac {\int \frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {c^2 x^2+1}}dx}{2 c^2}-\frac {i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )\right )}{8 c^3}+\frac {x \sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}{2 c^2}\right )\)

\(\Big \downarrow \) 6198

\(\displaystyle \frac {1}{2} x^2 (a+b \text {arcsinh}(c x))^{3/2}-\frac {3}{4} b c \left (-\frac {i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )\right )}{8 c^3}-\frac {(a+b \text {arcsinh}(c x))^{3/2}}{3 b c^3}+\frac {x \sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}{2 c^2}\right )\)

input

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

output

(x^2*(a + b*ArcSinh[c*x])^(3/2))/2 - (3*b*c*((x*Sqrt[1 + c^2*x^2]*Sqrt[a + 
 b*ArcSinh[c*x]])/(2*c^2) - (a + b*ArcSinh[c*x])^(3/2)/(3*b*c^3) - ((I/8)* 
((I/2)*Sqrt[b]*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x] 
])/Sqrt[b]] - ((I/2)*Sqrt[b]*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c 
*x]])/Sqrt[b]])/E^((2*a)/b)))/c^3))/4
 

3.2.40.3.1 Defintions of rubi rules used

rule 25

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

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 27

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

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 3789

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

rule 5971

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + 
(b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + 
b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & 
& IGtQ[p, 0]
 

rule 6192

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

rule 6195

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

rule 6198

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

rule 6227

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

3.2.40.4 Maple [F]

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

input

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

output

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

3.2.40.5 Fricas [F(-2)]

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

input

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

output

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

3.2.40.6 Sympy [F]

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

input

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

output

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

3.2.40.7 Maxima [F]

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

input

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

output

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

3.2.40.8 Giac [F]

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

input

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

output

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

3.2.40.9 Mupad [F(-1)]

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

input

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

output

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