3.3.13 \(\int \frac {c e+d e x}{(a+b \text {arcsinh}(c+d x))^{3/2}} \, dx\) [213]

3.3.13.1 Optimal result

Integrand size = 23, antiderivative size = 148 \[ \int \frac {c e+d e x}{(a+b \text {arcsinh}(c+d x))^{3/2}} \, dx=-\frac {2 e (c+d x) \sqrt {1+(c+d x)^2}}{b d \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {e e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d}+\frac {e e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d} \]

output

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

3.3.13.2 Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.99 \[ \int \frac {c e+d e x}{(a+b \text {arcsinh}(c+d x))^{3/2}} \, dx=\frac {e e^{-\frac {2 a}{b}} \left (\sqrt {2} \sqrt {-\frac {a+b \text {arcsinh}(c+d x)}{b}} \Gamma \left (\frac {1}{2},-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )-\sqrt {2} e^{\frac {4 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} \Gamma \left (\frac {1}{2},\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )-2 e^{\frac {2 a}{b}} \sinh (2 \text {arcsinh}(c+d x))\right )}{2 b d \sqrt {a+b \text {arcsinh}(c+d x)}} \]

input

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

output

(e*(Sqrt[2]*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[1/2, (-2*(a + b*ArcS 
inh[c + d*x]))/b] - Sqrt[2]*E^((4*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma 
[1/2, (2*(a + b*ArcSinh[c + d*x]))/b] - 2*E^((2*a)/b)*Sinh[2*ArcSinh[c + d 
*x]]))/(2*b*d*E^((2*a)/b)*Sqrt[a + b*ArcSinh[c + d*x]])
 

3.3.13.3 Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {6274, 27, 6193, 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.

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

\(\Big \downarrow \) 6274

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6193

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3788

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 2611

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

\(\Big \downarrow \) 2633

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

\(\Big \downarrow \) 2634

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

input

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

output

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

3.3.13.3.1 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 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 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 6193

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 
x^m*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1))), x] - Si 
mp[1/(b^2*c^(m + 1)*(n + 1))   Subst[Int[ExpandTrigReduce[x^(n + 1), Sinh[- 
a/b + x/b]^(m - 1)*(m + (m + 1)*Sinh[-a/b + x/b]^2), x], x], x, a + b*ArcSi 
nh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, - 
1]
 

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]
 

3.3.13.4 Maple [F]

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

input

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

output

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

3.3.13.5 Fricas [F(-2)]

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

input

integrate((d*e*x+c*e)/(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)
 

3.3.13.6 Sympy [F]

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

input

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

output

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

3.3.13.7 Maxima [F]

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

input

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

output

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

3.3.13.8 Giac [F]

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

input

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

output

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

3.3.13.9 Mupad [F(-1)]

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

input

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

output

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