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

Optimal result

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

[Out]

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)*Pi^(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)

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {5859, 12, 5778, 3388, 2211, 2236, 2235} \[ \int \frac {c e+d e x}{(a+b \text {arcsinh}(c+d x))^{3/2}} \, dx=\frac {\sqrt {\frac {\pi }{2}} e e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d}+\frac {\sqrt {\frac {\pi }{2}} e e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d}-\frac {2 e \sqrt {(c+d x)^2+1} (c+d x)}{b d \sqrt {a+b \text {arcsinh}(c+d x)}} \]

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2211

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[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 2235

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 2236

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] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 3388

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/(E^(
I*k*Pi)*E^(I*(e + f*x))), x], x] - Dist[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 5778

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

Rule 5859

Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[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
]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {e x}{(a+b \text {arcsinh}(x))^{3/2}} \, dx,x,c+d x\right )}{d} \\ & = \frac {e \text {Subst}\left (\int \frac {x}{(a+b \text {arcsinh}(x))^{3/2}} \, dx,x,c+d x\right )}{d} \\ & = -\frac {2 e (c+d x) \sqrt {1+(c+d x)^2}}{b d \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {(2 e) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{b^2 d} \\ & = -\frac {2 e (c+d x) \sqrt {1+(c+d x)^2}}{b d \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {e \text {Subst}\left (\int \frac {e^{-i \left (\frac {2 i a}{b}-\frac {2 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{b^2 d}+\frac {e \text {Subst}\left (\int \frac {e^{i \left (\frac {2 i a}{b}-\frac {2 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{b^2 d} \\ & = -\frac {2 e (c+d x) \sqrt {1+(c+d x)^2}}{b d \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {(2 e) \text {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{b^2 d}+\frac {(2 e) \text {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{b^2 d} \\ & = -\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} \\ \end{align*}

Mathematica [A] (verified)

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

[In]

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

[Out]

(e*(Sqrt[2]*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[1/2, (-2*(a + b*ArcSinh[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]])

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F]

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

[In]

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

[Out]

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))

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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