\(\int (c e+d e x) \sqrt {a+b \text {arcsinh}(c+d x)} \, dx\) [183]

Optimal result

Integrand size = 23, antiderivative size = 164 \[ \int (c e+d e x) \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=\frac {e \sqrt {a+b \text {arcsinh}(c+d x)}}{4 d}+\frac {e (c+d x)^2 \sqrt {a+b \text {arcsinh}(c+d x)}}{2 d}-\frac {\sqrt {b} e e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{16 d}-\frac {\sqrt {b} e e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{16 d} \]

[Out]

-1/32*e*exp(2*a/b)*erf(2^(1/2)*(a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*b^(1/2)*2^(1/2)*Pi^(1/2)/d-1/32*e*erfi(2^(1
/2)*(a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*b^(1/2)*2^(1/2)*Pi^(1/2)/d/exp(2*a/b)+1/4*e*(a+b*arcsinh(d*x+c))^(1/2)
/d+1/2*e*(d*x+c)^2*(a+b*arcsinh(d*x+c))^(1/2)/d

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {5859, 12, 5777, 5819, 3393, 3388, 2211, 2236, 2235} \[ \int (c e+d e x) \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{16 d}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{16 d}+\frac {e (c+d x)^2 \sqrt {a+b \text {arcsinh}(c+d x)}}{2 d}+\frac {e \sqrt {a+b \text {arcsinh}(c+d x)}}{4 d} \]

[In]

Int[(c*e + d*e*x)*Sqrt[a + b*ArcSinh[c + d*x]],x]

[Out]

(e*Sqrt[a + b*ArcSinh[c + d*x]])/(4*d) + (e*(c + d*x)^2*Sqrt[a + b*ArcSinh[c + d*x]])/(2*d) - (Sqrt[b]*e*E^((2
*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(16*d) - (Sqrt[b]*e*Sqrt[Pi/2]*Erfi[(Sq
rt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(16*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 3393

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 5777

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

Rule 5819

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

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 e x \sqrt {a+b \text {arcsinh}(x)} \, dx,x,c+d x\right )}{d} \\ & = \frac {e \text {Subst}\left (\int x \sqrt {a+b \text {arcsinh}(x)} \, dx,x,c+d x\right )}{d} \\ & = \frac {e (c+d x)^2 \sqrt {a+b \text {arcsinh}(c+d x)}}{2 d}-\frac {(b e) \text {Subst}\left (\int \frac {x^2}{\sqrt {1+x^2} \sqrt {a+b \text {arcsinh}(x)}} \, dx,x,c+d x\right )}{4 d} \\ & = \frac {e (c+d x)^2 \sqrt {a+b \text {arcsinh}(c+d x)}}{2 d}-\frac {e \text {Subst}\left (\int \frac {\sinh ^2\left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{4 d} \\ & = \frac {e (c+d x)^2 \sqrt {a+b \text {arcsinh}(c+d x)}}{2 d}+\frac {e \text {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}-\frac {\cosh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 \sqrt {x}}\right ) \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{4 d} \\ & = \frac {e \sqrt {a+b \text {arcsinh}(c+d x)}}{4 d}+\frac {e (c+d x)^2 \sqrt {a+b \text {arcsinh}(c+d x)}}{2 d}-\frac {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 )}{8 d} \\ & = \frac {e \sqrt {a+b \text {arcsinh}(c+d x)}}{4 d}+\frac {e (c+d x)^2 \sqrt {a+b \text {arcsinh}(c+d x)}}{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 )}{16 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 )}{16 d} \\ & = \frac {e \sqrt {a+b \text {arcsinh}(c+d x)}}{4 d}+\frac {e (c+d x)^2 \sqrt {a+b \text {arcsinh}(c+d x)}}{2 d}-\frac {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 )}{8 d}-\frac {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 )}{8 d} \\ & = \frac {e \sqrt {a+b \text {arcsinh}(c+d x)}}{4 d}+\frac {e (c+d x)^2 \sqrt {a+b \text {arcsinh}(c+d x)}}{2 d}-\frac {\sqrt {b} 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 )}{16 d}-\frac {\sqrt {b} 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 )}{16 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.74 \[ \int (c e+d e x) \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=\frac {e e^{-\frac {2 a}{b}} \left (-b \sqrt {-\frac {a+b \text {arcsinh}(c+d x)}{b}} \Gamma \left (\frac {3}{2},-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )+b e^{\frac {4 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} \Gamma \left (\frac {3}{2},\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )}{8 \sqrt {2} d \sqrt {a+b \text {arcsinh}(c+d x)}} \]

[In]

Integrate[(c*e + d*e*x)*Sqrt[a + b*ArcSinh[c + d*x]],x]

[Out]

(e*(-(b*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[3/2, (-2*(a + b*ArcSinh[c + d*x]))/b]) + b*E^((4*a)/b)*Sqrt[
a/b + ArcSinh[c + d*x]]*Gamma[3/2, (2*(a + b*ArcSinh[c + d*x]))/b]))/(8*Sqrt[2]*d*E^((2*a)/b)*Sqrt[a + b*ArcSi
nh[c + d*x]])

Maple [F]

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

[In]

int((d*e*x+c*e)*(a+b*arcsinh(d*x+c))^(1/2),x)

[Out]

int((d*e*x+c*e)*(a+b*arcsinh(d*x+c))^(1/2),x)

Fricas [F(-2)]

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

[In]

integrate((d*e*x+c*e)*(a+b*arcsinh(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

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

[In]

integrate((d*e*x+c*e)*(a+b*asinh(d*x+c))**(1/2),x)

[Out]

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

Maxima [F]

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

[In]

integrate((d*e*x+c*e)*(a+b*arcsinh(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate((d*e*x + c*e)*sqrt(b*arcsinh(d*x + c) + a), x)

Giac [F]

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

[In]

integrate((d*e*x+c*e)*(a+b*arcsinh(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)*sqrt(b*arcsinh(d*x + c) + a), x)

Mupad [F(-1)]

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

[In]

int((c*e + d*e*x)*(a + b*asinh(c + d*x))^(1/2),x)

[Out]

int((c*e + d*e*x)*(a + b*asinh(c + d*x))^(1/2), x)