3.630 \(\int (d+e x^2) \sqrt{a+b \sinh ^{-1}(c x)} \, dx\)

Optimal. Leaf size=322 \[ -\frac{\sqrt{\pi } \sqrt{b} e e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 c^3}+\frac{\sqrt{\frac{\pi }{3}} \sqrt{b} e e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{48 c^3}+\frac{\sqrt{\pi } \sqrt{b} e e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 c^3}-\frac{\sqrt{\frac{\pi }{3}} \sqrt{b} e e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{48 c^3}+\frac{\sqrt{\pi } \sqrt{b} d e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 c}-\frac{\sqrt{\pi } \sqrt{b} d e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 c}+d x \sqrt{a+b \sinh ^{-1}(c x)}+\frac{1}{3} e x^3 \sqrt{a+b \sinh ^{-1}(c x)} \]

[Out]

d*x*Sqrt[a + b*ArcSinh[c*x]] + (e*x^3*Sqrt[a + b*ArcSinh[c*x]])/3 + (Sqrt[b]*d*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b
*ArcSinh[c*x]]/Sqrt[b]])/(4*c) - (Sqrt[b]*e*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(16*c^3) +
(Sqrt[b]*e*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(48*c^3) - (Sqrt[b]*d*Sqrt
[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(4*c*E^(a/b)) + (Sqrt[b]*e*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]
/Sqrt[b]])/(16*c^3*E^(a/b)) - (Sqrt[b]*e*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(48*c^3*
E^((3*a)/b))

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Rubi [A]  time = 0.93161, antiderivative size = 322, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {5706, 5653, 5779, 3308, 2180, 2204, 2205, 5663, 3312} \[ -\frac{\sqrt{\pi } \sqrt{b} e e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 c^3}+\frac{\sqrt{\frac{\pi }{3}} \sqrt{b} e e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{48 c^3}+\frac{\sqrt{\pi } \sqrt{b} e e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 c^3}-\frac{\sqrt{\frac{\pi }{3}} \sqrt{b} e e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{48 c^3}+\frac{\sqrt{\pi } \sqrt{b} d e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 c}-\frac{\sqrt{\pi } \sqrt{b} d e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 c}+d x \sqrt{a+b \sinh ^{-1}(c x)}+\frac{1}{3} e x^3 \sqrt{a+b \sinh ^{-1}(c x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

d*x*Sqrt[a + b*ArcSinh[c*x]] + (e*x^3*Sqrt[a + b*ArcSinh[c*x]])/3 + (Sqrt[b]*d*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b
*ArcSinh[c*x]]/Sqrt[b]])/(4*c) - (Sqrt[b]*e*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(16*c^3) +
(Sqrt[b]*e*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(48*c^3) - (Sqrt[b]*d*Sqrt
[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(4*c*E^(a/b)) + (Sqrt[b]*e*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]
/Sqrt[b]])/(16*c^3*E^(a/b)) - (Sqrt[b]*e*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(48*c^3*
E^((3*a)/b))

Rule 5706

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a
+ b*ArcSinh[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[e, c^2*d] && IntegerQ[p] &&
(p > 0 || IGtQ[n, 0])

Rule 5653

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSinh[c*x])^n, x] - Dist[b*c*n, In
t[(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 5779

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

Rule 3308

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

Rule 2180

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] &&  !$UseGamma === True

Rule 2204

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 2205

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 5663

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 3312

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])
)

Rubi steps

\begin{align*} \int \left (d+e x^2\right ) \sqrt{a+b \sinh ^{-1}(c x)} \, dx &=\int \left (d \sqrt{a+b \sinh ^{-1}(c x)}+e x^2 \sqrt{a+b \sinh ^{-1}(c x)}\right ) \, dx\\ &=d \int \sqrt{a+b \sinh ^{-1}(c x)} \, dx+e \int x^2 \sqrt{a+b \sinh ^{-1}(c x)} \, dx\\ &=d x \sqrt{a+b \sinh ^{-1}(c x)}+\frac{1}{3} e x^3 \sqrt{a+b \sinh ^{-1}(c x)}-\frac{1}{2} (b c d) \int \frac{x}{\sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}} \, dx-\frac{1}{6} (b c e) \int \frac{x^3}{\sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}} \, dx\\ &=d x \sqrt{a+b \sinh ^{-1}(c x)}+\frac{1}{3} e x^3 \sqrt{a+b \sinh ^{-1}(c x)}-\frac{(b d) \operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{2 c}-\frac{(b e) \operatorname{Subst}\left (\int \frac{\sinh ^3(x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{6 c^3}\\ &=d x \sqrt{a+b \sinh ^{-1}(c x)}+\frac{1}{3} e x^3 \sqrt{a+b \sinh ^{-1}(c x)}+\frac{(b d) \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{4 c}-\frac{(b d) \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{4 c}-\frac{(i b e) \operatorname{Subst}\left (\int \left (\frac{3 i \sinh (x)}{4 \sqrt{a+b x}}-\frac{i \sinh (3 x)}{4 \sqrt{a+b x}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{6 c^3}\\ &=d x \sqrt{a+b \sinh ^{-1}(c x)}+\frac{1}{3} e x^3 \sqrt{a+b \sinh ^{-1}(c x)}+\frac{d \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{2 c}-\frac{d \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{2 c}-\frac{(b e) \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{24 c^3}+\frac{(b e) \operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{8 c^3}\\ &=d x \sqrt{a+b \sinh ^{-1}(c x)}+\frac{1}{3} e x^3 \sqrt{a+b \sinh ^{-1}(c x)}+\frac{\sqrt{b} d e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 c}-\frac{\sqrt{b} d e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 c}+\frac{(b e) \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{48 c^3}-\frac{(b e) \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{48 c^3}-\frac{(b e) \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{16 c^3}+\frac{(b e) \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{16 c^3}\\ &=d x \sqrt{a+b \sinh ^{-1}(c x)}+\frac{1}{3} e x^3 \sqrt{a+b \sinh ^{-1}(c x)}+\frac{\sqrt{b} d e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 c}-\frac{\sqrt{b} d e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 c}+\frac{e \operatorname{Subst}\left (\int e^{\frac{3 a}{b}-\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{24 c^3}-\frac{e \operatorname{Subst}\left (\int e^{-\frac{3 a}{b}+\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{24 c^3}-\frac{e \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{8 c^3}+\frac{e \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{8 c^3}\\ &=d x \sqrt{a+b \sinh ^{-1}(c x)}+\frac{1}{3} e x^3 \sqrt{a+b \sinh ^{-1}(c x)}+\frac{\sqrt{b} d e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 c}-\frac{\sqrt{b} e e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 c^3}+\frac{\sqrt{b} e e^{\frac{3 a}{b}} \sqrt{\frac{\pi }{3}} \text{erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{48 c^3}-\frac{\sqrt{b} d e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 c}+\frac{\sqrt{b} e e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 c^3}-\frac{\sqrt{b} e e^{-\frac{3 a}{b}} \sqrt{\frac{\pi }{3}} \text{erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{48 c^3}\\ \end{align*}

Mathematica [A]  time = 2.77962, size = 319, normalized size = 0.99 \[ \frac{e e^{-\frac{3 a}{b}} \sqrt{a+b \sinh ^{-1}(c x)} \left (9 e^{\frac{4 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \text{Gamma}\left (\frac{3}{2},\frac{a}{b}+\sinh ^{-1}(c x)\right )+\sqrt{3} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \text{Gamma}\left (\frac{3}{2},-\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )-9 e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \text{Gamma}\left (\frac{3}{2},-\frac{a+b \sinh ^{-1}(c x)}{b}\right )-\sqrt{3} e^{\frac{6 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \text{Gamma}\left (\frac{3}{2},\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )\right )}{72 c^3 \sqrt{-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}}}+\frac{d e^{-\frac{a}{b}} \sqrt{a+b \sinh ^{-1}(c x)} \left (\frac{\text{Gamma}\left (\frac{3}{2},-\frac{a+b \sinh ^{-1}(c x)}{b}\right )}{\sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}}}-\frac{e^{\frac{2 a}{b}} \text{Gamma}\left (\frac{3}{2},\frac{a}{b}+\sinh ^{-1}(c x)\right )}{\sqrt{\frac{a}{b}+\sinh ^{-1}(c x)}}\right )}{2 c} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(d*Sqrt[a + b*ArcSinh[c*x]]*(-((E^((2*a)/b)*Gamma[3/2, a/b + ArcSinh[c*x]])/Sqrt[a/b + ArcSinh[c*x]]) + Gamma[
3/2, -((a + b*ArcSinh[c*x])/b)]/Sqrt[-((a + b*ArcSinh[c*x])/b)]))/(2*c*E^(a/b)) + (e*Sqrt[a + b*ArcSinh[c*x]]*
(9*E^((4*a)/b)*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[3/2, a/b + ArcSinh[c*x]] + Sqrt[3]*Sqrt[a/b + ArcSinh[c*x
]]*Gamma[3/2, (-3*(a + b*ArcSinh[c*x]))/b] - 9*E^((2*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*Gamma[3/2, -((a + b*ArcSin
h[c*x])/b)] - Sqrt[3]*E^((6*a)/b)*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[3/2, (3*(a + b*ArcSinh[c*x]))/b]))/(72
*c^3*E^((3*a)/b)*Sqrt[-((a + b*ArcSinh[c*x])^2/b^2)])

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Maple [F]  time = 0.178, size = 0, normalized size = 0. \begin{align*} \int \left ( e{x}^{2}+d \right ) \sqrt{a+b{\it Arcsinh} \left ( cx \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x^{2} + d\right )} \sqrt{b \operatorname{arsinh}\left (c x\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a + b \operatorname{asinh}{\left (c x \right )}} \left (d + e x^{2}\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a + b*asinh(c*x))*(d + e*x**2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x^{2} + d\right )} \sqrt{b \operatorname{arsinh}\left (c x\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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