3.205 \(\int \sqrt{d x} (a+b \sin ^{-1}(c x)) \, dx\)

Optimal. Leaf size=88 \[ -\frac{4 b \sqrt{d} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right ),-1\right )}{9 c^{3/2}}+\frac{2 (d x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 d}+\frac{4 b \sqrt{1-c^2 x^2} \sqrt{d x}}{9 c} \]

[Out]

(4*b*Sqrt[d*x]*Sqrt[1 - c^2*x^2])/(9*c) + (2*(d*x)^(3/2)*(a + b*ArcSin[c*x]))/(3*d) - (4*b*Sqrt[d]*EllipticF[A
rcSin[(Sqrt[c]*Sqrt[d*x])/Sqrt[d]], -1])/(9*c^(3/2))

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Rubi [A]  time = 0.044585, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4627, 321, 329, 221} \[ \frac{2 (d x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 d}+\frac{4 b \sqrt{1-c^2 x^2} \sqrt{d x}}{9 c}-\frac{4 b \sqrt{d} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right )\right |-1\right )}{9 c^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[d*x]*(a + b*ArcSin[c*x]),x]

[Out]

(4*b*Sqrt[d*x]*Sqrt[1 - c^2*x^2])/(9*c) + (2*(d*x)^(3/2)*(a + b*ArcSin[c*x]))/(3*d) - (4*b*Sqrt[d]*EllipticF[A
rcSin[(Sqrt[c]*Sqrt[d*x])/Sqrt[d]], -1])/(9*c^(3/2))

Rule 4627

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

Rule 321

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

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[(Rt[-b, 4]*x)/Rt[a, 4]], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rubi steps

\begin{align*} \int \sqrt{d x} \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac{2 (d x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 d}-\frac{(2 b c) \int \frac{(d x)^{3/2}}{\sqrt{1-c^2 x^2}} \, dx}{3 d}\\ &=\frac{4 b \sqrt{d x} \sqrt{1-c^2 x^2}}{9 c}+\frac{2 (d x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 d}-\frac{(2 b d) \int \frac{1}{\sqrt{d x} \sqrt{1-c^2 x^2}} \, dx}{9 c}\\ &=\frac{4 b \sqrt{d x} \sqrt{1-c^2 x^2}}{9 c}+\frac{2 (d x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 d}-\frac{(4 b) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{c^2 x^4}{d^2}}} \, dx,x,\sqrt{d x}\right )}{9 c}\\ &=\frac{4 b \sqrt{d x} \sqrt{1-c^2 x^2}}{9 c}+\frac{2 (d x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 d}-\frac{4 b \sqrt{d} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right )\right |-1\right )}{9 c^{3/2}}\\ \end{align*}

Mathematica [C]  time = 0.0172088, size = 66, normalized size = 0.75 \[ \frac{2 \sqrt{d x} \left (-2 b \text{Hypergeometric2F1}\left (\frac{1}{4},\frac{1}{2},\frac{5}{4},c^2 x^2\right )+3 a c x+2 b \sqrt{1-c^2 x^2}+3 b c x \sin ^{-1}(c x)\right )}{9 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d*x]*(3*a*c*x + 2*b*Sqrt[1 - c^2*x^2] + 3*b*c*x*ArcSin[c*x] - 2*b*Hypergeometric2F1[1/4, 1/2, 5/4, c^2
*x^2]))/(9*c)

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Maple [A]  time = 0.008, size = 119, normalized size = 1.4 \begin{align*} 2\,{\frac{1}{d} \left ( 1/3\, \left ( dx \right ) ^{3/2}a+b \left ( 1/3\, \left ( dx \right ) ^{3/2}\arcsin \left ( cx \right ) -2/3\,{\frac{c}{d} \left ( -1/3\,{\frac{{d}^{2}\sqrt{dx}\sqrt{-{c}^{2}{x}^{2}+1}}{{c}^{2}}}+1/3\,{\frac{{d}^{2}\sqrt{-cx+1}\sqrt{cx+1}}{{c}^{2}\sqrt{-{c}^{2}{x}^{2}+1}}{\it EllipticF} \left ( \sqrt{dx}\sqrt{{\frac{c}{d}}},i \right ){\frac{1}{\sqrt{{\frac{c}{d}}}}}} \right ) } \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d*(1/3*(d*x)^(3/2)*a+b*(1/3*(d*x)^(3/2)*arcsin(c*x)-2/3*c/d*(-1/3/c^2*d^2*(d*x)^(1/2)*(-c^2*x^2+1)^(1/2)+1/3
/c^2*d^2/(c/d)^(1/2)*(-c*x+1)^(1/2)*(c*x+1)^(1/2)/(-c^2*x^2+1)^(1/2)*EllipticF((d*x)^(1/2)*(c/d)^(1/2),I))))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{d x}{\left (b \arcsin \left (c x\right ) + a\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(d*x)*(b*arcsin(c*x) + a), x)

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Sympy [A]  time = 4.41945, size = 76, normalized size = 0.86 \begin{align*} \frac{2 a \left (d x\right )^{\frac{3}{2}}}{3 d} - \frac{b c \left (d x\right )^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{c^{2} x^{2} e^{2 i \pi }} \right )}}{3 d^{2} \Gamma \left (\frac{9}{4}\right )} + \frac{2 b \left (d x\right )^{\frac{3}{2}} \operatorname{asin}{\left (c x \right )}}{3 d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*(d*x)**(3/2)/(3*d) - b*c*(d*x)**(5/2)*gamma(5/4)*hyper((1/2, 5/4), (9/4,), c**2*x**2*exp_polar(2*I*pi))/(3
*d**2*gamma(9/4)) + 2*b*(d*x)**(3/2)*asin(c*x)/(3*d)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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