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

Optimal. Leaf size=99 \[ -\frac{4 b \sqrt{e} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right ),-1\right )}{9 d}+\frac{2 (e (c+d x))^{3/2} \left (a+b \sin ^{-1}(c+d x)\right )}{3 d e}+\frac{4 b \sqrt{1-(c+d x)^2} \sqrt{e (c+d x)}}{9 d} \]

[Out]

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

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Rubi [A]  time = 0.0809253, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {4805, 4627, 321, 329, 221} \[ \frac{2 (e (c+d x))^{3/2} \left (a+b \sin ^{-1}(c+d x)\right )}{3 d e}+\frac{4 b \sqrt{1-(c+d x)^2} \sqrt{e (c+d x)}}{9 d}-\frac{4 b \sqrt{e} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right )\right |-1\right )}{9 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4805

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[I
nt[((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]

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

Mathematica [C]  time = 0.0315626, size = 87, normalized size = 0.88 \[ \frac{2 \sqrt{e (c+d x)} \left (-2 b \text{Hypergeometric2F1}\left (\frac{1}{4},\frac{1}{2},\frac{5}{4},(c+d x)^2\right )+3 a c+3 a d x+2 b \sqrt{1-(c+d x)^2}+3 b c \sin ^{-1}(c+d x)+3 b d x \sin ^{-1}(c+d x)\right )}{9 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.011, size = 172, normalized size = 1.7 \begin{align*} 2\,{\frac{1}{de} \left ( 1/3\, \left ( dex+ce \right ) ^{3/2}a+b \left ( 1/3\, \left ( dex+ce \right ) ^{3/2}\arcsin \left ({\frac{dex+ce}{e}} \right ) -2/3\,{\frac{1}{e} \left ( -1/3\,{e}^{2}\sqrt{dex+ce}\sqrt{-{\frac{ \left ( dex+ce \right ) ^{2}}{{e}^{2}}}+1}+1/3\,{\frac{{e}^{2}{\it EllipticF} \left ( \sqrt{dex+ce}\sqrt{{e}^{-1}},i \right ) }{\sqrt{{e}^{-1}}}\sqrt{1-{\frac{dex+ce}{e}}}\sqrt{{\frac{dex+ce}{e}}+1}{\frac{1}{\sqrt{-{\frac{ \left ( dex+ce \right ) ^{2}}{{e}^{2}}}+1}}}} \right ) } \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d/e*(1/3*(d*e*x+c*e)^(3/2)*a+b*(1/3*(d*e*x+c*e)^(3/2)*arcsin((d*e*x+c*e)/e)-2/3/e*(-1/3*e^2*(d*e*x+c*e)^(1/2
)*(-(d*e*x+c*e)^2/e^2+1)^(1/2)+1/3*e^2/(1/e)^(1/2)*(1-(d*e*x+c*e)/e)^(1/2)*((d*e*x+c*e)/e+1)^(1/2)/(-(d*e*x+c*
e)^2/e^2+1)^(1/2)*EllipticF((d*e*x+c*e)^(1/2)*(1/e)^(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*e*x+c*e)^(1/2)*(a+b*arcsin(d*x+c)),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 e x + c e}{\left (b \arcsin \left (d x + c\right ) + a\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 2.51641, size = 104, normalized size = 1.05 \begin{align*} \frac{2 a \left (c e + d e x\right )^{\frac{3}{2}}}{3 d e} + \frac{2 b \left (c e + d e x\right )^{\frac{3}{2}} \operatorname{asin}{\left (c + d x \right )}}{3 d e} - \frac{b \left (c e + d e 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 |{\frac{\left (c e + d e x\right )^{2} e^{2 i \pi }}{e^{2}}} \right )}}{3 d e^{2} \Gamma \left (\frac{9}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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