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

Optimal. Leaf size=156 \[ \frac{\sqrt{\pi } \sqrt{b} e \cos \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi } \sqrt{b}}\right )}{8 d}+\frac{\sqrt{\pi } \sqrt{b} e \sin \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right )}{8 d}+\frac{e (c+d x)^2 \sqrt{a+b \sin ^{-1}(c+d x)}}{2 d}-\frac{e \sqrt{a+b \sin ^{-1}(c+d x)}}{4 d} \]

[Out]

-(e*Sqrt[a + b*ArcSin[c + d*x]])/(4*d) + (e*(c + d*x)^2*Sqrt[a + b*ArcSin[c + d*x]])/(2*d) + (Sqrt[b]*e*Sqrt[P
i]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(8*d) + (Sqrt[b]*e*Sqrt[Pi]*Fres
nelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(8*d)

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Rubi [A]  time = 0.424398, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.435, Rules used = {4805, 12, 4629, 4723, 3312, 3306, 3305, 3351, 3304, 3352} \[ \frac{\sqrt{\pi } \sqrt{b} e \cos \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi } \sqrt{b}}\right )}{8 d}+\frac{\sqrt{\pi } \sqrt{b} e \sin \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right )}{8 d}+\frac{e (c+d x)^2 \sqrt{a+b \sin ^{-1}(c+d x)}}{2 d}-\frac{e \sqrt{a+b \sin ^{-1}(c+d x)}}{4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(e*Sqrt[a + b*ArcSin[c + d*x]])/(4*d) + (e*(c + d*x)^2*Sqrt[a + b*ArcSin[c + d*x]])/(2*d) + (Sqrt[b]*e*Sqrt[P
i]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(8*d) + (Sqrt[b]*e*Sqrt[Pi]*Fres
nelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(8*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 12

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

Rule 4629

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

Rule 4723

Int[((a_.) + ArcSin[(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*Sin[x]^m*Cos[x]^(2*p + 1), x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e, n},
x] && EqQ[c^2*d + e, 0] && IntegerQ[2*p] && GtQ[p, -1] && IGtQ[m, 0] && (IntegerQ[p] || GtQ[d, 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])
)

Rule 3306

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d +
f*x]/Sqrt[c + d*x], x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/Sqrt[c + d*x], x], x] /; FreeQ[{c
, d, e, f}, x] && ComplexFreeQ[f] && NeQ[d*e - c*f, 0]

Rule 3305

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Sin[(f*x^2)/d], x], x,
Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

Rule 3351

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rule 3304

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Cos[(f*x^2)/d],
x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

Rule 3352

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rubi steps

\begin{align*} \int (c e+d e x) \sqrt{a+b \sin ^{-1}(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int e x \sqrt{a+b \sin ^{-1}(x)} \, dx,x,c+d x\right )}{d}\\ &=\frac{e \operatorname{Subst}\left (\int x \sqrt{a+b \sin ^{-1}(x)} \, dx,x,c+d x\right )}{d}\\ &=\frac{e (c+d x)^2 \sqrt{a+b \sin ^{-1}(c+d x)}}{2 d}-\frac{(b e) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-x^2} \sqrt{a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{4 d}\\ &=\frac{e (c+d x)^2 \sqrt{a+b \sin ^{-1}(c+d x)}}{2 d}-\frac{(b e) \operatorname{Subst}\left (\int \frac{\sin ^2(x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{4 d}\\ &=\frac{e (c+d x)^2 \sqrt{a+b \sin ^{-1}(c+d x)}}{2 d}-\frac{(b e) \operatorname{Subst}\left (\int \left (\frac{1}{2 \sqrt{a+b x}}-\frac{\cos (2 x)}{2 \sqrt{a+b x}}\right ) \, dx,x,\sin ^{-1}(c+d x)\right )}{4 d}\\ &=-\frac{e \sqrt{a+b \sin ^{-1}(c+d x)}}{4 d}+\frac{e (c+d x)^2 \sqrt{a+b \sin ^{-1}(c+d x)}}{2 d}+\frac{(b e) \operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{8 d}\\ &=-\frac{e \sqrt{a+b \sin ^{-1}(c+d x)}}{4 d}+\frac{e (c+d x)^2 \sqrt{a+b \sin ^{-1}(c+d x)}}{2 d}+\frac{\left (b e \cos \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{2 a}{b}+2 x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{8 d}+\frac{\left (b e \sin \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{2 a}{b}+2 x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{8 d}\\ &=-\frac{e \sqrt{a+b \sin ^{-1}(c+d x)}}{4 d}+\frac{e (c+d x)^2 \sqrt{a+b \sin ^{-1}(c+d x)}}{2 d}+\frac{\left (e \cos \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{2 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{4 d}+\frac{\left (e \sin \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{2 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{4 d}\\ &=-\frac{e \sqrt{a+b \sin ^{-1}(c+d x)}}{4 d}+\frac{e (c+d x)^2 \sqrt{a+b \sin ^{-1}(c+d x)}}{2 d}+\frac{\sqrt{b} e \sqrt{\pi } \cos \left (\frac{2 a}{b}\right ) C\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right )}{8 d}+\frac{\sqrt{b} e \sqrt{\pi } S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right ) \sin \left (\frac{2 a}{b}\right )}{8 d}\\ \end{align*}

Mathematica [C]  time = 0.0745581, size = 154, normalized size = 0.99 \[ -\frac{e e^{-\frac{2 i a}{b}} \sqrt{a+b \sin ^{-1}(c+d x)} \left (\sqrt{\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{3}{2},-\frac{2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+e^{\frac{4 i a}{b}} \sqrt{-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{3}{2},\frac{2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )\right )}{8 \sqrt{2} d \sqrt{\frac{\left (a+b \sin ^{-1}(c+d x)\right )^2}{b^2}}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.074, size = 190, normalized size = 1.2 \begin{align*} -{\frac{e}{8\,d} \left ( -\sqrt{{b}^{-1}}\sqrt{\pi }\sqrt{a+b\arcsin \left ( dx+c \right ) }{\it FresnelS} \left ( 2\,{\frac{\sqrt{a+b\arcsin \left ( dx+c \right ) }}{\sqrt{\pi }\sqrt{{b}^{-1}}b}} \right ) \sin \left ( 2\,{\frac{a}{b}} \right ) b-\sqrt{{b}^{-1}}\sqrt{\pi }\sqrt{a+b\arcsin \left ( dx+c \right ) }\cos \left ( 2\,{\frac{a}{b}} \right ){\it FresnelC} \left ( 2\,{\frac{\sqrt{a+b\arcsin \left ( dx+c \right ) }}{\sqrt{\pi }\sqrt{{b}^{-1}}b}} \right ) b+2\,\arcsin \left ( dx+c \right ) \cos \left ( 2\,{\frac{a+b\arcsin \left ( dx+c \right ) }{b}}-2\,{\frac{a}{b}} \right ) b+2\,\cos \left ( 2\,{\frac{a+b\arcsin \left ( dx+c \right ) }{b}}-2\,{\frac{a}{b}} \right ) a \right ){\frac{1}{\sqrt{a+b\arcsin \left ( dx+c \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*e/d/(a+b*arcsin(d*x+c))^(1/2)*(-(1/b)^(1/2)*Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*FresnelS(2/Pi^(1/2)/(1/b)^
(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*sin(2*a/b)*b-(1/b)^(1/2)*Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*cos(2*a/b)*Fres
nelC(2/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b+2*arcsin(d*x+c)*cos(2*(a+b*arcsin(d*x+c))/b-2*a/b)*
b+2*cos(2*(a+b*arcsin(d*x+c))/b-2*a/b)*a)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.82442, size = 275, normalized size = 1.76 \begin{align*} -\frac{\sqrt{\pi } \sqrt{b} \operatorname{erf}\left (-\frac{\sqrt{b \arcsin \left (d x + c\right ) + a} \sqrt{b} i}{{\left | b \right |}} - \frac{\sqrt{b \arcsin \left (d x + c\right ) + a}}{\sqrt{b}}\right ) e^{\left (\frac{2 \, a i}{b} + 1\right )}}{16 \, d{\left (\frac{b i}{{\left | b \right |}} + 1\right )}} + \frac{\sqrt{\pi } \sqrt{b} \operatorname{erf}\left (\frac{\sqrt{b \arcsin \left (d x + c\right ) + a} \sqrt{b} i}{{\left | b \right |}} - \frac{\sqrt{b \arcsin \left (d x + c\right ) + a}}{\sqrt{b}}\right ) e^{\left (-\frac{2 \, a i}{b} + 1\right )}}{16 \, d{\left (\frac{b i}{{\left | b \right |}} - 1\right )}} - \frac{\sqrt{b \arcsin \left (d x + c\right ) + a} e^{\left (2 \, i \arcsin \left (d x + c\right ) + 1\right )}}{8 \, d} - \frac{\sqrt{b \arcsin \left (d x + c\right ) + a} e^{\left (-2 \, i \arcsin \left (d x + c\right ) + 1\right )}}{8 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*sqrt(pi)*sqrt(b)*erf(-sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(b*arcsin(d*x + c) + a)/sqrt(b)
)*e^(2*a*i/b + 1)/(d*(b*i/abs(b) + 1)) + 1/16*sqrt(pi)*sqrt(b)*erf(sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b
) - sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(-2*a*i/b + 1)/(d*(b*i/abs(b) - 1)) - 1/8*sqrt(b*arcsin(d*x + c) +
a)*e^(2*i*arcsin(d*x + c) + 1)/d - 1/8*sqrt(b*arcsin(d*x + c) + a)*e^(-2*i*arcsin(d*x + c) + 1)/d