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

Optimal. Leaf size=269 \[ -\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} c \sin \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{d^2}+\frac{\sqrt{\pi } \sqrt{b} \cos \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi } \sqrt{b}}\right )}{8 d^2}+\frac{\sqrt{\pi } \sqrt{b} \sin \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right )}{8 d^2}+\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} c \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{d^2}-\frac{c (c+d x) \sqrt{a+b \sin ^{-1}(c+d x)}}{d^2}-\frac{\cos \left (2 \sin ^{-1}(c+d x)\right ) \sqrt{a+b \sin ^{-1}(c+d x)}}{4 d^2} \]

[Out]

-((c*(c + d*x)*Sqrt[a + b*ArcSin[c + d*x]])/d^2) - (Sqrt[a + b*ArcSin[c + d*x]]*Cos[2*ArcSin[c + d*x]])/(4*d^2
) + (Sqrt[b]*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(8*d^2) + (Sq
rt[b]*c*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/d^2 - (Sqrt[b]*c*Sqrt[
Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/d^2 + (Sqrt[b]*Sqrt[Pi]*FresnelS[(2
*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(8*d^2)

________________________________________________________________________________________

Rubi [A]  time = 0.76976, antiderivative size = 269, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {4805, 4747, 6741, 6742, 3386, 3353, 3352, 3351, 3385, 3354} \[ -\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} c \sin \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{d^2}+\frac{\sqrt{\pi } \sqrt{b} \cos \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi } \sqrt{b}}\right )}{8 d^2}+\frac{\sqrt{\pi } \sqrt{b} \sin \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right )}{8 d^2}+\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} c \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{d^2}-\frac{c (c+d x) \sqrt{a+b \sin ^{-1}(c+d x)}}{d^2}-\frac{\cos \left (2 \sin ^{-1}(c+d x)\right ) \sqrt{a+b \sin ^{-1}(c+d x)}}{4 d^2} \]

Antiderivative was successfully verified.

[In]

Int[x*Sqrt[a + b*ArcSin[c + d*x]],x]

[Out]

-((c*(c + d*x)*Sqrt[a + b*ArcSin[c + d*x]])/d^2) - (Sqrt[a + b*ArcSin[c + d*x]]*Cos[2*ArcSin[c + d*x]])/(4*d^2
) + (Sqrt[b]*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(8*d^2) + (Sq
rt[b]*c*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/d^2 - (Sqrt[b]*c*Sqrt[
Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/d^2 + (Sqrt[b]*Sqrt[Pi]*FresnelS[(2
*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(8*d^2)

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 4747

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[I
nt[(a + b*x)^n*Cos[x]*(c*d + e*Sin[x])^m, x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[m, 0
]

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 3386

Int[Cos[(c_.) + (d_.)*(x_)^(n_)]*((e_.)*(x_))^(m_.), x_Symbol] :> Simp[(e^(n - 1)*(e*x)^(m - n + 1)*Sin[c + d*
x^n])/(d*n), x] - Dist[(e^n*(m - n + 1))/(d*n), Int[(e*x)^(m - n)*Sin[c + d*x^n], x], x] /; FreeQ[{c, d, e}, x
] && IGtQ[n, 0] && LtQ[n, m + 1]

Rule 3353

Int[Sin[(c_) + (d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Dist[Sin[c], Int[Cos[d*(e + f*x)^2], x], x] + Dist[
Cos[c], Int[Sin[d*(e + f*x)^2], x], x] /; FreeQ[{c, d, e, f}, x]

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]

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 3385

Int[((e_.)*(x_))^(m_.)*Sin[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> -Simp[(e^(n - 1)*(e*x)^(m - n + 1)*Cos[c + d
*x^n])/(d*n), x] + Dist[(e^n*(m - n + 1))/(d*n), Int[(e*x)^(m - n)*Cos[c + d*x^n], x], x] /; FreeQ[{c, d, e},
x] && IGtQ[n, 0] && LtQ[n, m + 1]

Rule 3354

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

Rubi steps

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

Mathematica [C]  time = 2.96324, size = 256, normalized size = 0.95 \[ \frac{\frac{2 \left (-2 b c e^{-\frac{i a}{b}} \sqrt{-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{3}{2},-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )-2 b c e^{\frac{i a}{b}} \sqrt{\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{3}{2},\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+\cos \left (2 \sin ^{-1}(c+d x)\right ) \left (-\left (a+b \sin ^{-1}(c+d x)\right )\right )\right )}{\sqrt{a+b \sin ^{-1}(c+d x)}}+\frac{\sqrt{\pi } \cos \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi }}\right )}{\sqrt{\frac{1}{b}}}+\frac{\sqrt{\pi } \sin \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi }}\right )}{\sqrt{\frac{1}{b}}}}{8 d^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[Pi]])/Sqrt[b^(-1)] + (2*(-(
(a + b*ArcSin[c + d*x])*Cos[2*ArcSin[c + d*x]]) - (2*b*c*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, ((-
I)*(a + b*ArcSin[c + d*x]))/b])/E^((I*a)/b) - 2*b*c*E^((I*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2,
(I*(a + b*ArcSin[c + d*x]))/b]))/Sqrt[a + b*ArcSin[c + d*x]] + (Sqrt[Pi]*FresnelS[(2*Sqrt[b^(-1)]*Sqrt[a + b*
ArcSin[c + d*x]])/Sqrt[Pi]]*Sin[(2*a)/b])/Sqrt[b^(-1)])/(8*d^2)

________________________________________________________________________________________

Maple [A]  time = 0.104, size = 369, normalized size = 1.4 \begin{align*} -{\frac{1}{8\,{d}^{2}} \left ( -4\,\sqrt{2}\sqrt{{b}^{-1}}\sqrt{\pi }\sqrt{a+b\arcsin \left ( dx+c \right ) }\cos \left ({\frac{a}{b}} \right ){\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{a+b\arcsin \left ( dx+c \right ) }}{\sqrt{\pi }\sqrt{{b}^{-1}}b}} \right ) bc+4\,\sqrt{2}\sqrt{{b}^{-1}}\sqrt{\pi }\sqrt{a+b\arcsin \left ( dx+c \right ) }\sin \left ({\frac{a}{b}} \right ){\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{a+b\arcsin \left ( dx+c \right ) }}{\sqrt{\pi }\sqrt{{b}^{-1}}b}} \right ) bc-\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+8\,\arcsin \left ( dx+c \right ) \sin \left ({\frac{a+b\arcsin \left ( dx+c \right ) }{b}}-{\frac{a}{b}} \right ) bc+2\,\arcsin \left ( dx+c \right ) \cos \left ( 2\,{\frac{a+b\arcsin \left ( dx+c \right ) }{b}}-2\,{\frac{a}{b}} \right ) b+8\,\sin \left ({\frac{a+b\arcsin \left ( dx+c \right ) }{b}}-{\frac{a}{b}} \right ) ac+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(x*(a+b*arcsin(d*x+c))^(1/2),x)

[Out]

-1/8/d^2/(a+b*arcsin(d*x+c))^(1/2)*(-4*2^(1/2)*(1/b)^(1/2)*Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*cos(a/b)*Fresnel
S(2^(1/2)/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b*c+4*2^(1/2)*(1/b)^(1/2)*Pi^(1/2)*(a+b*arcsin(d*x
+c))^(1/2)*sin(a/b)*FresnelC(2^(1/2)/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b*c-(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)*FresnelC(2/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)
*b+8*arcsin(d*x+c)*sin((a+b*arcsin(d*x+c))/b-a/b)*b*c+2*arcsin(d*x+c)*cos(2*(a+b*arcsin(d*x+c))/b-2*a/b)*b+8*s
in((a+b*arcsin(d*x+c))/b-a/b)*a*c+2*cos(2*(a+b*arcsin(d*x+c))/b-2*a/b)*a)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [B]  time = 1.93099, size = 593, normalized size = 2.2 \begin{align*} -\frac{\sqrt{2} \sqrt{\pi } b^{2} c i \operatorname{erf}\left (-\frac{\sqrt{2} \sqrt{b \arcsin \left (d x + c\right ) + a} i}{2 \, \sqrt{{\left | b \right |}}} - \frac{\sqrt{2} \sqrt{b \arcsin \left (d x + c\right ) + a} \sqrt{{\left | b \right |}}}{2 \, b}\right ) e^{\left (\frac{a i}{b}\right )}}{4 \,{\left (\frac{b^{2} i}{\sqrt{{\left | b \right |}}} + b \sqrt{{\left | b \right |}}\right )} d^{2}} - \frac{\sqrt{2} \sqrt{\pi } b^{2} c i \operatorname{erf}\left (\frac{\sqrt{2} \sqrt{b \arcsin \left (d x + c\right ) + a} i}{2 \, \sqrt{{\left | b \right |}}} - \frac{\sqrt{2} \sqrt{b \arcsin \left (d x + c\right ) + a} \sqrt{{\left | b \right |}}}{2 \, b}\right ) e^{\left (-\frac{a i}{b}\right )}}{4 \,{\left (\frac{b^{2} i}{\sqrt{{\left | b \right |}}} - b \sqrt{{\left | b \right |}}\right )} d^{2}} + \frac{\sqrt{b \arcsin \left (d x + c\right ) + a} c i e^{\left (i \arcsin \left (d x + c\right )\right )}}{2 \, d^{2}} - \frac{\sqrt{b \arcsin \left (d x + c\right ) + a} c i e^{\left (-i \arcsin \left (d x + c\right )\right )}}{2 \, d^{2}} - \frac{\sqrt{\pi } b^{\frac{3}{2}} \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}\right )}}{16 \,{\left (\frac{b^{2} i}{{\left | b \right |}} + b\right )} d^{2}} + \frac{\sqrt{\pi } b^{\frac{3}{2}} \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}\right )}}{16 \,{\left (\frac{b^{2} i}{{\left | b \right |}} - b\right )} d^{2}} - \frac{\sqrt{b \arcsin \left (d x + c\right ) + a} e^{\left (2 \, i \arcsin \left (d x + c\right )\right )}}{8 \, d^{2}} - \frac{\sqrt{b \arcsin \left (d x + c\right ) + a} e^{\left (-2 \, i \arcsin \left (d x + c\right )\right )}}{8 \, d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(2)*sqrt(pi)*b^2*c*i*erf(-1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b
*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(a*i/b)/((b^2*i/sqrt(abs(b)) + b*sqrt(abs(b)))*d^2) - 1/4*sqrt(2)*sqrt
(pi)*b^2*c*i*erf(1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) +
a)*sqrt(abs(b))/b)*e^(-a*i/b)/((b^2*i/sqrt(abs(b)) - b*sqrt(abs(b)))*d^2) + 1/2*sqrt(b*arcsin(d*x + c) + a)*c
*i*e^(i*arcsin(d*x + c))/d^2 - 1/2*sqrt(b*arcsin(d*x + c) + a)*c*i*e^(-i*arcsin(d*x + c))/d^2 - 1/16*sqrt(pi)*
b^(3/2)*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)/(
(b^2*i/abs(b) + b)*d^2) + 1/16*sqrt(pi)*b^(3/2)*erf(sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(b*arcs
in(d*x + c) + a)/sqrt(b))*e^(-2*a*i/b)/((b^2*i/abs(b) - b)*d^2) - 1/8*sqrt(b*arcsin(d*x + c) + a)*e^(2*i*arcsi
n(d*x + c))/d^2 - 1/8*sqrt(b*arcsin(d*x + c) + a)*e^(-2*i*arcsin(d*x + c))/d^2