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

Optimal. Leaf size=274 \[ \frac{\sqrt{\frac{\pi }{2}} \sqrt{b} e^2 \sin \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 d}-\frac{\sqrt{\frac{\pi }{6}} \sqrt{b} e^2 \sin \left (\frac{3 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{12 d}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} e^2 \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 d}+\frac{\sqrt{\frac{\pi }{6}} \sqrt{b} e^2 \cos \left (\frac{3 a}{b}\right ) S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{12 d}+\frac{e^2 (c+d x)^3 \sqrt{a+b \sin ^{-1}(c+d x)}}{3 d} \]

[Out]

(e^2*(c + d*x)^3*Sqrt[a + b*ArcSin[c + d*x]])/(3*d) - (Sqrt[b]*e^2*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sq
rt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(4*d) + (Sqrt[b]*e^2*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a
+ b*ArcSin[c + d*x]])/Sqrt[b]])/(12*d) + (Sqrt[b]*e^2*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*
x]])/Sqrt[b]]*Sin[a/b])/(4*d) - (Sqrt[b]*e^2*Sqrt[Pi/6]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt
[b]]*Sin[(3*a)/b])/(12*d)

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Rubi [A]  time = 0.748132, antiderivative size = 274, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 10, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4805, 12, 4629, 4723, 3312, 3306, 3305, 3351, 3304, 3352} \[ \frac{\sqrt{\frac{\pi }{2}} \sqrt{b} e^2 \sin \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 d}-\frac{\sqrt{\frac{\pi }{6}} \sqrt{b} e^2 \sin \left (\frac{3 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{12 d}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} e^2 \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 d}+\frac{\sqrt{\frac{\pi }{6}} \sqrt{b} e^2 \cos \left (\frac{3 a}{b}\right ) S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{12 d}+\frac{e^2 (c+d x)^3 \sqrt{a+b \sin ^{-1}(c+d x)}}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e^2*(c + d*x)^3*Sqrt[a + b*ArcSin[c + d*x]])/(3*d) - (Sqrt[b]*e^2*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sq
rt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(4*d) + (Sqrt[b]*e^2*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a
+ b*ArcSin[c + d*x]])/Sqrt[b]])/(12*d) + (Sqrt[b]*e^2*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*
x]])/Sqrt[b]]*Sin[a/b])/(4*d) - (Sqrt[b]*e^2*Sqrt[Pi/6]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt
[b]]*Sin[(3*a)/b])/(12*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)^2 \sqrt{a+b \sin ^{-1}(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int e^2 x^2 \sqrt{a+b \sin ^{-1}(x)} \, dx,x,c+d x\right )}{d}\\ &=\frac{e^2 \operatorname{Subst}\left (\int x^2 \sqrt{a+b \sin ^{-1}(x)} \, dx,x,c+d x\right )}{d}\\ &=\frac{e^2 (c+d x)^3 \sqrt{a+b \sin ^{-1}(c+d x)}}{3 d}-\frac{\left (b e^2\right ) \operatorname{Subst}\left (\int \frac{x^3}{\sqrt{1-x^2} \sqrt{a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{6 d}\\ &=\frac{e^2 (c+d x)^3 \sqrt{a+b \sin ^{-1}(c+d x)}}{3 d}-\frac{\left (b e^2\right ) \operatorname{Subst}\left (\int \frac{\sin ^3(x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{6 d}\\ &=\frac{e^2 (c+d x)^3 \sqrt{a+b \sin ^{-1}(c+d x)}}{3 d}-\frac{\left (b e^2\right ) \operatorname{Subst}\left (\int \left (\frac{3 \sin (x)}{4 \sqrt{a+b x}}-\frac{\sin (3 x)}{4 \sqrt{a+b x}}\right ) \, dx,x,\sin ^{-1}(c+d x)\right )}{6 d}\\ &=\frac{e^2 (c+d x)^3 \sqrt{a+b \sin ^{-1}(c+d x)}}{3 d}+\frac{\left (b e^2\right ) \operatorname{Subst}\left (\int \frac{\sin (3 x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{24 d}-\frac{\left (b e^2\right ) \operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{8 d}\\ &=\frac{e^2 (c+d x)^3 \sqrt{a+b \sin ^{-1}(c+d x)}}{3 d}-\frac{\left (b e^2 \cos \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{a}{b}+x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{8 d}+\frac{\left (b e^2 \cos \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{3 a}{b}+3 x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{24 d}+\frac{\left (b e^2 \sin \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{a}{b}+x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{8 d}-\frac{\left (b e^2 \sin \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{3 a}{b}+3 x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{24 d}\\ &=\frac{e^2 (c+d x)^3 \sqrt{a+b \sin ^{-1}(c+d x)}}{3 d}-\frac{\left (e^2 \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 )}{4 d}+\frac{\left (e^2 \cos \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{3 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{12 d}+\frac{\left (e^2 \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 )}{4 d}-\frac{\left (e^2 \sin \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{3 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{12 d}\\ &=\frac{e^2 (c+d x)^3 \sqrt{a+b \sin ^{-1}(c+d x)}}{3 d}-\frac{\sqrt{b} e^2 \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 )}{4 d}+\frac{\sqrt{b} e^2 \sqrt{\frac{\pi }{6}} \cos \left (\frac{3 a}{b}\right ) S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{12 d}+\frac{\sqrt{b} e^2 \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 )}{4 d}-\frac{\sqrt{b} e^2 \sqrt{\frac{\pi }{6}} C\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right ) \sin \left (\frac{3 a}{b}\right )}{12 d}\\ \end{align*}

Mathematica [C]  time = 0.287771, size = 269, normalized size = 0.98 \[ -\frac{i e^2 e^{-\frac{3 i a}{b}} \sqrt{a+b \sin ^{-1}(c+d x)} \left (9 e^{\frac{2 i a}{b}} \sqrt{\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{3}{2},-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )-9 e^{\frac{4 i a}{b}} \sqrt{-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{3}{2},\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+\sqrt{3} \left (e^{\frac{6 i a}{b}} \sqrt{-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{3}{2},\frac{3 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )-\sqrt{\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{3}{2},-\frac{3 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )\right )\right )}{72 d \sqrt{\frac{\left (a+b \sin ^{-1}(c+d x)\right )^2}{b^2}}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.103, size = 389, normalized size = 1.4 \begin{align*}{\frac{{e}^{2}}{72\,d} \left ( \sqrt{3}\cos \left ( 3\,{\frac{a}{b}} \right ){\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{3}}{\sqrt{\pi }b}\sqrt{a+b\arcsin \left ( dx+c \right ) }{\frac{1}{\sqrt{{b}^{-1}}}}} \right ) \sqrt{{b}^{-1}}\sqrt{\pi }\sqrt{2}\sqrt{a+b\arcsin \left ( dx+c \right ) }b-\sqrt{3}\sin \left ( 3\,{\frac{a}{b}} \right ){\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{3}}{\sqrt{\pi }b}\sqrt{a+b\arcsin \left ( dx+c \right ) }{\frac{1}{\sqrt{{b}^{-1}}}}} \right ) \sqrt{{b}^{-1}}\sqrt{\pi }\sqrt{2}\sqrt{a+b\arcsin \left ( dx+c \right ) }b-9\,\sqrt{2}\sqrt{\pi }\sqrt{{b}^{-1}}\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 ) b+9\,\sqrt{2}\sqrt{\pi }\sqrt{{b}^{-1}}\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 ) b+18\,\arcsin \left ( dx+c \right ) \sin \left ({\frac{a+b\arcsin \left ( dx+c \right ) }{b}}-{\frac{a}{b}} \right ) b+18\,\sin \left ({\frac{a+b\arcsin \left ( dx+c \right ) }{b}}-{\frac{a}{b}} \right ) a-6\,\arcsin \left ( dx+c \right ) \sin \left ( 3\,{\frac{a+b\arcsin \left ( dx+c \right ) }{b}}-3\,{\frac{a}{b}} \right ) b-6\,\sin \left ( 3\,{\frac{a+b\arcsin \left ( dx+c \right ) }{b}}-3\,{\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)^2*(a+b*arcsin(d*x+c))^(1/2),x)

[Out]

1/72/d*e^2/(a+b*arcsin(d*x+c))^(1/2)*(3^(1/2)*cos(3*a/b)*FresnelS(2^(1/2)/Pi^(1/2)*3^(1/2)/(1/b)^(1/2)*(a+b*ar
csin(d*x+c))^(1/2)/b)*(1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*b-3^(1/2)*sin(3*a/b)*FresnelC(2^(
1/2)/Pi^(1/2)*3^(1/2)/(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*(1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arcsin(d*x+c)
)^(1/2)*b-9*2^(1/2)*Pi^(1/2)*(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)/(1/b)^(1
/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b+9*2^(1/2)*Pi^(1/2)*(1/b)^(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+18*arcsin(d*x+c)*sin((a+b*arcsin(d*x+c))/b-a/b)*b+
18*sin((a+b*arcsin(d*x+c))/b-a/b)*a-6*arcsin(d*x+c)*sin(3*(a+b*arcsin(d*x+c))/b-3*a/b)*b-6*sin(3*(a+b*arcsin(d
*x+c))/b-3*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 )}^{2} \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)^2*(a+b*arcsin(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate((d*e*x + c*e)^2*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)^2*(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^{2} \left (\int c^{2} \sqrt{a + b \operatorname{asin}{\left (c + d x \right )}}\, dx + \int d^{2} x^{2} \sqrt{a + b \operatorname{asin}{\left (c + d x \right )}}\, dx + \int 2 c 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)**2*(a+b*asin(d*x+c))**(1/2),x)

[Out]

e**2*(Integral(c**2*sqrt(a + b*asin(c + d*x)), x) + Integral(d**2*x**2*sqrt(a + b*asin(c + d*x)), x) + Integra
l(2*c*d*x*sqrt(a + b*asin(c + d*x)), x))

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Giac [B]  time = 2.17925, size = 625, normalized size = 2.28 \begin{align*} \frac{\sqrt{2} \sqrt{\pi } b 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} + 2\right )}}{16 \,{\left (\frac{b i}{\sqrt{{\left | b \right |}}} + \sqrt{{\left | b \right |}}\right )} d} + \frac{\sqrt{2} \sqrt{\pi } b 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} + 2\right )}}{16 \,{\left (\frac{b i}{\sqrt{{\left | b \right |}}} - \sqrt{{\left | b \right |}}\right )} d} - \frac{\sqrt{\pi } \sqrt{b} i \operatorname{erf}\left (-\frac{\sqrt{6} \sqrt{b \arcsin \left (d x + c\right ) + a} \sqrt{b} i}{2 \,{\left | b \right |}} - \frac{\sqrt{6} \sqrt{b \arcsin \left (d x + c\right ) + a}}{2 \, \sqrt{b}}\right ) e^{\left (\frac{3 \, a i}{b} + 2\right )}}{24 \,{\left (\frac{\sqrt{6} b i}{{\left | b \right |}} + \sqrt{6}\right )} d} - \frac{\sqrt{\pi } \sqrt{b} i \operatorname{erf}\left (\frac{\sqrt{6} \sqrt{b \arcsin \left (d x + c\right ) + a} \sqrt{b} i}{2 \,{\left | b \right |}} - \frac{\sqrt{6} \sqrt{b \arcsin \left (d x + c\right ) + a}}{2 \, \sqrt{b}}\right ) e^{\left (-\frac{3 \, a i}{b} + 2\right )}}{24 \,{\left (\frac{\sqrt{6} b i}{{\left | b \right |}} - \sqrt{6}\right )} d} + \frac{\sqrt{b \arcsin \left (d x + c\right ) + a} i e^{\left (3 \, i \arcsin \left (d x + c\right ) + 2\right )}}{24 \, d} - \frac{\sqrt{b \arcsin \left (d x + c\right ) + a} i e^{\left (i \arcsin \left (d x + c\right ) + 2\right )}}{8 \, d} + \frac{\sqrt{b \arcsin \left (d x + c\right ) + a} i e^{\left (-i \arcsin \left (d x + c\right ) + 2\right )}}{8 \, d} - \frac{\sqrt{b \arcsin \left (d x + c\right ) + a} i e^{\left (-3 \, i \arcsin \left (d x + c\right ) + 2\right )}}{24 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*sqrt(2)*sqrt(pi)*b*i*erf(-1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arc
sin(d*x + c) + a)*sqrt(abs(b))/b)*e^(a*i/b + 2)/((b*i/sqrt(abs(b)) + sqrt(abs(b)))*d) + 1/16*sqrt(2)*sqrt(pi)*
b*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 + 2)/((b*i/sqrt(abs(b)) - sqrt(abs(b)))*d) - 1/24*sqrt(pi)*sqrt(b)*i*erf(-1/2*sqrt(6)*sqr
t(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - 1/2*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(3*a*i/b + 2)/(
(sqrt(6)*b*i/abs(b) + sqrt(6))*d) - 1/24*sqrt(pi)*sqrt(b)*i*erf(1/2*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b
)*i/abs(b) - 1/2*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(-3*a*i/b + 2)/((sqrt(6)*b*i/abs(b) - sqrt(6))
*d) + 1/24*sqrt(b*arcsin(d*x + c) + a)*i*e^(3*i*arcsin(d*x + c) + 2)/d - 1/8*sqrt(b*arcsin(d*x + c) + a)*i*e^(
i*arcsin(d*x + c) + 2)/d + 1/8*sqrt(b*arcsin(d*x + c) + a)*i*e^(-i*arcsin(d*x + c) + 2)/d - 1/24*sqrt(b*arcsin
(d*x + c) + a)*i*e^(-3*i*arcsin(d*x + c) + 2)/d