3.240 \(\int (c e+d e x)^3 \sqrt{a+b \sin ^{-1}(c+d x)} \, dx\)

Optimal. Leaf size=288 \[ -\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} e^3 \cos \left (\frac{4 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{64 d}+\frac{\sqrt{\pi } \sqrt{b} e^3 \cos \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi } \sqrt{b}}\right )}{16 d}+\frac{\sqrt{\pi } \sqrt{b} e^3 \sin \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right )}{16 d}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} e^3 \sin \left (\frac{4 a}{b}\right ) S\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{64 d}+\frac{e^3 (c+d x)^4 \sqrt{a+b \sin ^{-1}(c+d x)}}{4 d}-\frac{3 e^3 \sqrt{a+b \sin ^{-1}(c+d x)}}{32 d} \]

[Out]

(-3*e^3*Sqrt[a + b*ArcSin[c + d*x]])/(32*d) + (e^3*(c + d*x)^4*Sqrt[a + b*ArcSin[c + d*x]])/(4*d) - (Sqrt[b]*e
^3*Sqrt[Pi/2]*Cos[(4*a)/b]*FresnelC[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(64*d) + (Sqrt[b]*e^3
*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(16*d) + (Sqrt[b]*e^3*Sqr
t[Pi]*FresnelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(16*d) - (Sqrt[b]*e^3*Sqrt[Pi
/2]*FresnelS[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[(4*a)/b])/(64*d)

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Rubi [A]  time = 0.723537, antiderivative size = 288, 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} \[ -\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} e^3 \cos \left (\frac{4 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{64 d}+\frac{\sqrt{\pi } \sqrt{b} e^3 \cos \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi } \sqrt{b}}\right )}{16 d}+\frac{\sqrt{\pi } \sqrt{b} e^3 \sin \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right )}{16 d}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} e^3 \sin \left (\frac{4 a}{b}\right ) S\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{64 d}+\frac{e^3 (c+d x)^4 \sqrt{a+b \sin ^{-1}(c+d x)}}{4 d}-\frac{3 e^3 \sqrt{a+b \sin ^{-1}(c+d x)}}{32 d} \]

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)^3*Sqrt[a + b*ArcSin[c + d*x]],x]

[Out]

(-3*e^3*Sqrt[a + b*ArcSin[c + d*x]])/(32*d) + (e^3*(c + d*x)^4*Sqrt[a + b*ArcSin[c + d*x]])/(4*d) - (Sqrt[b]*e
^3*Sqrt[Pi/2]*Cos[(4*a)/b]*FresnelC[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(64*d) + (Sqrt[b]*e^3
*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(16*d) + (Sqrt[b]*e^3*Sqr
t[Pi]*FresnelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(16*d) - (Sqrt[b]*e^3*Sqrt[Pi
/2]*FresnelS[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[(4*a)/b])/(64*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)^3 \sqrt{a+b \sin ^{-1}(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int e^3 x^3 \sqrt{a+b \sin ^{-1}(x)} \, dx,x,c+d x\right )}{d}\\ &=\frac{e^3 \operatorname{Subst}\left (\int x^3 \sqrt{a+b \sin ^{-1}(x)} \, dx,x,c+d x\right )}{d}\\ &=\frac{e^3 (c+d x)^4 \sqrt{a+b \sin ^{-1}(c+d x)}}{4 d}-\frac{\left (b e^3\right ) \operatorname{Subst}\left (\int \frac{x^4}{\sqrt{1-x^2} \sqrt{a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{8 d}\\ &=\frac{e^3 (c+d x)^4 \sqrt{a+b \sin ^{-1}(c+d x)}}{4 d}-\frac{\left (b e^3\right ) \operatorname{Subst}\left (\int \frac{\sin ^4(x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{8 d}\\ &=\frac{e^3 (c+d x)^4 \sqrt{a+b \sin ^{-1}(c+d x)}}{4 d}-\frac{\left (b e^3\right ) \operatorname{Subst}\left (\int \left (\frac{3}{8 \sqrt{a+b x}}-\frac{\cos (2 x)}{2 \sqrt{a+b x}}+\frac{\cos (4 x)}{8 \sqrt{a+b x}}\right ) \, dx,x,\sin ^{-1}(c+d x)\right )}{8 d}\\ &=-\frac{3 e^3 \sqrt{a+b \sin ^{-1}(c+d x)}}{32 d}+\frac{e^3 (c+d x)^4 \sqrt{a+b \sin ^{-1}(c+d x)}}{4 d}-\frac{\left (b e^3\right ) \operatorname{Subst}\left (\int \frac{\cos (4 x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{64 d}+\frac{\left (b e^3\right ) \operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{16 d}\\ &=-\frac{3 e^3 \sqrt{a+b \sin ^{-1}(c+d x)}}{32 d}+\frac{e^3 (c+d x)^4 \sqrt{a+b \sin ^{-1}(c+d x)}}{4 d}+\frac{\left (b e^3 \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 )}{16 d}-\frac{\left (b e^3 \cos \left (\frac{4 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{4 a}{b}+4 x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{64 d}+\frac{\left (b e^3 \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 )}{16 d}-\frac{\left (b e^3 \sin \left (\frac{4 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{4 a}{b}+4 x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{64 d}\\ &=-\frac{3 e^3 \sqrt{a+b \sin ^{-1}(c+d x)}}{32 d}+\frac{e^3 (c+d x)^4 \sqrt{a+b \sin ^{-1}(c+d x)}}{4 d}+\frac{\left (e^3 \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 )}{8 d}-\frac{\left (e^3 \cos \left (\frac{4 a}{b}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{4 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{32 d}+\frac{\left (e^3 \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 )}{8 d}-\frac{\left (e^3 \sin \left (\frac{4 a}{b}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{4 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{32 d}\\ &=-\frac{3 e^3 \sqrt{a+b \sin ^{-1}(c+d x)}}{32 d}+\frac{e^3 (c+d x)^4 \sqrt{a+b \sin ^{-1}(c+d x)}}{4 d}-\frac{\sqrt{b} e^3 \sqrt{\frac{\pi }{2}} \cos \left (\frac{4 a}{b}\right ) C\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{64 d}+\frac{\sqrt{b} e^3 \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 )}{16 d}+\frac{\sqrt{b} e^3 \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 )}{16 d}-\frac{\sqrt{b} e^3 \sqrt{\frac{\pi }{2}} S\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right ) \sin \left (\frac{4 a}{b}\right )}{64 d}\\ \end{align*}

Mathematica [C]  time = 0.185268, size = 269, normalized size = 0.93 \[ \frac{e^3 e^{-\frac{4 i a}{b}} \sqrt{a+b \sin ^{-1}(c+d x)} \left (-4 \sqrt{2} e^{\frac{2 i a}{b}} \sqrt{\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{3}{2},-\frac{2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )-4 \sqrt{2} e^{\frac{6 i a}{b}} \sqrt{-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{3}{2},\frac{2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+\sqrt{\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{3}{2},-\frac{4 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+e^{\frac{8 i a}{b}} \sqrt{-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{3}{2},\frac{4 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )\right )}{128 d \sqrt{\frac{\left (a+b \sin ^{-1}(c+d x)\right )^2}{b^2}}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(c*e + d*e*x)^3*Sqrt[a + b*ArcSin[c + d*x]],x]

[Out]

(e^3*Sqrt[a + b*ArcSin[c + d*x]]*(-4*Sqrt[2]*E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, ((
-2*I)*(a + b*ArcSin[c + d*x]))/b] - 4*Sqrt[2]*E^(((6*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2
, ((2*I)*(a + b*ArcSin[c + d*x]))/b] + Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, ((-4*I)*(a + b*ArcSin[c
+ d*x]))/b] + E^(((8*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, ((4*I)*(a + b*ArcSin[c + d*x])
)/b]))/(128*d*E^(((4*I)*a)/b)*Sqrt[(a + b*ArcSin[c + d*x])^2/b^2])

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Maple [A]  time = 0.099, size = 374, normalized size = 1.3 \begin{align*}{\frac{{e}^{3}}{128\,d} \left ( -\sqrt{2}\sqrt{\pi }\sqrt{{b}^{-1}}\sqrt{a+b\arcsin \left ( dx+c \right ) }{\it FresnelC} \left ( 2\,{\frac{\sqrt{2}\sqrt{a+b\arcsin \left ( dx+c \right ) }}{\sqrt{\pi }\sqrt{{b}^{-1}}b}} \right ) \cos \left ( 4\,{\frac{a}{b}} \right ) b-\sqrt{2}\sqrt{\pi }\sqrt{{b}^{-1}}\sqrt{a+b\arcsin \left ( dx+c \right ) }{\it FresnelS} \left ( 2\,{\frac{\sqrt{2}\sqrt{a+b\arcsin \left ( dx+c \right ) }}{\sqrt{\pi }\sqrt{{b}^{-1}}b}} \right ) \sin \left ( 4\,{\frac{a}{b}} \right ) b+8\,\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\,\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+4\,\arcsin \left ( dx+c \right ) \cos \left ( 4\,{\frac{a+b\arcsin \left ( dx+c \right ) }{b}}-4\,{\frac{a}{b}} \right ) b+4\,\cos \left ( 4\,{\frac{a+b\arcsin \left ( dx+c \right ) }{b}}-4\,{\frac{a}{b}} \right ) a-16\,\arcsin \left ( dx+c \right ) \cos \left ( 2\,{\frac{a+b\arcsin \left ( dx+c \right ) }{b}}-2\,{\frac{a}{b}} \right ) b-16\,\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)^3*(a+b*arcsin(d*x+c))^(1/2),x)

[Out]

1/128/d*e^3/(a+b*arcsin(d*x+c))^(1/2)*(-2^(1/2)*Pi^(1/2)*(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*FresnelC(2*2^(1
/2)/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*cos(4*a/b)*b-2^(1/2)*Pi^(1/2)*(1/b)^(1/2)*(a+b*arcsin(d*
x+c))^(1/2)*FresnelS(2*2^(1/2)/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*sin(4*a/b)*b+8*(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*(
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+4*arcsin(d*x+c)*cos(4*(a+b*arcsin(d*x+c))/b-4*a/b)*b+4*cos(4*(a+b*arcsin(d*x+c))/b-4*a/b)*a-16*arcsi
n(d*x+c)*cos(2*(a+b*arcsin(d*x+c))/b-2*a/b)*b-16*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 )}^{3} \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)^3*(a+b*arcsin(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate((d*e*x + c*e)^3*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)^3*(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^{3} \left (\int c^{3} \sqrt{a + b \operatorname{asin}{\left (c + d x \right )}}\, dx + \int d^{3} x^{3} \sqrt{a + b \operatorname{asin}{\left (c + d x \right )}}\, dx + \int 3 c d^{2} x^{2} \sqrt{a + b \operatorname{asin}{\left (c + d x \right )}}\, dx + \int 3 c^{2} 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)**3*(a+b*asin(d*x+c))**(1/2),x)

[Out]

e**3*(Integral(c**3*sqrt(a + b*asin(c + d*x)), x) + Integral(d**3*x**3*sqrt(a + b*asin(c + d*x)), x) + Integra
l(3*c*d**2*x**2*sqrt(a + b*asin(c + d*x)), x) + Integral(3*c**2*d*x*sqrt(a + b*asin(c + d*x)), x))

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Giac [A]  time = 2.03682, size = 586, normalized size = 2.03 \begin{align*} -\frac{\sqrt{\pi } b \operatorname{erf}\left (\frac{\sqrt{2} \sqrt{b \arcsin \left (d x + c\right ) + a} \sqrt{b} i}{{\left | b \right |}} - \frac{\sqrt{2} \sqrt{b \arcsin \left (d x + c\right ) + a}}{\sqrt{b}}\right ) e^{\left (-\frac{4 \, a i}{b} + 3\right )}}{128 \,{\left (\frac{\sqrt{2} b^{\frac{3}{2}} i}{{\left | b \right |}} - \sqrt{2} \sqrt{b}\right )} d} + \frac{\sqrt{\pi } \sqrt{b} \operatorname{erf}\left (-\frac{\sqrt{2} \sqrt{b \arcsin \left (d x + c\right ) + a} \sqrt{b} i}{{\left | b \right |}} - \frac{\sqrt{2} \sqrt{b \arcsin \left (d x + c\right ) + a}}{\sqrt{b}}\right ) e^{\left (\frac{4 \, a i}{b} + 3\right )}}{128 \,{\left (\frac{\sqrt{2} b i}{{\left | b \right |}} + \sqrt{2}\right )} d} - \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} + 3\right )}}{32 \, 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} + 3\right )}}{32 \, d{\left (\frac{b i}{{\left | b \right |}} - 1\right )}} + \frac{\sqrt{b \arcsin \left (d x + c\right ) + a} e^{\left (4 \, i \arcsin \left (d x + c\right ) + 3\right )}}{64 \, d} - \frac{\sqrt{b \arcsin \left (d x + c\right ) + a} e^{\left (2 \, i \arcsin \left (d x + c\right ) + 3\right )}}{16 \, d} - \frac{\sqrt{b \arcsin \left (d x + c\right ) + a} e^{\left (-2 \, i \arcsin \left (d x + c\right ) + 3\right )}}{16 \, d} + \frac{\sqrt{b \arcsin \left (d x + c\right ) + a} e^{\left (-4 \, i \arcsin \left (d x + c\right ) + 3\right )}}{64 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^3*(a+b*arcsin(d*x+c))^(1/2),x, algorithm="giac")

[Out]

-1/128*sqrt(pi)*b*erf(sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(2)*sqrt(b*arcsin(d*x + c) +
a)/sqrt(b))*e^(-4*a*i/b + 3)/((sqrt(2)*b^(3/2)*i/abs(b) - sqrt(2)*sqrt(b))*d) + 1/128*sqrt(pi)*sqrt(b)*erf(-sq
rt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(4*a*i/b +
3)/((sqrt(2)*b*i/abs(b) + sqrt(2))*d) - 1/32*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 + 3)/(d*(b*i/abs(b) + 1)) + 1/32*sqrt(pi)*sqrt(b)*erf(sqr
t(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 + 3)/(d*(b*i/abs(
b) - 1)) + 1/64*sqrt(b*arcsin(d*x + c) + a)*e^(4*i*arcsin(d*x + c) + 3)/d - 1/16*sqrt(b*arcsin(d*x + c) + a)*e
^(2*i*arcsin(d*x + c) + 3)/d - 1/16*sqrt(b*arcsin(d*x + c) + a)*e^(-2*i*arcsin(d*x + c) + 3)/d + 1/64*sqrt(b*a
rcsin(d*x + c) + a)*e^(-4*i*arcsin(d*x + c) + 3)/d