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

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

[Out]

-1/128*e^3*cos(4*a/b)*FresnelC(2*2^(1/2)/Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2))*b^(1/2)*2^(1/2)*Pi^(1/2)/
d-1/128*e^3*FresnelS(2*2^(1/2)/Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2))*sin(4*a/b)*b^(1/2)*2^(1/2)*Pi^(1/2)
/d+1/16*e^3*cos(2*a/b)*FresnelC(2*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2)/Pi^(1/2))*b^(1/2)*Pi^(1/2)/d+1/16*e^3*Fres
nelS(2*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2)/Pi^(1/2))*sin(2*a/b)*b^(1/2)*Pi^(1/2)/d-3/32*e^3*(a+b*arcsin(d*x+c))^
(1/2)/d+1/4*e^3*(d*x+c)^4*(a+b*arcsin(d*x+c))^(1/2)/d

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Rubi [A]  time = 0.72, antiderivative size = 288, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, 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 12

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

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*}

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Mathematica [C]  time = 0.20, size = 269, normalized size = 0.93 \[ \frac {e^3 e^{-\frac {4 i a}{b}} \sqrt {a+b \sin ^{-1}(c+d x)} \left (-4 \sqrt {2} e^{\frac {2 i a}{b}} \sqrt {\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \Gamma \left (\frac {3}{2},-\frac {2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )-4 \sqrt {2} e^{\frac {6 i a}{b}} \sqrt {-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \Gamma \left (\frac {3}{2},\frac {2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+\sqrt {\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \Gamma \left (\frac {3}{2},-\frac {4 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+e^{\frac {8 i a}{b}} \sqrt {-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \Gamma \left (\frac {3}{2},\frac {4 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )\right )}{128 d \sqrt {\frac {\left (a+b \sin ^{-1}(c+d x)\right )^2}{b^2}}} \]

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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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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giac [B]  time = 2.19, size = 1111, normalized size = 3.86 \[ \text {result too large to display} \]

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]

-1/16*sqrt(pi)*a*b*i*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^(5/2)*i/abs(b) - sqrt(2)*b^(3/2))*d) - 1/16*sqrt(pi)*a*sqrt(b)*i*e
rf(-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^2*i/abs(b) + sqrt(2)*b)*d) + 1/8*sqrt(pi)*a*sqrt(b)*i*erf(-sqrt(b*arcsin(d*x + c) + a)*s
qrt(b)*i/abs(b) - sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(2*a*i/b + 3)/((b^2*i/abs(b) + b)*d) + 1/8*sqrt(pi)*a
*sqrt(b)*i*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)/((b^2*i/abs(b) - b)*d) + 1/16*sqrt(pi)*a*i*erf(-sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - s
qrt(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) -
1/8*sqrt(pi)*a*i*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)/((b^(3/2)*i/abs(b) - sqrt(b))*d) - 1/128*sqrt(pi)*b^2*erf(sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqr
t(b)*i/abs(b) - sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(-4*a*i/b + 3)/((sqrt(2)*b^(5/2)*i/abs(b) - sqr
t(2)*b^(3/2))*d) + 1/16*sqrt(pi)*a*i*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)*b^(3/2)*erf(-sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sq
rt(b))*e^(4*a*i/b + 3)/((sqrt(2)*b^2*i/abs(b) + sqrt(2)*b)*d) - 1/32*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 + 3)/((b^2*i/abs(b) + b)*d) - 1/8*
sqrt(pi)*a*i*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)/(sqrt(b)*d*(b*i/abs(b) + 1)) + 1/32*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 + 3)/((b^2*i/abs(b) - b)*d) + 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*arcsin(d*x + c) + a)*e^(-4*i*arcsin(d*x +
c) + 3)/d

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maple [A]  time = 0.39, size = 372, normalized size = 1.29 \[ -\frac {e^{3} \left (\sqrt {2}\, \sqrt {\pi }\, \sqrt {\frac {1}{b}}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {4 a}{b}\right ) \FresnelC \left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) b +\sqrt {2}\, \sqrt {\pi }\, \sqrt {\frac {1}{b}}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {4 a}{b}\right ) \mathrm {S}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) b -8 \cos \left (\frac {2 a}{b}\right ) \FresnelC \left (\frac {2 \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) \sqrt {a +b \arcsin \left (d x +c \right )}\, \sqrt {\frac {1}{b}}\, \sqrt {\pi }\, b -8 \sin \left (\frac {2 a}{b}\right ) \mathrm {S}\left (\frac {2 \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) \sqrt {a +b \arcsin \left (d x +c \right )}\, \sqrt {\frac {1}{b}}\, \sqrt {\pi }\, b +16 \arcsin \left (d x +c \right ) \cos \left (\frac {2 a +2 b \arcsin \left (d x +c \right )}{b}-\frac {2 a}{b}\right ) b +16 \cos \left (\frac {2 a +2 b \arcsin \left (d x +c \right )}{b}-\frac {2 a}{b}\right ) a -4 \arcsin \left (d x +c \right ) \cos \left (\frac {4 a +4 b \arcsin \left (d x +c \right )}{b}-\frac {4 a}{b}\right ) b -4 \cos \left (\frac {4 a +4 b \arcsin \left (d x +c \right )}{b}-\frac {4 a}{b}\right ) a \right )}{128 d \sqrt {a +b \arcsin \left (d x +c \right )}} \]

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)*cos(4*a/b)*Fres
nelC(2*2^(1/2)/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b+2^(1/2)*Pi^(1/2)*(1/b)^(1/2)*(a+b*arcsin(d*
x+c))^(1/2)*sin(4*a/b)*FresnelS(2*2^(1/2)/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b-8*cos(2*a/b)*Fre
snelC(2/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*(a+b*arcsin(d*x+c))^(1/2)*(1/b)^(1/2)*Pi^(1/2)*b-8*s
in(2*a/b)*FresnelS(2/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*(a+b*arcsin(d*x+c))^(1/2)*(1/b)^(1/2)*P
i^(1/2)*b+16*arcsin(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-4*arcs
in(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)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d e x + c e\right )}^{3} \sqrt {b \arcsin \left (d x + c\right ) + a}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (c\,e+d\,e\,x\right )}^3\,\sqrt {a+b\,\mathrm {asin}\left (c+d\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*e + d*e*x)^3*(a + b*asin(c + d*x))^(1/2),x)

[Out]

int((c*e + d*e*x)^3*(a + b*asin(c + d*x))^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ e^{3} \left (\int c^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int d^{3} x^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int 3 c d^{2} x^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int 3 c^{2} d x \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx\right ) \]

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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