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

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

[Out]

1/72*e^2*cos(3*a/b)*FresnelS(6^(1/2)/Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2))*b^(1/2)*6^(1/2)*Pi^(1/2)/d-1/
72*e^2*FresnelC(6^(1/2)/Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2))*sin(3*a/b)*b^(1/2)*6^(1/2)*Pi^(1/2)/d-1/8*
e^2*cos(a/b)*FresnelS(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+1/8*e^2*F
resnelC(2^(1/2)/Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2))*sin(a/b)*b^(1/2)*2^(1/2)*Pi^(1/2)/d+1/3*e^2*(d*x+c
)^3*(a+b*arcsin(d*x+c))^(1/2)/d

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Rubi [A]  time = 0.75, antiderivative size = 274, 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} \[ \frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e^2 \sin \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{4 d}-\frac {\sqrt {\frac {\pi }{6}} \sqrt {b} e^2 \sin \left (\frac {3 a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{12 d}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e^2 \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{4 d}+\frac {\sqrt {\frac {\pi }{6}} \sqrt {b} e^2 \cos \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{12 d}+\frac {e^2 (c+d x)^3 \sqrt {a+b \sin ^{-1}(c+d x)}}{3 d} \]

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 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)^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*}

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Mathematica [C]  time = 0.34, size = 269, normalized size = 0.98 \[ -\frac {i e^2 e^{-\frac {3 i a}{b}} \sqrt {a+b \sin ^{-1}(c+d x)} \left (9 e^{\frac {2 i a}{b}} \sqrt {\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \Gamma \left (\frac {3}{2},-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )-9 e^{\frac {4 i a}{b}} \sqrt {-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \Gamma \left (\frac {3}{2},\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+\sqrt {3} \left (e^{\frac {6 i a}{b}} \sqrt {-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \Gamma \left (\frac {3}{2},\frac {3 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )-\sqrt {\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \Gamma \left (\frac {3}{2},-\frac {3 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )\right )\right )}{72 d \sqrt {\frac {\left (a+b \sin ^{-1}(c+d x)\right )^2}{b^2}}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-1/72*I)*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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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)^2*(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 = 2.39, size = 1191, normalized size = 4.35 \[ \text {result too large to display} \]

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^2*i*erf(-1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*a
rcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(a*i/b + 2)/((b^2*i/sqrt(abs(b)) + b*sqrt(abs(b)))*d) + 1/16*sqrt(2)*sqr
t(pi)*b^2*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^2*i/sqrt(abs(b)) - b*sqrt(abs(b)))*d) - 1/24*sqrt(pi)*b^(3/2)*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^2*i/abs(b) + sqrt(6)*b)*d) + 1/8*sqrt(2)*sqrt(pi)*a*b*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^2*i/sqr
t(abs(b)) + b*sqrt(abs(b)))*d) - 1/8*sqrt(2)*sqrt(pi)*a*b*erf(1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*i/sqrt(a
bs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(-a*i/b + 2)/((b^2*i/sqrt(abs(b)) - b*sqrt(
abs(b)))*d) - 1/24*sqrt(pi)*b^(3/2)*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^2*i/abs(b) - sqrt(6)*b)*d) - 1/4*sqrt(pi)
*a*sqrt(b)*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^2*i/abs(b) + sqrt(6)*b)*d) + 1/4*sqrt(pi)*a*sqrt(b)*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^2*i/abs(b) - sqrt(6)*b)*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*arcsi
n(d*x + c) + 2)/d - 1/24*sqrt(b*arcsin(d*x + c) + a)*i*e^(-3*i*arcsin(d*x + c) + 2)/d + 1/4*sqrt(pi)*a*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^(3/2)*i/abs(b) + sqrt(6)*sqrt(b))*d) - 1/4*sqrt(pi)*a*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)/((sqrt(2)*
b*i/sqrt(abs(b)) + sqrt(2)*sqrt(abs(b)))*d) + 1/4*sqrt(pi)*a*erf(1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*i/sqr
t(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(-a*i/b + 2)/((sqrt(2)*b*i/sqrt(abs(b))
- sqrt(2)*sqrt(abs(b)))*d) - 1/4*sqrt(pi)*a*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^(3/2)*i/abs(b) - sqrt(6)*sqrt(b))*d
)

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maple [A]  time = 0.32, size = 389, normalized size = 1.42 \[ -\frac {e^{2} \left (\sqrt {\frac {1}{b}}\, \sqrt {2}\, \sqrt {\pi }\, \sin \left (\frac {3 a}{b}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) \sqrt {3}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, b -\sqrt {\frac {1}{b}}\, \sqrt {2}\, \sqrt {\pi }\, \cos \left (\frac {3 a}{b}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) \sqrt {3}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, b +9 \sqrt {2}\, \sqrt {\pi }\, \sqrt {\frac {1}{b}}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {a}{b}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) b -9 \sqrt {2}\, \sqrt {\pi }\, \sqrt {\frac {1}{b}}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \FresnelC \left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) \sin \left (\frac {a}{b}\right ) b +6 \sin \left (\frac {3 a +3 b \arcsin \left (d x +c \right )}{b}-\frac {3 a}{b}\right ) \arcsin \left (d x +c \right ) b +6 \sin \left (\frac {3 a +3 b \arcsin \left (d x +c \right )}{b}-\frac {3 a}{b}\right ) a -18 \sin \left (\frac {a +b \arcsin \left (d x +c \right )}{b}-\frac {a}{b}\right ) \arcsin \left (d x +c \right ) b -18 \sin \left (\frac {a +b \arcsin \left (d x +c \right )}{b}-\frac {a}{b}\right ) a \right )}{72 d \sqrt {a +b \arcsin \left (d x +c \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/72/d*e^2/(a+b*arcsin(d*x+c))^(1/2)*((1/b)^(1/2)*2^(1/2)*Pi^(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)*3^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*b-(1/b)^(1/2)*2^(1/2)*Pi^(1/2)*c
os(3*a/b)*FresnelS(2^(1/2)/Pi^(1/2)*3^(1/2)/(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*3^(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)*FresnelC(2^(1/2)/
Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*sin(a/b)*b+6*sin(3*(a+b*arcsin(d*x+c))/b-3*a/b)*arcsin(d*x+c
)*b+6*sin(3*(a+b*arcsin(d*x+c))/b-3*a/b)*a-18*sin((a+b*arcsin(d*x+c))/b-a/b)*arcsin(d*x+c)*b-18*sin((a+b*arcsi
n(d*x+c))/b-a/b)*a)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (c\,e+d\,e\,x\right )}^2\,\sqrt {a+b\,\mathrm {asin}\left (c+d\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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