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

Optimal. Leaf size=369 \[ \frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e \sin \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{4 c^3}-\frac {\sqrt {\frac {\pi }{6}} \sqrt {b} e \sin \left (\frac {3 a}{b}\right ) C\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{12 c^3}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{4 c^3}+\frac {\sqrt {\frac {\pi }{6}} \sqrt {b} e \cos \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{12 c^3}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} d \sin \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{c}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} d \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{c}+d x \sqrt {a+b \sin ^{-1}(c x)}+\frac {1}{3} e x^3 \sqrt {a+b \sin ^{-1}(c x)} \]

[Out]

1/72*e*cos(3*a/b)*FresnelS(6^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*b^(1/2)*6^(1/2)*Pi^(1/2)/c^3-1/72
*e*FresnelC(6^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*sin(3*a/b)*b^(1/2)*6^(1/2)*Pi^(1/2)/c^3-1/2*d*co
s(a/b)*FresnelS(2^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*b^(1/2)*2^(1/2)*Pi^(1/2)/c-1/8*e*cos(a/b)*Fr
esnelS(2^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*b^(1/2)*2^(1/2)*Pi^(1/2)/c^3+1/2*d*FresnelC(2^(1/2)/P
i^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*sin(a/b)*b^(1/2)*2^(1/2)*Pi^(1/2)/c+1/8*e*FresnelC(2^(1/2)/Pi^(1/2)*(
a+b*arcsin(c*x))^(1/2)/b^(1/2))*sin(a/b)*b^(1/2)*2^(1/2)*Pi^(1/2)/c^3+d*x*(a+b*arcsin(c*x))^(1/2)+1/3*e*x^3*(a
+b*arcsin(c*x))^(1/2)

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Rubi [A]  time = 1.03, antiderivative size = 369, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4667, 4619, 4723, 3306, 3305, 3351, 3304, 3352, 4629, 3312} \[ \frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e \sin \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{4 c^3}-\frac {\sqrt {\frac {\pi }{6}} \sqrt {b} e \sin \left (\frac {3 a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{12 c^3}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{4 c^3}+\frac {\sqrt {\frac {\pi }{6}} \sqrt {b} e \cos \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{12 c^3}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} d \sin \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{c}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} d \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{c}+d x \sqrt {a+b \sin ^{-1}(c x)}+\frac {1}{3} e x^3 \sqrt {a+b \sin ^{-1}(c x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 4619

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSin[c*x])^n, x] - Dist[b*c*n, Int[
(x*(a + b*ArcSin[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

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 4667

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a
+ b*ArcSin[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p]
&& (GtQ[p, 0] || IGtQ[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])

Rubi steps

\begin {align*} \int \left (d+e x^2\right ) \sqrt {a+b \sin ^{-1}(c x)} \, dx &=\int \left (d \sqrt {a+b \sin ^{-1}(c x)}+e x^2 \sqrt {a+b \sin ^{-1}(c x)}\right ) \, dx\\ &=d \int \sqrt {a+b \sin ^{-1}(c x)} \, dx+e \int x^2 \sqrt {a+b \sin ^{-1}(c x)} \, dx\\ &=d x \sqrt {a+b \sin ^{-1}(c x)}+\frac {1}{3} e x^3 \sqrt {a+b \sin ^{-1}(c x)}-\frac {1}{2} (b c d) \int \frac {x}{\sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}} \, dx-\frac {1}{6} (b c e) \int \frac {x^3}{\sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}} \, dx\\ &=d x \sqrt {a+b \sin ^{-1}(c x)}+\frac {1}{3} e x^3 \sqrt {a+b \sin ^{-1}(c x)}-\frac {(b d) \operatorname {Subst}\left (\int \frac {\sin (x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{2 c}-\frac {(b e) \operatorname {Subst}\left (\int \frac {\sin ^3(x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{6 c^3}\\ &=d x \sqrt {a+b \sin ^{-1}(c x)}+\frac {1}{3} e x^3 \sqrt {a+b \sin ^{-1}(c x)}-\frac {(b e) \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 x)\right )}{6 c^3}-\frac {\left (b d \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 x)\right )}{2 c}+\frac {\left (b d \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 x)\right )}{2 c}\\ &=d x \sqrt {a+b \sin ^{-1}(c x)}+\frac {1}{3} e x^3 \sqrt {a+b \sin ^{-1}(c x)}+\frac {(b e) \operatorname {Subst}\left (\int \frac {\sin (3 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{24 c^3}-\frac {(b e) \operatorname {Subst}\left (\int \frac {\sin (x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{8 c^3}-\frac {\left (d \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 x)}\right )}{c}+\frac {\left (d \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 x)}\right )}{c}\\ &=d x \sqrt {a+b \sin ^{-1}(c x)}+\frac {1}{3} e x^3 \sqrt {a+b \sin ^{-1}(c x)}-\frac {\sqrt {b} d \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{c}+\frac {\sqrt {b} d \sqrt {\frac {\pi }{2}} C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{c}-\frac {\left (b e \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 x)\right )}{8 c^3}+\frac {\left (b e \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 x)\right )}{24 c^3}+\frac {\left (b e \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 x)\right )}{8 c^3}-\frac {\left (b e \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 x)\right )}{24 c^3}\\ &=d x \sqrt {a+b \sin ^{-1}(c x)}+\frac {1}{3} e x^3 \sqrt {a+b \sin ^{-1}(c x)}-\frac {\sqrt {b} d \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{c}+\frac {\sqrt {b} d \sqrt {\frac {\pi }{2}} C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{c}-\frac {\left (e \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 x)}\right )}{4 c^3}+\frac {\left (e \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 x)}\right )}{12 c^3}+\frac {\left (e \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 x)}\right )}{4 c^3}-\frac {\left (e \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 x)}\right )}{12 c^3}\\ &=d x \sqrt {a+b \sin ^{-1}(c x)}+\frac {1}{3} e x^3 \sqrt {a+b \sin ^{-1}(c x)}-\frac {\sqrt {b} d \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{c}-\frac {\sqrt {b} e \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{4 c^3}+\frac {\sqrt {b} e \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{12 c^3}+\frac {\sqrt {b} d \sqrt {\frac {\pi }{2}} C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{c}+\frac {\sqrt {b} e \sqrt {\frac {\pi }{2}} C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{4 c^3}-\frac {\sqrt {b} e \sqrt {\frac {\pi }{6}} C\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{12 c^3}\\ \end {align*}

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Mathematica [C]  time = 0.67, size = 244, normalized size = 0.66 \[ \frac {b e^{-\frac {3 i a}{b}} \left (9 e^{\frac {2 i a}{b}} \left (4 c^2 d+e\right ) \sqrt {-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}} \Gamma \left (\frac {3}{2},-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+9 e^{\frac {4 i a}{b}} \left (4 c^2 d+e\right ) \sqrt {\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}} \Gamma \left (\frac {3}{2},\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )-\sqrt {3} e \left (\sqrt {-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}} \Gamma \left (\frac {3}{2},-\frac {3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+e^{\frac {6 i a}{b}} \sqrt {\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}} \Gamma \left (\frac {3}{2},\frac {3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )\right )\right )}{72 c^3 \sqrt {a+b \sin ^{-1}(c x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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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((e*x^2+d)*(a+b*arcsin(c*x))^(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 [C]  time = 3.90, size = 1677, normalized size = 4.54 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(2)*sqrt(pi)*a*b^2*d*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcs
in(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/((I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c) + 1/4*I*sqrt(2)*sqrt(pi)*b^
3*d*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))
/b)*e^(I*a/b)/((I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c) + 1/2*sqrt(2)*sqrt(pi)*a*b^2*d*erf(1/2*I*sqrt(2)*sqr
t(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/((-I*b^3/sq
rt(abs(b)) + b^2*sqrt(abs(b)))*c) - 1/4*I*sqrt(2)*sqrt(pi)*b^3*d*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqr
t(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/((-I*b^3/sqrt(abs(b)) + b^2*sqrt(ab
s(b)))*c) - sqrt(pi)*a*b*d*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin
(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/((I*sqrt(2)*b^2/sqrt(abs(b)) + sqrt(2)*b*sqrt(abs(b)))*c) - sqrt(pi)*a*b*
d*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)
*e^(-I*a/b)/((-I*sqrt(2)*b^2/sqrt(abs(b)) + sqrt(2)*b*sqrt(abs(b)))*c) + 1/8*sqrt(2)*sqrt(pi)*a*b^2*erf(-1/2*I
*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b +
1)/((I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c^3) + 1/16*I*sqrt(2)*sqrt(pi)*b^3*erf(-1/2*I*sqrt(2)*sqrt(b*arcs
in(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b + 1)/((I*b^3/sqrt(abs
(b)) + b^2*sqrt(abs(b)))*c^3) + 1/8*sqrt(2)*sqrt(pi)*a*b^2*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(
b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b + 1)/((-I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(
b)))*c^3) - 1/16*I*sqrt(2)*sqrt(pi)*b^3*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*s
qrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b + 1)/((-I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c^3) - 1/2*I*s
qrt(b*arcsin(c*x) + a)*d*e^(I*arcsin(c*x))/c + 1/2*I*sqrt(b*arcsin(c*x) + a)*d*e^(-I*arcsin(c*x))/c - 1/4*sqrt
(pi)*a*b^(3/2)*erf(-1/2*sqrt(6)*sqrt(b*arcsin(c*x) + a)/sqrt(b) - 1/2*I*sqrt(6)*sqrt(b*arcsin(c*x) + a)*sqrt(b
)/abs(b))*e^(3*I*a/b + 1)/((sqrt(6)*b^2 + I*sqrt(6)*b^3/abs(b))*c^3) - 1/24*I*sqrt(pi)*b^(5/2)*erf(-1/2*sqrt(6
)*sqrt(b*arcsin(c*x) + a)/sqrt(b) - 1/2*I*sqrt(6)*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(3*I*a/b + 1)/((sq
rt(6)*b^2 + I*sqrt(6)*b^3/abs(b))*c^3) - 1/4*sqrt(pi)*a*b^(3/2)*erf(-1/2*sqrt(6)*sqrt(b*arcsin(c*x) + a)/sqrt(
b) + 1/2*I*sqrt(6)*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(-3*I*a/b + 1)/((sqrt(6)*b^2 - I*sqrt(6)*b^3/abs(
b))*c^3) + 1/24*I*sqrt(pi)*b^(5/2)*erf(-1/2*sqrt(6)*sqrt(b*arcsin(c*x) + a)/sqrt(b) + 1/2*I*sqrt(6)*sqrt(b*arc
sin(c*x) + a)*sqrt(b)/abs(b))*e^(-3*I*a/b + 1)/((sqrt(6)*b^2 - I*sqrt(6)*b^3/abs(b))*c^3) + 1/4*sqrt(pi)*a*b*e
rf(-1/2*sqrt(6)*sqrt(b*arcsin(c*x) + a)/sqrt(b) - 1/2*I*sqrt(6)*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(3*I
*a/b + 1)/((sqrt(6)*b^(3/2) + I*sqrt(6)*b^(5/2)/abs(b))*c^3) - 1/4*sqrt(pi)*a*b*erf(-1/2*I*sqrt(2)*sqrt(b*arcs
in(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b + 1)/((I*sqrt(2)*b^2/
sqrt(abs(b)) + sqrt(2)*b*sqrt(abs(b)))*c^3) - 1/4*sqrt(pi)*a*b*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(
abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b + 1)/((-I*sqrt(2)*b^2/sqrt(abs(b)) + s
qrt(2)*b*sqrt(abs(b)))*c^3) + 1/4*sqrt(pi)*a*b*erf(-1/2*sqrt(6)*sqrt(b*arcsin(c*x) + a)/sqrt(b) + 1/2*I*sqrt(6
)*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(-3*I*a/b + 1)/((sqrt(6)*b^(3/2) - I*sqrt(6)*b^(5/2)/abs(b))*c^3)
+ 1/24*I*sqrt(b*arcsin(c*x) + a)*e^(3*I*arcsin(c*x) + 1)/c^3 - 1/8*I*sqrt(b*arcsin(c*x) + a)*e^(I*arcsin(c*x)
+ 1)/c^3 + 1/8*I*sqrt(b*arcsin(c*x) + a)*e^(-I*arcsin(c*x) + 1)/c^3 - 1/24*I*sqrt(b*arcsin(c*x) + a)*e^(-3*I*a
rcsin(c*x) + 1)/c^3

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maple [A]  time = 0.32, size = 542, normalized size = 1.47 \[ \frac {-36 \cos \left (\frac {a}{b}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) \sqrt {\frac {1}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \sqrt {\pi }\, \sqrt {2}\, b \,c^{2} d +36 \sin \left (\frac {a}{b}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) \sqrt {\frac {1}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \sqrt {\pi }\, \sqrt {2}\, b \,c^{2} d +\cos \left (\frac {3 a}{b}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) \sqrt {3}\, \sqrt {\frac {1}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \sqrt {\pi }\, \sqrt {2}\, b e -\sin \left (\frac {3 a}{b}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) \sqrt {3}\, \sqrt {\frac {1}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \sqrt {\pi }\, \sqrt {2}\, b e -9 \cos \left (\frac {a}{b}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) \sqrt {\frac {1}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \sqrt {\pi }\, \sqrt {2}\, b e +9 \sin \left (\frac {a}{b}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) \sqrt {\frac {1}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \sqrt {\pi }\, \sqrt {2}\, b e +72 \arcsin \left (c x \right ) \sin \left (\frac {a +b \arcsin \left (c x \right )}{b}-\frac {a}{b}\right ) b \,c^{2} d +72 \sin \left (\frac {a +b \arcsin \left (c x \right )}{b}-\frac {a}{b}\right ) a \,c^{2} d +18 \arcsin \left (c x \right ) \sin \left (\frac {a +b \arcsin \left (c x \right )}{b}-\frac {a}{b}\right ) b e -6 \arcsin \left (c x \right ) \sin \left (\frac {3 a +3 b \arcsin \left (c x \right )}{b}-\frac {3 a}{b}\right ) b e +18 \sin \left (\frac {a +b \arcsin \left (c x \right )}{b}-\frac {a}{b}\right ) a e -6 \sin \left (\frac {3 a +3 b \arcsin \left (c x \right )}{b}-\frac {3 a}{b}\right ) a e}{72 c^{3} \sqrt {a +b \arcsin \left (c x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/72/c^3*(-36*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*(1/b)^(1/2)*(a+b*arcsi
n(c*x))^(1/2)*Pi^(1/2)*2^(1/2)*b*c^2*d+36*sin(a/b)*FresnelC(2^(1/2)/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(c*x))^(1/
2)/b)*(1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)*Pi^(1/2)*2^(1/2)*b*c^2*d+cos(3*a/b)*FresnelS(2^(1/2)/Pi^(1/2)*3^(1/2
)/(1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*3^(1/2)*(1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)*Pi^(1/2)*2^(1/2)*b*e-sin(
3*a/b)*FresnelC(2^(1/2)/Pi^(1/2)*3^(1/2)/(1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*3^(1/2)*(1/b)^(1/2)*(a+b*arcsi
n(c*x))^(1/2)*Pi^(1/2)*2^(1/2)*b*e-9*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)
*(1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)*Pi^(1/2)*2^(1/2)*b*e+9*sin(a/b)*FresnelC(2^(1/2)/Pi^(1/2)/(1/b)^(1/2)*(a+
b*arcsin(c*x))^(1/2)/b)*(1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)*Pi^(1/2)*2^(1/2)*b*e+72*arcsin(c*x)*sin((a+b*arcsi
n(c*x))/b-a/b)*b*c^2*d+72*sin((a+b*arcsin(c*x))/b-a/b)*a*c^2*d+18*arcsin(c*x)*sin((a+b*arcsin(c*x))/b-a/b)*b*e
-6*arcsin(c*x)*sin(3*(a+b*arcsin(c*x))/b-3*a/b)*b*e+18*sin((a+b*arcsin(c*x))/b-a/b)*a*e-6*sin(3*(a+b*arcsin(c*
x))/b-3*a/b)*a*e)/(a+b*arcsin(c*x))^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((e*x^2 + d)*sqrt(b*arcsin(c*x) + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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