\(\int x^2 (a+b \arcsin (c x))^{3/2} \, dx\) [178]

Optimal result

Integrand size = 16, antiderivative size = 313 \[ \int x^2 (a+b \arcsin (c x))^{3/2} \, dx=\frac {b \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{3 c^3}+\frac {b x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{6 c}+\frac {1}{3} x^3 (a+b \arcsin (c x))^{3/2}-\frac {3 b^{3/2} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{8 c^3}+\frac {b^{3/2} \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{24 c^3}-\frac {3 b^{3/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{8 c^3}+\frac {b^{3/2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{24 c^3} \]

[Out]

1/3*x^3*(a+b*arcsin(c*x))^(3/2)+1/144*b^(3/2)*cos(3*a/b)*FresnelC(6^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(
1/2))*6^(1/2)*Pi^(1/2)/c^3+1/144*b^(3/2)*FresnelS(6^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*sin(3*a/b)
*6^(1/2)*Pi^(1/2)/c^3-3/16*b^(3/2)*cos(a/b)*FresnelC(2^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*2^(1/2)
*Pi^(1/2)/c^3-3/16*b^(3/2)*FresnelS(2^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*sin(a/b)*2^(1/2)*Pi^(1/2
)/c^3+1/3*b*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))^(1/2)/c^3+1/6*b*x^2*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))^(1/2
)/c

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {4725, 4795, 4767, 4719, 3387, 3386, 3432, 3385, 3433, 4731, 4491} \[ \int x^2 (a+b \arcsin (c x))^{3/2} \, dx=-\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{8 c^3}+\frac {\sqrt {\frac {\pi }{6}} b^{3/2} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{24 c^3}-\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{8 c^3}+\frac {\sqrt {\frac {\pi }{6}} b^{3/2} \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{24 c^3}+\frac {b x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{6 c}+\frac {b \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{3 c^3}+\frac {1}{3} x^3 (a+b \arcsin (c x))^{3/2} \]

[In]

Int[x^2*(a + b*ArcSin[c*x])^(3/2),x]

[Out]

(b*Sqrt[1 - c^2*x^2]*Sqrt[a + b*ArcSin[c*x]])/(3*c^3) + (b*x^2*Sqrt[1 - c^2*x^2]*Sqrt[a + b*ArcSin[c*x]])/(6*c
) + (x^3*(a + b*ArcSin[c*x])^(3/2))/3 - (3*b^(3/2)*Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[
c*x]])/Sqrt[b]])/(8*c^3) + (b^(3/2)*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt
[b]])/(24*c^3) - (3*b^(3/2)*Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(8*c^3
) + (b^(3/2)*Sqrt[Pi/6]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(24*c^3)

Rule 3385

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 3386

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 3387

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 3432

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2
]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]

Rule 3433

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2
]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]

Rule 4491

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rule 4719

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

Rule 4725

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 4731

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/(b*c^(m + 1)), Subst[Int[x^n*Sin[-
a/b + x/b]^m*Cos[-a/b + x/b], x], x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rule 4767

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

Rule 4795

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[f
*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(e*(m + 2*p + 1))), x] + (Dist[f^2*((m - 1)/(c^2*(m
+ 2*p + 1))), Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] + Dist[b*f*(n/(c*(m + 2*p + 1)))*S
imp[(d + e*x^2)^p/(1 - c^2*x^2)^p], Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x],
x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m, 1] && NeQ[m + 2*p + 1, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 (a+b \arcsin (c x))^{3/2}-\frac {1}{2} (b c) \int \frac {x^3 \sqrt {a+b \arcsin (c x)}}{\sqrt {1-c^2 x^2}} \, dx \\ & = \frac {b x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{6 c}+\frac {1}{3} x^3 (a+b \arcsin (c x))^{3/2}-\frac {1}{12} b^2 \int \frac {x^2}{\sqrt {a+b \arcsin (c x)}} \, dx-\frac {b \int \frac {x \sqrt {a+b \arcsin (c x)}}{\sqrt {1-c^2 x^2}} \, dx}{3 c} \\ & = \frac {b \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{3 c^3}+\frac {b x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{6 c}+\frac {1}{3} x^3 (a+b \arcsin (c x))^{3/2}-\frac {b \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right ) \sin ^2\left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{12 c^3}-\frac {b^2 \int \frac {1}{\sqrt {a+b \arcsin (c x)}} \, dx}{6 c^2} \\ & = \frac {b \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{3 c^3}+\frac {b x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{6 c}+\frac {1}{3} x^3 (a+b \arcsin (c x))^{3/2}-\frac {b \text {Subst}\left (\int \left (-\frac {\cos \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{4 \sqrt {x}}+\frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right )}{4 \sqrt {x}}\right ) \, dx,x,a+b \arcsin (c x)\right )}{12 c^3}-\frac {b \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{6 c^3} \\ & = \frac {b \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{3 c^3}+\frac {b x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{6 c}+\frac {1}{3} x^3 (a+b \arcsin (c x))^{3/2}+\frac {b \text {Subst}\left (\int \frac {\cos \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{48 c^3}-\frac {b \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{48 c^3}-\frac {\left (b \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{6 c^3}-\frac {\left (b \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{6 c^3} \\ & = \frac {b \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{3 c^3}+\frac {b x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{6 c}+\frac {1}{3} x^3 (a+b \arcsin (c x))^{3/2}-\frac {\left (b \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{48 c^3}-\frac {\left (b \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c x)}\right )}{3 c^3}+\frac {\left (b \cos \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {3 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{48 c^3}-\frac {\left (b \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{48 c^3}-\frac {\left (b \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c x)}\right )}{3 c^3}+\frac {\left (b \sin \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {3 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{48 c^3} \\ & = \frac {b \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{3 c^3}+\frac {b x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{6 c}+\frac {1}{3} x^3 (a+b \arcsin (c x))^{3/2}-\frac {b^{3/2} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{3 c^3}-\frac {b^{3/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{3 c^3}-\frac {\left (b \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c x)}\right )}{24 c^3}+\frac {\left (b \cos \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c x)}\right )}{24 c^3}-\frac {\left (b \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c x)}\right )}{24 c^3}+\frac {\left (b \sin \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c x)}\right )}{24 c^3} \\ & = \frac {b \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{3 c^3}+\frac {b x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{6 c}+\frac {1}{3} x^3 (a+b \arcsin (c x))^{3/2}-\frac {3 b^{3/2} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{8 c^3}+\frac {b^{3/2} \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{24 c^3}-\frac {3 b^{3/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{8 c^3}+\frac {b^{3/2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{24 c^3} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.22 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.78 \[ \int x^2 (a+b \arcsin (c x))^{3/2} \, dx=\frac {b e^{-\frac {3 i a}{b}} \sqrt {a+b \arcsin (c x)} \left (27 e^{\frac {2 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {5}{2},-\frac {i (a+b \arcsin (c x))}{b}\right )+27 e^{\frac {4 i a}{b}} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {5}{2},\frac {i (a+b \arcsin (c x))}{b}\right )-\sqrt {3} \left (\sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {5}{2},-\frac {3 i (a+b \arcsin (c x))}{b}\right )+e^{\frac {6 i a}{b}} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {5}{2},\frac {3 i (a+b \arcsin (c x))}{b}\right )\right )\right )}{216 c^3 \sqrt {\frac {(a+b \arcsin (c x))^2}{b^2}}} \]

[In]

Integrate[x^2*(a + b*ArcSin[c*x])^(3/2),x]

[Out]

(b*Sqrt[a + b*ArcSin[c*x]]*(27*E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[5/2, ((-I)*(a + b*ArcSin[
c*x]))/b] + 27*E^(((4*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[5/2, (I*(a + b*ArcSin[c*x]))/b] - Sqrt
[3]*(Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[5/2, ((-3*I)*(a + b*ArcSin[c*x]))/b] + E^(((6*I)*a)/b)*Sqrt[((-I)*(
a + b*ArcSin[c*x]))/b]*Gamma[5/2, ((3*I)*(a + b*ArcSin[c*x]))/b])))/(216*c^3*E^(((3*I)*a)/b)*Sqrt[(a + b*ArcSi
n[c*x])^2/b^2])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(546\) vs. \(2(241)=482\).

Time = 0.10 (sec) , antiderivative size = 547, normalized size of antiderivative = 1.75

method	result	size
default	\(-\frac {-\sqrt {-\frac {3}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) b^{2}+\sqrt {-\frac {3}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) b^{2}+27 \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {a}{b}\right ) \operatorname {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}}\, b^{2}-27 \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, b^{2}+36 \arcsin \left (c x \right )^{2} \sin \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) b^{2}-12 \arcsin \left (c x \right )^{2} \sin \left (-\frac {3 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) b^{2}+72 \arcsin \left (c x \right ) \sin \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) a b -54 \arcsin \left (c x \right ) \cos \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) b^{2}-24 \arcsin \left (c x \right ) \sin \left (-\frac {3 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) a b +6 \arcsin \left (c x \right ) \cos \left (-\frac {3 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) b^{2}+36 \sin \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) a^{2}-54 \cos \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) a b -12 \sin \left (-\frac {3 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) a^{2}+6 \cos \left (-\frac {3 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) a b}{144 c^{3} \sqrt {a +b \arcsin \left (c x \right )}}\)	\(547\)

[In]

int(x^2*(a+b*arcsin(c*x))^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/144/c^3*(-(-3/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arcsin(c*x))^(1/2)*cos(3*a/b)*FresnelC(3*2^(1/2)/Pi^(1/2)/(-3/
b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*b^2+(-3/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arcsin(c*x))^(1/2)*sin(3*a/b)*Fresn
elS(3*2^(1/2)/Pi^(1/2)/(-3/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*b^2+27*Pi^(1/2)*2^(1/2)*(a+b*arcsin(c*x))^(1/2)
*cos(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)*b^2-27*Pi^(1/2)*2^(1/
2)*(a+b*arcsin(c*x))^(1/2)*sin(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)*b^2+36*arcsin(c*x)^2*sin(-(a+b*arcsin(c*x))/b+a/b)*b^2-12*arcsin(c*x)^2*sin(-3*(a+b*arcsin(c*x))/b+3*a/b)
*b^2+72*arcsin(c*x)*sin(-(a+b*arcsin(c*x))/b+a/b)*a*b-54*arcsin(c*x)*cos(-(a+b*arcsin(c*x))/b+a/b)*b^2-24*arcs
in(c*x)*sin(-3*(a+b*arcsin(c*x))/b+3*a/b)*a*b+6*arcsin(c*x)*cos(-3*(a+b*arcsin(c*x))/b+3*a/b)*b^2+36*sin(-(a+b
*arcsin(c*x))/b+a/b)*a^2-54*cos(-(a+b*arcsin(c*x))/b+a/b)*a*b-12*sin(-3*(a+b*arcsin(c*x))/b+3*a/b)*a^2+6*cos(-
3*(a+b*arcsin(c*x))/b+3*a/b)*a*b)/(a+b*arcsin(c*x))^(1/2)

Fricas [F(-2)]

Exception generated. \[ \int x^2 (a+b \arcsin (c x))^{3/2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^2*(a+b*arcsin(c*x))^(3/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int x^2 (a+b \arcsin (c x))^{3/2} \, dx=\int x^{2} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{\frac {3}{2}}\, dx \]

[In]

integrate(x**2*(a+b*asin(c*x))**(3/2),x)

[Out]

Integral(x**2*(a + b*asin(c*x))**(3/2), x)

Maxima [F]

\[ \int x^2 (a+b \arcsin (c x))^{3/2} \, dx=\int { {\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {3}{2}} x^{2} \,d x } \]

[In]

integrate(x^2*(a+b*arcsin(c*x))^(3/2),x, algorithm="maxima")

[Out]

integrate((b*arcsin(c*x) + a)^(3/2)*x^2, x)

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.68 (sec) , antiderivative size = 1967, normalized size of antiderivative = 6.28 \[ \int x^2 (a+b \arcsin (c x))^{3/2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/8*sqrt(2)*sqrt(pi)*a^2*b^2*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^3) + 1/8*I*sqrt(2)*sqrt(pi)*
a*b^3*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^3) + 1/8*sqrt(2)*sqrt(pi)*a^2*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)/((-I*b^
3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c^3) - 1/8*I*sqrt(2)*sqrt(pi)*a*b^3*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*s
qrt(abs(b)))*c^3) - 1/4*sqrt(pi)*a^2*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)/((sqrt(6)*b^2 + I*sqrt(6)*b^3/abs(b))*c^3) - 1/12*I*sqrt(p
i)*a*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)/((sqrt(6)*b^2 + I*sqrt(6)*b^3/abs(b))*c^3) - 1/8*I*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)/((I*b^
2/sqrt(abs(b)) + b*sqrt(abs(b)))*c^3) + 3/32*sqrt(2)*sqrt(pi)*b^3*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/s
qrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/((I*b^2/sqrt(abs(b)) + b*sqrt(abs(
b)))*c^3) + 1/8*I*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)/((-I*b^2/sqrt(abs(b)) + b*sqrt(abs(b)))*c^3) + 3/32*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)*sqrt(b*arcsin(c*x) + a)*sqr
t(abs(b))/b)*e^(-I*a/b)/((-I*b^2/sqrt(abs(b)) + b*sqrt(abs(b)))*c^3) - 1/4*sqrt(pi)*a^2*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)/((sqrt
(6)*b^2 - I*sqrt(6)*b^3/abs(b))*c^3) + 1/12*I*sqrt(pi)*a*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)/((sqrt(6)*b^2 - I*sqrt(6)*b^3/abs(b))
*c^3) + 1/4*sqrt(pi)*a^2*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)/((sqrt(6)*b^(3/2) + I*sqrt(6)*b^(5/2)/abs(b))*c^3) + 1/12*I*sqrt(pi)*a*b^2*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)/((sqrt(6)*b^(3/2) + I*sqrt(6)*b^(5/2)/abs(b))*c^3) - 1/4*sqrt(pi)*a^2*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)/((I*sqrt(2)*b^2/sqrt(a
bs(b)) + sqrt(2)*b*sqrt(abs(b)))*c^3) - 1/4*sqrt(pi)*a^2*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)/((-I*sqrt(2)*b^2/sqrt(abs(b)) + sqrt(2)*b
*sqrt(abs(b)))*c^3) + 1/4*sqrt(pi)*a^2*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)/((sqrt(6)*b^(3/2) - I*sqrt(6)*b^(5/2)/abs(b))*c^3) - 1/12*I*s
qrt(pi)*a*b^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)/((sqrt(6)*b^(3/2) - I*sqrt(6)*b^(5/2)/abs(b))*c^3) - 1/48*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)/((sqrt
(6)*b + I*sqrt(6)*b^2/abs(b))*c^3) - 1/48*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)/((sqrt(6)*b - I*sqrt(6)*b^2/abs(b))*c^3) +
1/24*I*sqrt(b*arcsin(c*x) + a)*b*arcsin(c*x)*e^(3*I*arcsin(c*x))/c^3 - 1/8*I*sqrt(b*arcsin(c*x) + a)*b*arcsin(
c*x)*e^(I*arcsin(c*x))/c^3 + 1/8*I*sqrt(b*arcsin(c*x) + a)*b*arcsin(c*x)*e^(-I*arcsin(c*x))/c^3 - 1/24*I*sqrt(
b*arcsin(c*x) + a)*b*arcsin(c*x)*e^(-3*I*arcsin(c*x))/c^3 + 1/24*I*sqrt(b*arcsin(c*x) + a)*a*e^(3*I*arcsin(c*x
))/c^3 - 1/48*sqrt(b*arcsin(c*x) + a)*b*e^(3*I*arcsin(c*x))/c^3 - 1/8*I*sqrt(b*arcsin(c*x) + a)*a*e^(I*arcsin(
c*x))/c^3 + 3/16*sqrt(b*arcsin(c*x) + a)*b*e^(I*arcsin(c*x))/c^3 + 1/8*I*sqrt(b*arcsin(c*x) + a)*a*e^(-I*arcsi
n(c*x))/c^3 + 3/16*sqrt(b*arcsin(c*x) + a)*b*e^(-I*arcsin(c*x))/c^3 - 1/24*I*sqrt(b*arcsin(c*x) + a)*a*e^(-3*I
*arcsin(c*x))/c^3 - 1/48*sqrt(b*arcsin(c*x) + a)*b*e^(-3*I*arcsin(c*x))/c^3

Mupad [F(-1)]

Timed out. \[ \int x^2 (a+b \arcsin (c x))^{3/2} \, dx=\int x^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^{3/2} \,d x \]

[In]

int(x^2*(a + b*asin(c*x))^(3/2),x)

[Out]

int(x^2*(a + b*asin(c*x))^(3/2), x)