\(\int x^2 (a+b \arcsin (c x))^{5/2} \, dx\) [183]

Optimal result

Integrand size = 16, antiderivative size = 358 \[ \int x^2 (a+b \arcsin (c x))^{5/2} \, dx=-\frac {5 b^2 x \sqrt {a+b \arcsin (c x)}}{6 c^2}-\frac {5}{36} b^2 x^3 \sqrt {a+b \arcsin (c x)}+\frac {5 b \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{3/2}}{9 c^3}+\frac {5 b x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{3/2}}{18 c}+\frac {1}{3} x^3 (a+b \arcsin (c x))^{5/2}+\frac {15 b^{5/2} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{16 c^3}-\frac {5 b^{5/2} \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{144 c^3}-\frac {15 b^{5/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{16 c^3}+\frac {5 b^{5/2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{144 c^3} \]

[Out]

1/3*x^3*(a+b*arcsin(c*x))^(5/2)-5/864*b^(5/2)*cos(3*a/b)*FresnelS(6^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(
1/2))*6^(1/2)*Pi^(1/2)/c^3+5/864*b^(5/2)*FresnelC(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+15/32*b^(5/2)*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*2^(1/2
)*Pi^(1/2)/c^3-15/32*b^(5/2)*FresnelC(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+5/9*b*(a+b*arcsin(c*x))^(3/2)*(-c^2*x^2+1)^(1/2)/c^3+5/18*b*x^2*(a+b*arcsin(c*x))^(3/2)*(-c^2*x^2+1)^(
1/2)/c-5/6*b^2*x*(a+b*arcsin(c*x))^(1/2)/c^2-5/36*b^2*x^3*(a+b*arcsin(c*x))^(1/2)

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {4725, 4795, 4767, 4715, 4809, 3387, 3386, 3432, 3385, 3433, 3393} \[ \int x^2 (a+b \arcsin (c x))^{5/2} \, dx=-\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{16 c^3}+\frac {5 \sqrt {\frac {\pi }{6}} b^{5/2} \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{144 c^3}+\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{16 c^3}-\frac {5 \sqrt {\frac {\pi }{6}} b^{5/2} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{144 c^3}-\frac {5 b^2 x \sqrt {a+b \arcsin (c x)}}{6 c^2}-\frac {5}{36} b^2 x^3 \sqrt {a+b \arcsin (c x)}+\frac {5 b x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{3/2}}{18 c}+\frac {5 b \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{3/2}}{9 c^3}+\frac {1}{3} x^3 (a+b \arcsin (c x))^{5/2} \]

[In]

Int[x^2*(a + b*ArcSin[c*x])^(5/2),x]

[Out]

(-5*b^2*x*Sqrt[a + b*ArcSin[c*x]])/(6*c^2) - (5*b^2*x^3*Sqrt[a + b*ArcSin[c*x]])/36 + (5*b*Sqrt[1 - c^2*x^2]*(
a + b*ArcSin[c*x])^(3/2))/(9*c^3) + (5*b*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^(3/2))/(18*c) + (x^3*(a + b
*ArcSin[c*x])^(5/2))/3 + (15*b^(5/2)*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]
])/(16*c^3) - (5*b^(5/2)*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(144*
c^3) - (15*b^(5/2)*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(16*c^3) + (5*b
^(5/2)*Sqrt[Pi/6]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(144*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 3393

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 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 4715

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 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 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]

Rule 4809

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 (a+b \arcsin (c x))^{5/2}-\frac {1}{6} (5 b c) \int \frac {x^3 (a+b \arcsin (c x))^{3/2}}{\sqrt {1-c^2 x^2}} \, dx \\ & = \frac {5 b x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{3/2}}{18 c}+\frac {1}{3} x^3 (a+b \arcsin (c x))^{5/2}-\frac {1}{12} \left (5 b^2\right ) \int x^2 \sqrt {a+b \arcsin (c x)} \, dx-\frac {(5 b) \int \frac {x (a+b \arcsin (c x))^{3/2}}{\sqrt {1-c^2 x^2}} \, dx}{9 c} \\ & = -\frac {5}{36} b^2 x^3 \sqrt {a+b \arcsin (c x)}+\frac {5 b \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{3/2}}{9 c^3}+\frac {5 b x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{3/2}}{18 c}+\frac {1}{3} x^3 (a+b \arcsin (c x))^{5/2}-\frac {\left (5 b^2\right ) \int \sqrt {a+b \arcsin (c x)} \, dx}{6 c^2}+\frac {1}{72} \left (5 b^3 c\right ) \int \frac {x^3}{\sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}} \, dx \\ & = -\frac {5 b^2 x \sqrt {a+b \arcsin (c x)}}{6 c^2}-\frac {5}{36} b^2 x^3 \sqrt {a+b \arcsin (c x)}+\frac {5 b \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{3/2}}{9 c^3}+\frac {5 b x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{3/2}}{18 c}+\frac {1}{3} x^3 (a+b \arcsin (c x))^{5/2}-\frac {\left (5 b^2\right ) \text {Subst}\left (\int \frac {\sin ^3\left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{72 c^3}+\frac {\left (5 b^3\right ) \int \frac {x}{\sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}} \, dx}{12 c} \\ & = -\frac {5 b^2 x \sqrt {a+b \arcsin (c x)}}{6 c^2}-\frac {5}{36} b^2 x^3 \sqrt {a+b \arcsin (c x)}+\frac {5 b \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{3/2}}{9 c^3}+\frac {5 b x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{3/2}}{18 c}+\frac {1}{3} x^3 (a+b \arcsin (c x))^{5/2}-\frac {\left (5 b^2\right ) \text {Subst}\left (\int \left (-\frac {\sin \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{4 \sqrt {x}}+\frac {3 \sin \left (\frac {a}{b}-\frac {x}{b}\right )}{4 \sqrt {x}}\right ) \, dx,x,a+b \arcsin (c x)\right )}{72 c^3}-\frac {\left (5 b^2\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{12 c^3} \\ & = -\frac {5 b^2 x \sqrt {a+b \arcsin (c x)}}{6 c^2}-\frac {5}{36} b^2 x^3 \sqrt {a+b \arcsin (c x)}+\frac {5 b \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{3/2}}{9 c^3}+\frac {5 b x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{3/2}}{18 c}+\frac {1}{3} x^3 (a+b \arcsin (c x))^{5/2}+\frac {\left (5 b^2\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{288 c^3}-\frac {\left (5 b^2\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{96 c^3}+\frac {\left (5 b^2 \cos \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 )}{12 c^3}-\frac {\left (5 b^2 \sin \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 )}{12 c^3} \\ & = -\frac {5 b^2 x \sqrt {a+b \arcsin (c x)}}{6 c^2}-\frac {5}{36} b^2 x^3 \sqrt {a+b \arcsin (c x)}+\frac {5 b \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{3/2}}{9 c^3}+\frac {5 b x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{3/2}}{18 c}+\frac {1}{3} x^3 (a+b \arcsin (c x))^{5/2}+\frac {\left (5 b^2 \cos \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 )}{96 c^3}+\frac {\left (5 b^2 \cos \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 )}{6 c^3}-\frac {\left (5 b^2 \cos \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 )}{288 c^3}-\frac {\left (5 b^2 \sin \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 )}{96 c^3}-\frac {\left (5 b^2 \sin \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 )}{6 c^3}+\frac {\left (5 b^2 \sin \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 )}{288 c^3} \\ & = -\frac {5 b^2 x \sqrt {a+b \arcsin (c x)}}{6 c^2}-\frac {5}{36} b^2 x^3 \sqrt {a+b \arcsin (c x)}+\frac {5 b \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{3/2}}{9 c^3}+\frac {5 b x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{3/2}}{18 c}+\frac {1}{3} x^3 (a+b \arcsin (c x))^{5/2}+\frac {5 b^{5/2} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{6 c^3}-\frac {5 b^{5/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{6 c^3}+\frac {\left (5 b^2 \cos \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 )}{48 c^3}-\frac {\left (5 b^2 \cos \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 )}{144 c^3}-\frac {\left (5 b^2 \sin \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 )}{48 c^3}+\frac {\left (5 b^2 \sin \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 )}{144 c^3} \\ & = -\frac {5 b^2 x \sqrt {a+b \arcsin (c x)}}{6 c^2}-\frac {5}{36} b^2 x^3 \sqrt {a+b \arcsin (c x)}+\frac {5 b \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{3/2}}{9 c^3}+\frac {5 b x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{3/2}}{18 c}+\frac {1}{3} x^3 (a+b \arcsin (c x))^{5/2}+\frac {15 b^{5/2} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{16 c^3}-\frac {5 b^{5/2} \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{144 c^3}-\frac {15 b^{5/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{16 c^3}+\frac {5 b^{5/2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{144 c^3} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.20 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.64 \[ \int x^2 (a+b \arcsin (c x))^{5/2} \, dx=\frac {b^3 e^{-\frac {3 i a}{b}} \left (-81 e^{\frac {2 i a}{b}} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {7}{2},-\frac {i (a+b \arcsin (c x))}{b}\right )-81 e^{\frac {4 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {7}{2},\frac {i (a+b \arcsin (c x))}{b}\right )+\sqrt {3} \left (\sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {7}{2},-\frac {3 i (a+b \arcsin (c x))}{b}\right )+e^{\frac {6 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {7}{2},\frac {3 i (a+b \arcsin (c x))}{b}\right )\right )\right )}{648 c^3 \sqrt {a+b \arcsin (c x)}} \]

[In]

Integrate[x^2*(a + b*ArcSin[c*x])^(5/2),x]

[Out]

(b^3*(-81*E^(((2*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[7/2, ((-I)*(a + b*ArcSin[c*x]))/b] - 81*E^(
((4*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[7/2, (I*(a + b*ArcSin[c*x]))/b] + Sqrt[3]*(Sqrt[((-I)*(a +
b*ArcSin[c*x]))/b]*Gamma[7/2, ((-3*I)*(a + b*ArcSin[c*x]))/b] + E^(((6*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b
]*Gamma[7/2, ((3*I)*(a + b*ArcSin[c*x]))/b])))/(648*c^3*E^(((3*I)*a)/b)*Sqrt[a + b*ArcSin[c*x]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(818\) vs. \(2(278)=556\).

Time = 0.12 (sec) , antiderivative size = 819, normalized size of antiderivative = 2.29

method	result	size
default	\(\text {Expression too large to display}\)	\(819\)

[In]

int(x^2*(a+b*arcsin(c*x))^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/864/c^3*b*(36*arcsin(c*x)^2*2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)*sin(-3*(a+b*arcsin(c*x))/
b+3*a/b)*b^2-108*arcsin(c*x)^2*2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)*sin(-(a+b*arcsin(c*x))/b+
a/b)*b^2+72*arcsin(c*x)*2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)*sin(-3*(a+b*arcsin(c*x))/b+3*a/b
)*a*b-30*arcsin(c*x)*2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)*cos(-3*(a+b*arcsin(c*x))/b+3*a/b)*b
^2-216*arcsin(c*x)*2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)*sin(-(a+b*arcsin(c*x))/b+a/b)*a*b+270
*arcsin(c*x)*2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)*cos(-(a+b*arcsin(c*x))/b+a/b)*b^2+5*Pi*cos(
3*a/b)*FresnelS(3*2^(1/2)/Pi^(1/2)/(-3/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*(-1/b)^(1/2)*(-3/b)^(1/2)*b^3+5*Pi*
sin(3*a/b)*FresnelC(3*2^(1/2)/Pi^(1/2)/(-3/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*(-1/b)^(1/2)*(-3/b)^(1/2)*b^3+3
6*2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)*sin(-3*(a+b*arcsin(c*x))/b+3*a/b)*a^2-15*2^(1/2)*Pi^(1
/2)*(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)*sin(-3*(a+b*arcsin(c*x))/b+3*a/b)*b^2-30*2^(1/2)*Pi^(1/2)*(-1/b)^(1/2
)*(a+b*arcsin(c*x))^(1/2)*cos(-3*(a+b*arcsin(c*x))/b+3*a/b)*a*b-108*2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*(a+b*arcsin(
c*x))^(1/2)*sin(-(a+b*arcsin(c*x))/b+a/b)*a^2+405*2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)*sin(-(
a+b*arcsin(c*x))/b+a/b)*b^2+270*2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)*cos(-(a+b*arcsin(c*x))/b
+a/b)*a*b+405*Pi*b^2*sin(a/b)*FresnelC(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)+405*Pi*b^2*cos
(a/b)*FresnelS(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b))*2^(1/2)/Pi^(1/2)*(-1/b)^(1/2)

Fricas [F(-2)]

Exception generated. \[ \int x^2 (a+b \arcsin (c x))^{5/2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^2*(a+b*arcsin(c*x))^(5/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int x^2 (a+b \arcsin (c x))^{5/2} \, dx=\int x^{2} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{\frac {5}{2}}\, dx \]

[In]

integrate(x**2*(a+b*asin(c*x))**(5/2),x)

[Out]

Integral(x**2*(a + b*asin(c*x))**(5/2), x)

Maxima [F]

\[ \int x^2 (a+b \arcsin (c x))^{5/2} \, dx=\int { {\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {5}{2}} x^{2} \,d x } \]

[In]

integrate(x^2*(a+b*arcsin(c*x))^(5/2),x, algorithm="maxima")

[Out]

integrate((b*arcsin(c*x) + a)^(5/2)*x^2, x)

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.40 (sec) , antiderivative size = 2466, normalized size of antiderivative = 6.89 \[ \int x^2 (a+b \arcsin (c x))^{5/2} \, dx=\text {Too large to display} \]

[In]

integrate(x^2*(a+b*arcsin(c*x))^(5/2),x, algorithm="giac")

[Out]

1/576*(72*sqrt(2)*sqrt(pi)*a^3*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))) + 72*sqrt(2)*sqrt(pi)*a^3
*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))) + 216*I*sqrt(2)*sqrt(pi)*a^2*b^2*erf(-1/2*I*sqrt(2)*sq
rt(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))) - 216*I*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^2/sqrt(abs(b)) + b*sqrt(abs(b))) +
24*I*sqrt(b*arcsin(c*x) + a)*b^2*arcsin(c*x)^2*e^(3*I*arcsin(c*x)) - 72*I*sqrt(b*arcsin(c*x) + a)*b^2*arcsin(
c*x)^2*e^(I*arcsin(c*x)) + 72*I*sqrt(b*arcsin(c*x) + a)*b^2*arcsin(c*x)^2*e^(-I*arcsin(c*x)) - 24*I*sqrt(b*arc
sin(c*x) + a)*b^2*arcsin(c*x)^2*e^(-3*I*arcsin(c*x)) - 144*sqrt(pi)*a^3*sqrt(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 + I*sqrt(6)*
b^2/abs(b)) - 144*I*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 + I*sqrt(6)*b^2/abs(b)) - 216*I*sqrt(2)*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*b/sqrt(abs(b)) + sqrt(abs(b))) - 135*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)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/(I*b/sqrt(abs(b)) + sqr
t(abs(b))) + 216*I*sqrt(2)*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*b/sqrt(abs(b)) + sqrt(abs(b))) + 135*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)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b)
)/b)*e^(-I*a/b)/(-I*b/sqrt(abs(b)) + sqrt(abs(b))) - 144*sqrt(pi)*a^3*sqrt(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 - I*sqrt(6)*b
^2/abs(b)) + 144*I*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 - I*sqrt(6)*b^2/abs(b)) + 48*I*sqrt(b*arcsin(c*x) +
a)*a*b*arcsin(c*x)*e^(3*I*arcsin(c*x)) - 20*sqrt(b*arcsin(c*x) + a)*b^2*arcsin(c*x)*e^(3*I*arcsin(c*x)) - 144
*I*sqrt(b*arcsin(c*x) + a)*a*b*arcsin(c*x)*e^(I*arcsin(c*x)) + 180*sqrt(b*arcsin(c*x) + a)*b^2*arcsin(c*x)*e^(
I*arcsin(c*x)) + 144*I*sqrt(b*arcsin(c*x) + a)*a*b*arcsin(c*x)*e^(-I*arcsin(c*x)) + 180*sqrt(b*arcsin(c*x) + a
)*b^2*arcsin(c*x)*e^(-I*arcsin(c*x)) - 48*I*sqrt(b*arcsin(c*x) + a)*a*b*arcsin(c*x)*e^(-3*I*arcsin(c*x)) - 20*
sqrt(b*arcsin(c*x) + a)*b^2*arcsin(c*x)*e^(-3*I*arcsin(c*x)) + 144*sqrt(pi)*a^3*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)*sqrt(b) + I*sq
rt(6)*b^(3/2)/abs(b)) + 144*I*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)*sqrt(b) + I*sqrt(6)*b^(3/2)/abs(b)) + 36*sqrt(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)*sqrt(b) + I*sqrt(6)*b^(3/2)/abs(b)) - 144*sqrt(pi)*a^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*sqrt(2)*b/sqrt(abs(
b)) + sqrt(2)*sqrt(abs(b))) - 144*sqrt(pi)*a^3*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sq
rt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/(-I*sqrt(2)*b/sqrt(abs(b)) + sqrt(2)*sqrt(abs(b))) +
144*sqrt(pi)*a^3*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)*sqrt(b) - I*sqrt(6)*b^(3/2)/abs(b)) - 144*I*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)*
sqrt(b) - I*sqrt(6)*b^(3/2)/abs(b)) + 36*sqrt(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)*sqrt(b) - I*sqrt(6)*b^(3/2)/abs(b)) -
36*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)/(sqrt(6) + I*sqrt(6)*b/abs(b)) + 10*I*sqrt(pi)*b^(5/2)*erf(-1/2*sqrt(6)*sqrt(b*a
rcsin(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) + I*sqrt(
6)*b/abs(b)) - 36*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*a
rcsin(c*x) + a)*sqrt(b)/abs(b))*e^(-3*I*a/b)/(sqrt(6) - I*sqrt(6)*b/abs(b)) - 10*I*sqrt(pi)*b^(5/2)*erf(-1/2*s
qrt(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)/(s
qrt(6) - I*sqrt(6)*b/abs(b)) + 24*I*sqrt(b*arcsin(c*x) + a)*a^2*e^(3*I*arcsin(c*x)) - 20*sqrt(b*arcsin(c*x) +
a)*a*b*e^(3*I*arcsin(c*x)) - 10*I*sqrt(b*arcsin(c*x) + a)*b^2*e^(3*I*arcsin(c*x)) - 72*I*sqrt(b*arcsin(c*x) +
a)*a^2*e^(I*arcsin(c*x)) + 180*sqrt(b*arcsin(c*x) + a)*a*b*e^(I*arcsin(c*x)) + 270*I*sqrt(b*arcsin(c*x) + a)*b
^2*e^(I*arcsin(c*x)) + 72*I*sqrt(b*arcsin(c*x) + a)*a^2*e^(-I*arcsin(c*x)) + 180*sqrt(b*arcsin(c*x) + a)*a*b*e
^(-I*arcsin(c*x)) - 270*I*sqrt(b*arcsin(c*x) + a)*b^2*e^(-I*arcsin(c*x)) - 24*I*sqrt(b*arcsin(c*x) + a)*a^2*e^
(-3*I*arcsin(c*x)) - 20*sqrt(b*arcsin(c*x) + a)*a*b*e^(-3*I*arcsin(c*x)) + 10*I*sqrt(b*arcsin(c*x) + a)*b^2*e^
(-3*I*arcsin(c*x)))/c^3

Mupad [F(-1)]

Timed out. \[ \int x^2 (a+b \arcsin (c x))^{5/2} \, dx=\int x^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^{5/2} \,d x \]

[In]

int(x^2*(a + b*asin(c*x))^(5/2),x)

[Out]

int(x^2*(a + b*asin(c*x))^(5/2), x)