\(\int \frac {x^2}{(a+b \arcsin (c x))^{5/2}} \, dx\) [198]

Optimal result

Integrand size = 16, antiderivative size = 291 \[ \int \frac {x^2}{(a+b \arcsin (c x))^{5/2}} \, dx=-\frac {2 x^2 \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}}-\frac {8 x}{3 b^2 c^2 \sqrt {a+b \arcsin (c x)}}+\frac {4 x^3}{b^2 \sqrt {a+b \arcsin (c x)}}-\frac {\sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c^3}+\frac {\sqrt {6 \pi } \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{b^{5/2} c^3}-\frac {\sqrt {2 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{3 b^{5/2} c^3}+\frac {\sqrt {6 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{b^{5/2} c^3} \]

[Out]

-1/3*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)/b^(5/2)/c^3-1/3*Fres
nelS(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)/b^(5/2)/c^3+cos(3*a/b)*Fresne
lC(6^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*6^(1/2)*Pi^(1/2)/b^(5/2)/c^3+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)/b^(5/2)/c^3-2/3*x^2*(-c^2*x^2+1)^(1/2)/b/c/(a+b*arc
sin(c*x))^(3/2)-8/3*x/b^2/c^2/(a+b*arcsin(c*x))^(1/2)+4*x^3/b^2/(a+b*arcsin(c*x))^(1/2)

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {4729, 4807, 4731, 4491, 3387, 3386, 3432, 3385, 3433, 4719} \[ \int \frac {x^2}{(a+b \arcsin (c x))^{5/2}} \, dx=-\frac {\sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c^3}+\frac {\sqrt {6 \pi } \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{b^{5/2} c^3}-\frac {\sqrt {2 \pi } \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c^3}+\frac {\sqrt {6 \pi } \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{b^{5/2} c^3}-\frac {8 x}{3 b^2 c^2 \sqrt {a+b \arcsin (c x)}}+\frac {4 x^3}{b^2 \sqrt {a+b \arcsin (c x)}}-\frac {2 x^2 \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}} \]

[In]

Int[x^2/(a + b*ArcSin[c*x])^(5/2),x]

[Out]

(-2*x^2*Sqrt[1 - c^2*x^2])/(3*b*c*(a + b*ArcSin[c*x])^(3/2)) - (8*x)/(3*b^2*c^2*Sqrt[a + b*ArcSin[c*x]]) + (4*
x^3)/(b^2*Sqrt[a + b*ArcSin[c*x]]) - (Sqrt[2*Pi]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b
]])/(3*b^(5/2)*c^3) + (Sqrt[6*Pi]*Cos[(3*a)/b]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(b^(5/2
)*c^3) - (Sqrt[2*Pi]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(3*b^(5/2)*c^3) + (Sqrt[
6*Pi]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(b^(5/2)*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 4729

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

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 4807

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

Rubi steps \begin{align*} \text {integral}& = -\frac {2 x^2 \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}}+\frac {4 \int \frac {x}{\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{3/2}} \, dx}{3 b c}-\frac {(2 c) \int \frac {x^3}{\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{3/2}} \, dx}{b} \\ & = -\frac {2 x^2 \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}}-\frac {8 x}{3 b^2 c^2 \sqrt {a+b \arcsin (c x)}}+\frac {4 x^3}{b^2 \sqrt {a+b \arcsin (c x)}}-\frac {12 \int \frac {x^2}{\sqrt {a+b \arcsin (c x)}} \, dx}{b^2}+\frac {8 \int \frac {1}{\sqrt {a+b \arcsin (c x)}} \, dx}{3 b^2 c^2} \\ & = -\frac {2 x^2 \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}}-\frac {8 x}{3 b^2 c^2 \sqrt {a+b \arcsin (c x)}}+\frac {4 x^3}{b^2 \sqrt {a+b \arcsin (c x)}}+\frac {8 \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{3 b^3 c^3}-\frac {12 \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 )}{b^3 c^3} \\ & = -\frac {2 x^2 \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}}-\frac {8 x}{3 b^2 c^2 \sqrt {a+b \arcsin (c x)}}+\frac {4 x^3}{b^2 \sqrt {a+b \arcsin (c x)}}-\frac {12 \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 )}{b^3 c^3}+\frac {\left (8 \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 )}{3 b^3 c^3}+\frac {\left (8 \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 )}{3 b^3 c^3} \\ & = -\frac {2 x^2 \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}}-\frac {8 x}{3 b^2 c^2 \sqrt {a+b \arcsin (c x)}}+\frac {4 x^3}{b^2 \sqrt {a+b \arcsin (c x)}}+\frac {3 \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 )}{b^3 c^3}-\frac {3 \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{b^3 c^3}+\frac {\left (16 \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 b^3 c^3}+\frac {\left (16 \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 b^3 c^3} \\ & = -\frac {2 x^2 \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}}-\frac {8 x}{3 b^2 c^2 \sqrt {a+b \arcsin (c x)}}+\frac {4 x^3}{b^2 \sqrt {a+b \arcsin (c x)}}+\frac {8 \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c^3}+\frac {8 \sqrt {2 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{3 b^{5/2} c^3}-\frac {\left (3 \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 )}{b^3 c^3}+\frac {\left (3 \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 )}{b^3 c^3}-\frac {\left (3 \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 )}{b^3 c^3}+\frac {\left (3 \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 )}{b^3 c^3} \\ & = -\frac {2 x^2 \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}}-\frac {8 x}{3 b^2 c^2 \sqrt {a+b \arcsin (c x)}}+\frac {4 x^3}{b^2 \sqrt {a+b \arcsin (c x)}}+\frac {8 \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c^3}+\frac {8 \sqrt {2 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{3 b^{5/2} c^3}-\frac {\left (6 \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 )}{b^3 c^3}+\frac {\left (6 \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 )}{b^3 c^3}-\frac {\left (6 \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 )}{b^3 c^3}+\frac {\left (6 \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 )}{b^3 c^3} \\ & = -\frac {2 x^2 \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}}-\frac {8 x}{3 b^2 c^2 \sqrt {a+b \arcsin (c x)}}+\frac {4 x^3}{b^2 \sqrt {a+b \arcsin (c x)}}-\frac {\sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c^3}+\frac {\sqrt {6 \pi } \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{b^{5/2} c^3}-\frac {\sqrt {2 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{3 b^{5/2} c^3}+\frac {\sqrt {6 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{b^{5/2} c^3} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.53 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.27 \[ \int \frac {x^2}{(a+b \arcsin (c x))^{5/2}} \, dx=\frac {-6 i a e^{-3 i \arcsin (c x)}+b e^{-3 i \arcsin (c x)} (1-6 i \arcsin (c x))+e^{3 i \arcsin (c x)} (6 i a+b+6 i b \arcsin (c x))-i e^{i \arcsin (c x)} (2 a-i b+2 b \arcsin (c x))-2 b e^{-\frac {i a}{b}} \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {i (a+b \arcsin (c x))}{b}\right )+i e^{-i \arcsin (c x)} \left (2 a+i b+2 b \arcsin (c x)+2 i b e^{\frac {i (a+b \arcsin (c x))}{b}} \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},\frac {i (a+b \arcsin (c x))}{b}\right )\right )+6 \sqrt {3} b e^{-\frac {3 i a}{b}} \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {3 i (a+b \arcsin (c x))}{b}\right )+6 \sqrt {3} b e^{\frac {3 i a}{b}} \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},\frac {3 i (a+b \arcsin (c x))}{b}\right )}{12 b^2 c^3 (a+b \arcsin (c x))^{3/2}} \]

[In]

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

[Out]

(((-6*I)*a)/E^((3*I)*ArcSin[c*x]) + (b*(1 - (6*I)*ArcSin[c*x]))/E^((3*I)*ArcSin[c*x]) + E^((3*I)*ArcSin[c*x])*
((6*I)*a + b + (6*I)*b*ArcSin[c*x]) - I*E^(I*ArcSin[c*x])*(2*a - I*b + 2*b*ArcSin[c*x]) - (2*b*(((-I)*(a + b*A
rcSin[c*x]))/b)^(3/2)*Gamma[1/2, ((-I)*(a + b*ArcSin[c*x]))/b])/E^((I*a)/b) + (I*(2*a + I*b + 2*b*ArcSin[c*x]
+ (2*I)*b*E^((I*(a + b*ArcSin[c*x]))/b)*((I*(a + b*ArcSin[c*x]))/b)^(3/2)*Gamma[1/2, (I*(a + b*ArcSin[c*x]))/b
]))/E^(I*ArcSin[c*x]) + (6*Sqrt[3]*b*(((-I)*(a + b*ArcSin[c*x]))/b)^(3/2)*Gamma[1/2, ((-3*I)*(a + b*ArcSin[c*x
]))/b])/E^(((3*I)*a)/b) + 6*Sqrt[3]*b*E^(((3*I)*a)/b)*((I*(a + b*ArcSin[c*x]))/b)^(3/2)*Gamma[1/2, ((3*I)*(a +
b*ArcSin[c*x]))/b])/(12*b^2*c^3*(a + b*ArcSin[c*x])^(3/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(671\) vs. \(2(235)=470\).

Time = 0.10 (sec) , antiderivative size = 672, normalized size of antiderivative = 2.31

method	result	size
default	\(-\frac {-6 \arcsin \left (c x \right ) \sqrt {-\frac {3}{b}}\, \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 ) \sqrt {\pi }\, \sqrt {2}\, b +6 \arcsin \left (c x \right ) \sqrt {-\frac {3}{b}}\, \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 ) \sqrt {\pi }\, \sqrt {2}\, b +2 \arcsin \left (c x \right ) \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 {\pi }\, \sqrt {2}\, \sqrt {-\frac {1}{b}}\, b -2 \arcsin \left (c x \right ) \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 {\pi }\, \sqrt {2}\, \sqrt {-\frac {1}{b}}\, b -6 \sqrt {-\frac {3}{b}}\, \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 ) \sqrt {\pi }\, \sqrt {2}\, a +6 \sqrt {-\frac {3}{b}}\, \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 ) \sqrt {\pi }\, \sqrt {2}\, a +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 {\pi }\, \sqrt {2}\, \sqrt {-\frac {1}{b}}\, a -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 {\pi }\, \sqrt {2}\, \sqrt {-\frac {1}{b}}\, a +2 \arcsin \left (c x \right ) \sin \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) b -6 \arcsin \left (c x \right ) \sin \left (-\frac {3 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) b +\cos \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) b +2 \sin \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) a -\cos \left (-\frac {3 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) b -6 \sin \left (-\frac {3 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) a}{6 c^{3} b^{2} \left (a +b \arcsin \left (c x \right )\right )^{\frac {3}{2}}}\)	\(672\)

[In]

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

[Out]

-1/6/c^3/b^2*(-6*arcsin(c*x)*(-3/b)^(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)*Pi^(1/2)*2^(1/2)*b+6*arcsin(c*x)*(-3/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)*sin(3
*a/b)*FresnelS(3*2^(1/2)/Pi^(1/2)/(-3/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*Pi^(1/2)*2^(1/2)*b+2*arcsin(c*x)*(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)*Pi^(1/2)*2^(1/
2)*(-1/b)^(1/2)*b-2*arcsin(c*x)*(a+b*arcsin(c*x))^(1/2)*sin(a/b)*FresnelS(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*a
rcsin(c*x))^(1/2)/b)*Pi^(1/2)*2^(1/2)*(-1/b)^(1/2)*b-6*(-3/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)*cos(3*a/b)*Fresnel
C(3*2^(1/2)/Pi^(1/2)/(-3/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*Pi^(1/2)*2^(1/2)*a+6*(-3/b)^(1/2)*(a+b*arcsin(c*x
))^(1/2)*sin(3*a/b)*FresnelS(3*2^(1/2)/Pi^(1/2)/(-3/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*Pi^(1/2)*2^(1/2)*a+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)*Pi^(1/2)*2^(
1/2)*(-1/b)^(1/2)*a-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)*Pi^(1/2)*2^(1/2)*(-1/b)^(1/2)*a+2*arcsin(c*x)*sin(-(a+b*arcsin(c*x))/b+a/b)*b-6*arcsin(c*x)*sin(-3*
(a+b*arcsin(c*x))/b+3*a/b)*b+cos(-(a+b*arcsin(c*x))/b+a/b)*b+2*sin(-(a+b*arcsin(c*x))/b+a/b)*a-cos(-3*(a+b*arc
sin(c*x))/b+3*a/b)*b-6*sin(-3*(a+b*arcsin(c*x))/b+3*a/b)*a)/(a+b*arcsin(c*x))^(3/2)

Fricas [F(-2)]

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

[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]

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

[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]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(x^2/(a + b*asin(c*x))^(5/2),x)

[Out]

int(x^2/(a + b*asin(c*x))^(5/2), x)