\(\int \frac {x^2}{(a+b \arcsin (c x))^{5/2}} \, dx\) [166]

Optimal result

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

-2/3*x^2*(-c^2*x^2+1)^(1/2)/b/c/(a+b*arcsin(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)-1/3*2^(1/2)*Pi^(1/2) 
*cos(a/b)*FresnelC(2^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))/b^(5/ 
2)/c^3+6^(1/2)*Pi^(1/2)*cos(3*a/b)*FresnelC(6^(1/2)/Pi^(1/2)*(a+b*arcsin(c 
*x))^(1/2)/b^(1/2))/b^(5/2)/c^3-1/3*2^(1/2)*Pi^(1/2)*FresnelS(2^(1/2)/Pi^( 
1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*sin(a/b)/b^(5/2)/c^3+6^(1/2)*Pi^(1/2 
)*FresnelS(6^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*sin(3*a/b)/b^ 
(5/2)/c^3
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.29 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.27 \[ \int \frac {x^2}{(a+b \arcsin (c x))^{5/2}} \, dx=\frac {-6 i a e^{-3 i \arcsin (c x)}+b e^{-3 i \arcsin (c x)} (1-6 i \arcsin (c x))+e^{3 i \arcsin (c x)} (6 i a+b+6 i b \arcsin (c x))-i e^{i \arcsin (c x)} (2 a-i b+2 b \arcsin (c x))-2 b e^{-\frac {i a}{b}} \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {i (a+b \arcsin (c x))}{b}\right )+i e^{-i \arcsin (c x)} \left (2 a+i b+2 b \arcsin (c x)+2 i b e^{\frac {i (a+b \arcsin (c x))}{b}} \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},\frac {i (a+b \arcsin (c x))}{b}\right )\right )+6 \sqrt {3} b e^{-\frac {3 i a}{b}} \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {3 i (a+b \arcsin (c x))}{b}\right )+6 \sqrt {3} b e^{\frac {3 i a}{b}} \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},\frac {3 i (a+b \arcsin (c x))}{b}\right )}{12 b^2 c^3 (a+b \arcsin (c x))^{3/2}} \] Input:

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

Output:

(((-6*I)*a)/E^((3*I)*ArcSin[c*x]) + (b*(1 - (6*I)*ArcSin[c*x]))/E^((3*I)*A 
rcSin[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*Arc 
Sin[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)*Gamm 
a[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*Arc 
Sin[c*x]))/b])/(12*b^2*c^3*(a + b*ArcSin[c*x])^(3/2))
 

Rubi [A] (verified)

Time = 2.12 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.46, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {5144, 5222, 5134, 3042, 3787, 25, 3042, 3785, 3786, 3832, 3833, 5146, 4906, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 5144

\(\displaystyle \frac {4 \int \frac {x}{\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{3/2}}dx}{3 b c}-\frac {2 c \int \frac {x^3}{\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{3/2}}dx}{b}-\frac {2 x^2 \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}}\)

\(\Big \downarrow \) 5222

\(\displaystyle -\frac {2 c \left (\frac {6 \int \frac {x^2}{\sqrt {a+b \arcsin (c x)}}dx}{b c}-\frac {2 x^3}{b c \sqrt {a+b \arcsin (c x)}}\right )}{b}+\frac {4 \left (\frac {2 \int \frac {1}{\sqrt {a+b \arcsin (c x)}}dx}{b c}-\frac {2 x}{b c \sqrt {a+b \arcsin (c x)}}\right )}{3 b c}-\frac {2 x^2 \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}}\)

\(\Big \downarrow \) 5134

\(\displaystyle \frac {4 \left (\frac {2 \int \frac {\cos \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{b^2 c^2}-\frac {2 x}{b c \sqrt {a+b \arcsin (c x)}}\right )}{3 b c}-\frac {2 c \left (\frac {6 \int \frac {x^2}{\sqrt {a+b \arcsin (c x)}}dx}{b c}-\frac {2 x^3}{b c \sqrt {a+b \arcsin (c x)}}\right )}{b}-\frac {2 x^2 \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {4 \left (\frac {2 \int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{b^2 c^2}-\frac {2 x}{b c \sqrt {a+b \arcsin (c x)}}\right )}{3 b c}-\frac {2 c \left (\frac {6 \int \frac {x^2}{\sqrt {a+b \arcsin (c x)}}dx}{b c}-\frac {2 x^3}{b c \sqrt {a+b \arcsin (c x)}}\right )}{b}-\frac {2 x^2 \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}}\)

\(\Big \downarrow \) 3787

\(\displaystyle \frac {4 \left (\frac {2 \left (\cos \left (\frac {a}{b}\right ) \int \frac {\cos \left (\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))-\sin \left (\frac {a}{b}\right ) \int -\frac {\sin \left (\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))\right )}{b^2 c^2}-\frac {2 x}{b c \sqrt {a+b \arcsin (c x)}}\right )}{3 b c}-\frac {2 c \left (\frac {6 \int \frac {x^2}{\sqrt {a+b \arcsin (c x)}}dx}{b c}-\frac {2 x^3}{b c \sqrt {a+b \arcsin (c x)}}\right )}{b}-\frac {2 x^2 \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {4 \left (\frac {2 \left (\sin \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))+\cos \left (\frac {a}{b}\right ) \int \frac {\cos \left (\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))\right )}{b^2 c^2}-\frac {2 x}{b c \sqrt {a+b \arcsin (c x)}}\right )}{3 b c}-\frac {2 c \left (\frac {6 \int \frac {x^2}{\sqrt {a+b \arcsin (c x)}}dx}{b c}-\frac {2 x^3}{b c \sqrt {a+b \arcsin (c x)}}\right )}{b}-\frac {2 x^2 \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {4 \left (\frac {2 \left (\sin \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))+\cos \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arcsin (c x)}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))\right )}{b^2 c^2}-\frac {2 x}{b c \sqrt {a+b \arcsin (c x)}}\right )}{3 b c}-\frac {2 c \left (\frac {6 \int \frac {x^2}{\sqrt {a+b \arcsin (c x)}}dx}{b c}-\frac {2 x^3}{b c \sqrt {a+b \arcsin (c x)}}\right )}{b}-\frac {2 x^2 \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}}\)

\(\Big \downarrow \) 3785

\(\displaystyle \frac {4 \left (\frac {2 \left (\sin \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))+2 \cos \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arcsin (c x)}{b}\right )d\sqrt {a+b \arcsin (c x)}\right )}{b^2 c^2}-\frac {2 x}{b c \sqrt {a+b \arcsin (c x)}}\right )}{3 b c}-\frac {2 c \left (\frac {6 \int \frac {x^2}{\sqrt {a+b \arcsin (c x)}}dx}{b c}-\frac {2 x^3}{b c \sqrt {a+b \arcsin (c x)}}\right )}{b}-\frac {2 x^2 \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}}\)

\(\Big \downarrow \) 3786

\(\displaystyle \frac {4 \left (\frac {2 \left (2 \sin \left (\frac {a}{b}\right ) \int \sin \left (\frac {a+b \arcsin (c x)}{b}\right )d\sqrt {a+b \arcsin (c x)}+2 \cos \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arcsin (c x)}{b}\right )d\sqrt {a+b \arcsin (c x)}\right )}{b^2 c^2}-\frac {2 x}{b c \sqrt {a+b \arcsin (c x)}}\right )}{3 b c}-\frac {2 c \left (\frac {6 \int \frac {x^2}{\sqrt {a+b \arcsin (c x)}}dx}{b c}-\frac {2 x^3}{b c \sqrt {a+b \arcsin (c x)}}\right )}{b}-\frac {2 x^2 \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}}\)

\(\Big \downarrow \) 3832

\(\displaystyle \frac {4 \left (\frac {2 \left (2 \cos \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arcsin (c x)}{b}\right )d\sqrt {a+b \arcsin (c x)}+\sqrt {2 \pi } \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )\right )}{b^2 c^2}-\frac {2 x}{b c \sqrt {a+b \arcsin (c x)}}\right )}{3 b c}-\frac {2 c \left (\frac {6 \int \frac {x^2}{\sqrt {a+b \arcsin (c x)}}dx}{b c}-\frac {2 x^3}{b c \sqrt {a+b \arcsin (c x)}}\right )}{b}-\frac {2 x^2 \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}}\)

\(\Big \downarrow \) 3833

\(\displaystyle -\frac {2 c \left (\frac {6 \int \frac {x^2}{\sqrt {a+b \arcsin (c x)}}dx}{b c}-\frac {2 x^3}{b c \sqrt {a+b \arcsin (c x)}}\right )}{b}+\frac {4 \left (\frac {2 \left (\sqrt {2 \pi } \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\sqrt {2 \pi } \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )\right )}{b^2 c^2}-\frac {2 x}{b c \sqrt {a+b \arcsin (c x)}}\right )}{3 b c}-\frac {2 x^2 \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}}\)

\(\Big \downarrow \) 5146

\(\displaystyle -\frac {2 c \left (\frac {6 \int \frac {\cos \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin ^2\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{b^2 c^4}-\frac {2 x^3}{b c \sqrt {a+b \arcsin (c x)}}\right )}{b}+\frac {4 \left (\frac {2 \left (\sqrt {2 \pi } \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\sqrt {2 \pi } \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )\right )}{b^2 c^2}-\frac {2 x}{b c \sqrt {a+b \arcsin (c x)}}\right )}{3 b c}-\frac {2 x^2 \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}}\)

\(\Big \downarrow \) 4906

\(\displaystyle -\frac {2 c \left (\frac {6 \int \left (\frac {\cos \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{4 \sqrt {a+b \arcsin (c x)}}-\frac {\cos \left (\frac {3 a}{b}-\frac {3 (a+b \arcsin (c x))}{b}\right )}{4 \sqrt {a+b \arcsin (c x)}}\right )d(a+b \arcsin (c x))}{b^2 c^4}-\frac {2 x^3}{b c \sqrt {a+b \arcsin (c x)}}\right )}{b}+\frac {4 \left (\frac {2 \left (\sqrt {2 \pi } \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\sqrt {2 \pi } \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )\right )}{b^2 c^2}-\frac {2 x}{b c \sqrt {a+b \arcsin (c x)}}\right )}{3 b c}-\frac {2 x^2 \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 c \left (\frac {6 \left (\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )\right )}{b^2 c^4}-\frac {2 x^3}{b c \sqrt {a+b \arcsin (c x)}}\right )}{b}+\frac {4 \left (\frac {2 \left (\sqrt {2 \pi } \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\sqrt {2 \pi } \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )\right )}{b^2 c^2}-\frac {2 x}{b c \sqrt {a+b \arcsin (c x)}}\right )}{3 b c}-\frac {2 x^2 \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}}\)

Input:

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

Output:

(-2*x^2*Sqrt[1 - c^2*x^2])/(3*b*c*(a + b*ArcSin[c*x])^(3/2)) + (4*((-2*x)/ 
(b*c*Sqrt[a + b*ArcSin[c*x]]) + (2*(Sqrt[b]*Sqrt[2*Pi]*Cos[a/b]*FresnelC[( 
Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]] + Sqrt[b]*Sqrt[2*Pi]*FresnelS 
[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b]))/(b^2*c^2)))/(3*b 
*c) - (2*c*((-2*x^3)/(b*c*Sqrt[a + b*ArcSin[c*x]]) + (6*((Sqrt[b]*Sqrt[Pi/ 
2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/2 - (S 
qrt[b]*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x] 
])/Sqrt[b]])/2 + (Sqrt[b]*Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSi 
n[c*x]])/Sqrt[b]]*Sin[a/b])/2 - (Sqrt[b]*Sqrt[Pi/6]*FresnelS[(Sqrt[6/Pi]*S 
qrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/2))/(b^2*c^4)))/b
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3785

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> S 
imp[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 3786

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[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 3787

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Cos 
[(d*e - c*f)/d]   Int[Sin[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] + Simp[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 3832

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 3833

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 4906

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b 
_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(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] && IG 
tQ[p, 0]
 

rule 5134

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

rule 5144

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] + (Sim 
p[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] - Simp[m/(b*c*(n + 1))   Int[x^(m - 1)*((a + b*ArcSi 
n[c*x])^(n + 1)/Sqrt[1 - c^2*x^2]), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[ 
m, 0] && LtQ[n, -2]
 

rule 5146

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[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 5222

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] - Simp[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]
 

Maple [B] (verified)

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

Time = 0.15 (sec) , antiderivative size = 672, normalized size of antiderivative = 2.31

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

Input:

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

Output:

-1/6/c^3/b^2*(2*arcsin(c*x)*2^(1/2)*Pi^(1/2)*(-1/b)^(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)*b-2*arcsin(c*x)*2^(1/2)*Pi^(1/2)*(-1/b)^(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)*b-6*arcsin(c*x)*cos(3*a/b)*FresnelC(3*2^(1/2)/Pi^(1/2)/(-3/b)^(1/2)* 
(a+b*arcsin(c*x))^(1/2)/b)*(-3/b)^(1/2)*2^(1/2)*Pi^(1/2)*(a+b*arcsin(c*x)) 
^(1/2)*b+6*arcsin(c*x)*sin(3*a/b)*FresnelS(3*2^(1/2)/Pi^(1/2)/(-3/b)^(1/2) 
*(a+b*arcsin(c*x))^(1/2)/b)*(-3/b)^(1/2)*2^(1/2)*Pi^(1/2)*(a+b*arcsin(c*x) 
)^(1/2)*b+2*2^(1/2)*Pi^(1/2)*(-1/b)^(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)*a-2*2^( 
1/2)*Pi^(1/2)*(-1/b)^(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)*a-6*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)*(-3/b)^(1/2)* 
2^(1/2)*Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)*a+6*sin(3*a/b)*FresnelS(3*2^(1/2) 
/Pi^(1/2)/(-3/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*(-3/b)^(1/2)*2^(1/2)*Pi^ 
(1/2)*(a+b*arcsin(c*x))^(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*arcsin(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. \[ \int \frac {x^2}{(a+b \arcsin (c x))^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^2}{(a+b \arcsin (c x))^{5/2}} \, dx=\int \frac {\sqrt {\mathit {asin} \left (c x \right ) b +a}\, x^{2}}{\mathit {asin} \left (c x \right )^{3} b^{3}+3 \mathit {asin} \left (c x \right )^{2} a \,b^{2}+3 \mathit {asin} \left (c x \right ) a^{2} b +a^{3}}d x \] Input:

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

Output:

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