\(\int \frac {x^2 (d-c^2 d x^2)^2}{(a+b \arcsin (c x))^{3/2}} \, dx\) [394]

Optimal result

Integrand size = 29, antiderivative size = 511 \[ \int \frac {x^2 \left (d-c^2 d x^2\right )^2}{(a+b \arcsin (c x))^{3/2}} \, dx=-\frac {2 d^2 x^2 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \arcsin (c x)}}-\frac {5 d^2 \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^3}+\frac {d^2 \sqrt {\frac {3 \pi }{2}} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^3}+\frac {3 d^2 \sqrt {\frac {5 \pi }{2}} \cos \left (\frac {5 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^3}+\frac {d^2 \sqrt {\frac {7 \pi }{2}} \cos \left (\frac {7 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {14}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^3}+\frac {5 d^2 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{16 b^{3/2} c^3}-\frac {d^2 \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{16 b^{3/2} c^3}-\frac {3 d^2 \sqrt {\frac {5 \pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {5 a}{b}\right )}{16 b^{3/2} c^3}-\frac {d^2 \sqrt {\frac {7 \pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {14}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {7 a}{b}\right )}{16 b^{3/2} c^3} \] Output:

-2*d^2*x^2*(-c^2*x^2+1)^(5/2)/b/c/(a+b*arcsin(c*x))^(1/2)-5/32*d^2*2^(1/2) 
*Pi^(1/2)*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/ 
2))/b^(3/2)/c^3+1/32*d^2*6^(1/2)*Pi^(1/2)*cos(3*a/b)*FresnelS(6^(1/2)/Pi^( 
1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))/b^(3/2)/c^3+3/32*d^2*10^(1/2)*Pi^(1/ 
2)*cos(5*a/b)*FresnelS(10^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))/ 
b^(3/2)/c^3+1/32*d^2*14^(1/2)*Pi^(1/2)*cos(7*a/b)*FresnelS(14^(1/2)/Pi^(1/ 
2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))/b^(3/2)/c^3+5/32*d^2*2^(1/2)*Pi^(1/2)* 
FresnelC(2^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*sin(a/b)/b^(3/2 
)/c^3-1/32*d^2*6^(1/2)*Pi^(1/2)*FresnelC(6^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x) 
)^(1/2)/b^(1/2))*sin(3*a/b)/b^(3/2)/c^3-3/32*d^2*10^(1/2)*Pi^(1/2)*Fresnel 
C(10^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*sin(5*a/b)/b^(3/2)/c^ 
3-1/32*d^2*14^(1/2)*Pi^(1/2)*FresnelC(14^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^ 
(1/2)/b^(1/2))*sin(7*a/b)/b^(3/2)/c^3
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.99 (sec) , antiderivative size = 686, normalized size of antiderivative = 1.34 \[ \int \frac {x^2 \left (d-c^2 d x^2\right )^2}{(a+b \arcsin (c x))^{3/2}} \, dx=\frac {d^2 e^{-\frac {7 i (a+b \arcsin (c x))}{b}} \left (e^{\frac {7 i a}{b}}+3 e^{\frac {7 i a}{b}+2 i \arcsin (c x)}+e^{\frac {7 i a}{b}+4 i \arcsin (c x)}-5 e^{\frac {7 i a}{b}+6 i \arcsin (c x)}-5 e^{\frac {7 i a}{b}+8 i \arcsin (c x)}+e^{\frac {7 i a}{b}+10 i \arcsin (c x)}+3 e^{\frac {7 i a}{b}+12 i \arcsin (c x)}+e^{\frac {7 i (a+2 b \arcsin (c x))}{b}}+5 e^{\frac {6 i a}{b}+7 i \arcsin (c x)} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {i (a+b \arcsin (c x))}{b}\right )+5 e^{\frac {8 i a}{b}+7 i \arcsin (c x)} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {i (a+b \arcsin (c x))}{b}\right )-\sqrt {3} e^{\frac {4 i a}{b}+7 i \arcsin (c x)} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {3 i (a+b \arcsin (c x))}{b}\right )-\sqrt {3} e^{\frac {10 i a}{b}+7 i \arcsin (c x)} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {3 i (a+b \arcsin (c x))}{b}\right )-3 \sqrt {5} e^{\frac {2 i a}{b}+7 i \arcsin (c x)} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {5 i (a+b \arcsin (c x))}{b}\right )-3 \sqrt {5} e^{\frac {12 i a}{b}+7 i \arcsin (c x)} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {5 i (a+b \arcsin (c x))}{b}\right )-\sqrt {7} e^{7 i \arcsin (c x)} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {7 i (a+b \arcsin (c x))}{b}\right )-\sqrt {7} e^{\frac {7 i (2 a+b \arcsin (c x))}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {7 i (a+b \arcsin (c x))}{b}\right )\right )}{64 b c^3 \sqrt {a+b \arcsin (c x)}} \] Input:

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

Output:

(d^2*(E^(((7*I)*a)/b) + 3*E^(((7*I)*a)/b + (2*I)*ArcSin[c*x]) + E^(((7*I)* 
a)/b + (4*I)*ArcSin[c*x]) - 5*E^(((7*I)*a)/b + (6*I)*ArcSin[c*x]) - 5*E^(( 
(7*I)*a)/b + (8*I)*ArcSin[c*x]) + E^(((7*I)*a)/b + (10*I)*ArcSin[c*x]) + 3 
*E^(((7*I)*a)/b + (12*I)*ArcSin[c*x]) + E^(((7*I)*(a + 2*b*ArcSin[c*x]))/b 
) + 5*E^(((6*I)*a)/b + (7*I)*ArcSin[c*x])*Sqrt[((-I)*(a + b*ArcSin[c*x]))/ 
b]*Gamma[1/2, ((-I)*(a + b*ArcSin[c*x]))/b] + 5*E^(((8*I)*a)/b + (7*I)*Arc 
Sin[c*x])*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, (I*(a + b*ArcSin[c*x] 
))/b] - Sqrt[3]*E^(((4*I)*a)/b + (7*I)*ArcSin[c*x])*Sqrt[((-I)*(a + b*ArcS 
in[c*x]))/b]*Gamma[1/2, ((-3*I)*(a + b*ArcSin[c*x]))/b] - Sqrt[3]*E^(((10* 
I)*a)/b + (7*I)*ArcSin[c*x])*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, (( 
3*I)*(a + b*ArcSin[c*x]))/b] - 3*Sqrt[5]*E^(((2*I)*a)/b + (7*I)*ArcSin[c*x 
])*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-5*I)*(a + b*ArcSin[c*x 
]))/b] - 3*Sqrt[5]*E^(((12*I)*a)/b + (7*I)*ArcSin[c*x])*Sqrt[(I*(a + b*Arc 
Sin[c*x]))/b]*Gamma[1/2, ((5*I)*(a + b*ArcSin[c*x]))/b] - Sqrt[7]*E^((7*I) 
*ArcSin[c*x])*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-7*I)*(a + b 
*ArcSin[c*x]))/b] - Sqrt[7]*E^(((7*I)*(2*a + b*ArcSin[c*x]))/b)*Sqrt[(I*(a 
 + b*ArcSin[c*x]))/b]*Gamma[1/2, ((7*I)*(a + b*ArcSin[c*x]))/b]))/(64*b*c^ 
3*E^(((7*I)*(a + b*ArcSin[c*x]))/b)*Sqrt[a + b*ArcSin[c*x]])
 

Rubi [A] (verified)

Time = 2.08 (sec) , antiderivative size = 803, normalized size of antiderivative = 1.57, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {5214, 5224, 25, 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.

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

\(\Big \downarrow \) 5214

\(\displaystyle \frac {4 d^2 \int \frac {x \left (1-c^2 x^2\right )^{3/2}}{\sqrt {a+b \arcsin (c x)}}dx}{b c}-\frac {14 c d^2 \int \frac {x^3 \left (1-c^2 x^2\right )^{3/2}}{\sqrt {a+b \arcsin (c x)}}dx}{b}-\frac {2 d^2 x^2 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \arcsin (c x)}}\)

\(\Big \downarrow \) 5224

\(\displaystyle -\frac {14 d^2 \int -\frac {\cos ^4\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin ^3\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{b^2 c^3}+\frac {4 d^2 \int -\frac {\cos ^4\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{b^2 c^3}-\frac {2 d^2 x^2 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \arcsin (c x)}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {14 d^2 \int \frac {\cos ^4\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin ^3\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{b^2 c^3}-\frac {4 d^2 \int \frac {\cos ^4\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{b^2 c^3}-\frac {2 d^2 x^2 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \arcsin (c x)}}\)

\(\Big \downarrow \) 4906

\(\displaystyle \frac {14 d^2 \int \left (-\frac {\sin \left (\frac {7 a}{b}-\frac {7 (a+b \arcsin (c x))}{b}\right )}{64 \sqrt {a+b \arcsin (c x)}}-\frac {\sin \left (\frac {5 a}{b}-\frac {5 (a+b \arcsin (c x))}{b}\right )}{64 \sqrt {a+b \arcsin (c x)}}+\frac {3 \sin \left (\frac {3 a}{b}-\frac {3 (a+b \arcsin (c x))}{b}\right )}{64 \sqrt {a+b \arcsin (c x)}}+\frac {3 \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{64 \sqrt {a+b \arcsin (c x)}}\right )d(a+b \arcsin (c x))}{b^2 c^3}-\frac {4 d^2 \int \left (\frac {\sin \left (\frac {5 a}{b}-\frac {5 (a+b \arcsin (c x))}{b}\right )}{16 \sqrt {a+b \arcsin (c x)}}+\frac {3 \sin \left (\frac {3 a}{b}-\frac {3 (a+b \arcsin (c x))}{b}\right )}{16 \sqrt {a+b \arcsin (c x)}}+\frac {\sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{8 \sqrt {a+b \arcsin (c x)}}\right )d(a+b \arcsin (c x))}{b^2 c^3}-\frac {2 d^2 x^2 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \arcsin (c x)}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 d^2 x^2 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \arcsin (c x)}}+\frac {4 d^2 \left (\frac {1}{4} \sqrt {b} \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 )+\frac {1}{8} \sqrt {b} \sqrt {\frac {3 \pi }{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 )+\frac {1}{8} \sqrt {b} \sqrt {\frac {\pi }{10}} \cos \left (\frac {5 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-\frac {1}{4} \sqrt {b} \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 )-\frac {1}{8} \sqrt {b} \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )-\frac {1}{8} \sqrt {b} \sqrt {\frac {\pi }{10}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {5 a}{b}\right )\right )}{b^2 c^3}-\frac {14 d^2 \left (\frac {3}{32} \sqrt {b} \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 )+\frac {1}{32} \sqrt {b} \sqrt {\frac {3 \pi }{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 )-\frac {1}{32} \sqrt {b} \sqrt {\frac {\pi }{10}} \cos \left (\frac {5 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-\frac {1}{32} \sqrt {b} \sqrt {\frac {\pi }{14}} \cos \left (\frac {7 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {14}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-\frac {3}{32} \sqrt {b} \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 )-\frac {1}{32} \sqrt {b} \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )+\frac {1}{32} \sqrt {b} \sqrt {\frac {\pi }{10}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {5 a}{b}\right )+\frac {1}{32} \sqrt {b} \sqrt {\frac {\pi }{14}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {14}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {7 a}{b}\right )\right )}{b^2 c^3}\)

Input:

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

Output:

(-2*d^2*x^2*(1 - c^2*x^2)^(5/2))/(b*c*Sqrt[a + b*ArcSin[c*x]]) + (4*d^2*(( 
Sqrt[b]*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/ 
Sqrt[b]])/4 + (Sqrt[b]*Sqrt[(3*Pi)/2]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sq 
rt[a + b*ArcSin[c*x]])/Sqrt[b]])/8 + (Sqrt[b]*Sqrt[Pi/10]*Cos[(5*a)/b]*Fre 
snelS[(Sqrt[10/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/8 - (Sqrt[b]*Sqrt[Pi 
/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/4 - ( 
Sqrt[b]*Sqrt[(3*Pi)/2]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[ 
b]]*Sin[(3*a)/b])/8 - (Sqrt[b]*Sqrt[Pi/10]*FresnelC[(Sqrt[10/Pi]*Sqrt[a + 
b*ArcSin[c*x]])/Sqrt[b]]*Sin[(5*a)/b])/8))/(b^2*c^3) - (14*d^2*((3*Sqrt[b] 
*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b] 
])/32 + (Sqrt[b]*Sqrt[(3*Pi)/2]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + 
 b*ArcSin[c*x]])/Sqrt[b]])/32 - (Sqrt[b]*Sqrt[Pi/10]*Cos[(5*a)/b]*FresnelS 
[(Sqrt[10/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/32 - (Sqrt[b]*Sqrt[Pi/14] 
*Cos[(7*a)/b]*FresnelS[(Sqrt[14/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/32 
- (3*Sqrt[b]*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt 
[b]]*Sin[a/b])/32 - (Sqrt[b]*Sqrt[(3*Pi)/2]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + 
b*ArcSin[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/32 + (Sqrt[b]*Sqrt[Pi/10]*FresnelC[ 
(Sqrt[10/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(5*a)/b])/32 + (Sqrt[b] 
*Sqrt[Pi/14]*FresnelC[(Sqrt[14/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[( 
7*a)/b])/32))/(b^2*c^3)
 

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 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 5214

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.)*((d_) + (e_. 
)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^m*Sqrt[1 - c^2*x^2]*(d + e*x^2)^p* 
((a + b*ArcSin[c*x])^(n + 1)/(b*c*(n + 1))), x] + (-Simp[f*(m/(b*c*(n + 1)) 
)*Simp[(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] + Simp[c*((m + 2*p + 1)/(b*f*(n 
+ 1)))*Simp[(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 
}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && IGtQ[2*p, 0] && NeQ[m + 2*p + 1 
, 0] && IGtQ[m, -3]
 

rule 5224

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

Maple [A] (verified)

Time = 1.32 (sec) , antiderivative size = 594, normalized size of antiderivative = 1.16

method	result	size
default	\(-\frac {d^{2} \left (\cos \left (\frac {7 a}{b}\right ) \operatorname {FresnelS}\left (\frac {7 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {7}{b}}\, b}\right ) \sqrt {a +b \arcsin \left (c x \right )}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {-\frac {7}{b}}+\sin \left (\frac {7 a}{b}\right ) \operatorname {FresnelC}\left (\frac {7 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {7}{b}}\, b}\right ) \sqrt {a +b \arcsin \left (c x \right )}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {-\frac {7}{b}}+\sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}\, \cos \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 {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \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}}-5 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}\, \cos \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 )-5 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \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 )+3 \cos \left (\frac {5 a}{b}\right ) \operatorname {FresnelS}\left (\frac {5 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {5}{b}}\, b}\right ) \sqrt {a +b \arcsin \left (c x \right )}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {-\frac {5}{b}}+3 \sin \left (\frac {5 a}{b}\right ) \operatorname {FresnelC}\left (\frac {5 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {5}{b}}\, b}\right ) \sqrt {a +b \arcsin \left (c x \right )}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {-\frac {5}{b}}+5 \cos \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right )-\cos \left (-\frac {3 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {3 a}{b}\right )-3 \cos \left (-\frac {5 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {5 a}{b}\right )-\cos \left (-\frac {7 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {7 a}{b}\right )\right )}{32 c^{3} b \sqrt {a +b \arcsin \left (c x \right )}}\)	\(594\)

Input:

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

Output:

-1/32*d^2/c^3/b/(a+b*arcsin(c*x))^(1/2)*(cos(7*a/b)*FresnelS(7*2^(1/2)/Pi^ 
(1/2)/(-7/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*(a+b*arcsin(c*x))^(1/2)*Pi^( 
1/2)*2^(1/2)*(-7/b)^(1/2)+sin(7*a/b)*FresnelC(7*2^(1/2)/Pi^(1/2)/(-7/b)^(1 
/2)*(a+b*arcsin(c*x))^(1/2)/b)*(a+b*arcsin(c*x))^(1/2)*Pi^(1/2)*2^(1/2)*(- 
7/b)^(1/2)+Pi^(1/2)*2^(1/2)*(a+b*arcsin(c*x))^(1/2)*cos(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)+Pi^( 
1/2)*2^(1/2)*(a+b*arcsin(c*x))^(1/2)*sin(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)-5*(-1/b)^(1/2)*Pi^( 
1/2)*2^(1/2)*(a+b*arcsin(c*x))^(1/2)*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)/(- 
1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)-5*(-1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b 
*arcsin(c*x))^(1/2)*sin(a/b)*FresnelC(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*a 
rcsin(c*x))^(1/2)/b)+3*cos(5*a/b)*FresnelS(5*2^(1/2)/Pi^(1/2)/(-5/b)^(1/2) 
*(a+b*arcsin(c*x))^(1/2)/b)*(a+b*arcsin(c*x))^(1/2)*Pi^(1/2)*2^(1/2)*(-5/b 
)^(1/2)+3*sin(5*a/b)*FresnelC(5*2^(1/2)/Pi^(1/2)/(-5/b)^(1/2)*(a+b*arcsin( 
c*x))^(1/2)/b)*(a+b*arcsin(c*x))^(1/2)*Pi^(1/2)*2^(1/2)*(-5/b)^(1/2)+5*cos 
(-(a+b*arcsin(c*x))/b+a/b)-cos(-3*(a+b*arcsin(c*x))/b+3*a/b)-3*cos(-5*(a+b 
*arcsin(c*x))/b+5*a/b)-cos(-7*(a+b*arcsin(c*x))/b+7*a/b))
 

Fricas [F(-2)]

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

integrate(x^2*(-c^2*d*x^2+d)^2/(a+b*arcsin(c*x))^(3/2),x, algorithm="frica 
s")
 

Output:

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

Sympy [F]

\[ \int \frac {x^2 \left (d-c^2 d x^2\right )^2}{(a+b \arcsin (c x))^{3/2}} \, dx=d^{2} \left (\int \frac {x^{2}}{a \sqrt {a + b \operatorname {asin}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asin}{\left (c x \right )}} \operatorname {asin}{\left (c x \right )}}\, dx + \int \left (- \frac {2 c^{2} x^{4}}{a \sqrt {a + b \operatorname {asin}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asin}{\left (c x \right )}} \operatorname {asin}{\left (c x \right )}}\right )\, dx + \int \frac {c^{4} x^{6}}{a \sqrt {a + b \operatorname {asin}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asin}{\left (c x \right )}} \operatorname {asin}{\left (c x \right )}}\, dx\right ) \] Input:

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

Output:

d**2*(Integral(x**2/(a*sqrt(a + b*asin(c*x)) + b*sqrt(a + b*asin(c*x))*asi 
n(c*x)), x) + Integral(-2*c**2*x**4/(a*sqrt(a + b*asin(c*x)) + b*sqrt(a + 
b*asin(c*x))*asin(c*x)), x) + Integral(c**4*x**6/(a*sqrt(a + b*asin(c*x)) 
+ b*sqrt(a + b*asin(c*x))*asin(c*x)), x))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

int((x^2*(d - c^2*d*x^2)^2)/(a + b*asin(c*x))^(3/2),x)
 

Output:

int((x^2*(d - c^2*d*x^2)^2)/(a + b*asin(c*x))^(3/2), x)
 

Reduce [F]

\[ \int \frac {x^2 \left (d-c^2 d x^2\right )^2}{(a+b \arcsin (c x))^{3/2}} \, dx=d^{2} \left (\left (\int \frac {\sqrt {\mathit {asin} \left (c x \right ) b +a}\, x^{6}}{\mathit {asin} \left (c x \right )^{2} b^{2}+2 \mathit {asin} \left (c x \right ) a b +a^{2}}d x \right ) c^{4}-2 \left (\int \frac {\sqrt {\mathit {asin} \left (c x \right ) b +a}\, x^{4}}{\mathit {asin} \left (c x \right )^{2} b^{2}+2 \mathit {asin} \left (c x \right ) a b +a^{2}}d x \right ) c^{2}+\int \frac {\sqrt {\mathit {asin} \left (c x \right ) b +a}\, x^{2}}{\mathit {asin} \left (c x \right )^{2} b^{2}+2 \mathit {asin} \left (c x \right ) a b +a^{2}}d x \right ) \] Input:

int(x^2*(-c^2*d*x^2+d)^2/(a+b*asin(c*x))^(3/2),x)
 

Output:

d**2*(int((sqrt(asin(c*x)*b + a)*x**6)/(asin(c*x)**2*b**2 + 2*asin(c*x)*a* 
b + a**2),x)*c**4 - 2*int((sqrt(asin(c*x)*b + a)*x**4)/(asin(c*x)**2*b**2 
+ 2*asin(c*x)*a*b + a**2),x)*c**2 + int((sqrt(asin(c*x)*b + a)*x**2)/(asin 
(c*x)**2*b**2 + 2*asin(c*x)*a*b + a**2),x))