\(\int \frac {(d-c^2 d x^2)^3}{(a+b \arcsin (c x))^{3/2}} \, dx\) [96]

Optimal result

Integrand size = 26, antiderivative size = 508 \[ \int \frac {\left (d-c^2 d x^2\right )^3}{(a+b \arcsin (c x))^{3/2}} \, dx=-\frac {2 d^3 \left (1-c^2 x^2\right )^{7/2}}{b c \sqrt {a+b \arcsin (c x)}}-\frac {35 d^3 \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c}-\frac {21 d^3 \sqrt {\frac {3 \pi }{2}} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c}-\frac {7 d^3 \sqrt {\frac {5 \pi }{2}} \cos \left (\frac {5 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c}-\frac {d^3 \sqrt {\frac {7 \pi }{2}} \cos \left (\frac {7 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {14}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c}+\frac {35 d^3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{16 b^{3/2} c}+\frac {21 d^3 \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{16 b^{3/2} c}+\frac {7 d^3 \sqrt {\frac {5 \pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {5 a}{b}\right )}{16 b^{3/2} c}+\frac {d^3 \sqrt {\frac {7 \pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {14}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {7 a}{b}\right )}{16 b^{3/2} c} \] Output:

-2*d^3*(-c^2*x^2+1)^(7/2)/b/c/(a+b*arcsin(c*x))^(1/2)-35/32*d^3*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-21/32*d^3*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-7/32*d^3*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-1/32*d^3*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+35/32*d^3*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+21/32 
*d^3*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+7/32*d^3*10^(1/2)*Pi^(1/2)*FresnelC(10^(1/2)/P 
i^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*sin(5*a/b)/b^(3/2)/c+1/32*d^3*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
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.68 (sec) , antiderivative size = 692, normalized size of antiderivative = 1.36 \[ \int \frac {\left (d-c^2 d x^2\right )^3}{(a+b \arcsin (c x))^{3/2}} \, dx=\frac {d^3 e^{-\frac {7 i (a+b \arcsin (c x))}{b}} \left (-e^{\frac {7 i a}{b}}-7 e^{\frac {7 i a}{b}+2 i \arcsin (c x)}-21 e^{\frac {7 i a}{b}+4 i \arcsin (c x)}-35 e^{\frac {7 i a}{b}+6 i \arcsin (c x)}-35 e^{\frac {7 i a}{b}+8 i \arcsin (c x)}-21 e^{\frac {7 i a}{b}+10 i \arcsin (c x)}-7 e^{\frac {7 i a}{b}+12 i \arcsin (c x)}-e^{\frac {7 i (a+2 b \arcsin (c x))}{b}}+35 e^{\frac {6 i a}{b}+7 i \arcsin (c x)} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {i (a+b \arcsin (c x))}{b}\right )+35 e^{\frac {8 i a}{b}+7 i \arcsin (c x)} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {i (a+b \arcsin (c x))}{b}\right )+21 \sqrt {3} e^{\frac {4 i a}{b}+7 i \arcsin (c x)} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {3 i (a+b \arcsin (c x))}{b}\right )+21 \sqrt {3} e^{\frac {10 i a}{b}+7 i \arcsin (c x)} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {3 i (a+b \arcsin (c x))}{b}\right )+7 \sqrt {5} e^{\frac {2 i a}{b}+7 i \arcsin (c x)} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {5 i (a+b \arcsin (c x))}{b}\right )+7 \sqrt {5} e^{\frac {12 i a}{b}+7 i \arcsin (c x)} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {5 i (a+b \arcsin (c x))}{b}\right )+\sqrt {7} e^{7 i \arcsin (c x)} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {7 i (a+b \arcsin (c x))}{b}\right )+\sqrt {7} e^{\frac {7 i (2 a+b \arcsin (c x))}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {7 i (a+b \arcsin (c x))}{b}\right )\right )}{64 b c \sqrt {a+b \arcsin (c x)}} \] Input:

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

Output:

(d^3*(-E^(((7*I)*a)/b) - 7*E^(((7*I)*a)/b + (2*I)*ArcSin[c*x]) - 21*E^(((7 
*I)*a)/b + (4*I)*ArcSin[c*x]) - 35*E^(((7*I)*a)/b + (6*I)*ArcSin[c*x]) - 3 
5*E^(((7*I)*a)/b + (8*I)*ArcSin[c*x]) - 21*E^(((7*I)*a)/b + (10*I)*ArcSin[ 
c*x]) - 7*E^(((7*I)*a)/b + (12*I)*ArcSin[c*x]) - E^(((7*I)*(a + 2*b*ArcSin 
[c*x]))/b) + 35*E^(((6*I)*a)/b + (7*I)*ArcSin[c*x])*Sqrt[((-I)*(a + b*ArcS 
in[c*x]))/b]*Gamma[1/2, ((-I)*(a + b*ArcSin[c*x]))/b] + 35*E^(((8*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] + 21*Sqrt[3]*E^(((4*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] + 21 
*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] + 7*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] + 7*Sqrt[5]*E^(((12*I)*a)/b + (7*I)*ArcSin[c*x])*S 
qrt[(I*(a + b*ArcSin[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*E^(((7*I)*(a + b*ArcSin[c*x]))/b)*Sqrt[a + b*ArcSin[c*x]] 
)
 

Rubi [A] (verified)

Time = 1.05 (sec) , antiderivative size = 472, normalized size of antiderivative = 0.93, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {5166, 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.

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

\(\Big \downarrow \) 5166

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

\(\Big \downarrow \) 5224

\(\displaystyle -\frac {14 d^3 \int -\frac {\cos ^6\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{b^2 c}-\frac {2 d^3 \left (1-c^2 x^2\right )^{7/2}}{b c \sqrt {a+b \arcsin (c x)}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {14 d^3 \int \frac {\cos ^6\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{b^2 c}-\frac {2 d^3 \left (1-c^2 x^2\right )^{7/2}}{b c \sqrt {a+b \arcsin (c x)}}\)

\(\Big \downarrow \) 4906

\(\displaystyle \frac {14 d^3 \int \left (\frac {\sin \left (\frac {7 a}{b}-\frac {7 (a+b \arcsin (c x))}{b}\right )}{64 \sqrt {a+b \arcsin (c x)}}+\frac {5 \sin \left (\frac {5 a}{b}-\frac {5 (a+b \arcsin (c x))}{b}\right )}{64 \sqrt {a+b \arcsin (c x)}}+\frac {9 \sin \left (\frac {3 a}{b}-\frac {3 (a+b \arcsin (c x))}{b}\right )}{64 \sqrt {a+b \arcsin (c x)}}+\frac {5 \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{64 \sqrt {a+b \arcsin (c x)}}\right )d(a+b \arcsin (c x))}{b^2 c}-\frac {2 d^3 \left (1-c^2 x^2\right )^{7/2}}{b c \sqrt {a+b \arcsin (c x)}}\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(-2*d^3*(1 - c^2*x^2)^(7/2))/(b*c*Sqrt[a + b*ArcSin[c*x]]) - (14*d^3*((5*S 
qrt[b]*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/S 
qrt[b]])/32 + (3*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[(5*Pi)/2]*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]])/Sq 
rt[b]])/32 - (5*Sqrt[b]*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[ 
c*x]])/Sqrt[b]]*Sin[a/b])/32 - (3*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[(5* 
Pi)/2]*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)
 

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 5166

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_)^2)^(p_.), x_ 
Symbol] :> Simp[Sqrt[1 - c^2*x^2]*(d + e*x^2)^p*((a + b*ArcSin[c*x])^(n + 1 
)/(b*c*(n + 1))), x] + Simp[c*((2*p + 1)/(b*(n + 1)))*Simp[(d + e*x^2)^p/(1 
 - c^2*x^2)^p]   Int[x*(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] && LtQ[n, -1]
 

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 = 0.18 (sec) , antiderivative size = 596, normalized size of antiderivative = 1.17

method	result	size
default	\(-\frac {d^{3} \left (-21 \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}}-21 \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}}-35 \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 )-35 \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 )-7 \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}}-7 \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}}-\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}}+35 \cos \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right )+7 \cos \left (-\frac {5 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {5 a}{b}\right )+21 \cos \left (-\frac {3 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {3 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 b \sqrt {a +b \arcsin \left (c x \right )}}\)	\(596\)

Input:

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

Output:

-1/32/c*d^3/b/(a+b*arcsin(c*x))^(1/2)*(-21*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)-21*Pi^(1/2)*2^(1/2)*(a+b*arcsin(c*x))^(1/2)*si 
n(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)-35*(-1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arcsin(c*x))^(1/2)*c 
os(a/b)*FresnelS(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)- 
35*(-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*arcsin(c*x))^(1/2)/b)-7*cos(5*a/b)*Fre 
snelS(5*2^(1/2)/Pi^(1/2)/(-5/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*(a+b*arcs 
in(c*x))^(1/2)*Pi^(1/2)*2^(1/2)*(-5/b)^(1/2)-7*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)-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 
)+35*cos(-(a+b*arcsin(c*x))/b+a/b)+7*cos(-5*(a+b*arcsin(c*x))/b+5*a/b)+21* 
cos(-3*(a+b*arcsin(c*x))/b+3*a/b)+cos(-7*(a+b*arcsin(c*x))/b+7*a/b))
 

Fricas [F(-2)]

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

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

Output:

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

Sympy [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^3}{(a+b \arcsin (c x))^{3/2}} \, dx=- d^{3} \left (\int \frac {3 c^{2} x^{2}}{a \sqrt {a + b \operatorname {asin}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asin}{\left (c x \right )}} \operatorname {asin}{\left (c x \right )}}\, dx + \int \left (- \frac {3 c^{4} x^{4}}{a \sqrt {a + b \operatorname {asin}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asin}{\left (c x \right )}} \operatorname {asin}{\left (c x \right )}}\right )\, dx + \int \frac {c^{6} x^{6}}{a \sqrt {a + b \operatorname {asin}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asin}{\left (c x \right )}} \operatorname {asin}{\left (c x \right )}}\, dx + \int \left (- \frac {1}{a \sqrt {a + b \operatorname {asin}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asin}{\left (c x \right )}} \operatorname {asin}{\left (c x \right )}}\right )\, dx\right ) \] Input:

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

Output:

-d**3*(Integral(3*c**2*x**2/(a*sqrt(a + b*asin(c*x)) + b*sqrt(a + b*asin(c 
*x))*asin(c*x)), x) + Integral(-3*c**4*x**4/(a*sqrt(a + b*asin(c*x)) + b*s 
qrt(a + b*asin(c*x))*asin(c*x)), x) + Integral(c**6*x**6/(a*sqrt(a + b*asi 
n(c*x)) + b*sqrt(a + b*asin(c*x))*asin(c*x)), x) + Integral(-1/(a*sqrt(a + 
 b*asin(c*x)) + b*sqrt(a + b*asin(c*x))*asin(c*x)), x))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(d**3*(28*sqrt(asin(c*x)*b + a)*sqrt( - c**2*x**2 + 1)*asin(c*x)*b + 28*as 
in(c*x)*int(sqrt(asin(c*x)*b + a)/(asin(c*x)**2*b**2*c**2*x**2 - asin(c*x) 
**2*b**2 + 2*asin(c*x)*a*b*c**2*x**2 - 2*asin(c*x)*a*b + a**2*c**2*x**2 - 
a**2),x)*a*b**2*c - 8*asin(c*x)*int((sqrt(asin(c*x)*b + a)*x**8)/(asin(c*x 
)**2*b**2*c**2*x**2 - asin(c*x)**2*b**2 + 2*asin(c*x)*a*b*c**2*x**2 - 2*as 
in(c*x)*a*b + a**2*c**2*x**2 - a**2),x)*a*b**2*c**9 + 32*asin(c*x)*int((sq 
rt(asin(c*x)*b + a)*x**6)/(asin(c*x)**2*b**2*c**2*x**2 - asin(c*x)**2*b**2 
 + 2*asin(c*x)*a*b*c**2*x**2 - 2*asin(c*x)*a*b + a**2*c**2*x**2 - a**2),x) 
*a*b**2*c**7 - 52*asin(c*x)*int((sqrt(asin(c*x)*b + a)*x**4)/(asin(c*x)**2 
*b**2*c**2*x**2 - asin(c*x)**2*b**2 + 2*asin(c*x)*a*b*c**2*x**2 - 2*asin(c 
*x)*a*b + a**2*c**2*x**2 - a**2),x)*a*b**2*c**5 + 24*asin(c*x)*int((sqrt(a 
sin(c*x)*b + a)*sqrt( - c**2*x**2 + 1)*asin(c*x)*x**3)/(asin(c*x)**2*b**2* 
c**2*x**2 - asin(c*x)**2*b**2 + 2*asin(c*x)*a*b*c**2*x**2 - 2*asin(c*x)*a* 
b + a**2*c**2*x**2 - a**2),x)*a*b**2*c**4 - 28*asin(c*x)*int((sqrt(asin(c* 
x)*b + a)*sqrt( - c**2*x**2 + 1)*asin(c*x)**2*x)/(asin(c*x)**2*b**2*c**2*x 
**2 - asin(c*x)**2*b**2 + 2*asin(c*x)*a*b*c**2*x**2 - 2*asin(c*x)*a*b + a* 
*2*c**2*x**2 - a**2),x)*b**3*c**2 + 24*asin(c*x)*int((sqrt(asin(c*x)*b + a 
)*sqrt( - c**2*x**2 + 1)*x**3)/(asin(c*x)**2*b**2*c**2*x**2 - asin(c*x)**2 
*b**2 + 2*asin(c*x)*a*b*c**2*x**2 - 2*asin(c*x)*a*b + a**2*c**2*x**2 - a** 
2),x)*a**2*b*c**4 + 28*asin(c*x)*int((sqrt(asin(c*x)*b + a)*sqrt( - c**...