\(\int \frac {x^3 (d-c^2 d x^2)}{(a+b \arcsin (c x))^{3/2}} \, dx\) [387]

Optimal result

Integrand size = 27, antiderivative size = 251 \[ \int \frac {x^3 \left (d-c^2 d x^2\right )}{(a+b \arcsin (c x))^{3/2}} \, dx=-\frac {2 d x^3 \left (1-c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \arcsin (c x)}}-\frac {d \sqrt {3 \pi } \cos \left (\frac {6 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {3}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{8 b^{3/2} c^4}+\frac {3 d \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{8 b^{3/2} c^4}+\frac {3 d \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{8 b^{3/2} c^4}-\frac {d \sqrt {3 \pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {3}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {6 a}{b}\right )}{8 b^{3/2} c^4} \] Output:

-2*d*x^3*(-c^2*x^2+1)^(3/2)/b/c/(a+b*arcsin(c*x))^(1/2)-1/8*d*3^(1/2)*Pi^( 
1/2)*cos(6*a/b)*FresnelC(2*3^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2 
))/b^(3/2)/c^4+3/8*d*Pi^(1/2)*cos(2*a/b)*FresnelC(2*(a+b*arcsin(c*x))^(1/2 
)/b^(1/2)/Pi^(1/2))/b^(3/2)/c^4+3/8*d*Pi^(1/2)*FresnelS(2*(a+b*arcsin(c*x) 
)^(1/2)/b^(1/2)/Pi^(1/2))*sin(2*a/b)/b^(3/2)/c^4-1/8*d*3^(1/2)*Pi^(1/2)*Fr 
esnelS(2*3^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*sin(6*a/b)/b^(3 
/2)/c^4
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.35 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.14 \[ \int \frac {x^3 \left (d-c^2 d x^2\right )}{(a+b \arcsin (c x))^{3/2}} \, dx=-\frac {i d e^{-\frac {6 i a}{b}} \left (3 \sqrt {2} e^{\frac {4 i a}{b}} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {2 i (a+b \arcsin (c x))}{b}\right )-3 \sqrt {2} e^{\frac {8 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {2 i (a+b \arcsin (c x))}{b}\right )-\sqrt {6} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {6 i (a+b \arcsin (c x))}{b}\right )+\sqrt {6} e^{\frac {12 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {6 i (a+b \arcsin (c x))}{b}\right )-6 i e^{\frac {6 i a}{b}} \sin (2 \arcsin (c x))+2 i e^{\frac {6 i a}{b}} \sin (6 \arcsin (c x))\right )}{32 b c^4 \sqrt {a+b \arcsin (c x)}} \] Input:

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

Output:

((-1/32*I)*d*(3*Sqrt[2]*E^(((4*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b] 
*Gamma[1/2, ((-2*I)*(a + b*ArcSin[c*x]))/b] - 3*Sqrt[2]*E^(((8*I)*a)/b)*Sq 
rt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((2*I)*(a + b*ArcSin[c*x]))/b] - 
Sqrt[6]*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-6*I)*(a + b*ArcSi 
n[c*x]))/b] + Sqrt[6]*E^(((12*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gam 
ma[1/2, ((6*I)*(a + b*ArcSin[c*x]))/b] - (6*I)*E^(((6*I)*a)/b)*Sin[2*ArcSi 
n[c*x]] + (2*I)*E^(((6*I)*a)/b)*Sin[6*ArcSin[c*x]]))/(b*c^4*E^(((6*I)*a)/b 
)*Sqrt[a + b*ArcSin[c*x]])
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(507\) vs. \(2(251)=502\).

Time = 1.51 (sec) , antiderivative size = 507, normalized size of antiderivative = 2.02, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {5214, 5224, 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.

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

\(\Big \downarrow \) 5214

\(\displaystyle \frac {6 d \int \frac {x^2 \sqrt {1-c^2 x^2}}{\sqrt {a+b \arcsin (c x)}}dx}{b c}-\frac {12 c d \int \frac {x^4 \sqrt {1-c^2 x^2}}{\sqrt {a+b \arcsin (c x)}}dx}{b}-\frac {2 d x^3 \left (1-c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \arcsin (c x)}}\)

\(\Big \downarrow \) 5224

\(\displaystyle -\frac {12 d \int \frac {\cos ^2\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin ^4\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{b^2 c^4}+\frac {6 d \int \frac {\cos ^2\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin ^2\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{b^2 c^4}-\frac {2 d x^3 \left (1-c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \arcsin (c x)}}\)

\(\Big \downarrow \) 4906

\(\displaystyle \frac {6 d \int \left (\frac {1}{8 \sqrt {a+b \arcsin (c x)}}-\frac {\cos \left (\frac {4 a}{b}-\frac {4 (a+b \arcsin (c x))}{b}\right )}{8 \sqrt {a+b \arcsin (c x)}}\right )d(a+b \arcsin (c x))}{b^2 c^4}-\frac {12 d \int \left (\frac {\cos \left (\frac {6 a}{b}-\frac {6 (a+b \arcsin (c x))}{b}\right )}{32 \sqrt {a+b \arcsin (c x)}}-\frac {\cos \left (\frac {4 a}{b}-\frac {4 (a+b \arcsin (c x))}{b}\right )}{16 \sqrt {a+b \arcsin (c x)}}-\frac {\cos \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c x))}{b}\right )}{32 \sqrt {a+b \arcsin (c x)}}+\frac {1}{16 \sqrt {a+b \arcsin (c x)}}\right )d(a+b \arcsin (c x))}{b^2 c^4}-\frac {2 d x^3 \left (1-c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \arcsin (c x)}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {6 d \left (-\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\frac {1}{4} \sqrt {a+b \arcsin (c x)}\right )}{b^2 c^4}-\frac {12 d \left (-\frac {1}{16} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\frac {1}{32} \sqrt {\frac {\pi }{3}} \sqrt {b} \cos \left (\frac {6 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {3}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-\frac {1}{32} \sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )-\frac {1}{32} \sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )-\frac {1}{16} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\frac {1}{32} \sqrt {\frac {\pi }{3}} \sqrt {b} \sin \left (\frac {6 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {3}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {a+b \arcsin (c x)}\right )}{b^2 c^4}-\frac {2 d x^3 \left (1-c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \arcsin (c x)}}\)

Input:

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

Output:

(-2*d*x^3*(1 - c^2*x^2)^(3/2))/(b*c*Sqrt[a + b*ArcSin[c*x]]) + (6*d*(Sqrt[ 
a + b*ArcSin[c*x]]/4 - (Sqrt[b]*Sqrt[Pi/2]*Cos[(4*a)/b]*FresnelC[(2*Sqrt[2 
/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/8 - (Sqrt[b]*Sqrt[Pi/2]*FresnelS[( 
2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(4*a)/b])/8))/(b^2*c^4) 
 - (12*d*(Sqrt[a + b*ArcSin[c*x]]/8 - (Sqrt[b]*Sqrt[Pi/2]*Cos[(4*a)/b]*Fre 
snelC[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/16 + (Sqrt[b]*Sqrt[ 
Pi/3]*Cos[(6*a)/b]*FresnelC[(2*Sqrt[3/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b] 
])/32 - (Sqrt[b]*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c*x]] 
)/(Sqrt[b]*Sqrt[Pi])])/32 - (Sqrt[b]*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcSi 
n[c*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/32 - (Sqrt[b]*Sqrt[Pi/2]*Fresne 
lS[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(4*a)/b])/16 + (Sqr 
t[b]*Sqrt[Pi/3]*FresnelS[(2*Sqrt[3/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*S 
in[(6*a)/b])/32))/(b^2*c^4)
 

Defintions of rubi rules used

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 = 0.79 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.21

method	result	size
default	\(\frac {d \left (-\sqrt {-\frac {6}{b}}\, \operatorname {FresnelC}\left (\frac {6 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {6}{b}}\, b}\right ) \sqrt {a +b \arcsin \left (c x \right )}\, \sqrt {2}\, \cos \left (\frac {6 a}{b}\right ) \sqrt {\pi }+\sqrt {-\frac {6}{b}}\, \operatorname {FresnelS}\left (\frac {6 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {6}{b}}\, b}\right ) \sqrt {a +b \arcsin \left (c x \right )}\, \sqrt {2}\, \sin \left (\frac {6 a}{b}\right ) \sqrt {\pi }+6 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right )-6 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right )+3 \sin \left (-\frac {2 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {2 a}{b}\right )-\sin \left (-\frac {6 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {6 a}{b}\right )\right )}{16 c^{4} b \sqrt {a +b \arcsin \left (c x \right )}}\)	\(304\)

Input:

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

Output:

1/16*d/c^4/b*(-(-6/b)^(1/2)*FresnelC(6*2^(1/2)/Pi^(1/2)/(-6/b)^(1/2)*(a+b* 
arcsin(c*x))^(1/2)/b)*(a+b*arcsin(c*x))^(1/2)*2^(1/2)*cos(6*a/b)*Pi^(1/2)+ 
(-6/b)^(1/2)*FresnelS(6*2^(1/2)/Pi^(1/2)/(-6/b)^(1/2)*(a+b*arcsin(c*x))^(1 
/2)/b)*(a+b*arcsin(c*x))^(1/2)*2^(1/2)*sin(6*a/b)*Pi^(1/2)+6*(-1/b)^(1/2)* 
Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)*cos(2*a/b)*FresnelC(2*2^(1/2)/Pi^(1/2)/(- 
2/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)-6*(-1/b)^(1/2)*Pi^(1/2)*(a+b*arcsin( 
c*x))^(1/2)*sin(2*a/b)*FresnelS(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arcsi 
n(c*x))^(1/2)/b)+3*sin(-2*(a+b*arcsin(c*x))/b+2*a/b)-sin(-6*(a+b*arcsin(c* 
x))/b+6*a/b))/(a+b*arcsin(c*x))^(1/2)
 

Fricas [F(-2)]

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

integrate(x^3*(-c^2*d*x^2+d)/(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]

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

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

Output:

-d*(Integral(-x**3/(a*sqrt(a + b*asin(c*x)) + b*sqrt(a + b*asin(c*x))*asin 
(c*x)), x) + Integral(c**2*x**5/(a*sqrt(a + b*asin(c*x)) + b*sqrt(a + b*as 
in(c*x))*asin(c*x)), x))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

d*( - int((sqrt(asin(c*x)*b + a)*x**5)/(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**3)/(asin(c*x)**2*b**2 + 
2*asin(c*x)*a*b + a**2),x))