3.1.48 \(\int \frac {x^2 (a+b \arcsin (c x))}{(d-c^2 d x^2)^3} \, dx\) [48]

3.1.48.1 Optimal result

Integrand size = 25, antiderivative size = 202 \[ \int \frac {x^2 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^3} \, dx=-\frac {b}{12 c^3 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {b}{8 c^3 d^3 \sqrt {1-c^2 x^2}}+\frac {x (a+b \arcsin (c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {x (a+b \arcsin (c x))}{8 c^2 d^3 \left (1-c^2 x^2\right )}+\frac {i (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{4 c^3 d^3}-\frac {i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{8 c^3 d^3}+\frac {i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{8 c^3 d^3} \]

output

-1/12*b/c^3/d^3/(-c^2*x^2+1)^(3/2)+1/4*x*(a+b*arcsin(c*x))/c^2/d^3/(-c^2*x 
^2+1)^2-1/8*x*(a+b*arcsin(c*x))/c^2/d^3/(-c^2*x^2+1)+1/4*I*(a+b*arcsin(c*x 
))*arctan(I*c*x+(-c^2*x^2+1)^(1/2))/c^3/d^3-1/8*I*b*polylog(2,-I*(I*c*x+(- 
c^2*x^2+1)^(1/2)))/c^3/d^3+1/8*I*b*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2))) 
/c^3/d^3+1/8*b/c^3/d^3/(-c^2*x^2+1)^(1/2)
 

3.1.48.2 Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(445\) vs. \(2(202)=404\).

Time = 0.90 (sec) , antiderivative size = 445, normalized size of antiderivative = 2.20 \[ \int \frac {x^2 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {-\frac {2 b \sqrt {1-c^2 x^2}}{(-1+c x)^2}+\frac {b c x \sqrt {1-c^2 x^2}}{(-1+c x)^2}-\frac {3 b \sqrt {1-c^2 x^2}}{-1+c x}-\frac {2 b \sqrt {1-c^2 x^2}}{(1+c x)^2}-\frac {b c x \sqrt {1-c^2 x^2}}{(1+c x)^2}+\frac {3 b \sqrt {1-c^2 x^2}}{1+c x}+\frac {12 a c x}{\left (-1+c^2 x^2\right )^2}+\frac {6 a c x}{-1+c^2 x^2}+3 i b \pi \arcsin (c x)+\frac {3 b \arcsin (c x)}{(-1+c x)^2}+\frac {3 b \arcsin (c x)}{-1+c x}-\frac {3 b \arcsin (c x)}{(1+c x)^2}+\frac {3 b \arcsin (c x)}{1+c x}-3 b \pi \log \left (1-i e^{i \arcsin (c x)}\right )-6 b \arcsin (c x) \log \left (1-i e^{i \arcsin (c x)}\right )-3 b \pi \log \left (1+i e^{i \arcsin (c x)}\right )+6 b \arcsin (c x) \log \left (1+i e^{i \arcsin (c x)}\right )+3 a \log (1-c x)-3 a \log (1+c x)+3 b \pi \log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )+3 b \pi \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )-6 i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )+6 i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{48 c^3 d^3} \]

input

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

output

((-2*b*Sqrt[1 - c^2*x^2])/(-1 + c*x)^2 + (b*c*x*Sqrt[1 - c^2*x^2])/(-1 + c 
*x)^2 - (3*b*Sqrt[1 - c^2*x^2])/(-1 + c*x) - (2*b*Sqrt[1 - c^2*x^2])/(1 + 
c*x)^2 - (b*c*x*Sqrt[1 - c^2*x^2])/(1 + c*x)^2 + (3*b*Sqrt[1 - c^2*x^2])/( 
1 + c*x) + (12*a*c*x)/(-1 + c^2*x^2)^2 + (6*a*c*x)/(-1 + c^2*x^2) + (3*I)* 
b*Pi*ArcSin[c*x] + (3*b*ArcSin[c*x])/(-1 + c*x)^2 + (3*b*ArcSin[c*x])/(-1 
+ c*x) - (3*b*ArcSin[c*x])/(1 + c*x)^2 + (3*b*ArcSin[c*x])/(1 + c*x) - 3*b 
*Pi*Log[1 - I*E^(I*ArcSin[c*x])] - 6*b*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c 
*x])] - 3*b*Pi*Log[1 + I*E^(I*ArcSin[c*x])] + 6*b*ArcSin[c*x]*Log[1 + I*E^ 
(I*ArcSin[c*x])] + 3*a*Log[1 - c*x] - 3*a*Log[1 + c*x] + 3*b*Pi*Log[-Cos[( 
Pi + 2*ArcSin[c*x])/4]] + 3*b*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (6*I)* 
b*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (6*I)*b*PolyLog[2, I*E^(I*ArcSin[c* 
x])])/(48*c^3*d^3)
 

3.1.48.3 Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.93, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5206, 27, 241, 5162, 241, 5164, 3042, 4669, 2715, 2838}

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.

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

\(\Big \downarrow \) 5206

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 241

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

\(\Big \downarrow \) 5162

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

\(\Big \downarrow \) 241

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

\(\Big \downarrow \) 5164

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4669

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

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {\frac {i b \int e^{-i \arcsin (c x)} \log \left (1-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-i b \int e^{-i \arcsin (c x)} \log \left (1+i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}}{4 c^2 d^3}+\frac {x (a+b \arcsin (c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {b}{12 c^3 d^3 \left (1-c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}}{4 c^2 d^3}+\frac {x (a+b \arcsin (c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {b}{12 c^3 d^3 \left (1-c^2 x^2\right )^{3/2}}\)

input

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

output

-1/12*b/(c^3*d^3*(1 - c^2*x^2)^(3/2)) + (x*(a + b*ArcSin[c*x]))/(4*c^2*d^3 
*(1 - c^2*x^2)^2) - (-1/2*b/(c*Sqrt[1 - c^2*x^2]) + (x*(a + b*ArcSin[c*x]) 
)/(2*(1 - c^2*x^2)) + ((-2*I)*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x]) 
] + I*b*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - I*b*PolyLog[2, I*E^(I*ArcSin[ 
c*x])])/(2*c))/(4*c^2*d^3)
 

3.1.48.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 241

Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ 
(2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

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

rule 4669

Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol 
] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Si 
mp[d*(m/f)   Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))], x], 
 x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x 
))], x], x]) /; FreeQ[{c, d, e, f}, x] && IntegerQ[2*k] && IGtQ[m, 0]
 

rule 5162

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_ 
Symbol] :> Simp[(-x)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(2*d*(p + 1 
))), x] + (Simp[(2*p + 3)/(2*d*(p + 1))   Int[(d + e*x^2)^(p + 1)*(a + b*Ar 
cSin[c*x])^n, x], x] + Simp[b*c*(n/(2*(p + 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}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, 
 -1] && NeQ[p, -3/2]
 

rule 5164

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symbo 
l] :> Simp[1/(c*d)   Subst[Int[(a + b*x)^n*Sec[x], x], x, ArcSin[c*x]], x] 
/; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
 

rule 5206

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

3.1.48.4 Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.29

method	result	size
derivativedivides	\(\frac {-\frac {a \left (-\frac {1}{16 \left (c x -1\right )^{2}}-\frac {1}{16 \left (c x -1\right )}-\frac {\ln \left (c x -1\right )}{16}+\frac {1}{16 \left (c x +1\right )^{2}}-\frac {1}{16 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{16}\right )}{d^{3}}-\frac {b \left (-\frac {3 c^{3} x^{3} \arcsin \left (c x \right )-3 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}+3 c x \arcsin \left (c x \right )+\sqrt {-c^{2} x^{2}+1}}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {\arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {\arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}\right )}{d^{3}}}{c^{3}}\)	\(260\)
default	\(\frac {-\frac {a \left (-\frac {1}{16 \left (c x -1\right )^{2}}-\frac {1}{16 \left (c x -1\right )}-\frac {\ln \left (c x -1\right )}{16}+\frac {1}{16 \left (c x +1\right )^{2}}-\frac {1}{16 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{16}\right )}{d^{3}}-\frac {b \left (-\frac {3 c^{3} x^{3} \arcsin \left (c x \right )-3 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}+3 c x \arcsin \left (c x \right )+\sqrt {-c^{2} x^{2}+1}}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {\arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {\arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}\right )}{d^{3}}}{c^{3}}\)	\(260\)
parts	\(-\frac {a \left (-\frac {1}{16 c^{3} \left (c x -1\right )^{2}}-\frac {1}{16 c^{3} \left (c x -1\right )}-\frac {\ln \left (c x -1\right )}{16 c^{3}}+\frac {1}{16 c^{3} \left (c x +1\right )^{2}}-\frac {1}{16 c^{3} \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{16 c^{3}}\right )}{d^{3}}-\frac {b \left (-\frac {3 c^{3} x^{3} \arcsin \left (c x \right )-3 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}+3 c x \arcsin \left (c x \right )+\sqrt {-c^{2} x^{2}+1}}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {\arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {\arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}\right )}{d^{3} c^{3}}\)	\(277\)

input

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

output

1/c^3*(-a/d^3*(-1/16/(c*x-1)^2-1/16/(c*x-1)-1/16*ln(c*x-1)+1/16/(c*x+1)^2- 
1/16/(c*x+1)+1/16*ln(c*x+1))-b/d^3*(-1/24*(3*c^3*x^3*arcsin(c*x)-3*c^2*x^2 
*(-c^2*x^2+1)^(1/2)+3*c*x*arcsin(c*x)+(-c^2*x^2+1)^(1/2))/(c^4*x^4-2*c^2*x 
^2+1)-1/8*arcsin(c*x)*ln(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))+1/8*arcsin(c*x)*l 
n(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))+1/8*I*dilog(1+I*(I*c*x+(-c^2*x^2+1)^(1/2 
)))-1/8*I*dilog(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))))
 

3.1.48.5 Fricas [F]

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

input

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

output

integral(-(b*x^2*arcsin(c*x) + a*x^2)/(c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2 
*d^3*x^2 - d^3), x)
 

3.1.48.6 Sympy [F]

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

input

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

output

-(Integral(a*x**2/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1), x) + Integr 
al(b*x**2*asin(c*x)/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1), x))/d**3
 

3.1.48.7 Maxima [F]

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

input

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

output

1/16*a*(2*(c^2*x^3 + x)/(c^6*d^3*x^4 - 2*c^4*d^3*x^2 + c^2*d^3) - log(c*x 
+ 1)/(c^3*d^3) + log(c*x - 1)/(c^3*d^3)) - 1/16*((c^4*x^4 - 2*c^2*x^2 + 1) 
*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))*log(c*x + 1) - (c^4*x^4 - 2*c^ 
2*x^2 + 1)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))*log(-c*x + 1) - 2*(c 
^3*x^3 + c*x)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + 16*(c^7*d^3*x^4 
 - 2*c^5*d^3*x^2 + c^3*d^3)*integrate(-1/16*(2*c^3*x^3 + 2*c*x - (c^4*x^4 
- 2*c^2*x^2 + 1)*log(c*x + 1) + (c^4*x^4 - 2*c^2*x^2 + 1)*log(-c*x + 1))*s 
qrt(c*x + 1)*sqrt(-c*x + 1)/(c^8*d^3*x^6 - 3*c^6*d^3*x^4 + 3*c^4*d^3*x^2 - 
 c^2*d^3), x))*b/(c^7*d^3*x^4 - 2*c^5*d^3*x^2 + c^3*d^3)
 

3.1.48.8 Giac [F]

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

input

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

output

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

3.1.48.9 Mupad [F(-1)]

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

input

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

output

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