\(\int \frac {x^2 (a+b x+c x^2)}{\sqrt {1-d x} \sqrt {1+d x}} \, dx\) [1]

Optimal result

Integrand size = 33, antiderivative size = 121 \[ \int \frac {x^2 \left (a+b x+c x^2\right )}{\sqrt {1-d x} \sqrt {1+d x}} \, dx=-\frac {b \sqrt {1-d^2 x^2}}{d^4}-\frac {\left (3 c+4 a d^2\right ) x \sqrt {1-d^2 x^2}}{8 d^4}-\frac {c x^3 \sqrt {1-d^2 x^2}}{4 d^2}+\frac {b \left (1-d^2 x^2\right )^{3/2}}{3 d^4}+\frac {\left (3 c+4 a d^2\right ) \arcsin (d x)}{8 d^5} \] Output:

-b*(-d^2*x^2+1)^(1/2)/d^4-1/8*(4*a*d^2+3*c)*x*(-d^2*x^2+1)^(1/2)/d^4-1/4*c 
*x^3*(-d^2*x^2+1)^(1/2)/d^2+1/3*b*(-d^2*x^2+1)^(3/2)/d^4+1/8*(4*a*d^2+3*c) 
*arcsin(d*x)/d^5
 

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.78 \[ \int \frac {x^2 \left (a+b x+c x^2\right )}{\sqrt {1-d x} \sqrt {1+d x}} \, dx=\frac {\sqrt {1-d^2 x^2} \left (-16 b-9 c x-12 a d^2 x-8 b d^2 x^2-6 c d^2 x^3\right )}{24 d^4}+\frac {\left (3 c+4 a d^2\right ) \arctan \left (\frac {d x}{-1+\sqrt {1-d^2 x^2}}\right )}{4 d^5} \] Input:

Integrate[(x^2*(a + b*x + c*x^2))/(Sqrt[1 - d*x]*Sqrt[1 + d*x]),x]
 

Output:

(Sqrt[1 - d^2*x^2]*(-16*b - 9*c*x - 12*a*d^2*x - 8*b*d^2*x^2 - 6*c*d^2*x^3 
))/(24*d^4) + ((3*c + 4*a*d^2)*ArcTan[(d*x)/(-1 + Sqrt[1 - d^2*x^2])])/(4* 
d^5)
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.11, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {2112, 2340, 25, 533, 27, 533, 455, 223}

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.

\(\displaystyle \int \frac {x^2 \left (a+b x+c x^2\right )}{\sqrt {1-d x} \sqrt {d x+1}} \, dx\)

\(\Big \downarrow \) 2112

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

\(\Big \downarrow \) 2340

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 533

\(\displaystyle \frac {\frac {\int \frac {d^2 x \left (8 b+3 \left (4 a d^2+3 c\right ) x\right )}{\sqrt {1-d^2 x^2}}dx}{3 d^2}-\frac {4}{3} b x^2 \sqrt {1-d^2 x^2}}{4 d^2}-\frac {c x^3 \sqrt {1-d^2 x^2}}{4 d^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{3} \int \frac {x \left (8 b+3 \left (4 a d^2+3 c\right ) x\right )}{\sqrt {1-d^2 x^2}}dx-\frac {4}{3} b x^2 \sqrt {1-d^2 x^2}}{4 d^2}-\frac {c x^3 \sqrt {1-d^2 x^2}}{4 d^2}\)

\(\Big \downarrow \) 533

\(\displaystyle \frac {\frac {1}{3} \left (\frac {\int \frac {16 b x d^2+3 \left (4 a d^2+3 c\right )}{\sqrt {1-d^2 x^2}}dx}{2 d^2}-\frac {3}{2} x \sqrt {1-d^2 x^2} \left (4 a+\frac {3 c}{d^2}\right )\right )-\frac {4}{3} b x^2 \sqrt {1-d^2 x^2}}{4 d^2}-\frac {c x^3 \sqrt {1-d^2 x^2}}{4 d^2}\)

\(\Big \downarrow \) 455

\(\displaystyle \frac {\frac {1}{3} \left (\frac {3 \left (4 a d^2+3 c\right ) \int \frac {1}{\sqrt {1-d^2 x^2}}dx-16 b \sqrt {1-d^2 x^2}}{2 d^2}-\frac {3}{2} x \sqrt {1-d^2 x^2} \left (4 a+\frac {3 c}{d^2}\right )\right )-\frac {4}{3} b x^2 \sqrt {1-d^2 x^2}}{4 d^2}-\frac {c x^3 \sqrt {1-d^2 x^2}}{4 d^2}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {\frac {1}{3} \left (\frac {\frac {3 \left (4 a d^2+3 c\right ) \arcsin (d x)}{d}-16 b \sqrt {1-d^2 x^2}}{2 d^2}-\frac {3}{2} x \sqrt {1-d^2 x^2} \left (4 a+\frac {3 c}{d^2}\right )\right )-\frac {4}{3} b x^2 \sqrt {1-d^2 x^2}}{4 d^2}-\frac {c x^3 \sqrt {1-d^2 x^2}}{4 d^2}\)

Input:

Int[(x^2*(a + b*x + c*x^2))/(Sqrt[1 - d*x]*Sqrt[1 + d*x]),x]
 

Output:

-1/4*(c*x^3*Sqrt[1 - d^2*x^2])/d^2 + ((-4*b*x^2*Sqrt[1 - d^2*x^2])/3 + ((- 
3*(4*a + (3*c)/d^2)*x*Sqrt[1 - d^2*x^2])/2 + (-16*b*Sqrt[1 - d^2*x^2] + (3 
*(3*c + 4*a*d^2)*ArcSin[d*x])/d)/(2*d^2))/3)/(4*d^2)
 

Defintions of rubi rules used

rule 25

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

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 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt 
[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
 

rule 455

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

rule 533

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
 Simp[d*x^m*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 2))), x] - Simp[1/(b*(m + 2* 
p + 2))   Int[x^(m - 1)*(a + b*x^2)^p*Simp[a*d*m - b*c*(m + 2*p + 2)*x, x], 
 x], x] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[m, 0] && GtQ[p, -1] && Integer 
Q[2*p]
 

rule 2112

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f 
_.)*(x_))^(p_.), x_Symbol] :> Int[Px*(a*c + b*d*x^2)^m*(e + f*x)^p, x] /; F 
reeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[b*c + a*d, 0] & 
& EqQ[m, n] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))
 

rule 2340

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ 
{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1 
)*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(b*(m 
+ q + 2*p + 1))   Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1) 
*Pq - b*f*(m + q + 2*p + 1)*x^q - a*f*(m + q - 1)*x^(q - 2), x], x], x] /; 
GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ 
[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])
 

Maple [A] (verified)

Time = 0.68 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.30

method	result	size
risch	\(\frac {\left (6 c \,x^{3} d^{2}+8 b \,d^{2} x^{2}+12 x a \,d^{2}+9 c x +16 b \right ) \sqrt {x d +1}\, \left (x d -1\right ) \sqrt {\left (-x d +1\right ) \left (x d +1\right )}}{24 d^{4} \sqrt {-\left (x d +1\right ) \left (x d -1\right )}\, \sqrt {-x d +1}}+\frac {\left (4 a \,d^{2}+3 c \right ) \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+1}}\right ) \sqrt {\left (-x d +1\right ) \left (x d +1\right )}}{8 d^{4} \sqrt {d^{2}}\, \sqrt {-x d +1}\, \sqrt {x d +1}}\)	\(157\)
default	\(-\frac {\sqrt {-x d +1}\, \sqrt {x d +1}\, \left (6 \,\operatorname {csgn}\left (d \right ) c \,d^{3} x^{3} \sqrt {-d^{2} x^{2}+1}+8 \,\operatorname {csgn}\left (d \right ) b \,d^{3} x^{2} \sqrt {-d^{2} x^{2}+1}+12 \sqrt {-d^{2} x^{2}+1}\, \operatorname {csgn}\left (d \right ) d^{3} a x +9 \sqrt {-d^{2} x^{2}+1}\, \operatorname {csgn}\left (d \right ) d c x +16 \,\operatorname {csgn}\left (d \right ) d \sqrt {-d^{2} x^{2}+1}\, b -12 \arctan \left (\frac {\operatorname {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) a \,d^{2}-9 \arctan \left (\frac {\operatorname {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) c \right ) \operatorname {csgn}\left (d \right )}{24 d^{5} \sqrt {-d^{2} x^{2}+1}}\)	\(185\)

Input:

int(x^2*(c*x^2+b*x+a)/(-d*x+1)^(1/2)/(d*x+1)^(1/2),x,method=_RETURNVERBOSE 
)
 

Output:

1/24*(6*c*d^2*x^3+8*b*d^2*x^2+12*a*d^2*x+9*c*x+16*b)*(d*x+1)^(1/2)*(d*x-1) 
/d^4/(-(d*x+1)*(d*x-1))^(1/2)*((-d*x+1)*(d*x+1))^(1/2)/(-d*x+1)^(1/2)+1/8* 
(4*a*d^2+3*c)/d^4/(d^2)^(1/2)*arctan((d^2)^(1/2)*x/(-d^2*x^2+1)^(1/2))*((- 
d*x+1)*(d*x+1))^(1/2)/(-d*x+1)^(1/2)/(d*x+1)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.80 \[ \int \frac {x^2 \left (a+b x+c x^2\right )}{\sqrt {1-d x} \sqrt {1+d x}} \, dx=-\frac {{\left (6 \, c d^{3} x^{3} + 8 \, b d^{3} x^{2} + 16 \, b d + 3 \, {\left (4 \, a d^{3} + 3 \, c d\right )} x\right )} \sqrt {d x + 1} \sqrt {-d x + 1} + 6 \, {\left (4 \, a d^{2} + 3 \, c\right )} \arctan \left (\frac {\sqrt {d x + 1} \sqrt {-d x + 1} - 1}{d x}\right )}{24 \, d^{5}} \] Input:

integrate(x^2*(c*x^2+b*x+a)/(-d*x+1)^(1/2)/(d*x+1)^(1/2),x, algorithm="fri 
cas")
 

Output:

-1/24*((6*c*d^3*x^3 + 8*b*d^3*x^2 + 16*b*d + 3*(4*a*d^3 + 3*c*d)*x)*sqrt(d 
*x + 1)*sqrt(-d*x + 1) + 6*(4*a*d^2 + 3*c)*arctan((sqrt(d*x + 1)*sqrt(-d*x 
 + 1) - 1)/(d*x)))/d^5
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^2 \left (a+b x+c x^2\right )}{\sqrt {1-d x} \sqrt {1+d x}} \, dx=\text {Timed out} \] Input:

integrate(x**2*(c*x**2+b*x+a)/(-d*x+1)**(1/2)/(d*x+1)**(1/2),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.98 \[ \int \frac {x^2 \left (a+b x+c x^2\right )}{\sqrt {1-d x} \sqrt {1+d x}} \, dx=-\frac {\sqrt {-d^{2} x^{2} + 1} c x^{3}}{4 \, d^{2}} - \frac {\sqrt {-d^{2} x^{2} + 1} b x^{2}}{3 \, d^{2}} - \frac {\sqrt {-d^{2} x^{2} + 1} a x}{2 \, d^{2}} + \frac {a \arcsin \left (d x\right )}{2 \, d^{3}} - \frac {3 \, \sqrt {-d^{2} x^{2} + 1} c x}{8 \, d^{4}} - \frac {2 \, \sqrt {-d^{2} x^{2} + 1} b}{3 \, d^{4}} + \frac {3 \, c \arcsin \left (d x\right )}{8 \, d^{5}} \] Input:

integrate(x^2*(c*x^2+b*x+a)/(-d*x+1)^(1/2)/(d*x+1)^(1/2),x, algorithm="max 
ima")
 

Output:

-1/4*sqrt(-d^2*x^2 + 1)*c*x^3/d^2 - 1/3*sqrt(-d^2*x^2 + 1)*b*x^2/d^2 - 1/2 
*sqrt(-d^2*x^2 + 1)*a*x/d^2 + 1/2*a*arcsin(d*x)/d^3 - 3/8*sqrt(-d^2*x^2 + 
1)*c*x/d^4 - 2/3*sqrt(-d^2*x^2 + 1)*b/d^4 + 3/8*c*arcsin(d*x)/d^5
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.87 \[ \int \frac {x^2 \left (a+b x+c x^2\right )}{\sqrt {1-d x} \sqrt {1+d x}} \, dx=\frac {{\left (12 \, a d^{2} - {\left (12 \, a d^{2} + 2 \, {\left (3 \, {\left (d x + 1\right )} c + 4 \, b d - 9 \, c\right )} {\left (d x + 1\right )} - 16 \, b d + 27 \, c\right )} {\left (d x + 1\right )} - 24 \, b d + 15 \, c\right )} \sqrt {d x + 1} \sqrt {-d x + 1} + 6 \, {\left (4 \, a d^{2} + 3 \, c\right )} \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {d x + 1}\right )}{24 \, d^{5}} \] Input:

integrate(x^2*(c*x^2+b*x+a)/(-d*x+1)^(1/2)/(d*x+1)^(1/2),x, algorithm="gia 
c")
 

Output:

1/24*((12*a*d^2 - (12*a*d^2 + 2*(3*(d*x + 1)*c + 4*b*d - 9*c)*(d*x + 1) - 
16*b*d + 27*c)*(d*x + 1) - 24*b*d + 15*c)*sqrt(d*x + 1)*sqrt(-d*x + 1) + 6 
*(4*a*d^2 + 3*c)*arcsin(1/2*sqrt(2)*sqrt(d*x + 1)))/d^5
 

Mupad [B] (verification not implemented)

Time = 10.64 (sec) , antiderivative size = 485, normalized size of antiderivative = 4.01 \[ \int \frac {x^2 \left (a+b x+c x^2\right )}{\sqrt {1-d x} \sqrt {1+d x}} \, dx=\frac {\frac {23\,c\,{\left (\sqrt {1-d\,x}-1\right )}^3}{2\,{\left (\sqrt {d\,x+1}-1\right )}^3}-\frac {333\,c\,{\left (\sqrt {1-d\,x}-1\right )}^5}{2\,{\left (\sqrt {d\,x+1}-1\right )}^5}+\frac {671\,c\,{\left (\sqrt {1-d\,x}-1\right )}^7}{2\,{\left (\sqrt {d\,x+1}-1\right )}^7}-\frac {671\,c\,{\left (\sqrt {1-d\,x}-1\right )}^9}{2\,{\left (\sqrt {d\,x+1}-1\right )}^9}+\frac {333\,c\,{\left (\sqrt {1-d\,x}-1\right )}^{11}}{2\,{\left (\sqrt {d\,x+1}-1\right )}^{11}}-\frac {23\,c\,{\left (\sqrt {1-d\,x}-1\right )}^{13}}{2\,{\left (\sqrt {d\,x+1}-1\right )}^{13}}-\frac {3\,c\,{\left (\sqrt {1-d\,x}-1\right )}^{15}}{2\,{\left (\sqrt {d\,x+1}-1\right )}^{15}}+\frac {3\,c\,\left (\sqrt {1-d\,x}-1\right )}{2\,\left (\sqrt {d\,x+1}-1\right )}}{d^5\,{\left (\frac {{\left (\sqrt {1-d\,x}-1\right )}^2}{{\left (\sqrt {d\,x+1}-1\right )}^2}+1\right )}^8}-\frac {3\,c\,\mathrm {atan}\left (\frac {\sqrt {1-d\,x}-1}{\sqrt {d\,x+1}-1}\right )}{2\,d^5}-\frac {2\,a\,\mathrm {atan}\left (\frac {\sqrt {1-d\,x}-1}{\sqrt {d\,x+1}-1}\right )}{d^3}-\frac {\frac {14\,a\,{\left (\sqrt {1-d\,x}-1\right )}^3}{{\left (\sqrt {d\,x+1}-1\right )}^3}-\frac {14\,a\,{\left (\sqrt {1-d\,x}-1\right )}^5}{{\left (\sqrt {d\,x+1}-1\right )}^5}+\frac {2\,a\,{\left (\sqrt {1-d\,x}-1\right )}^7}{{\left (\sqrt {d\,x+1}-1\right )}^7}-\frac {2\,a\,\left (\sqrt {1-d\,x}-1\right )}{\sqrt {d\,x+1}-1}}{d^3\,{\left (\frac {{\left (\sqrt {1-d\,x}-1\right )}^2}{{\left (\sqrt {d\,x+1}-1\right )}^2}+1\right )}^4}-\frac {\sqrt {1-d\,x}\,\left (\frac {2\,b}{3\,d^4}+\frac {b\,x^3}{3\,d}+\frac {b\,x^2}{3\,d^2}+\frac {2\,b\,x}{3\,d^3}\right )}{\sqrt {d\,x+1}} \] Input:

int((x^2*(a + b*x + c*x^2))/((1 - d*x)^(1/2)*(d*x + 1)^(1/2)),x)
                                                                                    
                                                                                    
 

Output:

((23*c*((1 - d*x)^(1/2) - 1)^3)/(2*((d*x + 1)^(1/2) - 1)^3) - (333*c*((1 - 
 d*x)^(1/2) - 1)^5)/(2*((d*x + 1)^(1/2) - 1)^5) + (671*c*((1 - d*x)^(1/2) 
- 1)^7)/(2*((d*x + 1)^(1/2) - 1)^7) - (671*c*((1 - d*x)^(1/2) - 1)^9)/(2*( 
(d*x + 1)^(1/2) - 1)^9) + (333*c*((1 - d*x)^(1/2) - 1)^11)/(2*((d*x + 1)^( 
1/2) - 1)^11) - (23*c*((1 - d*x)^(1/2) - 1)^13)/(2*((d*x + 1)^(1/2) - 1)^1 
3) - (3*c*((1 - d*x)^(1/2) - 1)^15)/(2*((d*x + 1)^(1/2) - 1)^15) + (3*c*(( 
1 - d*x)^(1/2) - 1))/(2*((d*x + 1)^(1/2) - 1)))/(d^5*(((1 - d*x)^(1/2) - 1 
)^2/((d*x + 1)^(1/2) - 1)^2 + 1)^8) - (3*c*atan(((1 - d*x)^(1/2) - 1)/((d* 
x + 1)^(1/2) - 1)))/(2*d^5) - (2*a*atan(((1 - d*x)^(1/2) - 1)/((d*x + 1)^( 
1/2) - 1)))/d^3 - ((14*a*((1 - d*x)^(1/2) - 1)^3)/((d*x + 1)^(1/2) - 1)^3 
- (14*a*((1 - d*x)^(1/2) - 1)^5)/((d*x + 1)^(1/2) - 1)^5 + (2*a*((1 - d*x) 
^(1/2) - 1)^7)/((d*x + 1)^(1/2) - 1)^7 - (2*a*((1 - d*x)^(1/2) - 1))/((d*x 
 + 1)^(1/2) - 1))/(d^3*(((1 - d*x)^(1/2) - 1)^2/((d*x + 1)^(1/2) - 1)^2 + 
1)^4) - ((1 - d*x)^(1/2)*((2*b)/(3*d^4) + (b*x^3)/(3*d) + (b*x^2)/(3*d^2) 
+ (2*b*x)/(3*d^3)))/(d*x + 1)^(1/2)
 

Reduce [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.16 \[ \int \frac {x^2 \left (a+b x+c x^2\right )}{\sqrt {1-d x} \sqrt {1+d x}} \, dx=\frac {-24 \mathit {asin} \left (\frac {\sqrt {-d x +1}}{\sqrt {2}}\right ) a \,d^{2}-18 \mathit {asin} \left (\frac {\sqrt {-d x +1}}{\sqrt {2}}\right ) c -12 \sqrt {d x +1}\, \sqrt {-d x +1}\, a \,d^{3} x -8 \sqrt {d x +1}\, \sqrt {-d x +1}\, b \,d^{3} x^{2}-16 \sqrt {d x +1}\, \sqrt {-d x +1}\, b d -6 \sqrt {d x +1}\, \sqrt {-d x +1}\, c \,d^{3} x^{3}-9 \sqrt {d x +1}\, \sqrt {-d x +1}\, c d x}{24 d^{5}} \] Input:

int(x^2*(c*x^2+b*x+a)/(-d*x+1)^(1/2)/(d*x+1)^(1/2),x)
 

Output:

( - 24*asin(sqrt( - d*x + 1)/sqrt(2))*a*d**2 - 18*asin(sqrt( - d*x + 1)/sq 
rt(2))*c - 12*sqrt(d*x + 1)*sqrt( - d*x + 1)*a*d**3*x - 8*sqrt(d*x + 1)*sq 
rt( - d*x + 1)*b*d**3*x**2 - 16*sqrt(d*x + 1)*sqrt( - d*x + 1)*b*d - 6*sqr 
t(d*x + 1)*sqrt( - d*x + 1)*c*d**3*x**3 - 9*sqrt(d*x + 1)*sqrt( - d*x + 1) 
*c*d*x)/(24*d**5)