\(\int \frac {x^2 (a+b x^2)^2}{(c+d x^2)^3} \, dx\) [585]

Optimal result

Integrand size = 22, antiderivative size = 123 \[ \int \frac {x^2 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=\frac {b^2 x}{d^3}-\frac {(b c-a d)^2 x}{4 d^3 \left (c+d x^2\right )^2}+\frac {(b c-a d) (9 b c-a d) x}{8 c d^3 \left (c+d x^2\right )}-\frac {\left (15 b^2 c^2-6 a b c d-a^2 d^2\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} d^{7/2}} \] Output:

b^2*x/d^3-1/4*(-a*d+b*c)^2*x/d^3/(d*x^2+c)^2+1/8*(-a*d+b*c)*(-a*d+9*b*c)*x 
/c/d^3/(d*x^2+c)-1/8*(-a^2*d^2-6*a*b*c*d+15*b^2*c^2)*arctan(d^(1/2)*x/c^(1 
/2))/c^(3/2)/d^(7/2)
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.06 \[ \int \frac {x^2 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=\frac {x \left (a^2 d^2 \left (-c+d x^2\right )-2 a b c d \left (3 c+5 d x^2\right )+b^2 c \left (15 c^2+25 c d x^2+8 d^2 x^4\right )\right )}{8 c d^3 \left (c+d x^2\right )^2}-\frac {\left (15 b^2 c^2-6 a b c d-a^2 d^2\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} d^{7/2}} \] Input:

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

Output:

(x*(a^2*d^2*(-c + d*x^2) - 2*a*b*c*d*(3*c + 5*d*x^2) + b^2*c*(15*c^2 + 25* 
c*d*x^2 + 8*d^2*x^4)))/(8*c*d^3*(c + d*x^2)^2) - ((15*b^2*c^2 - 6*a*b*c*d 
- a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(3/2)*d^(7/2))
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.13, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {366, 25, 25, 360, 25, 27, 299, 218}

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.

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

\(\Big \downarrow \) 366

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 360

\(\displaystyle \frac {x^3 (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2}-\frac {-\frac {\int -\frac {d \left ((b c-a d) (7 b c+a d)-8 b^2 c d x^2\right )}{d x^2+c}dx}{2 d^2}-\frac {x (b c-a d) (a d+7 b c)}{2 d \left (c+d x^2\right )}}{4 c d^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {x^3 (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2}-\frac {\frac {\int \frac {d \left ((b c-a d) (7 b c+a d)-8 b^2 c d x^2\right )}{d x^2+c}dx}{2 d^2}-\frac {x (b c-a d) (a d+7 b c)}{2 d \left (c+d x^2\right )}}{4 c d^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^3 (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2}-\frac {\frac {\int \frac {(b c-a d) (7 b c+a d)-8 b^2 c d x^2}{d x^2+c}dx}{2 d}-\frac {x (b c-a d) (a d+7 b c)}{2 d \left (c+d x^2\right )}}{4 c d^2}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {x^3 (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2}-\frac {\frac {\left (-a^2 d^2-6 a b c d+15 b^2 c^2\right ) \int \frac {1}{d x^2+c}dx-8 b^2 c x}{2 d}-\frac {x (b c-a d) (a d+7 b c)}{2 d \left (c+d x^2\right )}}{4 c d^2}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {x^3 (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2}-\frac {\frac {\frac {\left (-a^2 d^2-6 a b c d+15 b^2 c^2\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} \sqrt {d}}-8 b^2 c x}{2 d}-\frac {x (b c-a d) (a d+7 b c)}{2 d \left (c+d x^2\right )}}{4 c d^2}\)

Input:

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

Output:

((b*c - a*d)^2*x^3)/(4*c*d^2*(c + d*x^2)^2) - (-1/2*((b*c - a*d)*(7*b*c + 
a*d)*x)/(d*(c + d*x^2)) + (-8*b^2*c*x + ((15*b^2*c^2 - 6*a*b*c*d - a^2*d^2 
)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*Sqrt[d]))/(2*d))/(4*c*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 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 299

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x 
*((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 
*p + 3))   Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && NeQ[2*p + 3, 0]
 

rule 360

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] : 
> Simp[(-a)^(m/2 - 1)*(b*c - a*d)*x*((a + b*x^2)^(p + 1)/(2*b^(m/2 + 1)*(p 
+ 1))), x] + Simp[1/(2*b^(m/2 + 1)*(p + 1))   Int[(a + b*x^2)^(p + 1)*Expan 
dToSum[2*b*(p + 1)*x^2*Together[(b^(m/2)*x^(m - 2)*(c + d*x^2) - (-a)^(m/2 
- 1)*(b*c - a*d))/(a + b*x^2)] - (-a)^(m/2 - 1)*(b*c - a*d), x], x], x] /; 
FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IGtQ[m/2, 0] & 
& (IntegerQ[p] || EqQ[m + 2*p + 1, 0])
 

rule 366

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

Maple [A] (verified)

Time = 0.45 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00

method	result	size
default	\(\frac {b^{2} x}{d^{3}}+\frac {\frac {\frac {d \left (a^{2} d^{2}-10 a b c d +9 b^{2} c^{2}\right ) x^{3}}{8 c}+\left (-\frac {1}{8} a^{2} d^{2}-\frac {3}{4} a b c d +\frac {7}{8} b^{2} c^{2}\right ) x}{\left (x^{2} d +c \right )^{2}}+\frac {\left (a^{2} d^{2}+6 a b c d -15 b^{2} c^{2}\right ) \arctan \left (\frac {x d}{\sqrt {c d}}\right )}{8 c \sqrt {c d}}}{d^{3}}\)	\(123\)
risch	\(\frac {b^{2} x}{d^{3}}+\frac {\frac {d \left (a^{2} d^{2}-10 a b c d +9 b^{2} c^{2}\right ) x^{3}}{8 c}+\left (-\frac {1}{8} a^{2} d^{2}-\frac {3}{4} a b c d +\frac {7}{8} b^{2} c^{2}\right ) x}{d^{3} \left (x^{2} d +c \right )^{2}}-\frac {\ln \left (x d +\sqrt {-c d}\right ) a^{2}}{16 d \sqrt {-c d}\, c}-\frac {3 \ln \left (x d +\sqrt {-c d}\right ) a b}{8 d^{2} \sqrt {-c d}}+\frac {15 c \ln \left (x d +\sqrt {-c d}\right ) b^{2}}{16 d^{3} \sqrt {-c d}}+\frac {\ln \left (-x d +\sqrt {-c d}\right ) a^{2}}{16 d \sqrt {-c d}\, c}+\frac {3 \ln \left (-x d +\sqrt {-c d}\right ) a b}{8 d^{2} \sqrt {-c d}}-\frac {15 c \ln \left (-x d +\sqrt {-c d}\right ) b^{2}}{16 d^{3} \sqrt {-c d}}\)	\(239\)

Input:

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

Output:

b^2*x/d^3+1/d^3*((1/8*d*(a^2*d^2-10*a*b*c*d+9*b^2*c^2)/c*x^3+(-1/8*a^2*d^2 
-3/4*a*b*c*d+7/8*b^2*c^2)*x)/(d*x^2+c)^2+1/8*(a^2*d^2+6*a*b*c*d-15*b^2*c^2 
)/c/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 227 vs. \(2 (109) = 218\).

Time = 0.10 (sec) , antiderivative size = 475, normalized size of antiderivative = 3.86 \[ \int \frac {x^2 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=\left [\frac {16 \, b^{2} c^{2} d^{3} x^{5} + 2 \, {\left (25 \, b^{2} c^{3} d^{2} - 10 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x^{3} + {\left (15 \, b^{2} c^{4} - 6 \, a b c^{3} d - a^{2} c^{2} d^{2} + {\left (15 \, b^{2} c^{2} d^{2} - 6 \, a b c d^{3} - a^{2} d^{4}\right )} x^{4} + 2 \, {\left (15 \, b^{2} c^{3} d - 6 \, a b c^{2} d^{2} - a^{2} c d^{3}\right )} x^{2}\right )} \sqrt {-c d} \log \left (\frac {d x^{2} - 2 \, \sqrt {-c d} x - c}{d x^{2} + c}\right ) + 2 \, {\left (15 \, b^{2} c^{4} d - 6 \, a b c^{3} d^{2} - a^{2} c^{2} d^{3}\right )} x}{16 \, {\left (c^{2} d^{6} x^{4} + 2 \, c^{3} d^{5} x^{2} + c^{4} d^{4}\right )}}, \frac {8 \, b^{2} c^{2} d^{3} x^{5} + {\left (25 \, b^{2} c^{3} d^{2} - 10 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x^{3} - {\left (15 \, b^{2} c^{4} - 6 \, a b c^{3} d - a^{2} c^{2} d^{2} + {\left (15 \, b^{2} c^{2} d^{2} - 6 \, a b c d^{3} - a^{2} d^{4}\right )} x^{4} + 2 \, {\left (15 \, b^{2} c^{3} d - 6 \, a b c^{2} d^{2} - a^{2} c d^{3}\right )} x^{2}\right )} \sqrt {c d} \arctan \left (\frac {\sqrt {c d} x}{c}\right ) + {\left (15 \, b^{2} c^{4} d - 6 \, a b c^{3} d^{2} - a^{2} c^{2} d^{3}\right )} x}{8 \, {\left (c^{2} d^{6} x^{4} + 2 \, c^{3} d^{5} x^{2} + c^{4} d^{4}\right )}}\right ] \] Input:

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

Output:

[1/16*(16*b^2*c^2*d^3*x^5 + 2*(25*b^2*c^3*d^2 - 10*a*b*c^2*d^3 + a^2*c*d^4 
)*x^3 + (15*b^2*c^4 - 6*a*b*c^3*d - a^2*c^2*d^2 + (15*b^2*c^2*d^2 - 6*a*b* 
c*d^3 - a^2*d^4)*x^4 + 2*(15*b^2*c^3*d - 6*a*b*c^2*d^2 - a^2*c*d^3)*x^2)*s 
qrt(-c*d)*log((d*x^2 - 2*sqrt(-c*d)*x - c)/(d*x^2 + c)) + 2*(15*b^2*c^4*d 
- 6*a*b*c^3*d^2 - a^2*c^2*d^3)*x)/(c^2*d^6*x^4 + 2*c^3*d^5*x^2 + c^4*d^4), 
 1/8*(8*b^2*c^2*d^3*x^5 + (25*b^2*c^3*d^2 - 10*a*b*c^2*d^3 + a^2*c*d^4)*x^ 
3 - (15*b^2*c^4 - 6*a*b*c^3*d - a^2*c^2*d^2 + (15*b^2*c^2*d^2 - 6*a*b*c*d^ 
3 - a^2*d^4)*x^4 + 2*(15*b^2*c^3*d - 6*a*b*c^2*d^2 - a^2*c*d^3)*x^2)*sqrt( 
c*d)*arctan(sqrt(c*d)*x/c) + (15*b^2*c^4*d - 6*a*b*c^3*d^2 - a^2*c^2*d^3)* 
x)/(c^2*d^6*x^4 + 2*c^3*d^5*x^2 + c^4*d^4)]
 

Sympy [A] (verification not implemented)

Time = 0.74 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.81 \[ \int \frac {x^2 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=\frac {b^{2} x}{d^{3}} - \frac {\sqrt {- \frac {1}{c^{3} d^{7}}} \left (a^{2} d^{2} + 6 a b c d - 15 b^{2} c^{2}\right ) \log {\left (- c^{2} d^{3} \sqrt {- \frac {1}{c^{3} d^{7}}} + x \right )}}{16} + \frac {\sqrt {- \frac {1}{c^{3} d^{7}}} \left (a^{2} d^{2} + 6 a b c d - 15 b^{2} c^{2}\right ) \log {\left (c^{2} d^{3} \sqrt {- \frac {1}{c^{3} d^{7}}} + x \right )}}{16} + \frac {x^{3} \left (a^{2} d^{3} - 10 a b c d^{2} + 9 b^{2} c^{2} d\right ) + x \left (- a^{2} c d^{2} - 6 a b c^{2} d + 7 b^{2} c^{3}\right )}{8 c^{3} d^{3} + 16 c^{2} d^{4} x^{2} + 8 c d^{5} x^{4}} \] Input:

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

Output:

b**2*x/d**3 - sqrt(-1/(c**3*d**7))*(a**2*d**2 + 6*a*b*c*d - 15*b**2*c**2)* 
log(-c**2*d**3*sqrt(-1/(c**3*d**7)) + x)/16 + sqrt(-1/(c**3*d**7))*(a**2*d 
**2 + 6*a*b*c*d - 15*b**2*c**2)*log(c**2*d**3*sqrt(-1/(c**3*d**7)) + x)/16 
 + (x**3*(a**2*d**3 - 10*a*b*c*d**2 + 9*b**2*c**2*d) + x*(-a**2*c*d**2 - 6 
*a*b*c**2*d + 7*b**2*c**3))/(8*c**3*d**3 + 16*c**2*d**4*x**2 + 8*c*d**5*x* 
*4)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.16 \[ \int \frac {x^2 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=\frac {{\left (9 \, b^{2} c^{2} d - 10 \, a b c d^{2} + a^{2} d^{3}\right )} x^{3} + {\left (7 \, b^{2} c^{3} - 6 \, a b c^{2} d - a^{2} c d^{2}\right )} x}{8 \, {\left (c d^{5} x^{4} + 2 \, c^{2} d^{4} x^{2} + c^{3} d^{3}\right )}} + \frac {b^{2} x}{d^{3}} - \frac {{\left (15 \, b^{2} c^{2} - 6 \, a b c d - a^{2} d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \, \sqrt {c d} c d^{3}} \] Input:

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

Output:

1/8*((9*b^2*c^2*d - 10*a*b*c*d^2 + a^2*d^3)*x^3 + (7*b^2*c^3 - 6*a*b*c^2*d 
 - a^2*c*d^2)*x)/(c*d^5*x^4 + 2*c^2*d^4*x^2 + c^3*d^3) + b^2*x/d^3 - 1/8*( 
15*b^2*c^2 - 6*a*b*c*d - a^2*d^2)*arctan(d*x/sqrt(c*d))/(sqrt(c*d)*c*d^3)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.08 \[ \int \frac {x^2 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=\frac {b^{2} x}{d^{3}} - \frac {{\left (15 \, b^{2} c^{2} - 6 \, a b c d - a^{2} d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \, \sqrt {c d} c d^{3}} + \frac {9 \, b^{2} c^{2} d x^{3} - 10 \, a b c d^{2} x^{3} + a^{2} d^{3} x^{3} + 7 \, b^{2} c^{3} x - 6 \, a b c^{2} d x - a^{2} c d^{2} x}{8 \, {\left (d x^{2} + c\right )}^{2} c d^{3}} \] Input:

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

Output:

b^2*x/d^3 - 1/8*(15*b^2*c^2 - 6*a*b*c*d - a^2*d^2)*arctan(d*x/sqrt(c*d))/( 
sqrt(c*d)*c*d^3) + 1/8*(9*b^2*c^2*d*x^3 - 10*a*b*c*d^2*x^3 + a^2*d^3*x^3 + 
 7*b^2*c^3*x - 6*a*b*c^2*d*x - a^2*c*d^2*x)/((d*x^2 + c)^2*c*d^3)
 

Mupad [B] (verification not implemented)

Time = 0.52 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.10 \[ \int \frac {x^2 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=\frac {b^2\,x}{d^3}-\frac {x\,\left (\frac {a^2\,d^2}{8}+\frac {3\,a\,b\,c\,d}{4}-\frac {7\,b^2\,c^2}{8}\right )-\frac {x^3\,\left (a^2\,d^3-10\,a\,b\,c\,d^2+9\,b^2\,c^2\,d\right )}{8\,c}}{c^2\,d^3+2\,c\,d^4\,x^2+d^5\,x^4}+\frac {\mathrm {atan}\left (\frac {\sqrt {d}\,x}{\sqrt {c}}\right )\,\left (a^2\,d^2+6\,a\,b\,c\,d-15\,b^2\,c^2\right )}{8\,c^{3/2}\,d^{7/2}} \] Input:

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

Output:

(b^2*x)/d^3 - (x*((a^2*d^2)/8 - (7*b^2*c^2)/8 + (3*a*b*c*d)/4) - (x^3*(a^2 
*d^3 + 9*b^2*c^2*d - 10*a*b*c*d^2))/(8*c))/(c^2*d^3 + d^5*x^4 + 2*c*d^4*x^ 
2) + (atan((d^(1/2)*x)/c^(1/2))*(a^2*d^2 - 15*b^2*c^2 + 6*a*b*c*d))/(8*c^( 
3/2)*d^(7/2))
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 356, normalized size of antiderivative = 2.89 \[ \int \frac {x^2 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=\frac {\sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) a^{2} c^{2} d^{2}+2 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) a^{2} c \,d^{3} x^{2}+\sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) a^{2} d^{4} x^{4}+6 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) a b \,c^{3} d +12 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) a b \,c^{2} d^{2} x^{2}+6 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) a b c \,d^{3} x^{4}-15 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) b^{2} c^{4}-30 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) b^{2} c^{3} d \,x^{2}-15 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) b^{2} c^{2} d^{2} x^{4}-a^{2} c^{2} d^{3} x +a^{2} c \,d^{4} x^{3}-6 a b \,c^{3} d^{2} x -10 a b \,c^{2} d^{3} x^{3}+15 b^{2} c^{4} d x +25 b^{2} c^{3} d^{2} x^{3}+8 b^{2} c^{2} d^{3} x^{5}}{8 c^{2} d^{4} \left (d^{2} x^{4}+2 c d \,x^{2}+c^{2}\right )} \] Input:

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

Output:

(sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**2*c**2*d**2 + 2*sqrt(d)* 
sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**2*c*d**3*x**2 + sqrt(d)*sqrt(c)*a 
tan((d*x)/(sqrt(d)*sqrt(c)))*a**2*d**4*x**4 + 6*sqrt(d)*sqrt(c)*atan((d*x) 
/(sqrt(d)*sqrt(c)))*a*b*c**3*d + 12*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sq 
rt(c)))*a*b*c**2*d**2*x**2 + 6*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c) 
))*a*b*c*d**3*x**4 - 15*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*b**2 
*c**4 - 30*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*b**2*c**3*d*x**2 
- 15*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*b**2*c**2*d**2*x**4 - a 
**2*c**2*d**3*x + a**2*c*d**4*x**3 - 6*a*b*c**3*d**2*x - 10*a*b*c**2*d**3* 
x**3 + 15*b**2*c**4*d*x + 25*b**2*c**3*d**2*x**3 + 8*b**2*c**2*d**3*x**5)/ 
(8*c**2*d**4*(c**2 + 2*c*d*x**2 + d**2*x**4))