\(\int \frac {A+B x^2+C x^4+D x^6}{x^4 (a+b x^2)^3} \, dx\) [185]

Optimal result

Integrand size = 30, antiderivative size = 176 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^4 \left (a+b x^2\right )^3} \, dx=-\frac {A}{3 a^3 x^3}+\frac {3 A b-a B}{a^4 x}+\frac {\left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) x}{4 a^3 b \left (a+b x^2\right )^2}+\frac {\left (11 A b^3-a \left (7 b^2 B-3 a b C-a^2 D\right )\right ) x}{8 a^4 b \left (a+b x^2\right )}+\frac {\left (35 A b^3-a \left (15 b^2 B-3 a b C-a^2 D\right )\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{9/2} b^{3/2}} \] Output:

-1/3*A/a^3/x^3+(3*A*b-B*a)/a^4/x+1/4*(A*b^3-a*(B*b^2-C*a*b+D*a^2))*x/a^3/b 
/(b*x^2+a)^2+1/8*(11*A*b^3-a*(7*B*b^2-3*C*a*b-D*a^2))*x/a^4/b/(b*x^2+a)+1/ 
8*(35*A*b^3-a*(15*B*b^2-3*C*a*b-D*a^2))*arctan(b^(1/2)*x/a^(1/2))/a^(9/2)/ 
b^(3/2)
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.96 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^4 \left (a+b x^2\right )^3} \, dx=\frac {-3 a^4 D x^4+105 A b^4 x^6+5 a b^3 x^4 \left (35 A-9 B x^2\right )+a^2 b^2 x^2 \left (56 A-75 B x^2+9 C x^4\right )+a^3 b \left (-8 A+3 x^2 \left (-8 B+5 C x^2+D x^4\right )\right )}{24 a^4 b x^3 \left (a+b x^2\right )^2}+\frac {\left (35 A b^3+a \left (-15 b^2 B+3 a b C+a^2 D\right )\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{9/2} b^{3/2}} \] Input:

Integrate[(A + B*x^2 + C*x^4 + D*x^6)/(x^4*(a + b*x^2)^3),x]
 

Output:

(-3*a^4*D*x^4 + 105*A*b^4*x^6 + 5*a*b^3*x^4*(35*A - 9*B*x^2) + a^2*b^2*x^2 
*(56*A - 75*B*x^2 + 9*C*x^4) + a^3*b*(-8*A + 3*x^2*(-8*B + 5*C*x^2 + D*x^4 
)))/(24*a^4*b*x^3*(a + b*x^2)^2) + ((35*A*b^3 + a*(-15*b^2*B + 3*a*b*C + a 
^2*D))*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(9/2)*b^(3/2))
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2336, 25, 1582, 1584, 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.

\(\displaystyle \int \frac {A+B x^2+C x^4+D x^6}{x^4 \left (a+b x^2\right )^3} \, dx\)

\(\Big \downarrow \) 2336

\(\displaystyle \frac {x \left (\frac {A b^2}{a^2}-\frac {b B}{a}-\frac {a D}{b}+C\right )}{4 a \left (a+b x^2\right )^2}-\frac {\int -\frac {\left (\frac {3 A b^2}{a^2}-\frac {3 B b}{a}+3 C+\frac {a D}{b}\right ) x^4-4 \left (\frac {A b}{a}-B\right ) x^2+4 A}{x^4 \left (b x^2+a\right )^2}dx}{4 a}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {\left (\frac {3 A b^2}{a^2}-\frac {3 B b}{a}+3 C+\frac {a D}{b}\right ) x^4-4 \left (\frac {A b}{a}-B\right ) x^2+4 A}{x^4 \left (b x^2+a\right )^2}dx}{4 a}+\frac {x \left (\frac {A b^2}{a^2}-\frac {b B}{a}-\frac {a D}{b}+C\right )}{4 a \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 1582

\(\displaystyle \frac {\frac {\int \frac {b \left (11 A b^3-a \left (-D a^2-3 b C a+7 b^2 B\right )\right ) x^4-8 a b^2 (2 A b-a B) x^2+8 a^2 A b^2}{x^4 \left (b x^2+a\right )}dx}{2 a^3 b^2}+\frac {x \left (11 A b^3-a \left (a^2 (-D)-3 a b C+7 b^2 B\right )\right )}{2 a^3 b \left (a+b x^2\right )}}{4 a}+\frac {x \left (\frac {A b^2}{a^2}-\frac {b B}{a}-\frac {a D}{b}+C\right )}{4 a \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 1584

\(\displaystyle \frac {\frac {\int \left (-\frac {8 (3 A b-a B) b^2}{x^2}+\frac {8 a A b^2}{x^4}+\frac {\left (35 A b^3-a \left (-D a^2-3 b C a+15 b^2 B\right )\right ) b}{b x^2+a}\right )dx}{2 a^3 b^2}+\frac {x \left (11 A b^3-a \left (a^2 (-D)-3 a b C+7 b^2 B\right )\right )}{2 a^3 b \left (a+b x^2\right )}}{4 a}+\frac {x \left (\frac {A b^2}{a^2}-\frac {b B}{a}-\frac {a D}{b}+C\right )}{4 a \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {x \left (\frac {A b^2}{a^2}-\frac {b B}{a}-\frac {a D}{b}+C\right )}{4 a \left (a+b x^2\right )^2}+\frac {\frac {\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (35 A b^3-a \left (a^2 (-D)-3 a b C+15 b^2 B\right )\right )}{\sqrt {a}}+\frac {8 b^2 (3 A b-a B)}{x}-\frac {8 a A b^2}{3 x^3}}{2 a^3 b^2}+\frac {x \left (11 A b^3-a \left (a^2 (-D)-3 a b C+7 b^2 B\right )\right )}{2 a^3 b \left (a+b x^2\right )}}{4 a}\)

Input:

Int[(A + B*x^2 + C*x^4 + D*x^6)/(x^4*(a + b*x^2)^3),x]
 

Output:

(((A*b^2)/a^2 - (b*B)/a + C - (a*D)/b)*x)/(4*a*(a + b*x^2)^2) + (((11*A*b^ 
3 - a*(7*b^2*B - 3*a*b*C - a^2*D))*x)/(2*a^3*b*(a + b*x^2)) + ((-8*a*A*b^2 
)/(3*x^3) + (8*b^2*(3*A*b - a*B))/x + (Sqrt[b]*(35*A*b^3 - a*(15*b^2*B - 3 
*a*b*C - a^2*D))*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[a])/(2*a^3*b^2))/(4*a)
 

Defintions of rubi rules used

rule 25

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

rule 1582

Int[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^ 
4)^(p_.), x_Symbol] :> Simp[(-d)^(m/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p*x*((d 
+ e*x^2)^(q + 1)/(2*e^(2*p + m/2)*(q + 1))), x] + Simp[(-d)^(m/2 - 1)/(2*e^ 
(2*p)*(q + 1))   Int[x^m*(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1/(d + e 
*x^2))*(2*(-d)^(-m/2 + 1)*e^(2*p)*(q + 1)*(a + b*x^2 + c*x^4)^p - ((c*d^2 - 
 b*d*e + a*e^2)^p/(e^(m/2)*x^m))*(d + e*(2*q + 3)*x^2))], x], x], x] /; Fre 
eQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && ILtQ[q, -1] 
&& ILtQ[m/2, 0]
 

rule 1584

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( 
c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q* 
(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[ 
b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -2]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2336

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ 
{Q = PolynomialQuotient[(c*x)^m*Pq, a + b*x^2, x], f = Coeff[PolynomialRema 
inder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[(c*x) 
^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b*f*x)*((a + b*x^2)^(p + 1)/(2*a* 
b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   Int[(c*x)^m*(a + b*x^2)^(p + 1)*Ex 
pandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x], x]] /; F 
reeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]
 

Maple [A] (verified)

Time = 0.48 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.87

method	result	size
default	\(\frac {\frac {\left (\frac {11}{8} b^{3} A -\frac {7}{8} a \,b^{2} B +\frac {3}{8} a^{2} b C +\frac {1}{8} a^{3} D\right ) x^{3}+\frac {a \left (13 b^{3} A -9 a \,b^{2} B +5 a^{2} b C -a^{3} D\right ) x}{8 b}}{\left (b \,x^{2}+a \right )^{2}}+\frac {\left (35 b^{3} A -15 a \,b^{2} B +3 a^{2} b C +a^{3} D\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 b \sqrt {a b}}}{a^{4}}-\frac {A}{3 a^{3} x^{3}}-\frac {-3 A b +B a}{a^{4} x}\)	\(153\)

Input:

int((D*x^6+C*x^4+B*x^2+A)/x^4/(b*x^2+a)^3,x,method=_RETURNVERBOSE)
 

Output:

1/a^4*(((11/8*b^3*A-7/8*a*b^2*B+3/8*a^2*b*C+1/8*a^3*D)*x^3+1/8*a*(13*A*b^3 
-9*B*a*b^2+5*C*a^2*b-D*a^3)/b*x)/(b*x^2+a)^2+1/8*(35*A*b^3-15*B*a*b^2+3*C* 
a^2*b+D*a^3)/b/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2)))-1/3*A/a^3/x^3-(-3*A*b+ 
B*a)/a^4/x
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 569, normalized size of antiderivative = 3.23 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^4 \left (a+b x^2\right )^3} \, dx=\left [-\frac {16 \, A a^{4} b^{2} - 6 \, {\left (D a^{4} b^{2} + 3 \, C a^{3} b^{3} - 15 \, B a^{2} b^{4} + 35 \, A a b^{5}\right )} x^{6} + 2 \, {\left (3 \, D a^{5} b - 15 \, C a^{4} b^{2} + 75 \, B a^{3} b^{3} - 175 \, A a^{2} b^{4}\right )} x^{4} + 16 \, {\left (3 \, B a^{4} b^{2} - 7 \, A a^{3} b^{3}\right )} x^{2} + 3 \, {\left ({\left (D a^{3} b^{2} + 3 \, C a^{2} b^{3} - 15 \, B a b^{4} + 35 \, A b^{5}\right )} x^{7} + 2 \, {\left (D a^{4} b + 3 \, C a^{3} b^{2} - 15 \, B a^{2} b^{3} + 35 \, A a b^{4}\right )} x^{5} + {\left (D a^{5} + 3 \, C a^{4} b - 15 \, B a^{3} b^{2} + 35 \, A a^{2} b^{3}\right )} x^{3}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{48 \, {\left (a^{5} b^{4} x^{7} + 2 \, a^{6} b^{3} x^{5} + a^{7} b^{2} x^{3}\right )}}, -\frac {8 \, A a^{4} b^{2} - 3 \, {\left (D a^{4} b^{2} + 3 \, C a^{3} b^{3} - 15 \, B a^{2} b^{4} + 35 \, A a b^{5}\right )} x^{6} + {\left (3 \, D a^{5} b - 15 \, C a^{4} b^{2} + 75 \, B a^{3} b^{3} - 175 \, A a^{2} b^{4}\right )} x^{4} + 8 \, {\left (3 \, B a^{4} b^{2} - 7 \, A a^{3} b^{3}\right )} x^{2} - 3 \, {\left ({\left (D a^{3} b^{2} + 3 \, C a^{2} b^{3} - 15 \, B a b^{4} + 35 \, A b^{5}\right )} x^{7} + 2 \, {\left (D a^{4} b + 3 \, C a^{3} b^{2} - 15 \, B a^{2} b^{3} + 35 \, A a b^{4}\right )} x^{5} + {\left (D a^{5} + 3 \, C a^{4} b - 15 \, B a^{3} b^{2} + 35 \, A a^{2} b^{3}\right )} x^{3}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{24 \, {\left (a^{5} b^{4} x^{7} + 2 \, a^{6} b^{3} x^{5} + a^{7} b^{2} x^{3}\right )}}\right ] \] Input:

integrate((D*x^6+C*x^4+B*x^2+A)/x^4/(b*x^2+a)^3,x, algorithm="fricas")
 

Output:

[-1/48*(16*A*a^4*b^2 - 6*(D*a^4*b^2 + 3*C*a^3*b^3 - 15*B*a^2*b^4 + 35*A*a* 
b^5)*x^6 + 2*(3*D*a^5*b - 15*C*a^4*b^2 + 75*B*a^3*b^3 - 175*A*a^2*b^4)*x^4 
 + 16*(3*B*a^4*b^2 - 7*A*a^3*b^3)*x^2 + 3*((D*a^3*b^2 + 3*C*a^2*b^3 - 15*B 
*a*b^4 + 35*A*b^5)*x^7 + 2*(D*a^4*b + 3*C*a^3*b^2 - 15*B*a^2*b^3 + 35*A*a* 
b^4)*x^5 + (D*a^5 + 3*C*a^4*b - 15*B*a^3*b^2 + 35*A*a^2*b^3)*x^3)*sqrt(-a* 
b)*log((b*x^2 - 2*sqrt(-a*b)*x - a)/(b*x^2 + a)))/(a^5*b^4*x^7 + 2*a^6*b^3 
*x^5 + a^7*b^2*x^3), -1/24*(8*A*a^4*b^2 - 3*(D*a^4*b^2 + 3*C*a^3*b^3 - 15* 
B*a^2*b^4 + 35*A*a*b^5)*x^6 + (3*D*a^5*b - 15*C*a^4*b^2 + 75*B*a^3*b^3 - 1 
75*A*a^2*b^4)*x^4 + 8*(3*B*a^4*b^2 - 7*A*a^3*b^3)*x^2 - 3*((D*a^3*b^2 + 3* 
C*a^2*b^3 - 15*B*a*b^4 + 35*A*b^5)*x^7 + 2*(D*a^4*b + 3*C*a^3*b^2 - 15*B*a 
^2*b^3 + 35*A*a*b^4)*x^5 + (D*a^5 + 3*C*a^4*b - 15*B*a^3*b^2 + 35*A*a^2*b^ 
3)*x^3)*sqrt(a*b)*arctan(sqrt(a*b)*x/a))/(a^5*b^4*x^7 + 2*a^6*b^3*x^5 + a^ 
7*b^2*x^3)]
 

Sympy [A] (verification not implemented)

Time = 22.48 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.53 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^4 \left (a+b x^2\right )^3} \, dx=- \frac {\sqrt {- \frac {1}{a^{9} b^{3}}} \cdot \left (35 A b^{3} - 15 B a b^{2} + 3 C a^{2} b + D a^{3}\right ) \log {\left (- a^{5} b \sqrt {- \frac {1}{a^{9} b^{3}}} + x \right )}}{16} + \frac {\sqrt {- \frac {1}{a^{9} b^{3}}} \cdot \left (35 A b^{3} - 15 B a b^{2} + 3 C a^{2} b + D a^{3}\right ) \log {\left (a^{5} b \sqrt {- \frac {1}{a^{9} b^{3}}} + x \right )}}{16} + \frac {- 8 A a^{3} b + x^{6} \cdot \left (105 A b^{4} - 45 B a b^{3} + 9 C a^{2} b^{2} + 3 D a^{3} b\right ) + x^{4} \cdot \left (175 A a b^{3} - 75 B a^{2} b^{2} + 15 C a^{3} b - 3 D a^{4}\right ) + x^{2} \cdot \left (56 A a^{2} b^{2} - 24 B a^{3} b\right )}{24 a^{6} b x^{3} + 48 a^{5} b^{2} x^{5} + 24 a^{4} b^{3} x^{7}} \] Input:

integrate((D*x**6+C*x**4+B*x**2+A)/x**4/(b*x**2+a)**3,x)
 

Output:

-sqrt(-1/(a**9*b**3))*(35*A*b**3 - 15*B*a*b**2 + 3*C*a**2*b + D*a**3)*log( 
-a**5*b*sqrt(-1/(a**9*b**3)) + x)/16 + sqrt(-1/(a**9*b**3))*(35*A*b**3 - 1 
5*B*a*b**2 + 3*C*a**2*b + D*a**3)*log(a**5*b*sqrt(-1/(a**9*b**3)) + x)/16 
+ (-8*A*a**3*b + x**6*(105*A*b**4 - 45*B*a*b**3 + 9*C*a**2*b**2 + 3*D*a**3 
*b) + x**4*(175*A*a*b**3 - 75*B*a**2*b**2 + 15*C*a**3*b - 3*D*a**4) + x**2 
*(56*A*a**2*b**2 - 24*B*a**3*b))/(24*a**6*b*x**3 + 48*a**5*b**2*x**5 + 24* 
a**4*b**3*x**7)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.03 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^4 \left (a+b x^2\right )^3} \, dx=\frac {3 \, {\left (D a^{3} b + 3 \, C a^{2} b^{2} - 15 \, B a b^{3} + 35 \, A b^{4}\right )} x^{6} - 8 \, A a^{3} b - {\left (3 \, D a^{4} - 15 \, C a^{3} b + 75 \, B a^{2} b^{2} - 175 \, A a b^{3}\right )} x^{4} - 8 \, {\left (3 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{2}}{24 \, {\left (a^{4} b^{3} x^{7} + 2 \, a^{5} b^{2} x^{5} + a^{6} b x^{3}\right )}} + \frac {{\left (D a^{3} + 3 \, C a^{2} b - 15 \, B a b^{2} + 35 \, A b^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{4} b} \] Input:

integrate((D*x^6+C*x^4+B*x^2+A)/x^4/(b*x^2+a)^3,x, algorithm="maxima")
 

Output:

1/24*(3*(D*a^3*b + 3*C*a^2*b^2 - 15*B*a*b^3 + 35*A*b^4)*x^6 - 8*A*a^3*b - 
(3*D*a^4 - 15*C*a^3*b + 75*B*a^2*b^2 - 175*A*a*b^3)*x^4 - 8*(3*B*a^3*b - 7 
*A*a^2*b^2)*x^2)/(a^4*b^3*x^7 + 2*a^5*b^2*x^5 + a^6*b*x^3) + 1/8*(D*a^3 + 
3*C*a^2*b - 15*B*a*b^2 + 35*A*b^3)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^4*b)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.94 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^4 \left (a+b x^2\right )^3} \, dx=\frac {{\left (D a^{3} + 3 \, C a^{2} b - 15 \, B a b^{2} + 35 \, A b^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{4} b} + \frac {D a^{3} b x^{3} + 3 \, C a^{2} b^{2} x^{3} - 7 \, B a b^{3} x^{3} + 11 \, A b^{4} x^{3} - D a^{4} x + 5 \, C a^{3} b x - 9 \, B a^{2} b^{2} x + 13 \, A a b^{3} x}{8 \, {\left (b x^{2} + a\right )}^{2} a^{4} b} - \frac {3 \, B a x^{2} - 9 \, A b x^{2} + A a}{3 \, a^{4} x^{3}} \] Input:

integrate((D*x^6+C*x^4+B*x^2+A)/x^4/(b*x^2+a)^3,x, algorithm="giac")
 

Output:

1/8*(D*a^3 + 3*C*a^2*b - 15*B*a*b^2 + 35*A*b^3)*arctan(b*x/sqrt(a*b))/(sqr 
t(a*b)*a^4*b) + 1/8*(D*a^3*b*x^3 + 3*C*a^2*b^2*x^3 - 7*B*a*b^3*x^3 + 11*A* 
b^4*x^3 - D*a^4*x + 5*C*a^3*b*x - 9*B*a^2*b^2*x + 13*A*a*b^3*x)/((b*x^2 + 
a)^2*a^4*b) - 1/3*(3*B*a*x^2 - 9*A*b*x^2 + A*a)/(a^4*x^3)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x^2+C x^4+D x^6}{x^4 \left (a+b x^2\right )^3} \, dx=\int \frac {A+B\,x^2+C\,x^4+x^6\,D}{x^4\,{\left (b\,x^2+a\right )}^3} \,d x \] Input:

int((A + B*x^2 + C*x^4 + x^6*D)/(x^4*(a + b*x^2)^3),x)
 

Output:

int((A + B*x^2 + C*x^4 + x^6*D)/(x^4*(a + b*x^2)^3), x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 352, normalized size of antiderivative = 2.00 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^4 \left (a+b x^2\right )^3} \, dx=\frac {3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{4} d \,x^{3}+9 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{3} b c \,x^{3}+6 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{3} b d \,x^{5}+60 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b^{3} x^{3}+18 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b^{2} c \,x^{5}+3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b^{2} d \,x^{7}+120 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{4} x^{5}+9 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{3} c \,x^{7}+60 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b^{5} x^{7}-8 a^{4} b^{2}-3 a^{4} b d \,x^{4}+32 a^{3} b^{3} x^{2}+15 a^{3} b^{2} c \,x^{4}+3 a^{3} b^{2} d \,x^{6}+100 a^{2} b^{4} x^{4}+9 a^{2} b^{3} c \,x^{6}+60 a \,b^{5} x^{6}}{24 a^{4} b^{2} x^{3} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )} \] Input:

int((D*x^6+C*x^4+B*x^2+A)/x^4/(b*x^2+a)^3,x)
 

Output:

(3*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**4*d*x**3 + 9*sqrt(b)*s 
qrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3*b*c*x**3 + 6*sqrt(b)*sqrt(a)*ata 
n((b*x)/(sqrt(b)*sqrt(a)))*a**3*b*d*x**5 + 60*sqrt(b)*sqrt(a)*atan((b*x)/( 
sqrt(b)*sqrt(a)))*a**2*b**3*x**3 + 18*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)* 
sqrt(a)))*a**2*b**2*c*x**5 + 3*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a) 
))*a**2*b**2*d*x**7 + 120*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a* 
b**4*x**5 + 9*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b**3*c*x**7 
+ 60*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**5*x**7 - 8*a**4*b**2 
 - 3*a**4*b*d*x**4 + 32*a**3*b**3*x**2 + 15*a**3*b**2*c*x**4 + 3*a**3*b**2 
*d*x**6 + 100*a**2*b**4*x**4 + 9*a**2*b**3*c*x**6 + 60*a*b**5*x**6)/(24*a* 
*4*b**2*x**3*(a**2 + 2*a*b*x**2 + b**2*x**4))