\(\int \frac {x^6 (A+B x^2+C x^4+D x^6)}{(a+b x^2)^3} \, dx\) [180]

Optimal result

Integrand size = 30, antiderivative size = 235 \[ \int \frac {x^6 \left (A+B x^2+C x^4+D x^6\right )}{\left (a+b x^2\right )^3} \, dx=\frac {\left (A b^3-a \left (3 b^2 B-6 a b C+10 a^2 D\right )\right ) x}{b^6}+\frac {\left (b^2 B-3 a b C+6 a^2 D\right ) x^3}{3 b^5}+\frac {(b C-3 a D) x^5}{5 b^4}+\frac {D x^7}{7 b^3}-\frac {a^2 \left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) x}{4 b^6 \left (a+b x^2\right )^2}+\frac {a \left (9 A b^3-a \left (13 b^2 B-17 a b C+21 a^2 D\right )\right ) x}{8 b^6 \left (a+b x^2\right )}-\frac {\sqrt {a} \left (15 A b^3-a \left (35 b^2 B-63 a b C+99 a^2 D\right )\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 b^{13/2}} \] Output:

(A*b^3-a*(3*B*b^2-6*C*a*b+10*D*a^2))*x/b^6+1/3*(B*b^2-3*C*a*b+6*D*a^2)*x^3 
/b^5+1/5*(C*b-3*D*a)*x^5/b^4+1/7*D*x^7/b^3-1/4*a^2*(A*b^3-a*(B*b^2-C*a*b+D 
*a^2))*x/b^6/(b*x^2+a)^2+1/8*a*(9*A*b^3-a*(13*B*b^2-17*C*a*b+21*D*a^2))*x/ 
b^6/(b*x^2+a)-1/8*a^(1/2)*(15*A*b^3-a*(35*B*b^2-63*C*a*b+99*D*a^2))*arctan 
(b^(1/2)*x/a^(1/2))/b^(13/2)
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.99 \[ \int \frac {x^6 \left (A+B x^2+C x^4+D x^6\right )}{\left (a+b x^2\right )^3} \, dx=\frac {\left (A b^3+a \left (-3 b^2 B+6 a b C-10 a^2 D\right )\right ) x}{b^6}+\frac {\left (b^2 B-3 a b C+6 a^2 D\right ) x^3}{3 b^5}+\frac {(b C-3 a D) x^5}{5 b^4}+\frac {D x^7}{7 b^3}+\frac {a^2 \left (-A b^3+a \left (b^2 B-a b C+a^2 D\right )\right ) x}{4 b^6 \left (a+b x^2\right )^2}+\frac {a \left (9 A b^3+a \left (-13 b^2 B+17 a b C-21 a^2 D\right )\right ) x}{8 b^6 \left (a+b x^2\right )}+\frac {\sqrt {a} \left (-15 A b^3+a \left (35 b^2 B-63 a b C+99 a^2 D\right )\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 b^{13/2}} \] Input:

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

Output:

((A*b^3 + a*(-3*b^2*B + 6*a*b*C - 10*a^2*D))*x)/b^6 + ((b^2*B - 3*a*b*C + 
6*a^2*D)*x^3)/(3*b^5) + ((b*C - 3*a*D)*x^5)/(5*b^4) + (D*x^7)/(7*b^3) + (a 
^2*(-(A*b^3) + a*(b^2*B - a*b*C + a^2*D))*x)/(4*b^6*(a + b*x^2)^2) + (a*(9 
*A*b^3 + a*(-13*b^2*B + 17*a*b*C - 21*a^2*D))*x)/(8*b^6*(a + b*x^2)) + (Sq 
rt[a]*(-15*A*b^3 + a*(35*b^2*B - 63*a*b*C + 99*a^2*D))*ArcTan[(Sqrt[b]*x)/ 
Sqrt[a]])/(8*b^(13/2))
 

Rubi [A] (verified)

Time = 0.91 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.11, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2335, 9, 1580, 25, 2341, 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.

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

\(\Big \downarrow \) 2335

\(\displaystyle \frac {x^7 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{4 a \left (a+b x^2\right )^2}-\frac {\int \frac {x^5 \left (-4 a D x^5-4 a \left (C-\frac {a D}{b}\right ) x^3+\left (3 A b-\frac {7 a \left (D a^2-b C a+b^2 B\right )}{b^2}\right ) x\right )}{\left (b x^2+a\right )^2}dx}{4 a b}\)

\(\Big \downarrow \) 9

\(\displaystyle \frac {x^7 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{4 a \left (a+b x^2\right )^2}-\frac {\int \frac {x^6 \left (-4 a D x^4-4 a \left (C-\frac {a D}{b}\right ) x^2+3 A b-\frac {7 a \left (D a^2-b C a+b^2 B\right )}{b^2}\right )}{\left (b x^2+a\right )^2}dx}{4 a b}\)

\(\Big \downarrow \) 1580

\(\displaystyle \frac {x^7 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{4 a \left (a+b x^2\right )^2}-\frac {-\frac {\int -\frac {-8 a b^4 D x^8-8 a b^3 (b C-2 a D) x^6+2 b^2 \left (3 A b^3-a \left (15 D a^2-11 b C a+7 b^2 B\right )\right ) x^4-2 a b \left (3 A b^3-a \left (15 D a^2-11 b C a+7 b^2 B\right )\right ) x^2+a^2 \left (3 A b^3-a \left (15 D a^2-11 b C a+7 b^2 B\right )\right )}{b x^2+a}dx}{2 b^5}-\frac {a^2 x \left (3 A b^3-a \left (15 a^2 D-11 a b C+7 b^2 B\right )\right )}{2 b^5 \left (a+b x^2\right )}}{4 a b}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {x^7 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{4 a \left (a+b x^2\right )^2}-\frac {\frac {\int \frac {-8 a b^4 D x^8-8 a b^3 (b C-2 a D) x^6+2 b^2 \left (3 A b^3-a \left (15 D a^2-11 b C a+7 b^2 B\right )\right ) x^4-2 a b \left (3 A b^3-a \left (15 D a^2-11 b C a+7 b^2 B\right )\right ) x^2+a^2 \left (3 A b^3-a \left (15 D a^2-11 b C a+7 b^2 B\right )\right )}{b x^2+a}dx}{2 b^5}-\frac {a^2 x \left (3 A b^3-a \left (15 a^2 D-11 a b C+7 b^2 B\right )\right )}{2 b^5 \left (a+b x^2\right )}}{4 a b}\)

\(\Big \downarrow \) 2341

\(\displaystyle \frac {x^7 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{4 a \left (a+b x^2\right )^2}-\frac {\frac {\int \left (-8 a b^3 D x^6-8 a b^2 (b C-3 a D) x^4+2 b \left (3 A b^3-a \left (27 D a^2-15 b C a+7 b^2 B\right )\right ) x^2-4 a \left (3 A b^3-a \left (21 D a^2-13 b C a+7 b^2 B\right )\right )+\frac {-99 D a^5+63 b C a^4-35 b^2 B a^3+15 A b^3 a^2}{b x^2+a}\right )dx}{2 b^5}-\frac {a^2 x \left (3 A b^3-a \left (15 a^2 D-11 a b C+7 b^2 B\right )\right )}{2 b^5 \left (a+b x^2\right )}}{4 a b}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {x^7 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{4 a \left (a+b x^2\right )^2}-\frac {\frac {\frac {2}{3} b x^3 \left (3 A b^3-a \left (27 a^2 D-15 a b C+7 b^2 B\right )\right )-4 a x \left (3 A b^3-a \left (21 a^2 D-13 a b C+7 b^2 B\right )\right )+\frac {a^{3/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (15 A b^3-a \left (99 a^2 D-63 a b C+35 b^2 B\right )\right )}{\sqrt {b}}-\frac {8}{7} a b^3 D x^7-\frac {8}{5} a b^2 x^5 (b C-3 a D)}{2 b^5}-\frac {a^2 x \left (3 A b^3-a \left (15 a^2 D-11 a b C+7 b^2 B\right )\right )}{2 b^5 \left (a+b x^2\right )}}{4 a b}\)

Input:

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

Output:

((A - (a*(b^2*B - a*b*C + a^2*D))/b^3)*x^7)/(4*a*(a + b*x^2)^2) - (-1/2*(a 
^2*(3*A*b^3 - a*(7*b^2*B - 11*a*b*C + 15*a^2*D))*x)/(b^5*(a + b*x^2)) + (- 
4*a*(3*A*b^3 - a*(7*b^2*B - 13*a*b*C + 21*a^2*D))*x + (2*b*(3*A*b^3 - a*(7 
*b^2*B - 15*a*b*C + 27*a^2*D))*x^3)/3 - (8*a*b^2*(b*C - 3*a*D)*x^5)/5 - (8 
*a*b^3*D*x^7)/7 + (a^(3/2)*(15*A*b^3 - a*(35*b^2*B - 63*a*b*C + 99*a^2*D)) 
*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[b])/(2*b^5))/(4*a*b)
 

Defintions of rubi rules used

rule 9

Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, 
x, Min]}, Simp[1/e^(p*r)   Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, 
x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && 
  !MonomialQ[Px, x]
 

rule 25

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

rule 1580

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[1/(2*e^(2*p + m/2)* 
(q + 1))   Int[(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1/(d + e*x^2))*(2* 
e^(2*p + m/2)*(q + 1)*x^m*(a + b*x^2 + c*x^4)^p - (-d)^(m/2 - 1)*(c*d^2 - b 
*d*e + a*e^2)^p*(d + e*(2*q + 3)*x^2))], x], x], x] /; FreeQ[{a, b, c, d, e 
}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && ILtQ[q, -1] && IGtQ[m/2, 0]
 

rule 2009

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

rule 2335

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

rule 2341

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq* 
(a + b*x^2)^p, x], x] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
 

Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.93

method	result	size
default	\(\frac {\frac {1}{7} b^{3} D x^{7}+\frac {1}{5} b^{3} C \,x^{5}-\frac {3}{5} D a \,b^{2} x^{5}+\frac {1}{3} b^{3} B \,x^{3}-C a \,b^{2} x^{3}+2 D x^{3} b \,a^{2}+A \,b^{3} x -3 B a \,b^{2} x +6 C \,a^{2} b x -10 D a^{3} x}{b^{6}}-\frac {a \left (\frac {\left (-\frac {9}{8} A \,b^{4}+\frac {13}{8} B a \,b^{3}-\frac {17}{8} C \,a^{2} b^{2}+\frac {21}{8} a^{3} D b \right ) x^{3}-\frac {a \left (7 b^{3} A -11 a \,b^{2} B +15 a^{2} b C -19 a^{3} D\right ) x}{8}}{\left (b \,x^{2}+a \right )^{2}}+\frac {\left (15 b^{3} A -35 a \,b^{2} B +63 a^{2} b C -99 a^{3} D\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{b^{6}}\)	\(219\)

Input:

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

Output:

1/b^6*(1/7*b^3*D*x^7+1/5*b^3*C*x^5-3/5*D*a*b^2*x^5+1/3*b^3*B*x^3-C*a*b^2*x 
^3+2*D*x^3*b*a^2+A*b^3*x-3*B*a*b^2*x+6*C*a^2*b*x-10*D*a^3*x)-a/b^6*(((-9/8 
*A*b^4+13/8*B*a*b^3-17/8*C*a^2*b^2+21/8*a^3*D*b)*x^3-1/8*a*(7*A*b^3-11*B*a 
*b^2+15*C*a^2*b-19*D*a^3)*x)/(b*x^2+a)^2+1/8*(15*A*b^3-35*B*a*b^2+63*C*a^2 
*b-99*D*a^3)/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 668, normalized size of antiderivative = 2.84 \[ \int \frac {x^6 \left (A+B x^2+C x^4+D x^6\right )}{\left (a+b x^2\right )^3} \, dx=\left [\frac {240 \, D b^{5} x^{11} - 48 \, {\left (11 \, D a b^{4} - 7 \, C b^{5}\right )} x^{9} + 16 \, {\left (99 \, D a^{2} b^{3} - 63 \, C a b^{4} + 35 \, B b^{5}\right )} x^{7} - 112 \, {\left (99 \, D a^{3} b^{2} - 63 \, C a^{2} b^{3} + 35 \, B a b^{4} - 15 \, A b^{5}\right )} x^{5} - 350 \, {\left (99 \, D a^{4} b - 63 \, C a^{3} b^{2} + 35 \, B a^{2} b^{3} - 15 \, A a b^{4}\right )} x^{3} - 105 \, {\left (99 \, D a^{5} - 63 \, C a^{4} b + 35 \, B a^{3} b^{2} - 15 \, A a^{2} b^{3} + {\left (99 \, D a^{3} b^{2} - 63 \, C a^{2} b^{3} + 35 \, B a b^{4} - 15 \, A b^{5}\right )} x^{4} + 2 \, {\left (99 \, D a^{4} b - 63 \, C a^{3} b^{2} + 35 \, B a^{2} b^{3} - 15 \, A a b^{4}\right )} x^{2}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} - 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) - 210 \, {\left (99 \, D a^{5} - 63 \, C a^{4} b + 35 \, B a^{3} b^{2} - 15 \, A a^{2} b^{3}\right )} x}{1680 \, {\left (b^{8} x^{4} + 2 \, a b^{7} x^{2} + a^{2} b^{6}\right )}}, \frac {120 \, D b^{5} x^{11} - 24 \, {\left (11 \, D a b^{4} - 7 \, C b^{5}\right )} x^{9} + 8 \, {\left (99 \, D a^{2} b^{3} - 63 \, C a b^{4} + 35 \, B b^{5}\right )} x^{7} - 56 \, {\left (99 \, D a^{3} b^{2} - 63 \, C a^{2} b^{3} + 35 \, B a b^{4} - 15 \, A b^{5}\right )} x^{5} - 175 \, {\left (99 \, D a^{4} b - 63 \, C a^{3} b^{2} + 35 \, B a^{2} b^{3} - 15 \, A a b^{4}\right )} x^{3} + 105 \, {\left (99 \, D a^{5} - 63 \, C a^{4} b + 35 \, B a^{3} b^{2} - 15 \, A a^{2} b^{3} + {\left (99 \, D a^{3} b^{2} - 63 \, C a^{2} b^{3} + 35 \, B a b^{4} - 15 \, A b^{5}\right )} x^{4} + 2 \, {\left (99 \, D a^{4} b - 63 \, C a^{3} b^{2} + 35 \, B a^{2} b^{3} - 15 \, A a b^{4}\right )} x^{2}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) - 105 \, {\left (99 \, D a^{5} - 63 \, C a^{4} b + 35 \, B a^{3} b^{2} - 15 \, A a^{2} b^{3}\right )} x}{840 \, {\left (b^{8} x^{4} + 2 \, a b^{7} x^{2} + a^{2} b^{6}\right )}}\right ] \] Input:

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

Output:

[1/1680*(240*D*b^5*x^11 - 48*(11*D*a*b^4 - 7*C*b^5)*x^9 + 16*(99*D*a^2*b^3 
 - 63*C*a*b^4 + 35*B*b^5)*x^7 - 112*(99*D*a^3*b^2 - 63*C*a^2*b^3 + 35*B*a* 
b^4 - 15*A*b^5)*x^5 - 350*(99*D*a^4*b - 63*C*a^3*b^2 + 35*B*a^2*b^3 - 15*A 
*a*b^4)*x^3 - 105*(99*D*a^5 - 63*C*a^4*b + 35*B*a^3*b^2 - 15*A*a^2*b^3 + ( 
99*D*a^3*b^2 - 63*C*a^2*b^3 + 35*B*a*b^4 - 15*A*b^5)*x^4 + 2*(99*D*a^4*b - 
 63*C*a^3*b^2 + 35*B*a^2*b^3 - 15*A*a*b^4)*x^2)*sqrt(-a/b)*log((b*x^2 - 2* 
b*x*sqrt(-a/b) - a)/(b*x^2 + a)) - 210*(99*D*a^5 - 63*C*a^4*b + 35*B*a^3*b 
^2 - 15*A*a^2*b^3)*x)/(b^8*x^4 + 2*a*b^7*x^2 + a^2*b^6), 1/840*(120*D*b^5* 
x^11 - 24*(11*D*a*b^4 - 7*C*b^5)*x^9 + 8*(99*D*a^2*b^3 - 63*C*a*b^4 + 35*B 
*b^5)*x^7 - 56*(99*D*a^3*b^2 - 63*C*a^2*b^3 + 35*B*a*b^4 - 15*A*b^5)*x^5 - 
 175*(99*D*a^4*b - 63*C*a^3*b^2 + 35*B*a^2*b^3 - 15*A*a*b^4)*x^3 + 105*(99 
*D*a^5 - 63*C*a^4*b + 35*B*a^3*b^2 - 15*A*a^2*b^3 + (99*D*a^3*b^2 - 63*C*a 
^2*b^3 + 35*B*a*b^4 - 15*A*b^5)*x^4 + 2*(99*D*a^4*b - 63*C*a^3*b^2 + 35*B* 
a^2*b^3 - 15*A*a*b^4)*x^2)*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a) - 105*(99*D*a 
^5 - 63*C*a^4*b + 35*B*a^3*b^2 - 15*A*a^2*b^3)*x)/(b^8*x^4 + 2*a*b^7*x^2 + 
 a^2*b^6)]
 

Sympy [A] (verification not implemented)

Time = 6.97 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.34 \[ \int \frac {x^6 \left (A+B x^2+C x^4+D x^6\right )}{\left (a+b x^2\right )^3} \, dx=\frac {D x^{7}}{7 b^{3}} + x^{5} \left (\frac {C}{5 b^{3}} - \frac {3 D a}{5 b^{4}}\right ) + x^{3} \left (\frac {B}{3 b^{3}} - \frac {C a}{b^{4}} + \frac {2 D a^{2}}{b^{5}}\right ) + x \left (\frac {A}{b^{3}} - \frac {3 B a}{b^{4}} + \frac {6 C a^{2}}{b^{5}} - \frac {10 D a^{3}}{b^{6}}\right ) - \frac {\sqrt {- \frac {a}{b^{13}}} \left (- 15 A b^{3} + 35 B a b^{2} - 63 C a^{2} b + 99 D a^{3}\right ) \log {\left (- b^{6} \sqrt {- \frac {a}{b^{13}}} + x \right )}}{16} + \frac {\sqrt {- \frac {a}{b^{13}}} \left (- 15 A b^{3} + 35 B a b^{2} - 63 C a^{2} b + 99 D a^{3}\right ) \log {\left (b^{6} \sqrt {- \frac {a}{b^{13}}} + x \right )}}{16} + \frac {x^{3} \cdot \left (9 A a b^{4} - 13 B a^{2} b^{3} + 17 C a^{3} b^{2} - 21 D a^{4} b\right ) + x \left (7 A a^{2} b^{3} - 11 B a^{3} b^{2} + 15 C a^{4} b - 19 D a^{5}\right )}{8 a^{2} b^{6} + 16 a b^{7} x^{2} + 8 b^{8} x^{4}} \] Input:

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

Output:

D*x**7/(7*b**3) + x**5*(C/(5*b**3) - 3*D*a/(5*b**4)) + x**3*(B/(3*b**3) - 
C*a/b**4 + 2*D*a**2/b**5) + x*(A/b**3 - 3*B*a/b**4 + 6*C*a**2/b**5 - 10*D* 
a**3/b**6) - sqrt(-a/b**13)*(-15*A*b**3 + 35*B*a*b**2 - 63*C*a**2*b + 99*D 
*a**3)*log(-b**6*sqrt(-a/b**13) + x)/16 + sqrt(-a/b**13)*(-15*A*b**3 + 35* 
B*a*b**2 - 63*C*a**2*b + 99*D*a**3)*log(b**6*sqrt(-a/b**13) + x)/16 + (x** 
3*(9*A*a*b**4 - 13*B*a**2*b**3 + 17*C*a**3*b**2 - 21*D*a**4*b) + x*(7*A*a* 
*2*b**3 - 11*B*a**3*b**2 + 15*C*a**4*b - 19*D*a**5))/(8*a**2*b**6 + 16*a*b 
**7*x**2 + 8*b**8*x**4)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.02 \[ \int \frac {x^6 \left (A+B x^2+C x^4+D x^6\right )}{\left (a+b x^2\right )^3} \, dx=-\frac {{\left (21 \, D a^{4} b - 17 \, C a^{3} b^{2} + 13 \, B a^{2} b^{3} - 9 \, A a b^{4}\right )} x^{3} + {\left (19 \, D a^{5} - 15 \, C a^{4} b + 11 \, B a^{3} b^{2} - 7 \, A a^{2} b^{3}\right )} x}{8 \, {\left (b^{8} x^{4} + 2 \, a b^{7} x^{2} + a^{2} b^{6}\right )}} + \frac {{\left (99 \, D a^{4} - 63 \, C a^{3} b + 35 \, B a^{2} b^{2} - 15 \, A a b^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} b^{6}} + \frac {15 \, D b^{3} x^{7} - 21 \, {\left (3 \, D a b^{2} - C b^{3}\right )} x^{5} + 35 \, {\left (6 \, D a^{2} b - 3 \, C a b^{2} + B b^{3}\right )} x^{3} - 105 \, {\left (10 \, D a^{3} - 6 \, C a^{2} b + 3 \, B a b^{2} - A b^{3}\right )} x}{105 \, b^{6}} \] Input:

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

Output:

-1/8*((21*D*a^4*b - 17*C*a^3*b^2 + 13*B*a^2*b^3 - 9*A*a*b^4)*x^3 + (19*D*a 
^5 - 15*C*a^4*b + 11*B*a^3*b^2 - 7*A*a^2*b^3)*x)/(b^8*x^4 + 2*a*b^7*x^2 + 
a^2*b^6) + 1/8*(99*D*a^4 - 63*C*a^3*b + 35*B*a^2*b^2 - 15*A*a*b^3)*arctan( 
b*x/sqrt(a*b))/(sqrt(a*b)*b^6) + 1/105*(15*D*b^3*x^7 - 21*(3*D*a*b^2 - C*b 
^3)*x^5 + 35*(6*D*a^2*b - 3*C*a*b^2 + B*b^3)*x^3 - 105*(10*D*a^3 - 6*C*a^2 
*b + 3*B*a*b^2 - A*b^3)*x)/b^6
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.04 \[ \int \frac {x^6 \left (A+B x^2+C x^4+D x^6\right )}{\left (a+b x^2\right )^3} \, dx=\frac {{\left (99 \, D a^{4} - 63 \, C a^{3} b + 35 \, B a^{2} b^{2} - 15 \, A a b^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} b^{6}} - \frac {21 \, D a^{4} b x^{3} - 17 \, C a^{3} b^{2} x^{3} + 13 \, B a^{2} b^{3} x^{3} - 9 \, A a b^{4} x^{3} + 19 \, D a^{5} x - 15 \, C a^{4} b x + 11 \, B a^{3} b^{2} x - 7 \, A a^{2} b^{3} x}{8 \, {\left (b x^{2} + a\right )}^{2} b^{6}} + \frac {15 \, D b^{18} x^{7} - 63 \, D a b^{17} x^{5} + 21 \, C b^{18} x^{5} + 210 \, D a^{2} b^{16} x^{3} - 105 \, C a b^{17} x^{3} + 35 \, B b^{18} x^{3} - 1050 \, D a^{3} b^{15} x + 630 \, C a^{2} b^{16} x - 315 \, B a b^{17} x + 105 \, A b^{18} x}{105 \, b^{21}} \] Input:

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

Output:

1/8*(99*D*a^4 - 63*C*a^3*b + 35*B*a^2*b^2 - 15*A*a*b^3)*arctan(b*x/sqrt(a* 
b))/(sqrt(a*b)*b^6) - 1/8*(21*D*a^4*b*x^3 - 17*C*a^3*b^2*x^3 + 13*B*a^2*b^ 
3*x^3 - 9*A*a*b^4*x^3 + 19*D*a^5*x - 15*C*a^4*b*x + 11*B*a^3*b^2*x - 7*A*a 
^2*b^3*x)/((b*x^2 + a)^2*b^6) + 1/105*(15*D*b^18*x^7 - 63*D*a*b^17*x^5 + 2 
1*C*b^18*x^5 + 210*D*a^2*b^16*x^3 - 105*C*a*b^17*x^3 + 35*B*b^18*x^3 - 105 
0*D*a^3*b^15*x + 630*C*a^2*b^16*x - 315*B*a*b^17*x + 105*A*b^18*x)/b^21
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.74 \[ \int \frac {x^6 \left (A+B x^2+C x^4+D x^6\right )}{\left (a+b x^2\right )^3} \, dx=\frac {10395 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{5} d -6615 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{4} b c +20790 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{4} b d \,x^{2}+2100 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{3} b^{3}-13230 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{3} b^{2} c \,x^{2}+10395 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{3} b^{2} d \,x^{4}+4200 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b^{4} x^{2}-6615 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b^{3} c \,x^{4}+2100 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{5} x^{4}-10395 a^{5} b d x +6615 a^{4} b^{2} c x -17325 a^{4} b^{2} d \,x^{3}-2100 a^{3} b^{4} x +11025 a^{3} b^{3} c \,x^{3}-5544 a^{3} b^{3} d \,x^{5}-3500 a^{2} b^{5} x^{3}+3528 a^{2} b^{4} c \,x^{5}+792 a^{2} b^{4} d \,x^{7}-1120 a \,b^{6} x^{5}-504 a \,b^{5} c \,x^{7}-264 a \,b^{5} d \,x^{9}+280 b^{7} x^{7}+168 b^{6} c \,x^{9}+120 b^{6} d \,x^{11}}{840 b^{7} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )} \] Input:

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

Output:

(10395*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**5*d - 6615*sqrt(b) 
*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**4*b*c + 20790*sqrt(b)*sqrt(a)*at 
an((b*x)/(sqrt(b)*sqrt(a)))*a**4*b*d*x**2 + 2100*sqrt(b)*sqrt(a)*atan((b*x 
)/(sqrt(b)*sqrt(a)))*a**3*b**3 - 13230*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b) 
*sqrt(a)))*a**3*b**2*c*x**2 + 10395*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sq 
rt(a)))*a**3*b**2*d*x**4 + 4200*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a 
)))*a**2*b**4*x**2 - 6615*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a* 
*2*b**3*c*x**4 + 2100*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b**5 
*x**4 - 10395*a**5*b*d*x + 6615*a**4*b**2*c*x - 17325*a**4*b**2*d*x**3 - 2 
100*a**3*b**4*x + 11025*a**3*b**3*c*x**3 - 5544*a**3*b**3*d*x**5 - 3500*a* 
*2*b**5*x**3 + 3528*a**2*b**4*c*x**5 + 792*a**2*b**4*d*x**7 - 1120*a*b**6* 
x**5 - 504*a*b**5*c*x**7 - 264*a*b**5*d*x**9 + 280*b**7*x**7 + 168*b**6*c* 
x**9 + 120*b**6*d*x**11)/(840*b**7*(a**2 + 2*a*b*x**2 + b**2*x**4))