\(\int \frac {A+B x^2+C x^4+D x^6}{(a+b x^2)^{9/2}} \, dx\) [139]

Optimal result

Integrand size = 29, antiderivative size = 190 \[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {\left (\frac {A}{a}-\frac {b^2 B-a b C+a^2 D}{b^3}\right ) x}{7 \left (a+b x^2\right )^{7/2}}+\frac {\left (6 A b^3+a \left (b^2 B-8 a b C+15 a^2 D\right )\right ) x}{35 a^2 b^3 \left (a+b x^2\right )^{5/2}}+\frac {\left (24 A b^3+a \left (4 b^2 B+3 a b C-45 a^2 D\right )\right ) x}{105 a^3 b^3 \left (a+b x^2\right )^{3/2}}+\frac {\left (48 A b^3+a \left (8 b^2 B+6 a b C+15 a^2 D\right )\right ) x}{105 a^4 b^3 \sqrt {a+b x^2}} \] Output:

1/7*(A/a-(B*b^2-C*a*b+D*a^2)/b^3)*x/(b*x^2+a)^(7/2)+1/35*(6*A*b^3+a*(B*b^2 
-8*C*a*b+15*D*a^2))*x/a^2/b^3/(b*x^2+a)^(5/2)+1/105*(24*A*b^3+a*(4*B*b^2+3 
*C*a*b-45*D*a^2))*x/a^3/b^3/(b*x^2+a)^(3/2)+1/105*(48*A*b^3+a*(8*B*b^2+6*C 
*a*b+15*D*a^2))*x/a^4/b^3/(b*x^2+a)^(1/2)
 

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.52 \[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {48 A b^3 x^7+8 a b^2 x^5 \left (21 A+B x^2\right )+2 a^2 b x^3 \left (105 A+14 B x^2+3 C x^4\right )+a^3 \left (105 A x+35 B x^3+21 C x^5+15 D x^7\right )}{105 a^4 \left (a+b x^2\right )^{7/2}} \] Input:

Integrate[(A + B*x^2 + C*x^4 + D*x^6)/(a + b*x^2)^(9/2),x]
 

Output:

(48*A*b^3*x^7 + 8*a*b^2*x^5*(21*A + B*x^2) + 2*a^2*b*x^3*(105*A + 14*B*x^2 
 + 3*C*x^4) + a^3*(105*A*x + 35*B*x^3 + 21*C*x^5 + 15*D*x^7))/(105*a^4*(a 
+ b*x^2)^(7/2))
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.96, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2344, 2089, 1586, 9, 25, 27, 362, 242}

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.

\(\displaystyle \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{9/2}} \, dx\)

\(\Big \downarrow \) 2344

\(\displaystyle \frac {\int \frac {x^2 \left (6 A b+a \left (D x^4+C x^2+B\right )\right )}{\left (b x^2+a\right )^{9/2}}dx}{a}+\frac {A x}{a \left (a+b x^2\right )^{7/2}}\)

\(\Big \downarrow \) 2089

\(\displaystyle \frac {\int \frac {x^2 \left (a D x^4+a C x^2+6 A b+a B\right )}{\left (b x^2+a\right )^{9/2}}dx}{a}+\frac {A x}{a \left (a+b x^2\right )^{7/2}}\)

\(\Big \downarrow \) 1586

\(\displaystyle \frac {\frac {x^3 \left (\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^2}+6 A b\right )}{7 a \left (a+b x^2\right )^{7/2}}-\frac {\int -\frac {x \left (\frac {7 a^2 D x^3}{b}+\left (24 A b+\frac {a \left (-3 D a^2+3 b C a+4 b^2 B\right )}{b^2}\right ) x\right )}{\left (b x^2+a\right )^{7/2}}dx}{7 a}}{a}+\frac {A x}{a \left (a+b x^2\right )^{7/2}}\)

\(\Big \downarrow \) 9

\(\displaystyle \frac {\frac {x^3 \left (\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^2}+6 A b\right )}{7 a \left (a+b x^2\right )^{7/2}}-\frac {\int -\frac {x^2 \left (7 a^2 D x^2+b \left (24 A b+\frac {a \left (-3 D a^2+3 b C a+4 b^2 B\right )}{b^2}\right )\right )}{b \left (b x^2+a\right )^{7/2}}dx}{7 a}}{a}+\frac {A x}{a \left (a+b x^2\right )^{7/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {x^2 \left (24 A b^2+7 a^2 D x^2+a \left (-\frac {3 D a^2}{b}+3 C a+4 b B\right )\right )}{b \left (b x^2+a\right )^{7/2}}dx}{7 a}+\frac {x^3 \left (\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^2}+6 A b\right )}{7 a \left (a+b x^2\right )^{7/2}}}{a}+\frac {A x}{a \left (a+b x^2\right )^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {x^2 \left (24 A b^2+7 a^2 D x^2+a \left (-\frac {3 D a^2}{b}+3 C a+4 b B\right )\right )}{\left (b x^2+a\right )^{7/2}}dx}{7 a b}+\frac {x^3 \left (\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^2}+6 A b\right )}{7 a \left (a+b x^2\right )^{7/2}}}{a}+\frac {A x}{a \left (a+b x^2\right )^{7/2}}\)

\(\Big \downarrow \) 362

\(\displaystyle \frac {\frac {\frac {\left (a \left (15 a^2 D+6 a b C+8 b^2 B\right )+48 A b^3\right ) \int \frac {x^2}{\left (b x^2+a\right )^{5/2}}dx}{5 a b}+\frac {x^3 \left (a \left (-10 a^2 D+3 a b C+4 b^2 B\right )+24 A b^3\right )}{5 a b \left (a+b x^2\right )^{5/2}}}{7 a b}+\frac {x^3 \left (\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^2}+6 A b\right )}{7 a \left (a+b x^2\right )^{7/2}}}{a}+\frac {A x}{a \left (a+b x^2\right )^{7/2}}\)

\(\Big \downarrow \) 242

\(\displaystyle \frac {\frac {x^3 \left (\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^2}+6 A b\right )}{7 a \left (a+b x^2\right )^{7/2}}+\frac {\frac {x^3 \left (a \left (15 a^2 D+6 a b C+8 b^2 B\right )+48 A b^3\right )}{15 a^2 b \left (a+b x^2\right )^{3/2}}+\frac {x^3 \left (a \left (-10 a^2 D+3 a b C+4 b^2 B\right )+24 A b^3\right )}{5 a b \left (a+b x^2\right )^{5/2}}}{7 a b}}{a}+\frac {A x}{a \left (a+b x^2\right )^{7/2}}\)

Input:

Int[(A + B*x^2 + C*x^4 + D*x^6)/(a + b*x^2)^(9/2),x]
 

Output:

(A*x)/(a*(a + b*x^2)^(7/2)) + (((6*A*b + (a*(b^2*B - a*b*C + a^2*D))/b^2)* 
x^3)/(7*a*(a + b*x^2)^(7/2)) + (((24*A*b^3 + a*(4*b^2*B + 3*a*b*C - 10*a^2 
*D))*x^3)/(5*a*b*(a + b*x^2)^(5/2)) + ((48*A*b^3 + a*(8*b^2*B + 6*a*b*C + 
15*a^2*D))*x^3)/(15*a^2*b*(a + b*x^2)^(3/2)))/(7*a*b))/a
 

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 27

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

rule 242

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

rule 362

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x 
_Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*b*e 
*(p + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(2*a*b*(p + 1))   I 
nt[(e*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && N 
eQ[b*c - a*d, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) || 
  !RationalQ[m] || (ILtQ[p + 1/2, 0] && LeQ[-1, m, -2*(p + 1)]))
 

rule 1586

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c 
_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2 + 
c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^ 
p, d + e*x^2, x], x, 0]}, Simp[(-R)*(f*x)^(m + 1)*((d + e*x^2)^(q + 1)/(2*d 
*f*(q + 1))), x] + Simp[f/(2*d*(q + 1))   Int[(f*x)^(m - 1)*(d + e*x^2)^(q 
+ 1)*ExpandToSum[2*d*(q + 1)*x*Qx + R*(m + 2*q + 3)*x, x], x], x]] /; FreeQ 
[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && LtQ[q, -1] 
&& GtQ[m, 0]
 

rule 2089

Int[(u_)^(p_.)*((f_.)*(x_))^(m_.)*(z_)^(q_.), x_Symbol] :> Int[(f*x)^m*Expa 
ndToSum[z, x]^q*ExpandToSum[u, x]^p, x] /; FreeQ[{f, m, p, q}, x] && Binomi 
alQ[z, x] && TrinomialQ[u, x] &&  !(BinomialMatchQ[z, x] && TrinomialMatchQ 
[u, x])
 

rule 2344

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{A = Coeff[Pq, x, 0 
], Q = PolynomialQuotient[Pq - Coeff[Pq, x, 0], x^2, x]}, Simp[A*x*((a + b* 
x^2)^(p + 1)/a), x] + Simp[1/a   Int[x^2*(a + b*x^2)^p*(a*Q - A*b*(2*p + 3) 
), x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x^2] && ILtQ[p + 1/2, 0] && LtQ 
[Expon[Pq, x] + 2*p + 1, 0]
 

Maple [A] (verified)

Time = 0.53 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.47

method	result	size
pseudoelliptic	\(\frac {x \left (\left (\frac {1}{7} D x^{6}+\frac {1}{5} C \,x^{4}+\frac {1}{3} x^{2} B +A \right ) a^{3}+2 \left (\frac {1}{35} C \,x^{4}+\frac {2}{15} x^{2} B +A \right ) x^{2} b \,a^{2}+\frac {8 b^{2} \left (\frac {x^{2} B}{21}+A \right ) x^{4} a}{5}+\frac {16 A \,b^{3} x^{6}}{35}\right )}{\left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{4}}\)	\(89\)
gosper	\(\frac {x \left (48 A \,b^{3} x^{6}+8 B a \,b^{2} x^{6}+6 C \,a^{2} b \,x^{6}+15 D a^{3} x^{6}+168 a A \,b^{2} x^{4}+28 B \,a^{2} b \,x^{4}+21 C \,a^{3} x^{4}+210 a^{2} A b \,x^{2}+35 B \,a^{3} x^{2}+105 a^{3} A \right )}{105 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{4}}\)	\(109\)
trager	\(\frac {x \left (48 A \,b^{3} x^{6}+8 B a \,b^{2} x^{6}+6 C \,a^{2} b \,x^{6}+15 D a^{3} x^{6}+168 a A \,b^{2} x^{4}+28 B \,a^{2} b \,x^{4}+21 C \,a^{3} x^{4}+210 a^{2} A b \,x^{2}+35 B \,a^{3} x^{2}+105 a^{3} A \right )}{105 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{4}}\)	\(109\)
orering	\(\frac {x \left (48 A \,b^{3} x^{6}+8 B a \,b^{2} x^{6}+6 C \,a^{2} b \,x^{6}+15 D a^{3} x^{6}+168 a A \,b^{2} x^{4}+28 B \,a^{2} b \,x^{4}+21 C \,a^{3} x^{4}+210 a^{2} A b \,x^{2}+35 B \,a^{3} x^{2}+105 a^{3} A \right )}{105 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{4}}\)	\(109\)
default	\(A \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )+C \left (-\frac {x^{3}}{4 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {3 a \left (-\frac {x}{6 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {a \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{6 b}\right )}{4 b}\right )+D \left (-\frac {x^{5}}{2 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {5 a \left (-\frac {x^{3}}{4 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {3 a \left (-\frac {x}{6 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {a \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{6 b}\right )}{4 b}\right )}{2 b}\right )+B \left (-\frac {x}{6 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {a \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{6 b}\right )\)	\(440\)

Input:

int((D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(9/2),x,method=_RETURNVERBOSE)
 

Output:

1/(b*x^2+a)^(7/2)*x*((1/7*D*x^6+1/5*C*x^4+1/3*x^2*B+A)*a^3+2*(1/35*C*x^4+2 
/15*x^2*B+A)*x^2*b*a^2+8/5*b^2*(1/21*x^2*B+A)*x^4*a+16/35*A*b^3*x^6)/a^4
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.74 \[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {{\left ({\left (15 \, D a^{3} + 6 \, C a^{2} b + 8 \, B a b^{2} + 48 \, A b^{3}\right )} x^{7} + 7 \, {\left (3 \, C a^{3} + 4 \, B a^{2} b + 24 \, A a b^{2}\right )} x^{5} + 105 \, A a^{3} x + 35 \, {\left (B a^{3} + 6 \, A a^{2} b\right )} x^{3}\right )} \sqrt {b x^{2} + a}}{105 \, {\left (a^{4} b^{4} x^{8} + 4 \, a^{5} b^{3} x^{6} + 6 \, a^{6} b^{2} x^{4} + 4 \, a^{7} b x^{2} + a^{8}\right )}} \] Input:

integrate((D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(9/2),x, algorithm="fricas")
 

Output:

1/105*((15*D*a^3 + 6*C*a^2*b + 8*B*a*b^2 + 48*A*b^3)*x^7 + 7*(3*C*a^3 + 4* 
B*a^2*b + 24*A*a*b^2)*x^5 + 105*A*a^3*x + 35*(B*a^3 + 6*A*a^2*b)*x^3)*sqrt 
(b*x^2 + a)/(a^4*b^4*x^8 + 4*a^5*b^3*x^6 + 6*a^6*b^2*x^4 + 4*a^7*b*x^2 + a 
^8)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2088 vs. \(2 (187) = 374\).

Time = 53.23 (sec) , antiderivative size = 2088, normalized size of antiderivative = 10.99 \[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{9/2}} \, dx=\text {Too large to display} \] Input:

integrate((D*x**6+C*x**4+B*x**2+A)/(b*x**2+a)**(9/2),x)
 

Output:

A*(35*a**14*x/(35*a**(37/2)*sqrt(1 + b*x**2/a) + 210*a**(35/2)*b*x**2*sqrt 
(1 + b*x**2/a) + 525*a**(33/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 700*a**(31/2 
)*b**3*x**6*sqrt(1 + b*x**2/a) + 525*a**(29/2)*b**4*x**8*sqrt(1 + b*x**2/a 
) + 210*a**(27/2)*b**5*x**10*sqrt(1 + b*x**2/a) + 35*a**(25/2)*b**6*x**12* 
sqrt(1 + b*x**2/a)) + 175*a**13*b*x**3/(35*a**(37/2)*sqrt(1 + b*x**2/a) + 
210*a**(35/2)*b*x**2*sqrt(1 + b*x**2/a) + 525*a**(33/2)*b**2*x**4*sqrt(1 + 
 b*x**2/a) + 700*a**(31/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 525*a**(29/2)*b* 
*4*x**8*sqrt(1 + b*x**2/a) + 210*a**(27/2)*b**5*x**10*sqrt(1 + b*x**2/a) + 
 35*a**(25/2)*b**6*x**12*sqrt(1 + b*x**2/a)) + 371*a**12*b**2*x**5/(35*a** 
(37/2)*sqrt(1 + b*x**2/a) + 210*a**(35/2)*b*x**2*sqrt(1 + b*x**2/a) + 525* 
a**(33/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 700*a**(31/2)*b**3*x**6*sqrt(1 + 
b*x**2/a) + 525*a**(29/2)*b**4*x**8*sqrt(1 + b*x**2/a) + 210*a**(27/2)*b** 
5*x**10*sqrt(1 + b*x**2/a) + 35*a**(25/2)*b**6*x**12*sqrt(1 + b*x**2/a)) + 
 429*a**11*b**3*x**7/(35*a**(37/2)*sqrt(1 + b*x**2/a) + 210*a**(35/2)*b*x* 
*2*sqrt(1 + b*x**2/a) + 525*a**(33/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 700*a 
**(31/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 525*a**(29/2)*b**4*x**8*sqrt(1 + b 
*x**2/a) + 210*a**(27/2)*b**5*x**10*sqrt(1 + b*x**2/a) + 35*a**(25/2)*b**6 
*x**12*sqrt(1 + b*x**2/a)) + 286*a**10*b**4*x**9/(35*a**(37/2)*sqrt(1 + b* 
x**2/a) + 210*a**(35/2)*b*x**2*sqrt(1 + b*x**2/a) + 525*a**(33/2)*b**2*x** 
4*sqrt(1 + b*x**2/a) + 700*a**(31/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 525...
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.76 \[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{9/2}} \, dx=-\frac {D x^{5}}{2 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} - \frac {5 \, D a x^{3}}{8 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} - \frac {C x^{3}}{4 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} + \frac {16 \, A x}{35 \, \sqrt {b x^{2} + a} a^{4}} + \frac {8 \, A x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3}} + \frac {6 \, A x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2}} + \frac {A x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a} + \frac {D x}{14 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{3}} + \frac {D x}{7 \, \sqrt {b x^{2} + a} a b^{3}} + \frac {3 \, D a x}{56 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{3}} - \frac {15 \, D a^{2} x}{56 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}} + \frac {3 \, C x}{140 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{2}} + \frac {2 \, C x}{35 \, \sqrt {b x^{2} + a} a^{2} b^{2}} + \frac {C x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a b^{2}} - \frac {3 \, C a x}{28 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} - \frac {B x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} + \frac {8 \, B x}{105 \, \sqrt {b x^{2} + a} a^{3} b} + \frac {4 \, B x}{105 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} b} + \frac {B x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a b} \] Input:

integrate((D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(9/2),x, algorithm="maxima")
 

Output:

-1/2*D*x^5/((b*x^2 + a)^(7/2)*b) - 5/8*D*a*x^3/((b*x^2 + a)^(7/2)*b^2) - 1 
/4*C*x^3/((b*x^2 + a)^(7/2)*b) + 16/35*A*x/(sqrt(b*x^2 + a)*a^4) + 8/35*A* 
x/((b*x^2 + a)^(3/2)*a^3) + 6/35*A*x/((b*x^2 + a)^(5/2)*a^2) + 1/7*A*x/((b 
*x^2 + a)^(7/2)*a) + 1/14*D*x/((b*x^2 + a)^(3/2)*b^3) + 1/7*D*x/(sqrt(b*x^ 
2 + a)*a*b^3) + 3/56*D*a*x/((b*x^2 + a)^(5/2)*b^3) - 15/56*D*a^2*x/((b*x^2 
 + a)^(7/2)*b^3) + 3/140*C*x/((b*x^2 + a)^(5/2)*b^2) + 2/35*C*x/(sqrt(b*x^ 
2 + a)*a^2*b^2) + 1/35*C*x/((b*x^2 + a)^(3/2)*a*b^2) - 3/28*C*a*x/((b*x^2 
+ a)^(7/2)*b^2) - 1/7*B*x/((b*x^2 + a)^(7/2)*b) + 8/105*B*x/(sqrt(b*x^2 + 
a)*a^3*b) + 4/105*B*x/((b*x^2 + a)^(3/2)*a^2*b) + 1/35*B*x/((b*x^2 + a)^(5 
/2)*a*b)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.69 \[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {{\left ({\left (x^{2} {\left (\frac {{\left (15 \, D a^{3} b^{3} + 6 \, C a^{2} b^{4} + 8 \, B a b^{5} + 48 \, A b^{6}\right )} x^{2}}{a^{4} b^{3}} + \frac {7 \, {\left (3 \, C a^{3} b^{3} + 4 \, B a^{2} b^{4} + 24 \, A a b^{5}\right )}}{a^{4} b^{3}}\right )} + \frac {35 \, {\left (B a^{3} b^{3} + 6 \, A a^{2} b^{4}\right )}}{a^{4} b^{3}}\right )} x^{2} + \frac {105 \, A}{a}\right )} x}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} \] Input:

integrate((D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(9/2),x, algorithm="giac")
 

Output:

1/105*((x^2*((15*D*a^3*b^3 + 6*C*a^2*b^4 + 8*B*a*b^5 + 48*A*b^6)*x^2/(a^4* 
b^3) + 7*(3*C*a^3*b^3 + 4*B*a^2*b^4 + 24*A*a*b^5)/(a^4*b^3)) + 35*(B*a^3*b 
^3 + 6*A*a^2*b^4)/(a^4*b^3))*x^2 + 105*A/a)*x/(b*x^2 + a)^(7/2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{9/2}} \, dx=\int \frac {A+B\,x^2+C\,x^4+x^6\,D}{{\left (b\,x^2+a\right )}^{9/2}} \,d x \] Input:

int((A + B*x^2 + C*x^4 + x^6*D)/(a + b*x^2)^(9/2),x)
 

Output:

int((A + B*x^2 + C*x^4 + x^6*D)/(a + b*x^2)^(9/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.89 \[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {105 \sqrt {b \,x^{2}+a}\, a^{3} b^{4} x +245 \sqrt {b \,x^{2}+a}\, a^{2} b^{5} x^{3}+21 \sqrt {b \,x^{2}+a}\, a^{2} b^{4} c \,x^{5}+15 \sqrt {b \,x^{2}+a}\, a^{2} b^{4} d \,x^{7}+196 \sqrt {b \,x^{2}+a}\, a \,b^{6} x^{5}+6 \sqrt {b \,x^{2}+a}\, a \,b^{5} c \,x^{7}+56 \sqrt {b \,x^{2}+a}\, b^{7} x^{7}+15 \sqrt {b}\, a^{6} d -6 \sqrt {b}\, a^{5} b c +60 \sqrt {b}\, a^{5} b d \,x^{2}-56 \sqrt {b}\, a^{4} b^{3}-24 \sqrt {b}\, a^{4} b^{2} c \,x^{2}+90 \sqrt {b}\, a^{4} b^{2} d \,x^{4}-224 \sqrt {b}\, a^{3} b^{4} x^{2}-36 \sqrt {b}\, a^{3} b^{3} c \,x^{4}+60 \sqrt {b}\, a^{3} b^{3} d \,x^{6}-336 \sqrt {b}\, a^{2} b^{5} x^{4}-24 \sqrt {b}\, a^{2} b^{4} c \,x^{6}+15 \sqrt {b}\, a^{2} b^{4} d \,x^{8}-224 \sqrt {b}\, a \,b^{6} x^{6}-6 \sqrt {b}\, a \,b^{5} c \,x^{8}-56 \sqrt {b}\, b^{7} x^{8}}{105 a^{3} b^{4} \left (b^{4} x^{8}+4 a \,b^{3} x^{6}+6 a^{2} b^{2} x^{4}+4 a^{3} b \,x^{2}+a^{4}\right )} \] Input:

int((D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(9/2),x)
 

Output:

(105*sqrt(a + b*x**2)*a**3*b**4*x + 245*sqrt(a + b*x**2)*a**2*b**5*x**3 + 
21*sqrt(a + b*x**2)*a**2*b**4*c*x**5 + 15*sqrt(a + b*x**2)*a**2*b**4*d*x** 
7 + 196*sqrt(a + b*x**2)*a*b**6*x**5 + 6*sqrt(a + b*x**2)*a*b**5*c*x**7 + 
56*sqrt(a + b*x**2)*b**7*x**7 + 15*sqrt(b)*a**6*d - 6*sqrt(b)*a**5*b*c + 6 
0*sqrt(b)*a**5*b*d*x**2 - 56*sqrt(b)*a**4*b**3 - 24*sqrt(b)*a**4*b**2*c*x* 
*2 + 90*sqrt(b)*a**4*b**2*d*x**4 - 224*sqrt(b)*a**3*b**4*x**2 - 36*sqrt(b) 
*a**3*b**3*c*x**4 + 60*sqrt(b)*a**3*b**3*d*x**6 - 336*sqrt(b)*a**2*b**5*x* 
*4 - 24*sqrt(b)*a**2*b**4*c*x**6 + 15*sqrt(b)*a**2*b**4*d*x**8 - 224*sqrt( 
b)*a*b**6*x**6 - 6*sqrt(b)*a*b**5*c*x**8 - 56*sqrt(b)*b**7*x**8)/(105*a**3 
*b**4*(a**4 + 4*a**3*b*x**2 + 6*a**2*b**2*x**4 + 4*a*b**3*x**6 + b**4*x**8 
))