\(\int \frac {A+B x^2+C x^4+D x^6}{(a+b x^2)^{11/2}} \, dx\) [140]

Optimal result

Integrand size = 29, antiderivative size = 238 \[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{11/2}} \, dx=\frac {\left (\frac {A}{a}-\frac {b^2 B-a b C+a^2 D}{b^3}\right ) x}{9 \left (a+b x^2\right )^{9/2}}+\frac {\left (8 A b^3+a \left (b^2 B-10 a b C+19 a^2 D\right )\right ) x}{63 a^2 b^3 \left (a+b x^2\right )^{7/2}}+\frac {\left (16 A b^3+a \left (2 b^2 B+a b C-25 a^2 D\right )\right ) x}{105 a^3 b^3 \left (a+b x^2\right )^{5/2}}+\frac {\left (64 A b^3+a \left (8 b^2 B+4 a b C+5 a^2 D\right )\right ) x}{315 a^4 b^3 \left (a+b x^2\right )^{3/2}}+\frac {2 \left (64 A b^3+a \left (8 b^2 B+4 a b C+5 a^2 D\right )\right ) x}{315 a^5 b^3 \sqrt {a+b x^2}} \] Output:

1/9*(A/a-(B*b^2-C*a*b+D*a^2)/b^3)*x/(b*x^2+a)^(9/2)+1/63*(8*A*b^3+a*(B*b^2 
-10*C*a*b+19*D*a^2))*x/a^2/b^3/(b*x^2+a)^(7/2)+1/105*(16*A*b^3+a*(2*B*b^2+ 
C*a*b-25*D*a^2))*x/a^3/b^3/(b*x^2+a)^(5/2)+1/315*(64*A*b^3+a*(8*B*b^2+4*C* 
a*b+5*D*a^2))*x/a^4/b^3/(b*x^2+a)^(3/2)+2/315*(64*A*b^3+a*(8*B*b^2+4*C*a*b 
+5*D*a^2))*x/a^5/b^3/(b*x^2+a)^(1/2)
 

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.55 \[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{11/2}} \, dx=\frac {128 A b^4 x^9+16 a b^3 x^7 \left (36 A+B x^2\right )+8 a^2 b^2 x^5 \left (126 A+9 B x^2+C x^4\right )+2 a^3 b x^3 \left (420 A+63 B x^2+18 C x^4+5 D x^6\right )+3 a^4 \left (105 A x+35 B x^3+21 C x^5+15 D x^7\right )}{315 a^5 \left (a+b x^2\right )^{9/2}} \] Input:

Integrate[(A + B*x^2 + C*x^4 + D*x^6)/(a + b*x^2)^(11/2),x]
 

Output:

(128*A*b^4*x^9 + 16*a*b^3*x^7*(36*A + B*x^2) + 8*a^2*b^2*x^5*(126*A + 9*B* 
x^2 + C*x^4) + 2*a^3*b*x^3*(420*A + 63*B*x^2 + 18*C*x^4 + 5*D*x^6) + 3*a^4 
*(105*A*x + 35*B*x^3 + 21*C*x^5 + 15*D*x^7))/(315*a^5*(a + b*x^2)^(9/2))
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.89, 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, 27, 362, 245, 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.

\(\displaystyle \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{11/2}} \, dx\)

\(\Big \downarrow \) 2344

\(\displaystyle \frac {\int \frac {x^2 \left (8 A b+a \left (D x^4+C x^2+B\right )\right )}{\left (b x^2+a\right )^{11/2}}dx}{a}+\frac {A x}{a \left (a+b x^2\right )^{9/2}}\)

\(\Big \downarrow \) 2089

\(\displaystyle \frac {\int \frac {x^2 \left (a D x^4+a C x^2+8 A b+a B\right )}{\left (b x^2+a\right )^{11/2}}dx}{a}+\frac {A x}{a \left (a+b x^2\right )^{9/2}}\)

\(\Big \downarrow \) 1586

\(\displaystyle \frac {\frac {x^3 \left (\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^2}+8 A b\right )}{9 a \left (a+b x^2\right )^{9/2}}-\frac {\int -\frac {3 x \left (\frac {3 a^2 D x^3}{b}+\left (16 A b+\frac {a \left (-D a^2+b C a+2 b^2 B\right )}{b^2}\right ) x\right )}{\left (b x^2+a\right )^{9/2}}dx}{9 a}}{a}+\frac {A x}{a \left (a+b x^2\right )^{9/2}}\)

\(\Big \downarrow \) 9

\(\displaystyle \frac {\frac {x^3 \left (\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^2}+8 A b\right )}{9 a \left (a+b x^2\right )^{9/2}}-\frac {\int -\frac {3 x^2 \left (3 a^2 D x^2+b \left (16 A b+\frac {a \left (-D a^2+b C a+2 b^2 B\right )}{b^2}\right )\right )}{b \left (b x^2+a\right )^{9/2}}dx}{9 a}}{a}+\frac {A x}{a \left (a+b x^2\right )^{9/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {x^2 \left (16 A b^2+3 a^2 D x^2+a \left (-\frac {D a^2}{b}+C a+2 b B\right )\right )}{\left (b x^2+a\right )^{9/2}}dx}{3 a b}+\frac {x^3 \left (\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^2}+8 A b\right )}{9 a \left (a+b x^2\right )^{9/2}}}{a}+\frac {A x}{a \left (a+b x^2\right )^{9/2}}\)

\(\Big \downarrow \) 362

\(\displaystyle \frac {\frac {\frac {\left (a \left (5 a^2 D+4 a b C+8 b^2 B\right )+64 A b^3\right ) \int \frac {x^2}{\left (b x^2+a\right )^{7/2}}dx}{7 a b}+\frac {x^3 \left (a \left (-4 a^2 D+a b C+2 b^2 B\right )+16 A b^3\right )}{7 a b \left (a+b x^2\right )^{7/2}}}{3 a b}+\frac {x^3 \left (\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^2}+8 A b\right )}{9 a \left (a+b x^2\right )^{9/2}}}{a}+\frac {A x}{a \left (a+b x^2\right )^{9/2}}\)

\(\Big \downarrow \) 245

\(\displaystyle \frac {\frac {\frac {\left (a \left (5 a^2 D+4 a b C+8 b^2 B\right )+64 A b^3\right ) \left (\frac {2 b \int \frac {x^4}{\left (b x^2+a\right )^{7/2}}dx}{3 a}+\frac {x^3}{3 a \left (a+b x^2\right )^{5/2}}\right )}{7 a b}+\frac {x^3 \left (a \left (-4 a^2 D+a b C+2 b^2 B\right )+16 A b^3\right )}{7 a b \left (a+b x^2\right )^{7/2}}}{3 a b}+\frac {x^3 \left (\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^2}+8 A b\right )}{9 a \left (a+b x^2\right )^{9/2}}}{a}+\frac {A x}{a \left (a+b x^2\right )^{9/2}}\)

\(\Big \downarrow \) 242

\(\displaystyle \frac {\frac {x^3 \left (\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^2}+8 A b\right )}{9 a \left (a+b x^2\right )^{9/2}}+\frac {\frac {x^3 \left (a \left (-4 a^2 D+a b C+2 b^2 B\right )+16 A b^3\right )}{7 a b \left (a+b x^2\right )^{7/2}}+\frac {\left (\frac {2 b x^5}{15 a^2 \left (a+b x^2\right )^{5/2}}+\frac {x^3}{3 a \left (a+b x^2\right )^{5/2}}\right ) \left (a \left (5 a^2 D+4 a b C+8 b^2 B\right )+64 A b^3\right )}{7 a b}}{3 a b}}{a}+\frac {A x}{a \left (a+b x^2\right )^{9/2}}\)

Input:

Int[(A + B*x^2 + C*x^4 + D*x^6)/(a + b*x^2)^(11/2),x]
 

Output:

(A*x)/(a*(a + b*x^2)^(9/2)) + (((8*A*b + (a*(b^2*B - a*b*C + a^2*D))/b^2)* 
x^3)/(9*a*(a + b*x^2)^(9/2)) + (((16*A*b^3 + a*(2*b^2*B + a*b*C - 4*a^2*D) 
)*x^3)/(7*a*b*(a + b*x^2)^(7/2)) + ((64*A*b^3 + a*(8*b^2*B + 4*a*b*C + 5*a 
^2*D))*(x^3/(3*a*(a + b*x^2)^(5/2)) + (2*b*x^5)/(15*a^2*(a + b*x^2)^(5/2)) 
))/(7*a*b))/(3*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 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 245

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + 
 b*x^2)^(p + 1)/(a*(m + 1))), x] - Simp[b*((m + 2*(p + 1) + 1)/(a*(m + 1))) 
   Int[x^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, m, p}, x] && ILtQ[Si 
mplify[(m + 1)/2 + p + 1], 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 = 120, normalized size of antiderivative = 0.50

method	result	size
pseudoelliptic	\(\frac {x \left (\left (\frac {1}{7} D x^{6}+\frac {1}{5} C \,x^{4}+\frac {1}{3} x^{2} B +A \right ) a^{4}+\frac {8 \left (\frac {1}{84} D x^{6}+\frac {3}{70} C \,x^{4}+\frac {3}{20} x^{2} B +A \right ) x^{2} b \,a^{3}}{3}+\frac {16 \left (\frac {1}{126} C \,x^{4}+\frac {1}{14} x^{2} B +A \right ) x^{4} b^{2} a^{2}}{5}+\frac {64 \left (\frac {x^{2} B}{36}+A \right ) x^{6} b^{3} a}{35}+\frac {128 A \,x^{8} b^{4}}{315}\right )}{\left (b \,x^{2}+a \right )^{\frac {9}{2}} a^{5}}\)	\(120\)
gosper	\(\frac {x \left (128 A \,x^{8} b^{4}+16 B \,x^{8} a \,b^{3}+8 C \,a^{2} b^{2} x^{8}+10 D a^{3} b \,x^{8}+576 A \,x^{6} a \,b^{3}+72 B \,x^{6} a^{2} b^{2}+36 C \,a^{3} b \,x^{6}+45 D a^{4} x^{6}+1008 A \,x^{4} a^{2} b^{2}+126 B \,x^{4} a^{3} b +63 C \,a^{4} x^{4}+840 A \,x^{2} a^{3} b +105 B \,x^{2} a^{4}+315 A \,a^{4}\right )}{315 \left (b \,x^{2}+a \right )^{\frac {9}{2}} a^{5}}\)	\(155\)
trager	\(\frac {x \left (128 A \,x^{8} b^{4}+16 B \,x^{8} a \,b^{3}+8 C \,a^{2} b^{2} x^{8}+10 D a^{3} b \,x^{8}+576 A \,x^{6} a \,b^{3}+72 B \,x^{6} a^{2} b^{2}+36 C \,a^{3} b \,x^{6}+45 D a^{4} x^{6}+1008 A \,x^{4} a^{2} b^{2}+126 B \,x^{4} a^{3} b +63 C \,a^{4} x^{4}+840 A \,x^{2} a^{3} b +105 B \,x^{2} a^{4}+315 A \,a^{4}\right )}{315 \left (b \,x^{2}+a \right )^{\frac {9}{2}} a^{5}}\)	\(155\)
orering	\(\frac {x \left (128 A \,x^{8} b^{4}+16 B \,x^{8} a \,b^{3}+8 C \,a^{2} b^{2} x^{8}+10 D a^{3} b \,x^{8}+576 A \,x^{6} a \,b^{3}+72 B \,x^{6} a^{2} b^{2}+36 C \,a^{3} b \,x^{6}+45 D a^{4} x^{6}+1008 A \,x^{4} a^{2} b^{2}+126 B \,x^{4} a^{3} b +63 C \,a^{4} x^{4}+840 A \,x^{2} a^{3} b +105 B \,x^{2} a^{4}+315 A \,a^{4}\right )}{315 \left (b \,x^{2}+a \right )^{\frac {9}{2}} a^{5}}\)	\(155\)
default	\(A \left (\frac {x}{9 a \left (b \,x^{2}+a \right )^{\frac {9}{2}}}+\frac {\frac {8 x}{63 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {8 \left (\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}\right )}{9 a}}{a}\right )+C \left (-\frac {x^{3}}{6 b \left (b \,x^{2}+a \right )^{\frac {9}{2}}}+\frac {a \left (-\frac {x}{8 b \left (b \,x^{2}+a \right )^{\frac {9}{2}}}+\frac {a \left (\frac {x}{9 a \left (b \,x^{2}+a \right )^{\frac {9}{2}}}+\frac {\frac {8 x}{63 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {8 \left (\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}\right )}{9 a}}{a}\right )}{8 b}\right )}{2 b}\right )+D \left (-\frac {x^{5}}{4 b \left (b \,x^{2}+a \right )^{\frac {9}{2}}}+\frac {5 a \left (-\frac {x^{3}}{6 b \left (b \,x^{2}+a \right )^{\frac {9}{2}}}+\frac {a \left (-\frac {x}{8 b \left (b \,x^{2}+a \right )^{\frac {9}{2}}}+\frac {a \left (\frac {x}{9 a \left (b \,x^{2}+a \right )^{\frac {9}{2}}}+\frac {\frac {8 x}{63 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {8 \left (\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}\right )}{9 a}}{a}\right )}{8 b}\right )}{2 b}\right )}{4 b}\right )+B \left (-\frac {x}{8 b \left (b \,x^{2}+a \right )^{\frac {9}{2}}}+\frac {a \left (\frac {x}{9 a \left (b \,x^{2}+a \right )^{\frac {9}{2}}}+\frac {\frac {8 x}{63 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {8 \left (\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}\right )}{9 a}}{a}\right )}{8 b}\right )\)	\(524\)

Input:

int((D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(11/2),x,method=_RETURNVERBOSE)
 

Output:

1/(b*x^2+a)^(9/2)*x*((1/7*D*x^6+1/5*C*x^4+1/3*x^2*B+A)*a^4+8/3*(1/84*D*x^6 
+3/70*C*x^4+3/20*x^2*B+A)*x^2*b*a^3+16/5*(1/126*C*x^4+1/14*x^2*B+A)*x^4*b^ 
2*a^2+64/35*(1/36*x^2*B+A)*x^6*b^3*a+128/315*A*x^8*b^4)/a^5
 

Fricas [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.81 \[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{11/2}} \, dx=\frac {{\left (2 \, {\left (5 \, D a^{3} b + 4 \, C a^{2} b^{2} + 8 \, B a b^{3} + 64 \, A b^{4}\right )} x^{9} + 9 \, {\left (5 \, D a^{4} + 4 \, C a^{3} b + 8 \, B a^{2} b^{2} + 64 \, A a b^{3}\right )} x^{7} + 315 \, A a^{4} x + 63 \, {\left (C a^{4} + 2 \, B a^{3} b + 16 \, A a^{2} b^{2}\right )} x^{5} + 105 \, {\left (B a^{4} + 8 \, A a^{3} b\right )} x^{3}\right )} \sqrt {b x^{2} + a}}{315 \, {\left (a^{5} b^{5} x^{10} + 5 \, a^{6} b^{4} x^{8} + 10 \, a^{7} b^{3} x^{6} + 10 \, a^{8} b^{2} x^{4} + 5 \, a^{9} b x^{2} + a^{10}\right )}} \] Input:

integrate((D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(11/2),x, algorithm="fricas")
 

Output:

1/315*(2*(5*D*a^3*b + 4*C*a^2*b^2 + 8*B*a*b^3 + 64*A*b^4)*x^9 + 9*(5*D*a^4 
 + 4*C*a^3*b + 8*B*a^2*b^2 + 64*A*a*b^3)*x^7 + 315*A*a^4*x + 63*(C*a^4 + 2 
*B*a^3*b + 16*A*a^2*b^2)*x^5 + 105*(B*a^4 + 8*A*a^3*b)*x^3)*sqrt(b*x^2 + a 
)/(a^5*b^5*x^10 + 5*a^6*b^4*x^8 + 10*a^7*b^3*x^6 + 10*a^8*b^2*x^4 + 5*a^9* 
b*x^2 + a^10)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5440 vs. \(2 (238) = 476\).

Time = 120.47 (sec) , antiderivative size = 5440, normalized size of antiderivative = 22.86 \[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{11/2}} \, dx=\text {Too large to display} \] Input:

integrate((D*x**6+C*x**4+B*x**2+A)/(b*x**2+a)**(11/2),x)
 

Output:

A*(315*a**30*x/(315*a**(71/2)*sqrt(1 + b*x**2/a) + 3150*a**(69/2)*b*x**2*s 
qrt(1 + b*x**2/a) + 14175*a**(67/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 37800*a 
**(65/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 66150*a**(63/2)*b**4*x**8*sqrt(1 + 
 b*x**2/a) + 79380*a**(61/2)*b**5*x**10*sqrt(1 + b*x**2/a) + 66150*a**(59/ 
2)*b**6*x**12*sqrt(1 + b*x**2/a) + 37800*a**(57/2)*b**7*x**14*sqrt(1 + b*x 
**2/a) + 14175*a**(55/2)*b**8*x**16*sqrt(1 + b*x**2/a) + 3150*a**(53/2)*b* 
*9*x**18*sqrt(1 + b*x**2/a) + 315*a**(51/2)*b**10*x**20*sqrt(1 + b*x**2/a) 
) + 2730*a**29*b*x**3/(315*a**(71/2)*sqrt(1 + b*x**2/a) + 3150*a**(69/2)*b 
*x**2*sqrt(1 + b*x**2/a) + 14175*a**(67/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 
37800*a**(65/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 66150*a**(63/2)*b**4*x**8*s 
qrt(1 + b*x**2/a) + 79380*a**(61/2)*b**5*x**10*sqrt(1 + b*x**2/a) + 66150* 
a**(59/2)*b**6*x**12*sqrt(1 + b*x**2/a) + 37800*a**(57/2)*b**7*x**14*sqrt( 
1 + b*x**2/a) + 14175*a**(55/2)*b**8*x**16*sqrt(1 + b*x**2/a) + 3150*a**(5 
3/2)*b**9*x**18*sqrt(1 + b*x**2/a) + 315*a**(51/2)*b**10*x**20*sqrt(1 + b* 
x**2/a)) + 10773*a**28*b**2*x**5/(315*a**(71/2)*sqrt(1 + b*x**2/a) + 3150* 
a**(69/2)*b*x**2*sqrt(1 + b*x**2/a) + 14175*a**(67/2)*b**2*x**4*sqrt(1 + b 
*x**2/a) + 37800*a**(65/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 66150*a**(63/2)* 
b**4*x**8*sqrt(1 + b*x**2/a) + 79380*a**(61/2)*b**5*x**10*sqrt(1 + b*x**2/ 
a) + 66150*a**(59/2)*b**6*x**12*sqrt(1 + b*x**2/a) + 37800*a**(57/2)*b**7* 
x**14*sqrt(1 + b*x**2/a) + 14175*a**(55/2)*b**8*x**16*sqrt(1 + b*x**2/a...
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.71 \[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{11/2}} \, dx=-\frac {D x^{5}}{4 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} b} - \frac {5 \, D a x^{3}}{24 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} b^{2}} - \frac {C x^{3}}{6 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} b} + \frac {128 \, A x}{315 \, \sqrt {b x^{2} + a} a^{5}} + \frac {64 \, A x}{315 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{4}} + \frac {16 \, A x}{105 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{3}} + \frac {8 \, A x}{63 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2}} + \frac {A x}{9 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} a} + \frac {D x}{84 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{3}} + \frac {2 \, D x}{63 \, \sqrt {b x^{2} + a} a^{2} b^{3}} + \frac {D x}{63 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a b^{3}} + \frac {5 \, D a x}{504 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}} - \frac {5 \, D a^{2} x}{72 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} b^{3}} + \frac {C x}{126 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} + \frac {8 \, C x}{315 \, \sqrt {b x^{2} + a} a^{3} b^{2}} + \frac {4 \, C x}{315 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} b^{2}} + \frac {C x}{105 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a b^{2}} - \frac {C a x}{18 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} b^{2}} - \frac {B x}{9 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} b} + \frac {16 \, B x}{315 \, \sqrt {b x^{2} + a} a^{4} b} + \frac {8 \, B x}{315 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3} b} + \frac {2 \, B x}{105 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2} b} + \frac {B x}{63 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a b} \] Input:

integrate((D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(11/2),x, algorithm="maxima")
 

Output:

-1/4*D*x^5/((b*x^2 + a)^(9/2)*b) - 5/24*D*a*x^3/((b*x^2 + a)^(9/2)*b^2) - 
1/6*C*x^3/((b*x^2 + a)^(9/2)*b) + 128/315*A*x/(sqrt(b*x^2 + a)*a^5) + 64/3 
15*A*x/((b*x^2 + a)^(3/2)*a^4) + 16/105*A*x/((b*x^2 + a)^(5/2)*a^3) + 8/63 
*A*x/((b*x^2 + a)^(7/2)*a^2) + 1/9*A*x/((b*x^2 + a)^(9/2)*a) + 1/84*D*x/(( 
b*x^2 + a)^(5/2)*b^3) + 2/63*D*x/(sqrt(b*x^2 + a)*a^2*b^3) + 1/63*D*x/((b* 
x^2 + a)^(3/2)*a*b^3) + 5/504*D*a*x/((b*x^2 + a)^(7/2)*b^3) - 5/72*D*a^2*x 
/((b*x^2 + a)^(9/2)*b^3) + 1/126*C*x/((b*x^2 + a)^(7/2)*b^2) + 8/315*C*x/( 
sqrt(b*x^2 + a)*a^3*b^2) + 4/315*C*x/((b*x^2 + a)^(3/2)*a^2*b^2) + 1/105*C 
*x/((b*x^2 + a)^(5/2)*a*b^2) - 1/18*C*a*x/((b*x^2 + a)^(9/2)*b^2) - 1/9*B* 
x/((b*x^2 + a)^(9/2)*b) + 16/315*B*x/(sqrt(b*x^2 + a)*a^4*b) + 8/315*B*x/( 
(b*x^2 + a)^(3/2)*a^3*b) + 2/105*B*x/((b*x^2 + a)^(5/2)*a^2*b) + 1/63*B*x/ 
((b*x^2 + a)^(7/2)*a*b)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.76 \[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{11/2}} \, dx=\frac {{\left ({\left ({\left (x^{2} {\left (\frac {2 \, {\left (5 \, D a^{3} b^{5} + 4 \, C a^{2} b^{6} + 8 \, B a b^{7} + 64 \, A b^{8}\right )} x^{2}}{a^{5} b^{4}} + \frac {9 \, {\left (5 \, D a^{4} b^{4} + 4 \, C a^{3} b^{5} + 8 \, B a^{2} b^{6} + 64 \, A a b^{7}\right )}}{a^{5} b^{4}}\right )} + \frac {63 \, {\left (C a^{4} b^{4} + 2 \, B a^{3} b^{5} + 16 \, A a^{2} b^{6}\right )}}{a^{5} b^{4}}\right )} x^{2} + \frac {105 \, {\left (B a^{4} b^{4} + 8 \, A a^{3} b^{5}\right )}}{a^{5} b^{4}}\right )} x^{2} + \frac {315 \, A}{a}\right )} x}{315 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}}} \] Input:

integrate((D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(11/2),x, algorithm="giac")
 

Output:

1/315*(((x^2*(2*(5*D*a^3*b^5 + 4*C*a^2*b^6 + 8*B*a*b^7 + 64*A*b^8)*x^2/(a^ 
5*b^4) + 9*(5*D*a^4*b^4 + 4*C*a^3*b^5 + 8*B*a^2*b^6 + 64*A*a*b^7)/(a^5*b^4 
)) + 63*(C*a^4*b^4 + 2*B*a^3*b^5 + 16*A*a^2*b^6)/(a^5*b^4))*x^2 + 105*(B*a 
^4*b^4 + 8*A*a^3*b^5)/(a^5*b^4))*x^2 + 315*A/a)*x/(b*x^2 + a)^(9/2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{11/2}} \, dx=\int \frac {A+B\,x^2+C\,x^4+x^6\,D}{{\left (b\,x^2+a\right )}^{11/2}} \,d x \] Input:

int((A + B*x^2 + C*x^4 + x^6*D)/(a + b*x^2)^(11/2),x)
 

Output:

int((A + B*x^2 + C*x^4 + x^6*D)/(a + b*x^2)^(11/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 471, normalized size of antiderivative = 1.98 \[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{11/2}} \, dx=\frac {144 \sqrt {b \,x^{2}+a}\, b^{8} x^{9}-10 \sqrt {b}\, a^{7} d -144 \sqrt {b}\, a^{5} b^{3}-144 \sqrt {b}\, b^{8} x^{10}+315 \sqrt {b \,x^{2}+a}\, a^{4} b^{4} x +945 \sqrt {b \,x^{2}+a}\, a^{3} b^{5} x^{3}+1134 \sqrt {b \,x^{2}+a}\, a^{2} b^{6} x^{5}+648 \sqrt {b \,x^{2}+a}\, a \,b^{7} x^{7}-8 \sqrt {b}\, a^{6} b c -720 \sqrt {b}\, a^{4} b^{4} x^{2}-1440 \sqrt {b}\, a^{3} b^{5} x^{4}-1440 \sqrt {b}\, a^{2} b^{6} x^{6}-720 \sqrt {b}\, a \,b^{7} x^{8}+63 \sqrt {b \,x^{2}+a}\, a^{3} b^{4} c \,x^{5}+45 \sqrt {b \,x^{2}+a}\, a^{3} b^{4} d \,x^{7}+36 \sqrt {b \,x^{2}+a}\, a^{2} b^{5} c \,x^{7}+10 \sqrt {b \,x^{2}+a}\, a^{2} b^{5} d \,x^{9}+8 \sqrt {b \,x^{2}+a}\, a \,b^{6} c \,x^{9}-50 \sqrt {b}\, a^{6} b d \,x^{2}-40 \sqrt {b}\, a^{5} b^{2} c \,x^{2}-100 \sqrt {b}\, a^{5} b^{2} d \,x^{4}-80 \sqrt {b}\, a^{4} b^{3} c \,x^{4}-100 \sqrt {b}\, a^{4} b^{3} d \,x^{6}-80 \sqrt {b}\, a^{3} b^{4} c \,x^{6}-50 \sqrt {b}\, a^{3} b^{4} d \,x^{8}-40 \sqrt {b}\, a^{2} b^{5} c \,x^{8}-10 \sqrt {b}\, a^{2} b^{5} d \,x^{10}-8 \sqrt {b}\, a \,b^{6} c \,x^{10}}{315 a^{4} b^{4} \left (b^{5} x^{10}+5 a \,b^{4} x^{8}+10 a^{2} b^{3} x^{6}+10 a^{3} b^{2} x^{4}+5 a^{4} b \,x^{2}+a^{5}\right )} \] Input:

int((D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(11/2),x)
 

Output:

(315*sqrt(a + b*x**2)*a**4*b**4*x + 945*sqrt(a + b*x**2)*a**3*b**5*x**3 + 
63*sqrt(a + b*x**2)*a**3*b**4*c*x**5 + 45*sqrt(a + b*x**2)*a**3*b**4*d*x** 
7 + 1134*sqrt(a + b*x**2)*a**2*b**6*x**5 + 36*sqrt(a + b*x**2)*a**2*b**5*c 
*x**7 + 10*sqrt(a + b*x**2)*a**2*b**5*d*x**9 + 648*sqrt(a + b*x**2)*a*b**7 
*x**7 + 8*sqrt(a + b*x**2)*a*b**6*c*x**9 + 144*sqrt(a + b*x**2)*b**8*x**9 
- 10*sqrt(b)*a**7*d - 8*sqrt(b)*a**6*b*c - 50*sqrt(b)*a**6*b*d*x**2 - 144* 
sqrt(b)*a**5*b**3 - 40*sqrt(b)*a**5*b**2*c*x**2 - 100*sqrt(b)*a**5*b**2*d* 
x**4 - 720*sqrt(b)*a**4*b**4*x**2 - 80*sqrt(b)*a**4*b**3*c*x**4 - 100*sqrt 
(b)*a**4*b**3*d*x**6 - 1440*sqrt(b)*a**3*b**5*x**4 - 80*sqrt(b)*a**3*b**4* 
c*x**6 - 50*sqrt(b)*a**3*b**4*d*x**8 - 1440*sqrt(b)*a**2*b**6*x**6 - 40*sq 
rt(b)*a**2*b**5*c*x**8 - 10*sqrt(b)*a**2*b**5*d*x**10 - 720*sqrt(b)*a*b**7 
*x**8 - 8*sqrt(b)*a*b**6*c*x**10 - 144*sqrt(b)*b**8*x**10)/(315*a**4*b**4* 
(a**5 + 5*a**4*b*x**2 + 10*a**3*b**2*x**4 + 10*a**2*b**3*x**6 + 5*a*b**4*x 
**8 + b**5*x**10))