Integrand size = 25, antiderivative size = 132 \[ \int \frac {x^5 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=-\frac {x^5 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {x^4 (A b+6 a C-5 b B x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}+\frac {4 (A b+6 a C)}{105 b^4 \left (a+b x^2\right )^{3/2}}-\frac {4 (A b+6 a C)}{35 a b^4 \sqrt {a+b x^2}} \]
-1/7*x^5*(B*a-(A*b-C*a)*x)/a/b/(b*x^2+a)^(7/2)-1/35*x^4*(-5*B*b*x+A*b+6*C* a)/a/b^2/(b*x^2+a)^(5/2)+4/105*(A*b+6*C*a)/b^4/(b*x^2+a)^(3/2)-4/35*(A*b+6 *C*a)/a/b^4/(b*x^2+a)^(1/2)
Time = 0.69 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.67 \[ \int \frac {x^5 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {-48 a^4 C+15 b^4 B x^7-35 a b^3 x^4 \left (A+3 C x^2\right )-14 a^2 b^2 x^2 \left (2 A+15 C x^2\right )-8 a^3 b \left (A+21 C x^2\right )}{105 a b^4 \left (a+b x^2\right )^{7/2}} \]
(-48*a^4*C + 15*b^4*B*x^7 - 35*a*b^3*x^4*(A + 3*C*x^2) - 14*a^2*b^2*x^2*(2 *A + 15*C*x^2) - 8*a^3*b*(A + 21*C*x^2))/(105*a*b^4*(a + b*x^2)^(7/2))
Time = 0.50 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.17, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {2335, 25, 530, 27, 2345, 27, 453}
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.
| \(\displaystyle \int \frac {x^5 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx\) |
| \(\Big \downarrow \) 2335 |
| \(\displaystyle -\frac {\int -\frac {x^4 (5 a B+(A b+6 a C) x)}{\left (b x^2+a\right )^{7/2}}dx}{7 a b}-\frac {x^5 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {x^4 (5 a B+(A b+6 a C) x)}{\left (b x^2+a\right )^{7/2}}dx}{7 a b}-\frac {x^5 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 530 |
| \(\displaystyle \frac {-\frac {\int \frac {5 \left (\frac {B a^3}{b^2}-\frac {5 B x^2 a^2}{b}+\frac {(A b+6 a C) x a^2}{b^2}-\left (A+\frac {6 a C}{b}\right ) x^3 a\right )}{\left (b x^2+a\right )^{5/2}}dx}{5 a}-\frac {a^2 (6 a C+A b-5 b B x)}{5 b^3 \left (a+b x^2\right )^{5/2}}}{7 a b}-\frac {x^5 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int \frac {\frac {B a^3}{b^2}-\frac {5 B x^2 a^2}{b}+\frac {(A b+6 a C) x a^2}{b^2}-\left (A+\frac {6 a C}{b}\right ) x^3 a}{\left (b x^2+a\right )^{5/2}}dx}{a}-\frac {a^2 (6 a C+A b-5 b B x)}{5 b^3 \left (a+b x^2\right )^{5/2}}}{7 a b}-\frac {x^5 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle \frac {-\frac {-\frac {\int \frac {3 a^2 (a B+(A b+6 a C) x)}{b^2 \left (b x^2+a\right )^{3/2}}dx}{3 a}-\frac {2 a^2 (6 a C+A b-3 b B x)}{3 b^3 \left (a+b x^2\right )^{3/2}}}{a}-\frac {a^2 (6 a C+A b-5 b B x)}{5 b^3 \left (a+b x^2\right )^{5/2}}}{7 a b}-\frac {x^5 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {a \int \frac {a B+(A b+6 a C) x}{\left (b x^2+a\right )^{3/2}}dx}{b^2}-\frac {2 a^2 (6 a C+A b-3 b B x)}{3 b^3 \left (a+b x^2\right )^{3/2}}}{a}-\frac {a^2 (6 a C+A b-5 b B x)}{5 b^3 \left (a+b x^2\right )^{5/2}}}{7 a b}-\frac {x^5 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 453 |
| \(\displaystyle \frac {-\frac {a^2 (6 a C+A b-5 b B x)}{5 b^3 \left (a+b x^2\right )^{5/2}}-\frac {\frac {a (6 a C+A b-b B x)}{b^3 \sqrt {a+b x^2}}-\frac {2 a^2 (6 a C+A b-3 b B x)}{3 b^3 \left (a+b x^2\right )^{3/2}}}{a}}{7 a b}-\frac {x^5 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}}\) |
-1/7*(x^5*(a*B - (A*b - a*C)*x))/(a*b*(a + b*x^2)^(7/2)) + (-1/5*(a^2*(A*b + 6*a*C - 5*b*B*x))/(b^3*(a + b*x^2)^(5/2)) - ((-2*a^2*(A*b + 6*a*C - 3*b *B*x))/(3*b^3*(a + b*x^2)^(3/2)) + (a*(A*b + 6*a*C - b*B*x))/(b^3*Sqrt[a + b*x^2]))/a)/(7*a*b)
3.1.49.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_) + (d_.)*(x_))/((a_) + (b_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[-(a* d - b*c*x)/(a*b*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b, c, d}, x]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symb ol] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Co eff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Po lynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x )*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Qx + e*(2*p + 3), x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && IGtQ[m, 0] && LtQ[p, -1] && EqQ[n, 1] && IntegerQ[2*p]
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]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot ient[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[(a*g - b *f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) In t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
Time = 3.56 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.72
| method | result | size |
| gosper | \(-\frac {-15 x^{7} B \,b^{4}+105 C \,x^{6} a \,b^{3}+35 A a \,b^{3} x^{4}+210 C \,a^{2} b^{2} x^{4}+28 A \,a^{2} b^{2} x^{2}+168 C \,a^{3} b \,x^{2}+8 A \,a^{3} b +48 C \,a^{4}}{105 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a \,b^{4}}\) | \(95\) |
| trager | \(-\frac {-15 x^{7} B \,b^{4}+105 C \,x^{6} a \,b^{3}+35 A a \,b^{3} x^{4}+210 C \,a^{2} b^{2} x^{4}+28 A \,a^{2} b^{2} x^{2}+168 C \,a^{3} b \,x^{2}+8 A \,a^{3} b +48 C \,a^{4}}{105 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a \,b^{4}}\) | \(95\) |
| default | \(C \left (-\frac {x^{6}}{b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {6 a \left (-\frac {x^{4}}{3 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {4 a \left (-\frac {x^{2}}{5 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {2 a}{35 b^{2} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}\right )}{3 b}\right )}{b}\right )+B \left (-\frac {x^{5}}{2 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {5 a \left (-\frac {x^{3}}{4 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {3 a \left (-\frac {x}{6 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {a \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{6 b}\right )}{4 b}\right )}{2 b}\right )+A \left (-\frac {x^{4}}{3 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {4 a \left (-\frac {x^{2}}{5 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {2 a}{35 b^{2} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}\right )}{3 b}\right )\) | \(289\) |
-1/105*(-15*B*b^4*x^7+105*C*a*b^3*x^6+35*A*a*b^3*x^4+210*C*a^2*b^2*x^4+28* A*a^2*b^2*x^2+168*C*a^3*b*x^2+8*A*a^3*b+48*C*a^4)/(b*x^2+a)^(7/2)/a/b^4
Time = 0.29 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.04 \[ \int \frac {x^5 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {{\left (15 \, B b^{4} x^{7} - 105 \, C a b^{3} x^{6} - 48 \, C a^{4} - 8 \, A a^{3} b - 35 \, {\left (6 \, C a^{2} b^{2} + A a b^{3}\right )} x^{4} - 28 \, {\left (6 \, C a^{3} b + A a^{2} b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{105 \, {\left (a b^{8} x^{8} + 4 \, a^{2} b^{7} x^{6} + 6 \, a^{3} b^{6} x^{4} + 4 \, a^{4} b^{5} x^{2} + a^{5} b^{4}\right )}} \]
1/105*(15*B*b^4*x^7 - 105*C*a*b^3*x^6 - 48*C*a^4 - 8*A*a^3*b - 35*(6*C*a^2 *b^2 + A*a*b^3)*x^4 - 28*(6*C*a^3*b + A*a^2*b^2)*x^2)*sqrt(b*x^2 + a)/(a*b ^8*x^8 + 4*a^2*b^7*x^6 + 6*a^3*b^6*x^4 + 4*a^4*b^5*x^2 + a^5*b^4)
Leaf count of result is larger than twice the leaf count of optimal. 274 vs. \(2 (121) = 242\).
Time = 20.66 (sec) , antiderivative size = 740, normalized size of antiderivative = 5.61 \[ \int \frac {x^5 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=A \left (\begin {cases} - \frac {8 a^{2}}{105 a^{3} b^{3} \sqrt {a + b x^{2}} + 315 a^{2} b^{4} x^{2} \sqrt {a + b x^{2}} + 315 a b^{5} x^{4} \sqrt {a + b x^{2}} + 105 b^{6} x^{6} \sqrt {a + b x^{2}}} - \frac {28 a b x^{2}}{105 a^{3} b^{3} \sqrt {a + b x^{2}} + 315 a^{2} b^{4} x^{2} \sqrt {a + b x^{2}} + 315 a b^{5} x^{4} \sqrt {a + b x^{2}} + 105 b^{6} x^{6} \sqrt {a + b x^{2}}} - \frac {35 b^{2} x^{4}}{105 a^{3} b^{3} \sqrt {a + b x^{2}} + 315 a^{2} b^{4} x^{2} \sqrt {a + b x^{2}} + 315 a b^{5} x^{4} \sqrt {a + b x^{2}} + 105 b^{6} x^{6} \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{6}}{6 a^{\frac {9}{2}}} & \text {otherwise} \end {cases}\right ) + \frac {B x^{7}}{7 a^{\frac {9}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 21 a^{\frac {7}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}} + 21 a^{\frac {5}{2}} b^{2} x^{4} \sqrt {1 + \frac {b x^{2}}{a}} + 7 a^{\frac {3}{2}} b^{3} x^{6} \sqrt {1 + \frac {b x^{2}}{a}}} + C \left (\begin {cases} - \frac {16 a^{3}}{35 a^{3} b^{4} \sqrt {a + b x^{2}} + 105 a^{2} b^{5} x^{2} \sqrt {a + b x^{2}} + 105 a b^{6} x^{4} \sqrt {a + b x^{2}} + 35 b^{7} x^{6} \sqrt {a + b x^{2}}} - \frac {56 a^{2} b x^{2}}{35 a^{3} b^{4} \sqrt {a + b x^{2}} + 105 a^{2} b^{5} x^{2} \sqrt {a + b x^{2}} + 105 a b^{6} x^{4} \sqrt {a + b x^{2}} + 35 b^{7} x^{6} \sqrt {a + b x^{2}}} - \frac {70 a b^{2} x^{4}}{35 a^{3} b^{4} \sqrt {a + b x^{2}} + 105 a^{2} b^{5} x^{2} \sqrt {a + b x^{2}} + 105 a b^{6} x^{4} \sqrt {a + b x^{2}} + 35 b^{7} x^{6} \sqrt {a + b x^{2}}} - \frac {35 b^{3} x^{6}}{35 a^{3} b^{4} \sqrt {a + b x^{2}} + 105 a^{2} b^{5} x^{2} \sqrt {a + b x^{2}} + 105 a b^{6} x^{4} \sqrt {a + b x^{2}} + 35 b^{7} x^{6} \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{8}}{8 a^{\frac {9}{2}}} & \text {otherwise} \end {cases}\right ) \]
A*Piecewise((-8*a**2/(105*a**3*b**3*sqrt(a + b*x**2) + 315*a**2*b**4*x**2* sqrt(a + b*x**2) + 315*a*b**5*x**4*sqrt(a + b*x**2) + 105*b**6*x**6*sqrt(a + b*x**2)) - 28*a*b*x**2/(105*a**3*b**3*sqrt(a + b*x**2) + 315*a**2*b**4* x**2*sqrt(a + b*x**2) + 315*a*b**5*x**4*sqrt(a + b*x**2) + 105*b**6*x**6*s qrt(a + b*x**2)) - 35*b**2*x**4/(105*a**3*b**3*sqrt(a + b*x**2) + 315*a**2 *b**4*x**2*sqrt(a + b*x**2) + 315*a*b**5*x**4*sqrt(a + b*x**2) + 105*b**6* x**6*sqrt(a + b*x**2)), Ne(b, 0)), (x**6/(6*a**(9/2)), True)) + B*x**7/(7* a**(9/2)*sqrt(1 + b*x**2/a) + 21*a**(7/2)*b*x**2*sqrt(1 + b*x**2/a) + 21*a **(5/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 7*a**(3/2)*b**3*x**6*sqrt(1 + b*x** 2/a)) + C*Piecewise((-16*a**3/(35*a**3*b**4*sqrt(a + b*x**2) + 105*a**2*b* *5*x**2*sqrt(a + b*x**2) + 105*a*b**6*x**4*sqrt(a + b*x**2) + 35*b**7*x**6 *sqrt(a + b*x**2)) - 56*a**2*b*x**2/(35*a**3*b**4*sqrt(a + b*x**2) + 105*a **2*b**5*x**2*sqrt(a + b*x**2) + 105*a*b**6*x**4*sqrt(a + b*x**2) + 35*b** 7*x**6*sqrt(a + b*x**2)) - 70*a*b**2*x**4/(35*a**3*b**4*sqrt(a + b*x**2) + 105*a**2*b**5*x**2*sqrt(a + b*x**2) + 105*a*b**6*x**4*sqrt(a + b*x**2) + 35*b**7*x**6*sqrt(a + b*x**2)) - 35*b**3*x**6/(35*a**3*b**4*sqrt(a + b*x** 2) + 105*a**2*b**5*x**2*sqrt(a + b*x**2) + 105*a*b**6*x**4*sqrt(a + b*x**2 ) + 35*b**7*x**6*sqrt(a + b*x**2)), Ne(b, 0)), (x**8/(8*a**(9/2)), True))
Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (116) = 232\).
Time = 0.21 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.82 \[ \int \frac {x^5 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=-\frac {C x^{6}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b} - \frac {B x^{5}}{2 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} - \frac {2 \, C a x^{4}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} - \frac {A x^{4}}{3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} - \frac {5 \, B a x^{3}}{8 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} - \frac {8 \, C a^{2} x^{2}}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}} - \frac {4 \, A a x^{2}}{15 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} + \frac {B x}{14 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{3}} + \frac {B x}{7 \, \sqrt {b x^{2} + a} a b^{3}} + \frac {3 \, B a x}{56 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{3}} - \frac {15 \, B a^{2} x}{56 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}} - \frac {16 \, C a^{3}}{35 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{4}} - \frac {8 \, A a^{2}}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}} \]
-C*x^6/((b*x^2 + a)^(7/2)*b) - 1/2*B*x^5/((b*x^2 + a)^(7/2)*b) - 2*C*a*x^4 /((b*x^2 + a)^(7/2)*b^2) - 1/3*A*x^4/((b*x^2 + a)^(7/2)*b) - 5/8*B*a*x^3/( (b*x^2 + a)^(7/2)*b^2) - 8/5*C*a^2*x^2/((b*x^2 + a)^(7/2)*b^3) - 4/15*A*a* x^2/((b*x^2 + a)^(7/2)*b^2) + 1/14*B*x/((b*x^2 + a)^(3/2)*b^3) + 1/7*B*x/( sqrt(b*x^2 + a)*a*b^3) + 3/56*B*a*x/((b*x^2 + a)^(5/2)*b^3) - 15/56*B*a^2* x/((b*x^2 + a)^(7/2)*b^3) - 16/35*C*a^3/((b*x^2 + a)^(7/2)*b^4) - 8/105*A* a^2/((b*x^2 + a)^(7/2)*b^3)
Time = 0.31 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.85 \[ \int \frac {x^5 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {{\left (5 \, {\left (3 \, {\left (\frac {B x}{a} - \frac {7 \, C}{b}\right )} x^{2} - \frac {7 \, {\left (6 \, C a^{4} b^{2} + A a^{3} b^{3}\right )}}{a^{3} b^{4}}\right )} x^{2} - \frac {28 \, {\left (6 \, C a^{5} b + A a^{4} b^{2}\right )}}{a^{3} b^{4}}\right )} x^{2} - \frac {8 \, {\left (6 \, C a^{6} + A a^{5} b\right )}}{a^{3} b^{4}}}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} \]
1/105*((5*(3*(B*x/a - 7*C/b)*x^2 - 7*(6*C*a^4*b^2 + A*a^3*b^3)/(a^3*b^4))* x^2 - 28*(6*C*a^5*b + A*a^4*b^2)/(a^3*b^4))*x^2 - 8*(6*C*a^6 + A*a^5*b)/(a ^3*b^4))/(b*x^2 + a)^(7/2)
Time = 5.67 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.48 \[ \int \frac {x^5 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {\frac {a\,\left (\frac {C}{3\,b^3}-\frac {7\,A\,b-14\,C\,a}{21\,a\,b^3}\right )}{b}-\frac {3\,B\,x}{7\,b^3}}{{\left (b\,x^2+a\right )}^{3/2}}-\frac {\frac {a^2\,\left (\frac {A}{7\,b}-\frac {C\,a}{7\,b^2}\right )}{b^2}+\frac {B\,a^2\,x}{7\,b^3}}{{\left (b\,x^2+a\right )}^{7/2}}-\frac {\frac {C}{b^4}-\frac {B\,x}{7\,a\,b^3}}{\sqrt {b\,x^2+a}}-\frac {\frac {a\,\left (\frac {7\,C\,a^2-7\,A\,a\,b}{35\,a\,b^3}+\frac {a\,\left (\frac {C}{5\,b^2}-\frac {7\,A\,b^2-7\,C\,a\,b}{35\,a\,b^3}\right )}{b}\right )}{b}-\frac {3\,B\,a\,x}{7\,b^3}}{{\left (b\,x^2+a\right )}^{5/2}} \]
((a*(C/(3*b^3) - (7*A*b - 14*C*a)/(21*a*b^3)))/b - (3*B*x)/(7*b^3))/(a + b *x^2)^(3/2) - ((a^2*(A/(7*b) - (C*a)/(7*b^2)))/b^2 + (B*a^2*x)/(7*b^3))/(a + b*x^2)^(7/2) - (C/b^4 - (B*x)/(7*a*b^3))/(a + b*x^2)^(1/2) - ((a*((7*C* a^2 - 7*A*a*b)/(35*a*b^3) + (a*(C/(5*b^2) - (7*A*b^2 - 7*C*a*b)/(35*a*b^3) ))/b))/b - (3*B*a*x)/(7*b^3))/(a + b*x^2)^(5/2)