Integrand size = 32, antiderivative size = 242 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^4 \left (a+b x^2\right )^{9/2}} \, dx=-\frac {A}{3 a x^3 \left (a+b x^2\right )^{7/2}}+\frac {10 A b-3 a B}{3 a^2 x \left (a+b x^2\right )^{7/2}}+\frac {\left (80 A b^2-3 a (8 b B-a C)\right ) x}{3 a^3 \left (a+b x^2\right )^{7/2}}+\frac {\left (160 A b^3-a \left (48 b^2 B-6 a b C-a^2 D\right )\right ) x^3}{3 a^4 \left (a+b x^2\right )^{7/2}}+\frac {4 b \left (160 A b^3-a \left (48 b^2 B-6 a b C-a^2 D\right )\right ) x^5}{15 a^5 \left (a+b x^2\right )^{7/2}}+\frac {8 b^2 \left (160 A b^3-a \left (48 b^2 B-6 a b C-a^2 D\right )\right ) x^7}{105 a^6 \left (a+b x^2\right )^{7/2}} \]
-1/3*A/a/x^3/(b*x^2+a)^(7/2)+1/3*(10*A*b-3*B*a)/a^2/x/(b*x^2+a)^(7/2)+1/3* (80*A*b^2-3*a*(8*B*b-C*a))*x/a^3/(b*x^2+a)^(7/2)+1/3*(160*A*b^3-a*(48*B*b^ 2-6*C*a*b-D*a^2))*x^3/a^4/(b*x^2+a)^(7/2)+4/15*b*(160*A*b^3-a*(48*B*b^2-6* C*a*b-D*a^2))*x^5/a^5/(b*x^2+a)^(7/2)+8/105*b^2*(160*A*b^3-a*(48*B*b^2-6*C *a*b-D*a^2))*x^7/a^6/(b*x^2+a)^(7/2)
Time = 0.51 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.68 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^4 \left (a+b x^2\right )^{9/2}} \, dx=\frac {1280 A b^5 x^{10}+128 a b^4 x^8 \left (35 A-3 B x^2\right )+16 a^2 b^3 x^6 \left (350 A-84 B x^2+3 C x^4\right )-35 a^5 \left (A+3 B x^2-3 C x^4-D x^6\right )+8 a^3 b^2 x^4 \left (350 A-210 B x^2+21 C x^4+D x^6\right )+14 a^4 b x^2 \left (25 A-60 B x^2+15 C x^4+2 D x^6\right )}{105 a^6 x^3 \left (a+b x^2\right )^{7/2}} \]
(1280*A*b^5*x^10 + 128*a*b^4*x^8*(35*A - 3*B*x^2) + 16*a^2*b^3*x^6*(350*A - 84*B*x^2 + 3*C*x^4) - 35*a^5*(A + 3*B*x^2 - 3*C*x^4 - D*x^6) + 8*a^3*b^2 *x^4*(350*A - 210*B*x^2 + 21*C*x^4 + D*x^6) + 14*a^4*b*x^2*(25*A - 60*B*x^ 2 + 15*C*x^4 + 2*D*x^6))/(105*a^6*x^3*(a + b*x^2)^(7/2))
Time = 0.52 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.90, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {2334, 2089, 1588, 298, 209, 209, 208}
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.
| \(\displaystyle \int \frac {A+B x^2+C x^4+D x^6}{x^4 \left (a+b x^2\right )^{9/2}} \, dx\) |
| \(\Big \downarrow \) 2334 |
| \(\displaystyle -\frac {\int \frac {10 A b-3 a \left (D x^4+C x^2+B\right )}{x^2 \left (b x^2+a\right )^{9/2}}dx}{3 a}-\frac {A}{3 a x^3 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 2089 |
| \(\displaystyle -\frac {\int \frac {-3 a D x^4-3 a C x^2+10 A b-3 a B}{x^2 \left (b x^2+a\right )^{9/2}}dx}{3 a}-\frac {A}{3 a x^3 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 1588 |
| \(\displaystyle -\frac {-\frac {\int \frac {80 A b^2+3 a^2 D x^2-3 a (8 b B-a C)}{\left (b x^2+a\right )^{9/2}}dx}{a}-\frac {10 A b-3 a B}{a x \left (a+b x^2\right )^{7/2}}}{3 a}-\frac {A}{3 a x^3 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 298 |
| \(\displaystyle -\frac {-\frac {\frac {3 \left (160 A b^3-a \left (a^2 (-D)-6 a b C+48 b^2 B\right )\right ) \int \frac {1}{\left (b x^2+a\right )^{7/2}}dx}{7 a b}+\frac {x \left (80 A b^3-3 a \left (a^2 D-a b C+8 b^2 B\right )\right )}{7 a b \left (a+b x^2\right )^{7/2}}}{a}-\frac {10 A b-3 a B}{a x \left (a+b x^2\right )^{7/2}}}{3 a}-\frac {A}{3 a x^3 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle -\frac {-\frac {\frac {3 \left (160 A b^3-a \left (a^2 (-D)-6 a b C+48 b^2 B\right )\right ) \left (\frac {4 \int \frac {1}{\left (b x^2+a\right )^{5/2}}dx}{5 a}+\frac {x}{5 a \left (a+b x^2\right )^{5/2}}\right )}{7 a b}+\frac {x \left (80 A b^3-3 a \left (a^2 D-a b C+8 b^2 B\right )\right )}{7 a b \left (a+b x^2\right )^{7/2}}}{a}-\frac {10 A b-3 a B}{a x \left (a+b x^2\right )^{7/2}}}{3 a}-\frac {A}{3 a x^3 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle -\frac {-\frac {\frac {3 \left (160 A b^3-a \left (a^2 (-D)-6 a b C+48 b^2 B\right )\right ) \left (\frac {4 \left (\frac {2 \int \frac {1}{\left (b x^2+a\right )^{3/2}}dx}{3 a}+\frac {x}{3 a \left (a+b x^2\right )^{3/2}}\right )}{5 a}+\frac {x}{5 a \left (a+b x^2\right )^{5/2}}\right )}{7 a b}+\frac {x \left (80 A b^3-3 a \left (a^2 D-a b C+8 b^2 B\right )\right )}{7 a b \left (a+b x^2\right )^{7/2}}}{a}-\frac {10 A b-3 a B}{a x \left (a+b x^2\right )^{7/2}}}{3 a}-\frac {A}{3 a x^3 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 208 |
| \(\displaystyle -\frac {-\frac {\frac {x \left (80 A b^3-3 a \left (a^2 D-a b C+8 b^2 B\right )\right )}{7 a b \left (a+b x^2\right )^{7/2}}+\frac {3 \left (\frac {4 \left (\frac {2 x}{3 a^2 \sqrt {a+b x^2}}+\frac {x}{3 a \left (a+b x^2\right )^{3/2}}\right )}{5 a}+\frac {x}{5 a \left (a+b x^2\right )^{5/2}}\right ) \left (160 A b^3-a \left (a^2 (-D)-6 a b C+48 b^2 B\right )\right )}{7 a b}}{a}-\frac {10 A b-3 a B}{a x \left (a+b x^2\right )^{7/2}}}{3 a}-\frac {A}{3 a x^3 \left (a+b x^2\right )^{7/2}}\) |
-1/3*A/(a*x^3*(a + b*x^2)^(7/2)) - (-((10*A*b - 3*a*B)/(a*x*(a + b*x^2)^(7 /2))) - (((80*A*b^3 - 3*a*(8*b^2*B - a*b*C + a^2*D))*x)/(7*a*b*(a + b*x^2) ^(7/2)) + (3*(160*A*b^3 - a*(48*b^2*B - 6*a*b*C - a^2*D))*(x/(5*a*(a + b*x ^2)^(5/2)) + (4*(x/(3*a*(a + b*x^2)^(3/2)) + (2*x)/(3*a^2*Sqrt[a + b*x^2]) ))/(5*a)))/(7*a*b))/a)/(3*a)
3.2.65.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && ILtQ[p + 3/2, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-( b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 2*p + 3))/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 0])
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, f*x, x], R = PolynomialRemainder[(a + b*x^2 + c*x^4)^p, f*x, x]}, Simp[R*(f*x)^(m + 1)*((d + e*x^2)^(q + 1)/(d*f*(m + 1))), x] + Simp[1/(d*f ^2*(m + 1)) Int[(f*x)^(m + 2)*(d + e*x^2)^q*ExpandToSum[d*f*(m + 1)*(Qx/x ) - e*R*(m + 2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && Ne Q[b^2 - 4*a*c, 0] && IGtQ[p, 0] && LtQ[m, -1]
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])
Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{A = Coef f[Pq, x, 0], Q = PolynomialQuotient[Pq - Coeff[Pq, x, 0], x^2, x]}, Simp[A* x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] + Simp[1/(a*(m + 1)) Int[ x^(m + 2)*(a + b*x^2)^p*(a*(m + 1)*Q - A*b*(m + 2*(p + 1) + 1)), x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x^2] && IntegerQ[m/2] && ILtQ[(m + 1)/2 + p, 0] && LtQ[m + Expon[Pq, x] + 2*p + 1, 0]
Time = 3.55 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.64
| method | result | size |
| pseudoelliptic | \(\frac {7 \left (D x^{6}+3 C \,x^{4}-3 x^{2} B -A \right ) a^{5}+70 b \,x^{2} \left (\frac {2}{25} D x^{6}+\frac {3}{5} C \,x^{4}-\frac {12}{5} x^{2} B +A \right ) a^{4}+560 b^{2} x^{4} \left (\frac {1}{350} D x^{6}+\frac {3}{50} C \,x^{4}-\frac {3}{5} x^{2} B +A \right ) a^{3}+1120 \left (\frac {3}{350} C \,x^{4}-\frac {6}{25} x^{2} B +A \right ) b^{3} x^{6} a^{2}+896 \left (-\frac {3 x^{2} B}{35}+A \right ) b^{4} x^{8} a +256 A \,b^{5} x^{10}}{21 \left (b \,x^{2}+a \right )^{\frac {7}{2}} x^{3} a^{6}}\) | \(156\) |
| gosper | \(-\frac {-1280 A \,b^{5} x^{10}+384 B a \,b^{4} x^{10}-48 C \,a^{2} b^{3} x^{10}-8 D a^{3} b^{2} x^{10}-4480 a A \,b^{4} x^{8}+1344 B \,a^{2} b^{3} x^{8}-168 C \,a^{3} b^{2} x^{8}-28 D a^{4} b \,x^{8}-5600 a^{2} A \,b^{3} x^{6}+1680 B \,a^{3} b^{2} x^{6}-210 C \,a^{4} b \,x^{6}-35 D a^{5} x^{6}-2800 a^{3} A \,b^{2} x^{4}+840 B \,a^{4} b \,x^{4}-105 C \,a^{5} x^{4}-350 a^{4} A b \,x^{2}+105 B \,a^{5} x^{2}+35 a^{5} A}{105 x^{3} \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{6}}\) | \(205\) |
| trager | \(-\frac {-1280 A \,b^{5} x^{10}+384 B a \,b^{4} x^{10}-48 C \,a^{2} b^{3} x^{10}-8 D a^{3} b^{2} x^{10}-4480 a A \,b^{4} x^{8}+1344 B \,a^{2} b^{3} x^{8}-168 C \,a^{3} b^{2} x^{8}-28 D a^{4} b \,x^{8}-5600 a^{2} A \,b^{3} x^{6}+1680 B \,a^{3} b^{2} x^{6}-210 C \,a^{4} b \,x^{6}-35 D a^{5} x^{6}-2800 a^{3} A \,b^{2} x^{4}+840 B \,a^{4} b \,x^{4}-105 C \,a^{5} x^{4}-350 a^{4} A b \,x^{2}+105 B \,a^{5} x^{2}+35 a^{5} A}{105 x^{3} \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{6}}\) | \(205\) |
| default | \(C \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )+D \left (-\frac {x}{6 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {a \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{6 b}\right )+A \left (-\frac {1}{3 a \,x^{3} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {10 b \left (-\frac {1}{a x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {8 b \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{a}\right )}{3 a}\right )+B \left (-\frac {1}{a x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {8 b \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{a}\right )\) | \(396\) |
1/21*(7*(D*x^6+3*C*x^4-3*B*x^2-A)*a^5+70*b*x^2*(2/25*D*x^6+3/5*C*x^4-12/5* x^2*B+A)*a^4+560*b^2*x^4*(1/350*D*x^6+3/50*C*x^4-3/5*x^2*B+A)*a^3+1120*(3/ 350*C*x^4-6/25*x^2*B+A)*b^3*x^6*a^2+896*(-3/35*x^2*B+A)*b^4*x^8*a+256*A*b^ 5*x^10)/(b*x^2+a)^(7/2)/x^3/a^6
Time = 0.39 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.93 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^4 \left (a+b x^2\right )^{9/2}} \, dx=\frac {{\left (8 \, {\left (D a^{3} b^{2} + 6 \, C a^{2} b^{3} - 48 \, B a b^{4} + 160 \, A b^{5}\right )} x^{10} + 28 \, {\left (D a^{4} b + 6 \, C a^{3} b^{2} - 48 \, B a^{2} b^{3} + 160 \, A a b^{4}\right )} x^{8} + 35 \, {\left (D a^{5} + 6 \, C a^{4} b - 48 \, B a^{3} b^{2} + 160 \, A a^{2} b^{3}\right )} x^{6} - 35 \, A a^{5} + 35 \, {\left (3 \, C a^{5} - 24 \, B a^{4} b + 80 \, A a^{3} b^{2}\right )} x^{4} - 35 \, {\left (3 \, B a^{5} - 10 \, A a^{4} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{105 \, {\left (a^{6} b^{4} x^{11} + 4 \, a^{7} b^{3} x^{9} + 6 \, a^{8} b^{2} x^{7} + 4 \, a^{9} b x^{5} + a^{10} x^{3}\right )}} \]
1/105*(8*(D*a^3*b^2 + 6*C*a^2*b^3 - 48*B*a*b^4 + 160*A*b^5)*x^10 + 28*(D*a ^4*b + 6*C*a^3*b^2 - 48*B*a^2*b^3 + 160*A*a*b^4)*x^8 + 35*(D*a^5 + 6*C*a^4 *b - 48*B*a^3*b^2 + 160*A*a^2*b^3)*x^6 - 35*A*a^5 + 35*(3*C*a^5 - 24*B*a^4 *b + 80*A*a^3*b^2)*x^4 - 35*(3*B*a^5 - 10*A*a^4*b)*x^2)*sqrt(b*x^2 + a)/(a ^6*b^4*x^11 + 4*a^7*b^3*x^9 + 6*a^8*b^2*x^7 + 4*a^9*b*x^5 + a^10*x^3)
Leaf count of result is larger than twice the leaf count of optimal. 2861 vs. \(2 (224) = 448\).
Time = 120.75 (sec) , antiderivative size = 2861, normalized size of antiderivative = 11.82 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^4 \left (a+b x^2\right )^{9/2}} \, dx=\text {Too large to display} \]
A*(-7*a**6*b**(51/2)*sqrt(a/(b*x**2) + 1)/(21*a**11*b**25*x**2 + 105*a**10 *b**26*x**4 + 210*a**9*b**27*x**6 + 210*a**8*b**28*x**8 + 105*a**7*b**29*x **10 + 21*a**6*b**30*x**12) + 63*a**5*b**(53/2)*x**2*sqrt(a/(b*x**2) + 1)/ (21*a**11*b**25*x**2 + 105*a**10*b**26*x**4 + 210*a**9*b**27*x**6 + 210*a* *8*b**28*x**8 + 105*a**7*b**29*x**10 + 21*a**6*b**30*x**12) + 630*a**4*b** (55/2)*x**4*sqrt(a/(b*x**2) + 1)/(21*a**11*b**25*x**2 + 105*a**10*b**26*x* *4 + 210*a**9*b**27*x**6 + 210*a**8*b**28*x**8 + 105*a**7*b**29*x**10 + 21 *a**6*b**30*x**12) + 1680*a**3*b**(57/2)*x**6*sqrt(a/(b*x**2) + 1)/(21*a** 11*b**25*x**2 + 105*a**10*b**26*x**4 + 210*a**9*b**27*x**6 + 210*a**8*b**2 8*x**8 + 105*a**7*b**29*x**10 + 21*a**6*b**30*x**12) + 2016*a**2*b**(59/2) *x**8*sqrt(a/(b*x**2) + 1)/(21*a**11*b**25*x**2 + 105*a**10*b**26*x**4 + 2 10*a**9*b**27*x**6 + 210*a**8*b**28*x**8 + 105*a**7*b**29*x**10 + 21*a**6* b**30*x**12) + 1152*a*b**(61/2)*x**10*sqrt(a/(b*x**2) + 1)/(21*a**11*b**25 *x**2 + 105*a**10*b**26*x**4 + 210*a**9*b**27*x**6 + 210*a**8*b**28*x**8 + 105*a**7*b**29*x**10 + 21*a**6*b**30*x**12) + 256*b**(63/2)*x**12*sqrt(a/ (b*x**2) + 1)/(21*a**11*b**25*x**2 + 105*a**10*b**26*x**4 + 210*a**9*b**27 *x**6 + 210*a**8*b**28*x**8 + 105*a**7*b**29*x**10 + 21*a**6*b**30*x**12)) + B*(-35*a**4*b**(33/2)*sqrt(a/(b*x**2) + 1)/(35*a**9*b**16 + 140*a**8*b* *17*x**2 + 210*a**7*b**18*x**4 + 140*a**6*b**19*x**6 + 35*a**5*b**20*x**8) - 280*a**3*b**(35/2)*x**2*sqrt(a/(b*x**2) + 1)/(35*a**9*b**16 + 140*a*...
Time = 0.22 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.39 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^4 \left (a+b x^2\right )^{9/2}} \, dx=\frac {16 \, C x}{35 \, \sqrt {b x^{2} + a} a^{4}} + \frac {8 \, C x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3}} + \frac {6 \, C x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2}} + \frac {C x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a} - \frac {D x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} + \frac {8 \, D x}{105 \, \sqrt {b x^{2} + a} a^{3} b} + \frac {4 \, D x}{105 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} b} + \frac {D x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a b} - \frac {128 \, B b x}{35 \, \sqrt {b x^{2} + a} a^{5}} - \frac {64 \, B b x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{4}} - \frac {48 \, B b x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{3}} - \frac {8 \, B b x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2}} + \frac {256 \, A b^{2} x}{21 \, \sqrt {b x^{2} + a} a^{6}} + \frac {128 \, A b^{2} x}{21 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{5}} + \frac {32 \, A b^{2} x}{7 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{4}} + \frac {80 \, A b^{2} x}{21 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{3}} - \frac {B}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} a x} + \frac {10 \, A b}{3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2} x} - \frac {A}{3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a x^{3}} \]
16/35*C*x/(sqrt(b*x^2 + a)*a^4) + 8/35*C*x/((b*x^2 + a)^(3/2)*a^3) + 6/35* C*x/((b*x^2 + a)^(5/2)*a^2) + 1/7*C*x/((b*x^2 + a)^(7/2)*a) - 1/7*D*x/((b* x^2 + a)^(7/2)*b) + 8/105*D*x/(sqrt(b*x^2 + a)*a^3*b) + 4/105*D*x/((b*x^2 + a)^(3/2)*a^2*b) + 1/35*D*x/((b*x^2 + a)^(5/2)*a*b) - 128/35*B*b*x/(sqrt( b*x^2 + a)*a^5) - 64/35*B*b*x/((b*x^2 + a)^(3/2)*a^4) - 48/35*B*b*x/((b*x^ 2 + a)^(5/2)*a^3) - 8/7*B*b*x/((b*x^2 + a)^(7/2)*a^2) + 256/21*A*b^2*x/(sq rt(b*x^2 + a)*a^6) + 128/21*A*b^2*x/((b*x^2 + a)^(3/2)*a^5) + 32/7*A*b^2*x /((b*x^2 + a)^(5/2)*a^4) + 80/21*A*b^2*x/((b*x^2 + a)^(7/2)*a^3) - B/((b*x ^2 + a)^(7/2)*a*x) + 10/3*A*b/((b*x^2 + a)^(7/2)*a^2*x) - 1/3*A/((b*x^2 + a)^(7/2)*a*x^3)
Time = 0.29 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.44 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^4 \left (a+b x^2\right )^{9/2}} \, dx=\frac {{\left ({\left (x^{2} {\left (\frac {{\left (8 \, D a^{15} b^{5} + 48 \, C a^{14} b^{6} - 279 \, B a^{13} b^{7} + 790 \, A a^{12} b^{8}\right )} x^{2}}{a^{18} b^{3}} + \frac {7 \, {\left (4 \, D a^{16} b^{4} + 24 \, C a^{15} b^{5} - 132 \, B a^{14} b^{6} + 365 \, A a^{13} b^{7}\right )}}{a^{18} b^{3}}\right )} + \frac {35 \, {\left (D a^{17} b^{3} + 6 \, C a^{16} b^{4} - 30 \, B a^{15} b^{5} + 80 \, A a^{14} b^{6}\right )}}{a^{18} b^{3}}\right )} x^{2} + \frac {105 \, {\left (C a^{17} b^{3} - 4 \, B a^{16} b^{4} + 10 \, A a^{15} b^{5}\right )}}{a^{18} b^{3}}\right )} x}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} + \frac {2 \, {\left (3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a \sqrt {b} - 12 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A b^{\frac {3}{2}} - 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{2} \sqrt {b} + 30 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a b^{\frac {3}{2}} + 3 \, B a^{3} \sqrt {b} - 14 \, A a^{2} b^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{3} a^{5}} \]
1/105*((x^2*((8*D*a^15*b^5 + 48*C*a^14*b^6 - 279*B*a^13*b^7 + 790*A*a^12*b ^8)*x^2/(a^18*b^3) + 7*(4*D*a^16*b^4 + 24*C*a^15*b^5 - 132*B*a^14*b^6 + 36 5*A*a^13*b^7)/(a^18*b^3)) + 35*(D*a^17*b^3 + 6*C*a^16*b^4 - 30*B*a^15*b^5 + 80*A*a^14*b^6)/(a^18*b^3))*x^2 + 105*(C*a^17*b^3 - 4*B*a^16*b^4 + 10*A*a ^15*b^5)/(a^18*b^3))*x/(b*x^2 + a)^(7/2) + 2/3*(3*(sqrt(b)*x - sqrt(b*x^2 + a))^4*B*a*sqrt(b) - 12*(sqrt(b)*x - sqrt(b*x^2 + a))^4*A*b^(3/2) - 6*(sq rt(b)*x - sqrt(b*x^2 + a))^2*B*a^2*sqrt(b) + 30*(sqrt(b)*x - sqrt(b*x^2 + a))^2*A*a*b^(3/2) + 3*B*a^3*sqrt(b) - 14*A*a^2*b^(3/2))/(((sqrt(b)*x - sqr t(b*x^2 + a))^2 - a)^3*a^5)
Timed out. \[ \int \frac {A+B x^2+C x^4+D x^6}{x^4 \left (a+b x^2\right )^{9/2}} \, dx=\int \frac {A+B\,x^2+C\,x^4+x^6\,D}{x^4\,{\left (b\,x^2+a\right )}^{9/2}} \,d x \]