Integrand size = 32, antiderivative size = 281 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^6 \left (a+b x^2\right )^{9/2}} \, dx=-\frac {A}{5 a x^5 \left (a+b x^2\right )^{7/2}}+\frac {12 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{7/2}}-\frac {24 A b^2-a (10 b B-3 a C)}{3 a^3 x \left (a+b x^2\right )^{7/2}}-\frac {\left (192 A b^3-a \left (80 b^2 B-24 a b C+3 a^2 D\right )\right ) x}{21 a^4 \left (a+b x^2\right )^{7/2}}-\frac {2 \left (192 A b^3-a \left (80 b^2 B-24 a b C+3 a^2 D\right )\right ) x}{35 a^5 \left (a+b x^2\right )^{5/2}}-\frac {8 \left (192 A b^3-a \left (80 b^2 B-24 a b C+3 a^2 D\right )\right ) x}{105 a^6 \left (a+b x^2\right )^{3/2}}-\frac {16 \left (192 A b^3-a \left (80 b^2 B-24 a b C+3 a^2 D\right )\right ) x}{105 a^7 \sqrt {a+b x^2}} \]
-1/5*A/a/x^5/(b*x^2+a)^(7/2)+1/15*(12*A*b-5*B*a)/a^2/x^3/(b*x^2+a)^(7/2)+1 /3*(-24*A*b^2+a*(10*B*b-3*C*a))/a^3/x/(b*x^2+a)^(7/2)-1/21*(192*A*b^3-a*(8 0*B*b^2-24*C*a*b+3*D*a^2))*x/a^4/(b*x^2+a)^(7/2)-2/35*(192*A*b^3-a*(80*B*b ^2-24*C*a*b+3*D*a^2))*x/a^5/(b*x^2+a)^(5/2)-8/105*(192*A*b^3-a*(80*B*b^2-2 4*C*a*b+3*D*a^2))*x/a^6/(b*x^2+a)^(3/2)-16/105*(192*A*b^3-a*(80*B*b^2-24*C *a*b+3*D*a^2))*x/a^7/(b*x^2+a)^(1/2)
Time = 0.57 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.72 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^6 \left (a+b x^2\right )^{9/2}} \, dx=\frac {-3072 A b^6 x^{12}+256 a b^5 x^{10} \left (-42 A+5 B x^2\right )-128 a^2 b^4 x^8 \left (105 A-35 B x^2+3 C x^4\right )+16 a^3 b^3 x^6 \left (-420 A+350 B x^2-84 C x^4+3 D x^6\right )+56 a^4 b^2 x^4 \left (-15 A+50 B x^2-30 C x^4+3 D x^6\right )+14 a^5 b x^2 \left (6 A+25 B x^2-60 C x^4+15 D x^6\right )-7 a^6 \left (3 A+5 x^2 \left (B+3 C x^2-3 D x^4\right )\right )}{105 a^7 x^5 \left (a+b x^2\right )^{7/2}} \]
(-3072*A*b^6*x^12 + 256*a*b^5*x^10*(-42*A + 5*B*x^2) - 128*a^2*b^4*x^8*(10 5*A - 35*B*x^2 + 3*C*x^4) + 16*a^3*b^3*x^6*(-420*A + 350*B*x^2 - 84*C*x^4 + 3*D*x^6) + 56*a^4*b^2*x^4*(-15*A + 50*B*x^2 - 30*C*x^4 + 3*D*x^6) + 14*a ^5*b*x^2*(6*A + 25*B*x^2 - 60*C*x^4 + 15*D*x^6) - 7*a^6*(3*A + 5*x^2*(B + 3*C*x^2 - 3*D*x^4)))/(105*a^7*x^5*(a + b*x^2)^(7/2))
Time = 0.51 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.83, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {2334, 2089, 1588, 27, 359, 209, 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.
| \(\displaystyle \int \frac {A+B x^2+C x^4+D x^6}{x^6 \left (a+b x^2\right )^{9/2}} \, dx\) |
| \(\Big \downarrow \) 2334 |
| \(\displaystyle -\frac {\int \frac {12 A b-5 a \left (D x^4+C x^2+B\right )}{x^4 \left (b x^2+a\right )^{9/2}}dx}{5 a}-\frac {A}{5 a x^5 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 2089 |
| \(\displaystyle -\frac {\int \frac {-5 a D x^4-5 a C x^2+12 A b-5 a B}{x^4 \left (b x^2+a\right )^{9/2}}dx}{5 a}-\frac {A}{5 a x^5 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 1588 |
| \(\displaystyle -\frac {-\frac {\int \frac {5 \left (24 A b^2+3 a^2 D x^2-a (10 b B-3 a C)\right )}{x^2 \left (b x^2+a\right )^{9/2}}dx}{3 a}-\frac {12 A b-5 a B}{3 a x^3 \left (a+b x^2\right )^{7/2}}}{5 a}-\frac {A}{5 a x^5 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {5 \int \frac {24 A b^2+3 a^2 D x^2-a (10 b B-3 a C)}{x^2 \left (b x^2+a\right )^{9/2}}dx}{3 a}-\frac {12 A b-5 a B}{3 a x^3 \left (a+b x^2\right )^{7/2}}}{5 a}-\frac {A}{5 a x^5 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 359 |
| \(\displaystyle -\frac {-\frac {5 \left (-\frac {\left (192 A b^3-a \left (3 a^2 D-24 a b C+80 b^2 B\right )\right ) \int \frac {1}{\left (b x^2+a\right )^{9/2}}dx}{a}-\frac {24 A b^2-a (10 b B-3 a C)}{a x \left (a+b x^2\right )^{7/2}}\right )}{3 a}-\frac {12 A b-5 a B}{3 a x^3 \left (a+b x^2\right )^{7/2}}}{5 a}-\frac {A}{5 a x^5 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle -\frac {-\frac {5 \left (-\frac {\left (192 A b^3-a \left (3 a^2 D-24 a b C+80 b^2 B\right )\right ) \left (\frac {6 \int \frac {1}{\left (b x^2+a\right )^{7/2}}dx}{7 a}+\frac {x}{7 a \left (a+b x^2\right )^{7/2}}\right )}{a}-\frac {24 A b^2-a (10 b B-3 a C)}{a x \left (a+b x^2\right )^{7/2}}\right )}{3 a}-\frac {12 A b-5 a B}{3 a x^3 \left (a+b x^2\right )^{7/2}}}{5 a}-\frac {A}{5 a x^5 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle -\frac {-\frac {5 \left (-\frac {\left (192 A b^3-a \left (3 a^2 D-24 a b C+80 b^2 B\right )\right ) \left (\frac {6 \left (\frac {4 \int \frac {1}{\left (b x^2+a\right )^{5/2}}dx}{5 a}+\frac {x}{5 a \left (a+b x^2\right )^{5/2}}\right )}{7 a}+\frac {x}{7 a \left (a+b x^2\right )^{7/2}}\right )}{a}-\frac {24 A b^2-a (10 b B-3 a C)}{a x \left (a+b x^2\right )^{7/2}}\right )}{3 a}-\frac {12 A b-5 a B}{3 a x^3 \left (a+b x^2\right )^{7/2}}}{5 a}-\frac {A}{5 a x^5 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle -\frac {-\frac {5 \left (-\frac {\left (192 A b^3-a \left (3 a^2 D-24 a b C+80 b^2 B\right )\right ) \left (\frac {6 \left (\frac {4 \left (\frac {2 \int \frac {1}{\left (b x^2+a\right )^{3/2}}dx}{3 a}+\frac {x}{3 a \left (a+b x^2\right )^{3/2}}\right )}{5 a}+\frac {x}{5 a \left (a+b x^2\right )^{5/2}}\right )}{7 a}+\frac {x}{7 a \left (a+b x^2\right )^{7/2}}\right )}{a}-\frac {24 A b^2-a (10 b B-3 a C)}{a x \left (a+b x^2\right )^{7/2}}\right )}{3 a}-\frac {12 A b-5 a B}{3 a x^3 \left (a+b x^2\right )^{7/2}}}{5 a}-\frac {A}{5 a x^5 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 208 |
| \(\displaystyle -\frac {-\frac {5 \left (-\frac {\left (\frac {6 \left (\frac {4 \left (\frac {2 x}{3 a^2 \sqrt {a+b x^2}}+\frac {x}{3 a \left (a+b x^2\right )^{3/2}}\right )}{5 a}+\frac {x}{5 a \left (a+b x^2\right )^{5/2}}\right )}{7 a}+\frac {x}{7 a \left (a+b x^2\right )^{7/2}}\right ) \left (192 A b^3-a \left (3 a^2 D-24 a b C+80 b^2 B\right )\right )}{a}-\frac {24 A b^2-a (10 b B-3 a C)}{a x \left (a+b x^2\right )^{7/2}}\right )}{3 a}-\frac {12 A b-5 a B}{3 a x^3 \left (a+b x^2\right )^{7/2}}}{5 a}-\frac {A}{5 a x^5 \left (a+b x^2\right )^{7/2}}\) |
-1/5*A/(a*x^5*(a + b*x^2)^(7/2)) - (-1/3*(12*A*b - 5*a*B)/(a*x^3*(a + b*x^ 2)^(7/2)) - (5*(-((24*A*b^2 - a*(10*b*B - 3*a*C))/(a*x*(a + b*x^2)^(7/2))) - ((192*A*b^3 - a*(80*b^2*B - 24*a*b*C + 3*a^2*D))*(x/(7*a*(a + b*x^2)^(7 /2)) + (6*(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)))/a))/(3*a))/(5*a)
3.2.66.3.1 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[((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[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)* (a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !ILtQ[p, -1]
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 = 3.64 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.66
| method | result | size |
| pseudoelliptic | \(-\frac {\left (-5 D x^{6}+5 C \,x^{4}+\frac {5}{3} x^{2} B +A \right ) a^{6}-4 \left (\frac {5}{2} D x^{6}-10 C \,x^{4}+\frac {25}{6} x^{2} B +A \right ) b \,x^{2} a^{5}+40 \left (-\frac {1}{5} D x^{6}+2 C \,x^{4}-\frac {10}{3} x^{2} B +A \right ) b^{2} x^{4} a^{4}+320 b^{3} x^{6} \left (-\frac {1}{140} D x^{6}+\frac {1}{5} C \,x^{4}-\frac {5}{6} x^{2} B +A \right ) a^{3}+640 \left (\frac {1}{35} C \,x^{4}-\frac {1}{3} x^{2} B +A \right ) b^{4} x^{8} a^{2}+512 \left (-\frac {5 x^{2} B}{42}+A \right ) b^{5} x^{10} a +\frac {1024 A \,b^{6} x^{12}}{7}}{5 \left (b \,x^{2}+a \right )^{\frac {7}{2}} x^{5} a^{7}}\) | \(185\) |
| gosper | \(-\frac {3072 A \,b^{6} x^{12}-1280 B a \,b^{5} x^{12}+384 C \,a^{2} b^{4} x^{12}-48 D a^{3} b^{3} x^{12}+10752 A a \,b^{5} x^{10}-4480 B \,a^{2} b^{4} x^{10}+1344 C \,a^{3} b^{3} x^{10}-168 D a^{4} b^{2} x^{10}+13440 A \,a^{2} b^{4} x^{8}-5600 B \,a^{3} b^{3} x^{8}+1680 C \,a^{4} b^{2} x^{8}-210 D a^{5} b \,x^{8}+6720 A \,a^{3} b^{3} x^{6}-2800 B \,a^{4} b^{2} x^{6}+840 C \,a^{5} b \,x^{6}-105 D a^{6} x^{6}+840 A \,a^{4} b^{2} x^{4}-350 B \,a^{5} b \,x^{4}+105 C \,a^{6} x^{4}-84 A \,a^{5} b \,x^{2}+35 B \,a^{6} x^{2}+21 A \,a^{6}}{105 x^{5} \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{7}}\) | \(253\) |
| trager | \(-\frac {3072 A \,b^{6} x^{12}-1280 B a \,b^{5} x^{12}+384 C \,a^{2} b^{4} x^{12}-48 D a^{3} b^{3} x^{12}+10752 A a \,b^{5} x^{10}-4480 B \,a^{2} b^{4} x^{10}+1344 C \,a^{3} b^{3} x^{10}-168 D a^{4} b^{2} x^{10}+13440 A \,a^{2} b^{4} x^{8}-5600 B \,a^{3} b^{3} x^{8}+1680 C \,a^{4} b^{2} x^{8}-210 D a^{5} b \,x^{8}+6720 A \,a^{3} b^{3} x^{6}-2800 B \,a^{4} b^{2} x^{6}+840 C \,a^{5} b \,x^{6}-105 D a^{6} x^{6}+840 A \,a^{4} b^{2} x^{4}-350 B \,a^{5} b \,x^{4}+105 C \,a^{6} x^{4}-84 A \,a^{5} b \,x^{2}+35 B \,a^{6} x^{2}+21 A \,a^{6}}{105 x^{5} \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{7}}\) | \(253\) |
| default | \(D \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )+B \left (-\frac {1}{3 a \,x^{3} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {10 b \left (-\frac {1}{a x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {8 b \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{a}\right )}{3 a}\right )+C \left (-\frac {1}{a x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {8 b \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{a}\right )+A \left (-\frac {1}{5 a \,x^{5} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {12 b \left (-\frac {1}{3 a \,x^{3} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {10 b \left (-\frac {1}{a x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {8 b \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{a}\right )}{3 a}\right )}{5 a}\right )\) | \(446\) |
-1/5*((-5*D*x^6+5*C*x^4+5/3*x^2*B+A)*a^6-4*(5/2*D*x^6-10*C*x^4+25/6*x^2*B+ A)*b*x^2*a^5+40*(-1/5*D*x^6+2*C*x^4-10/3*x^2*B+A)*b^2*x^4*a^4+320*b^3*x^6* (-1/140*D*x^6+1/5*C*x^4-5/6*x^2*B+A)*a^3+640*(1/35*C*x^4-1/3*x^2*B+A)*b^4* x^8*a^2+512*(-5/42*x^2*B+A)*b^5*x^10*a+1024/7*A*b^6*x^12)/(b*x^2+a)^(7/2)/ x^5/a^7
Time = 0.52 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.96 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^6 \left (a+b x^2\right )^{9/2}} \, dx=\frac {{\left (16 \, {\left (3 \, D a^{3} b^{3} - 24 \, C a^{2} b^{4} + 80 \, B a b^{5} - 192 \, A b^{6}\right )} x^{12} + 56 \, {\left (3 \, D a^{4} b^{2} - 24 \, C a^{3} b^{3} + 80 \, B a^{2} b^{4} - 192 \, A a b^{5}\right )} x^{10} + 70 \, {\left (3 \, D a^{5} b - 24 \, C a^{4} b^{2} + 80 \, B a^{3} b^{3} - 192 \, A a^{2} b^{4}\right )} x^{8} - 21 \, A a^{6} + 35 \, {\left (3 \, D a^{6} - 24 \, C a^{5} b + 80 \, B a^{4} b^{2} - 192 \, A a^{3} b^{3}\right )} x^{6} - 35 \, {\left (3 \, C a^{6} - 10 \, B a^{5} b + 24 \, A a^{4} b^{2}\right )} x^{4} - 7 \, {\left (5 \, B a^{6} - 12 \, A a^{5} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{105 \, {\left (a^{7} b^{4} x^{13} + 4 \, a^{8} b^{3} x^{11} + 6 \, a^{9} b^{2} x^{9} + 4 \, a^{10} b x^{7} + a^{11} x^{5}\right )}} \]
1/105*(16*(3*D*a^3*b^3 - 24*C*a^2*b^4 + 80*B*a*b^5 - 192*A*b^6)*x^12 + 56* (3*D*a^4*b^2 - 24*C*a^3*b^3 + 80*B*a^2*b^4 - 192*A*a*b^5)*x^10 + 70*(3*D*a ^5*b - 24*C*a^4*b^2 + 80*B*a^3*b^3 - 192*A*a^2*b^4)*x^8 - 21*A*a^6 + 35*(3 *D*a^6 - 24*C*a^5*b + 80*B*a^4*b^2 - 192*A*a^3*b^3)*x^6 - 35*(3*C*a^6 - 10 *B*a^5*b + 24*A*a^4*b^2)*x^4 - 7*(5*B*a^6 - 12*A*a^5*b)*x^2)*sqrt(b*x^2 + a)/(a^7*b^4*x^13 + 4*a^8*b^3*x^11 + 6*a^9*b^2*x^9 + 4*a^10*b*x^7 + a^11*x^ 5)
Timed out. \[ \int \frac {A+B x^2+C x^4+D x^6}{x^6 \left (a+b x^2\right )^{9/2}} \, dx=\text {Timed out} \]
Time = 0.21 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.42 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^6 \left (a+b x^2\right )^{9/2}} \, dx=\frac {16 \, D x}{35 \, \sqrt {b x^{2} + a} a^{4}} + \frac {8 \, D x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3}} + \frac {6 \, D x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2}} + \frac {D x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a} - \frac {128 \, C b x}{35 \, \sqrt {b x^{2} + a} a^{5}} - \frac {64 \, C b x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{4}} - \frac {48 \, C b x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{3}} - \frac {8 \, C b x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2}} + \frac {256 \, B b^{2} x}{21 \, \sqrt {b x^{2} + a} a^{6}} + \frac {128 \, B b^{2} x}{21 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{5}} + \frac {32 \, B b^{2} x}{7 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{4}} + \frac {80 \, B b^{2} x}{21 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{3}} - \frac {1024 \, A b^{3} x}{35 \, \sqrt {b x^{2} + a} a^{7}} - \frac {512 \, A b^{3} x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{6}} - \frac {384 \, A b^{3} x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{5}} - \frac {64 \, A b^{3} x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{4}} - \frac {C}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} a x} + \frac {10 \, B b}{3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2} x} - \frac {8 \, A b^{2}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{3} x} - \frac {B}{3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a x^{3}} + \frac {4 \, A b}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2} x^{3}} - \frac {A}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a x^{5}} \]
16/35*D*x/(sqrt(b*x^2 + a)*a^4) + 8/35*D*x/((b*x^2 + a)^(3/2)*a^3) + 6/35* D*x/((b*x^2 + a)^(5/2)*a^2) + 1/7*D*x/((b*x^2 + a)^(7/2)*a) - 128/35*C*b*x /(sqrt(b*x^2 + a)*a^5) - 64/35*C*b*x/((b*x^2 + a)^(3/2)*a^4) - 48/35*C*b*x /((b*x^2 + a)^(5/2)*a^3) - 8/7*C*b*x/((b*x^2 + a)^(7/2)*a^2) + 256/21*B*b^ 2*x/(sqrt(b*x^2 + a)*a^6) + 128/21*B*b^2*x/((b*x^2 + a)^(3/2)*a^5) + 32/7* B*b^2*x/((b*x^2 + a)^(5/2)*a^4) + 80/21*B*b^2*x/((b*x^2 + a)^(7/2)*a^3) - 1024/35*A*b^3*x/(sqrt(b*x^2 + a)*a^7) - 512/35*A*b^3*x/((b*x^2 + a)^(3/2)* a^6) - 384/35*A*b^3*x/((b*x^2 + a)^(5/2)*a^5) - 64/7*A*b^3*x/((b*x^2 + a)^ (7/2)*a^4) - C/((b*x^2 + a)^(7/2)*a*x) + 10/3*B*b/((b*x^2 + a)^(7/2)*a^2*x ) - 8*A*b^2/((b*x^2 + a)^(7/2)*a^3*x) - 1/3*B/((b*x^2 + a)^(7/2)*a*x^3) + 4/5*A*b/((b*x^2 + a)^(7/2)*a^2*x^3) - 1/5*A/((b*x^2 + a)^(7/2)*a*x^5)
Leaf count of result is larger than twice the leaf count of optimal. 592 vs. \(2 (252) = 504\).
Time = 0.30 (sec) , antiderivative size = 592, normalized size of antiderivative = 2.11 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^6 \left (a+b x^2\right )^{9/2}} \, dx=\frac {{\left ({\left (x^{2} {\left (\frac {{\left (48 \, D a^{18} b^{6} - 279 \, C a^{17} b^{7} + 790 \, B a^{16} b^{8} - 1686 \, A a^{15} b^{9}\right )} x^{2}}{a^{22} b^{3}} + \frac {7 \, {\left (24 \, D a^{19} b^{5} - 132 \, C a^{18} b^{6} + 365 \, B a^{17} b^{7} - 768 \, A a^{16} b^{8}\right )}}{a^{22} b^{3}}\right )} + \frac {35 \, {\left (6 \, D a^{20} b^{4} - 30 \, C a^{19} b^{5} + 80 \, B a^{18} b^{6} - 165 \, A a^{17} b^{7}\right )}}{a^{22} b^{3}}\right )} x^{2} + \frac {105 \, {\left (D a^{21} b^{3} - 4 \, C a^{20} b^{4} + 10 \, B a^{19} b^{5} - 20 \, A a^{18} b^{6}\right )}}{a^{22} b^{3}}\right )} x}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} + \frac {2 \, {\left (15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} C a^{2} \sqrt {b} - 60 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} B a b^{\frac {3}{2}} + 150 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} A b^{\frac {5}{2}} - 60 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} C a^{3} \sqrt {b} + 270 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} B a^{2} b^{\frac {3}{2}} - 720 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} A a b^{\frac {5}{2}} + 90 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} C a^{4} \sqrt {b} - 430 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a^{3} b^{\frac {3}{2}} + 1260 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A a^{2} b^{\frac {5}{2}} - 60 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} C a^{5} \sqrt {b} + 290 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{4} b^{\frac {3}{2}} - 840 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a^{3} b^{\frac {5}{2}} + 15 \, C a^{6} \sqrt {b} - 70 \, B a^{5} b^{\frac {3}{2}} + 198 \, A a^{4} b^{\frac {5}{2}}\right )}}{15 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{5} a^{6}} \]
1/105*((x^2*((48*D*a^18*b^6 - 279*C*a^17*b^7 + 790*B*a^16*b^8 - 1686*A*a^1 5*b^9)*x^2/(a^22*b^3) + 7*(24*D*a^19*b^5 - 132*C*a^18*b^6 + 365*B*a^17*b^7 - 768*A*a^16*b^8)/(a^22*b^3)) + 35*(6*D*a^20*b^4 - 30*C*a^19*b^5 + 80*B*a ^18*b^6 - 165*A*a^17*b^7)/(a^22*b^3))*x^2 + 105*(D*a^21*b^3 - 4*C*a^20*b^4 + 10*B*a^19*b^5 - 20*A*a^18*b^6)/(a^22*b^3))*x/(b*x^2 + a)^(7/2) + 2/15*( 15*(sqrt(b)*x - sqrt(b*x^2 + a))^8*C*a^2*sqrt(b) - 60*(sqrt(b)*x - sqrt(b* x^2 + a))^8*B*a*b^(3/2) + 150*(sqrt(b)*x - sqrt(b*x^2 + a))^8*A*b^(5/2) - 60*(sqrt(b)*x - sqrt(b*x^2 + a))^6*C*a^3*sqrt(b) + 270*(sqrt(b)*x - sqrt(b *x^2 + a))^6*B*a^2*b^(3/2) - 720*(sqrt(b)*x - sqrt(b*x^2 + a))^6*A*a*b^(5/ 2) + 90*(sqrt(b)*x - sqrt(b*x^2 + a))^4*C*a^4*sqrt(b) - 430*(sqrt(b)*x - s qrt(b*x^2 + a))^4*B*a^3*b^(3/2) + 1260*(sqrt(b)*x - sqrt(b*x^2 + a))^4*A*a ^2*b^(5/2) - 60*(sqrt(b)*x - sqrt(b*x^2 + a))^2*C*a^5*sqrt(b) + 290*(sqrt( b)*x - sqrt(b*x^2 + a))^2*B*a^4*b^(3/2) - 840*(sqrt(b)*x - sqrt(b*x^2 + a) )^2*A*a^3*b^(5/2) + 15*C*a^6*sqrt(b) - 70*B*a^5*b^(3/2) + 198*A*a^4*b^(5/2 ))/(((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^5*a^6)
Time = 7.41 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.44 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^6 \left (a+b x^2\right )^{9/2}} \, dx=\frac {\frac {61\,A\,b}{35\,a^3}+\frac {78\,A\,b^2\,x^2}{35\,a^4}}{x^3\,{\left (b\,x^2+a\right )}^{5/2}}+\frac {\frac {128\,B\,b}{21\,a^5}+\frac {256\,B\,b^2\,x^2}{21\,a^6}}{x\,\sqrt {b\,x^2+a}}+\frac {x\,D}{{\left (b\,x^2+a\right )}^{9/2}}-\frac {\frac {B}{3\,a^2}+\frac {19\,B\,b\,x^2}{21\,a^3}}{x^3\,{\left (b\,x^2+a\right )}^{5/2}}-\frac {\frac {C}{a^4}+\frac {128\,C\,b\,x^2}{35\,a^5}}{x\,\sqrt {b\,x^2+a}}-\frac {\frac {512\,A\,b^2}{35\,a^6}+\frac {1024\,A\,b^3\,x^2}{35\,a^7}}{x\,\sqrt {b\,x^2+a}}-\frac {A\,\sqrt {b\,x^2+a}}{5\,a^5\,x^5}+\frac {18\,b^2\,x^5\,D}{5\,a^2\,{\left (b\,x^2+a\right )}^{9/2}}+\frac {72\,b^3\,x^7\,D}{35\,a^3\,{\left (b\,x^2+a\right )}^{9/2}}+\frac {16\,b^4\,x^9\,D}{35\,a^4\,{\left (b\,x^2+a\right )}^{9/2}}-\frac {A\,b}{7\,a^2\,x^3\,{\left (b\,x^2+a\right )}^{7/2}}-\frac {32\,B\,b}{21\,a^4\,x\,{\left (b\,x^2+a\right )}^{3/2}}+\frac {B\,b^2\,x}{7\,a^3\,{\left (b\,x^2+a\right )}^{7/2}}+\frac {27\,A\,b^2}{7\,a^5\,x\,{\left (b\,x^2+a\right )}^{3/2}}+\frac {3\,b\,x^3\,D}{a\,{\left (b\,x^2+a\right )}^{9/2}}-\frac {29\,C\,b\,x}{35\,a^4\,{\left (b\,x^2+a\right )}^{3/2}}-\frac {13\,C\,b\,x}{35\,a^3\,{\left (b\,x^2+a\right )}^{5/2}}-\frac {C\,b\,x}{7\,a^2\,{\left (b\,x^2+a\right )}^{7/2}} \]
((61*A*b)/(35*a^3) + (78*A*b^2*x^2)/(35*a^4))/(x^3*(a + b*x^2)^(5/2)) + (( 128*B*b)/(21*a^5) + (256*B*b^2*x^2)/(21*a^6))/(x*(a + b*x^2)^(1/2)) + (x*D )/(a + b*x^2)^(9/2) - (B/(3*a^2) + (19*B*b*x^2)/(21*a^3))/(x^3*(a + b*x^2) ^(5/2)) - (C/a^4 + (128*C*b*x^2)/(35*a^5))/(x*(a + b*x^2)^(1/2)) - ((512*A *b^2)/(35*a^6) + (1024*A*b^3*x^2)/(35*a^7))/(x*(a + b*x^2)^(1/2)) - (A*(a + b*x^2)^(1/2))/(5*a^5*x^5) + (18*b^2*x^5*D)/(5*a^2*(a + b*x^2)^(9/2)) + ( 72*b^3*x^7*D)/(35*a^3*(a + b*x^2)^(9/2)) + (16*b^4*x^9*D)/(35*a^4*(a + b*x ^2)^(9/2)) - (A*b)/(7*a^2*x^3*(a + b*x^2)^(7/2)) - (32*B*b)/(21*a^4*x*(a + b*x^2)^(3/2)) + (B*b^2*x)/(7*a^3*(a + b*x^2)^(7/2)) + (27*A*b^2)/(7*a^5*x *(a + b*x^2)^(3/2)) + (3*b*x^3*D)/(a*(a + b*x^2)^(9/2)) - (29*C*b*x)/(35*a ^4*(a + b*x^2)^(3/2)) - (13*C*b*x)/(35*a^3*(a + b*x^2)^(5/2)) - (C*b*x)/(7 *a^2*(a + b*x^2)^(7/2))