Integrand size = 32, antiderivative size = 334 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^8 \left (a+b x^2\right )^{9/2}} \, dx=-\frac {A}{7 a x^7 \left (a+b x^2\right )^{7/2}}+\frac {2 A b-a B}{5 a^2 x^5 \left (a+b x^2\right )^{7/2}}-\frac {24 A b^2-a (12 b B-5 a C)}{15 a^3 x^3 \left (a+b x^2\right )^{7/2}}+\frac {48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )}{3 a^4 x \left (a+b x^2\right )^{7/2}}+\frac {8 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{21 a^5 \left (a+b x^2\right )^{7/2}}+\frac {16 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{35 a^6 \left (a+b x^2\right )^{5/2}}+\frac {64 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{105 a^7 \left (a+b x^2\right )^{3/2}}+\frac {128 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{105 a^8 \sqrt {a+b x^2}} \]
-1/7*A/a/x^7/(b*x^2+a)^(7/2)+1/5*(2*A*b-B*a)/a^2/x^5/(b*x^2+a)^(7/2)+1/15* (-24*A*b^2+a*(12*B*b-5*C*a))/a^3/x^3/(b*x^2+a)^(7/2)+1/3*(48*A*b^3-a*(24*B *b^2-10*C*a*b+3*D*a^2))/a^4/x/(b*x^2+a)^(7/2)+8/21*b*(48*A*b^3-a*(24*B*b^2 -10*C*a*b+3*D*a^2))*x/a^5/(b*x^2+a)^(7/2)+16/35*b*(48*A*b^3-a*(24*B*b^2-10 *C*a*b+3*D*a^2))*x/a^6/(b*x^2+a)^(5/2)+64/105*b*(48*A*b^3-a*(24*B*b^2-10*C *a*b+3*D*a^2))*x/a^7/(b*x^2+a)^(3/2)+128/105*b*(48*A*b^3-a*(24*B*b^2-10*C* a*b+3*D*a^2))*x/a^8/(b*x^2+a)^(1/2)
Time = 0.63 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.70 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^8 \left (a+b x^2\right )^{9/2}} \, dx=\frac {6144 A b^7 x^{14}-3072 a b^6 x^{12} \left (-7 A+B x^2\right )+256 a^2 b^5 x^{10} \left (105 A-42 B x^2+5 C x^4\right )+14 a^6 b x^2 \left (3 A+6 B x^2+25 C x^4-60 D x^6\right )+112 a^4 b^3 x^6 \left (15 A-60 B x^2+50 C x^4-12 D x^6\right )+128 a^3 b^4 x^8 \left (105 A-105 B x^2+35 C x^4-3 D x^6\right )-56 a^5 b^2 x^4 \left (3 A+15 B x^2-50 C x^4+30 D x^6\right )-a^7 \left (15 A+21 B x^2+35 x^4 \left (C+3 D x^2\right )\right )}{105 a^8 x^7 \left (a+b x^2\right )^{7/2}} \]
(6144*A*b^7*x^14 - 3072*a*b^6*x^12*(-7*A + B*x^2) + 256*a^2*b^5*x^10*(105* A - 42*B*x^2 + 5*C*x^4) + 14*a^6*b*x^2*(3*A + 6*B*x^2 + 25*C*x^4 - 60*D*x^ 6) + 112*a^4*b^3*x^6*(15*A - 60*B*x^2 + 50*C*x^4 - 12*D*x^6) + 128*a^3*b^4 *x^8*(105*A - 105*B*x^2 + 35*C*x^4 - 3*D*x^6) - 56*a^5*b^2*x^4*(3*A + 15*B *x^2 - 50*C*x^4 + 30*D*x^6) - a^7*(15*A + 21*B*x^2 + 35*x^4*(C + 3*D*x^2)) )/(105*a^8*x^7*(a + b*x^2)^(7/2))
Time = 0.55 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.78, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {2334, 27, 2089, 1588, 359, 245, 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.
| \(\displaystyle \int \frac {A+B x^2+C x^4+D x^6}{x^8 \left (a+b x^2\right )^{9/2}} \, dx\) |
| \(\Big \downarrow \) 2334 |
| \(\displaystyle -\frac {\int \frac {7 \left (2 A b-a \left (D x^4+C x^2+B\right )\right )}{x^6 \left (b x^2+a\right )^{9/2}}dx}{7 a}-\frac {A}{7 a x^7 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {2 A b-a \left (D x^4+C x^2+B\right )}{x^6 \left (b x^2+a\right )^{9/2}}dx}{a}-\frac {A}{7 a x^7 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 2089 |
| \(\displaystyle -\frac {\int \frac {-a D x^4-a C x^2+2 A b-a B}{x^6 \left (b x^2+a\right )^{9/2}}dx}{a}-\frac {A}{7 a x^7 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 1588 |
| \(\displaystyle -\frac {-\frac {\int \frac {24 A b^2+5 a^2 D x^2-a (12 b B-5 a C)}{x^4 \left (b x^2+a\right )^{9/2}}dx}{5 a}-\frac {2 A b-a B}{5 a x^5 \left (a+b x^2\right )^{7/2}}}{a}-\frac {A}{7 a x^7 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 359 |
| \(\displaystyle -\frac {-\frac {-\frac {5 \left (48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )\right ) \int \frac {1}{x^2 \left (b x^2+a\right )^{9/2}}dx}{3 a}-\frac {24 A b^2-a (12 b B-5 a C)}{3 a x^3 \left (a+b x^2\right )^{7/2}}}{5 a}-\frac {2 A b-a B}{5 a x^5 \left (a+b x^2\right )^{7/2}}}{a}-\frac {A}{7 a x^7 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle -\frac {-\frac {-\frac {5 \left (48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )\right ) \left (-\frac {8 b \int \frac {1}{\left (b x^2+a\right )^{9/2}}dx}{a}-\frac {1}{a x \left (a+b x^2\right )^{7/2}}\right )}{3 a}-\frac {24 A b^2-a (12 b B-5 a C)}{3 a x^3 \left (a+b x^2\right )^{7/2}}}{5 a}-\frac {2 A b-a B}{5 a x^5 \left (a+b x^2\right )^{7/2}}}{a}-\frac {A}{7 a x^7 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle -\frac {-\frac {-\frac {5 \left (48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )\right ) \left (-\frac {8 b \left (\frac {6 \int \frac {1}{\left (b x^2+a\right )^{7/2}}dx}{7 a}+\frac {x}{7 a \left (a+b x^2\right )^{7/2}}\right )}{a}-\frac {1}{a x \left (a+b x^2\right )^{7/2}}\right )}{3 a}-\frac {24 A b^2-a (12 b B-5 a C)}{3 a x^3 \left (a+b x^2\right )^{7/2}}}{5 a}-\frac {2 A b-a B}{5 a x^5 \left (a+b x^2\right )^{7/2}}}{a}-\frac {A}{7 a x^7 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle -\frac {-\frac {-\frac {5 \left (48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )\right ) \left (-\frac {8 b \left (\frac {6 \left (\frac {4 \int \frac {1}{\left (b x^2+a\right )^{5/2}}dx}{5 a}+\frac {x}{5 a \left (a+b x^2\right )^{5/2}}\right )}{7 a}+\frac {x}{7 a \left (a+b x^2\right )^{7/2}}\right )}{a}-\frac {1}{a x \left (a+b x^2\right )^{7/2}}\right )}{3 a}-\frac {24 A b^2-a (12 b B-5 a C)}{3 a x^3 \left (a+b x^2\right )^{7/2}}}{5 a}-\frac {2 A b-a B}{5 a x^5 \left (a+b x^2\right )^{7/2}}}{a}-\frac {A}{7 a x^7 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle -\frac {-\frac {-\frac {5 \left (48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )\right ) \left (-\frac {8 b \left (\frac {6 \left (\frac {4 \left (\frac {2 \int \frac {1}{\left (b x^2+a\right )^{3/2}}dx}{3 a}+\frac {x}{3 a \left (a+b x^2\right )^{3/2}}\right )}{5 a}+\frac {x}{5 a \left (a+b x^2\right )^{5/2}}\right )}{7 a}+\frac {x}{7 a \left (a+b x^2\right )^{7/2}}\right )}{a}-\frac {1}{a x \left (a+b x^2\right )^{7/2}}\right )}{3 a}-\frac {24 A b^2-a (12 b B-5 a C)}{3 a x^3 \left (a+b x^2\right )^{7/2}}}{5 a}-\frac {2 A b-a B}{5 a x^5 \left (a+b x^2\right )^{7/2}}}{a}-\frac {A}{7 a x^7 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 208 |
| \(\displaystyle -\frac {-\frac {-\frac {5 \left (-\frac {8 b \left (\frac {6 \left (\frac {4 \left (\frac {2 x}{3 a^2 \sqrt {a+b x^2}}+\frac {x}{3 a \left (a+b x^2\right )^{3/2}}\right )}{5 a}+\frac {x}{5 a \left (a+b x^2\right )^{5/2}}\right )}{7 a}+\frac {x}{7 a \left (a+b x^2\right )^{7/2}}\right )}{a}-\frac {1}{a x \left (a+b x^2\right )^{7/2}}\right ) \left (48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )\right )}{3 a}-\frac {24 A b^2-a (12 b B-5 a C)}{3 a x^3 \left (a+b x^2\right )^{7/2}}}{5 a}-\frac {2 A b-a B}{5 a x^5 \left (a+b x^2\right )^{7/2}}}{a}-\frac {A}{7 a x^7 \left (a+b x^2\right )^{7/2}}\) |
-1/7*A/(a*x^7*(a + b*x^2)^(7/2)) - (-1/5*(2*A*b - a*B)/(a*x^5*(a + b*x^2)^ (7/2)) - (-1/3*(24*A*b^2 - a*(12*b*B - 5*a*C))/(a*x^3*(a + b*x^2)^(7/2)) - (5*(48*A*b^3 - a*(24*b^2*B - 10*a*b*C + 3*a^2*D))*(-(1/(a*x*(a + b*x^2)^( 7/2))) - (8*b*(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))/a
3.2.67.3.1 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[(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]
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 = 3.58 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.65
| method | result | size |
| pseudoelliptic | \(\frac {\left (-105 D x^{6}-35 C \,x^{4}-21 x^{2} B -15 A \right ) a^{7}+42 b \,x^{2} \left (-20 D x^{6}+\frac {25}{3} C \,x^{4}+2 x^{2} B +A \right ) a^{6}-168 b^{2} x^{4} \left (10 D x^{6}-\frac {50}{3} C \,x^{4}+5 x^{2} B +A \right ) a^{5}+1680 \left (-\frac {4}{5} D x^{6}+\frac {10}{3} C \,x^{4}-4 x^{2} B +A \right ) b^{3} x^{6} a^{4}+13440 \left (-\frac {1}{35} D x^{6}+\frac {1}{3} C \,x^{4}-x^{2} B +A \right ) b^{4} x^{8} a^{3}+26880 \left (\frac {1}{21} C \,x^{4}-\frac {2}{5} x^{2} B +A \right ) b^{5} x^{10} a^{2}+21504 b^{6} x^{12} \left (-\frac {x^{2} B}{7}+A \right ) a +6144 A \,b^{7} x^{14}}{105 \left (b \,x^{2}+a \right )^{\frac {7}{2}} x^{7} a^{8}}\) | \(218\) |
| gosper | \(-\frac {-6144 A \,b^{7} x^{14}+3072 B a \,b^{6} x^{14}-1280 C \,a^{2} b^{5} x^{14}+384 D a^{3} b^{4} x^{14}-21504 A a \,b^{6} x^{12}+10752 B \,a^{2} b^{5} x^{12}-4480 C \,a^{3} b^{4} x^{12}+1344 D a^{4} b^{3} x^{12}-26880 A \,a^{2} b^{5} x^{10}+13440 B \,a^{3} b^{4} x^{10}-5600 C \,a^{4} b^{3} x^{10}+1680 D a^{5} b^{2} x^{10}-13440 A \,a^{3} b^{4} x^{8}+6720 B \,a^{4} b^{3} x^{8}-2800 C \,a^{5} b^{2} x^{8}+840 D a^{6} b \,x^{8}-1680 A \,a^{4} b^{3} x^{6}+840 B \,a^{5} b^{2} x^{6}-350 C \,a^{6} b \,x^{6}+105 D a^{7} x^{6}+168 A \,a^{5} b^{2} x^{4}-84 B \,a^{6} b \,x^{4}+35 C \,a^{7} x^{4}-42 A \,a^{6} b \,x^{2}+21 B \,a^{7} x^{2}+15 A \,a^{7}}{105 x^{7} \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{8}}\) | \(301\) |
| trager | \(-\frac {-6144 A \,b^{7} x^{14}+3072 B a \,b^{6} x^{14}-1280 C \,a^{2} b^{5} x^{14}+384 D a^{3} b^{4} x^{14}-21504 A a \,b^{6} x^{12}+10752 B \,a^{2} b^{5} x^{12}-4480 C \,a^{3} b^{4} x^{12}+1344 D a^{4} b^{3} x^{12}-26880 A \,a^{2} b^{5} x^{10}+13440 B \,a^{3} b^{4} x^{10}-5600 C \,a^{4} b^{3} x^{10}+1680 D a^{5} b^{2} x^{10}-13440 A \,a^{3} b^{4} x^{8}+6720 B \,a^{4} b^{3} x^{8}-2800 C \,a^{5} b^{2} x^{8}+840 D a^{6} b \,x^{8}-1680 A \,a^{4} b^{3} x^{6}+840 B \,a^{5} b^{2} x^{6}-350 C \,a^{6} b \,x^{6}+105 D a^{7} x^{6}+168 A \,a^{5} b^{2} x^{4}-84 B \,a^{6} b \,x^{4}+35 C \,a^{7} x^{4}-42 A \,a^{6} b \,x^{2}+21 B \,a^{7} x^{2}+15 A \,a^{7}}{105 x^{7} \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{8}}\) | \(301\) |
| default | \(A \left (-\frac {1}{7 a \,x^{7} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {2 b \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 )}{a}\right )+C \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 )+D \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 )+B \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 )\) | \(542\) |
1/105*((-105*D*x^6-35*C*x^4-21*B*x^2-15*A)*a^7+42*b*x^2*(-20*D*x^6+25/3*C* x^4+2*x^2*B+A)*a^6-168*b^2*x^4*(10*D*x^6-50/3*C*x^4+5*x^2*B+A)*a^5+1680*(- 4/5*D*x^6+10/3*C*x^4-4*x^2*B+A)*b^3*x^6*a^4+13440*(-1/35*D*x^6+1/3*C*x^4-x ^2*B+A)*b^4*x^8*a^3+26880*(1/21*C*x^4-2/5*x^2*B+A)*b^5*x^10*a^2+21504*b^6* x^12*(-1/7*x^2*B+A)*a+6144*A*b^7*x^14)/(b*x^2+a)^(7/2)/x^7/a^8
Time = 0.70 (sec) , antiderivative size = 311, normalized size of antiderivative = 0.93 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^8 \left (a+b x^2\right )^{9/2}} \, dx=-\frac {{\left (128 \, {\left (3 \, D a^{3} b^{4} - 10 \, C a^{2} b^{5} + 24 \, B a b^{6} - 48 \, A b^{7}\right )} x^{14} + 448 \, {\left (3 \, D a^{4} b^{3} - 10 \, C a^{3} b^{4} + 24 \, B a^{2} b^{5} - 48 \, A a b^{6}\right )} x^{12} + 560 \, {\left (3 \, D a^{5} b^{2} - 10 \, C a^{4} b^{3} + 24 \, B a^{3} b^{4} - 48 \, A a^{2} b^{5}\right )} x^{10} + 280 \, {\left (3 \, D a^{6} b - 10 \, C a^{5} b^{2} + 24 \, B a^{4} b^{3} - 48 \, A a^{3} b^{4}\right )} x^{8} + 15 \, A a^{7} + 35 \, {\left (3 \, D a^{7} - 10 \, C a^{6} b + 24 \, B a^{5} b^{2} - 48 \, A a^{4} b^{3}\right )} x^{6} + 7 \, {\left (5 \, C a^{7} - 12 \, B a^{6} b + 24 \, A a^{5} b^{2}\right )} x^{4} + 21 \, {\left (B a^{7} - 2 \, A a^{6} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{105 \, {\left (a^{8} b^{4} x^{15} + 4 \, a^{9} b^{3} x^{13} + 6 \, a^{10} b^{2} x^{11} + 4 \, a^{11} b x^{9} + a^{12} x^{7}\right )}} \]
-1/105*(128*(3*D*a^3*b^4 - 10*C*a^2*b^5 + 24*B*a*b^6 - 48*A*b^7)*x^14 + 44 8*(3*D*a^4*b^3 - 10*C*a^3*b^4 + 24*B*a^2*b^5 - 48*A*a*b^6)*x^12 + 560*(3*D *a^5*b^2 - 10*C*a^4*b^3 + 24*B*a^3*b^4 - 48*A*a^2*b^5)*x^10 + 280*(3*D*a^6 *b - 10*C*a^5*b^2 + 24*B*a^4*b^3 - 48*A*a^3*b^4)*x^8 + 15*A*a^7 + 35*(3*D* a^7 - 10*C*a^6*b + 24*B*a^5*b^2 - 48*A*a^4*b^3)*x^6 + 7*(5*C*a^7 - 12*B*a^ 6*b + 24*A*a^5*b^2)*x^4 + 21*(B*a^7 - 2*A*a^6*b)*x^2)*sqrt(b*x^2 + a)/(a^8 *b^4*x^15 + 4*a^9*b^3*x^13 + 6*a^10*b^2*x^11 + 4*a^11*b*x^9 + a^12*x^7)
Timed out. \[ \int \frac {A+B x^2+C x^4+D x^6}{x^8 \left (a+b x^2\right )^{9/2}} \, dx=\text {Timed out} \]
Time = 0.21 (sec) , antiderivative size = 489, normalized size of antiderivative = 1.46 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^8 \left (a+b x^2\right )^{9/2}} \, dx=-\frac {128 \, D b x}{35 \, \sqrt {b x^{2} + a} a^{5}} - \frac {64 \, D b x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{4}} - \frac {48 \, D b x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{3}} - \frac {8 \, D b x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2}} + \frac {256 \, C b^{2} x}{21 \, \sqrt {b x^{2} + a} a^{6}} + \frac {128 \, C b^{2} x}{21 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{5}} + \frac {32 \, C b^{2} x}{7 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{4}} + \frac {80 \, C b^{2} x}{21 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{3}} - \frac {1024 \, B b^{3} x}{35 \, \sqrt {b x^{2} + a} a^{7}} - \frac {512 \, B b^{3} x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{6}} - \frac {384 \, B b^{3} x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{5}} - \frac {64 \, B b^{3} x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{4}} + \frac {2048 \, A b^{4} x}{35 \, \sqrt {b x^{2} + a} a^{8}} + \frac {1024 \, A b^{4} x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{7}} + \frac {768 \, A b^{4} x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{6}} + \frac {128 \, A b^{4} x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{5}} - \frac {D}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} a x} + \frac {10 \, C b}{3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2} x} - \frac {8 \, B b^{2}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{3} x} + \frac {16 \, A b^{3}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{4} x} - \frac {C}{3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a x^{3}} + \frac {4 \, B b}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2} x^{3}} - \frac {8 \, A b^{2}}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{3} x^{3}} - \frac {B}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a x^{5}} + \frac {2 \, A b}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2} x^{5}} - \frac {A}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a x^{7}} \]
-128/35*D*b*x/(sqrt(b*x^2 + a)*a^5) - 64/35*D*b*x/((b*x^2 + a)^(3/2)*a^4) - 48/35*D*b*x/((b*x^2 + a)^(5/2)*a^3) - 8/7*D*b*x/((b*x^2 + a)^(7/2)*a^2) + 256/21*C*b^2*x/(sqrt(b*x^2 + a)*a^6) + 128/21*C*b^2*x/((b*x^2 + a)^(3/2) *a^5) + 32/7*C*b^2*x/((b*x^2 + a)^(5/2)*a^4) + 80/21*C*b^2*x/((b*x^2 + a)^ (7/2)*a^3) - 1024/35*B*b^3*x/(sqrt(b*x^2 + a)*a^7) - 512/35*B*b^3*x/((b*x^ 2 + a)^(3/2)*a^6) - 384/35*B*b^3*x/((b*x^2 + a)^(5/2)*a^5) - 64/7*B*b^3*x/ ((b*x^2 + a)^(7/2)*a^4) + 2048/35*A*b^4*x/(sqrt(b*x^2 + a)*a^8) + 1024/35* A*b^4*x/((b*x^2 + a)^(3/2)*a^7) + 768/35*A*b^4*x/((b*x^2 + a)^(5/2)*a^6) + 128/7*A*b^4*x/((b*x^2 + a)^(7/2)*a^5) - D/((b*x^2 + a)^(7/2)*a*x) + 10/3* C*b/((b*x^2 + a)^(7/2)*a^2*x) - 8*B*b^2/((b*x^2 + a)^(7/2)*a^3*x) + 16*A*b ^3/((b*x^2 + a)^(7/2)*a^4*x) - 1/3*C/((b*x^2 + a)^(7/2)*a*x^3) + 4/5*B*b/( (b*x^2 + a)^(7/2)*a^2*x^3) - 8/5*A*b^2/((b*x^2 + a)^(7/2)*a^3*x^3) - 1/5*B /((b*x^2 + a)^(7/2)*a*x^5) + 2/5*A*b/((b*x^2 + a)^(7/2)*a^2*x^5) - 1/7*A/( (b*x^2 + a)^(7/2)*a*x^7)
Leaf count of result is larger than twice the leaf count of optimal. 938 vs. \(2 (300) = 600\).
Time = 0.32 (sec) , antiderivative size = 938, normalized size of antiderivative = 2.81 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^8 \left (a+b x^2\right )^{9/2}} \, dx=-\frac {{\left ({\left (x^{2} {\left (\frac {{\left (279 \, D a^{21} b^{7} - 790 \, C a^{20} b^{8} + 1686 \, B a^{19} b^{9} - 3072 \, A a^{18} b^{10}\right )} x^{2}}{a^{26} b^{3}} + \frac {7 \, {\left (132 \, D a^{22} b^{6} - 365 \, C a^{21} b^{7} + 768 \, B a^{20} b^{8} - 1386 \, A a^{19} b^{9}\right )}}{a^{26} b^{3}}\right )} + \frac {35 \, {\left (30 \, D a^{23} b^{5} - 80 \, C a^{22} b^{6} + 165 \, B a^{21} b^{7} - 294 \, A a^{20} b^{8}\right )}}{a^{26} b^{3}}\right )} x^{2} + \frac {105 \, {\left (4 \, D a^{24} b^{4} - 10 \, C a^{23} b^{5} + 20 \, B a^{22} b^{6} - 35 \, A a^{21} b^{7}\right )}}{a^{26} b^{3}}\right )} x}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} + \frac {2 \, {\left (105 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} D a^{3} \sqrt {b} - 420 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} C a^{2} b^{\frac {3}{2}} + 1050 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} B a b^{\frac {5}{2}} - 2100 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} A b^{\frac {7}{2}} - 630 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} D a^{4} \sqrt {b} + 2730 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} C a^{3} b^{\frac {3}{2}} - 7140 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} B a^{2} b^{\frac {5}{2}} + 14700 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} A a b^{\frac {7}{2}} + 1575 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} D a^{5} \sqrt {b} - 7210 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} C a^{4} b^{\frac {3}{2}} + 19950 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} B a^{3} b^{\frac {5}{2}} - 42840 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} A a^{2} b^{\frac {7}{2}} - 2100 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} D a^{6} \sqrt {b} + 9940 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} C a^{5} b^{\frac {3}{2}} - 28560 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} B a^{4} b^{\frac {5}{2}} + 64680 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} A a^{3} b^{\frac {7}{2}} + 1575 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} D a^{7} \sqrt {b} - 7560 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} C a^{6} b^{\frac {3}{2}} + 21966 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a^{5} b^{\frac {5}{2}} - 49812 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A a^{4} b^{\frac {7}{2}} - 630 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} D a^{8} \sqrt {b} + 3010 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} C a^{7} b^{\frac {3}{2}} - 8652 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{6} b^{\frac {5}{2}} + 19404 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a^{5} b^{\frac {7}{2}} + 105 \, D a^{9} \sqrt {b} - 490 \, C a^{8} b^{\frac {3}{2}} + 1386 \, B a^{7} b^{\frac {5}{2}} - 3072 \, A a^{6} b^{\frac {7}{2}}\right )}}{105 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{7} a^{7}} \]
-1/105*((x^2*((279*D*a^21*b^7 - 790*C*a^20*b^8 + 1686*B*a^19*b^9 - 3072*A* a^18*b^10)*x^2/(a^26*b^3) + 7*(132*D*a^22*b^6 - 365*C*a^21*b^7 + 768*B*a^2 0*b^8 - 1386*A*a^19*b^9)/(a^26*b^3)) + 35*(30*D*a^23*b^5 - 80*C*a^22*b^6 + 165*B*a^21*b^7 - 294*A*a^20*b^8)/(a^26*b^3))*x^2 + 105*(4*D*a^24*b^4 - 10 *C*a^23*b^5 + 20*B*a^22*b^6 - 35*A*a^21*b^7)/(a^26*b^3))*x/(b*x^2 + a)^(7/ 2) + 2/105*(105*(sqrt(b)*x - sqrt(b*x^2 + a))^12*D*a^3*sqrt(b) - 420*(sqrt (b)*x - sqrt(b*x^2 + a))^12*C*a^2*b^(3/2) + 1050*(sqrt(b)*x - sqrt(b*x^2 + a))^12*B*a*b^(5/2) - 2100*(sqrt(b)*x - sqrt(b*x^2 + a))^12*A*b^(7/2) - 63 0*(sqrt(b)*x - sqrt(b*x^2 + a))^10*D*a^4*sqrt(b) + 2730*(sqrt(b)*x - sqrt( b*x^2 + a))^10*C*a^3*b^(3/2) - 7140*(sqrt(b)*x - sqrt(b*x^2 + a))^10*B*a^2 *b^(5/2) + 14700*(sqrt(b)*x - sqrt(b*x^2 + a))^10*A*a*b^(7/2) + 1575*(sqrt (b)*x - sqrt(b*x^2 + a))^8*D*a^5*sqrt(b) - 7210*(sqrt(b)*x - sqrt(b*x^2 + a))^8*C*a^4*b^(3/2) + 19950*(sqrt(b)*x - sqrt(b*x^2 + a))^8*B*a^3*b^(5/2) - 42840*(sqrt(b)*x - sqrt(b*x^2 + a))^8*A*a^2*b^(7/2) - 2100*(sqrt(b)*x - sqrt(b*x^2 + a))^6*D*a^6*sqrt(b) + 9940*(sqrt(b)*x - sqrt(b*x^2 + a))^6*C* a^5*b^(3/2) - 28560*(sqrt(b)*x - sqrt(b*x^2 + a))^6*B*a^4*b^(5/2) + 64680* (sqrt(b)*x - sqrt(b*x^2 + a))^6*A*a^3*b^(7/2) + 1575*(sqrt(b)*x - sqrt(b*x ^2 + a))^4*D*a^7*sqrt(b) - 7560*(sqrt(b)*x - sqrt(b*x^2 + a))^4*C*a^6*b^(3 /2) + 21966*(sqrt(b)*x - sqrt(b*x^2 + a))^4*B*a^5*b^(5/2) - 49812*(sqrt(b) *x - sqrt(b*x^2 + a))^4*A*a^4*b^(7/2) - 630*(sqrt(b)*x - sqrt(b*x^2 + a...
Time = 7.74 (sec) , antiderivative size = 421, normalized size of antiderivative = 1.26 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^8 \left (a+b x^2\right )^{9/2}} \, dx=\frac {\frac {61\,B\,b}{35\,a^3}+\frac {78\,B\,b^2\,x^2}{35\,a^4}}{x^3\,{\left (b\,x^2+a\right )}^{5/2}}+\frac {\frac {128\,C\,b}{21\,a^5}+\frac {256\,C\,b^2\,x^2}{21\,a^6}}{x\,\sqrt {b\,x^2+a}}-\frac {\frac {C}{3\,a^2}+\frac {19\,C\,b\,x^2}{21\,a^3}}{x^3\,{\left (b\,x^2+a\right )}^{5/2}}-\frac {\frac {167\,A\,b^2}{35\,a^4}+\frac {191\,A\,b^3\,x^2}{35\,a^5}}{x^3\,{\left (b\,x^2+a\right )}^{5/2}}+\frac {\frac {1024\,A\,b^3}{35\,a^7}+\frac {2048\,A\,b^4\,x^2}{35\,a^8}}{x\,\sqrt {b\,x^2+a}}-\frac {\frac {512\,B\,b^2}{35\,a^6}+\frac {1024\,B\,b^3\,x^2}{35\,a^7}}{x\,\sqrt {b\,x^2+a}}-\frac {A\,\sqrt {b\,x^2+a}}{7\,a^5\,x^7}-\frac {B\,\sqrt {b\,x^2+a}}{5\,a^5\,x^5}-\frac {{\left (\frac {a}{b\,x^2}+1\right )}^{9/2}\,D\,{{}}_2{\mathrm {F}}_1\left (\frac {9}{2},5;\ 6;\ -\frac {a}{b\,x^2}\right )}{10\,x\,{\left (b\,x^2+a\right )}^{9/2}}+\frac {34\,A\,b\,\sqrt {b\,x^2+a}}{35\,a^6\,x^5}-\frac {B\,b}{7\,a^2\,x^3\,{\left (b\,x^2+a\right )}^{7/2}}-\frac {32\,C\,b}{21\,a^4\,x\,{\left (b\,x^2+a\right )}^{3/2}}+\frac {C\,b^2\,x}{7\,a^3\,{\left (b\,x^2+a\right )}^{7/2}}-\frac {58\,A\,b^3}{7\,a^6\,x\,{\left (b\,x^2+a\right )}^{3/2}}+\frac {A\,b^2}{7\,a^3\,x^3\,{\left (b\,x^2+a\right )}^{7/2}}+\frac {27\,B\,b^2}{7\,a^5\,x\,{\left (b\,x^2+a\right )}^{3/2}} \]
((61*B*b)/(35*a^3) + (78*B*b^2*x^2)/(35*a^4))/(x^3*(a + b*x^2)^(5/2)) + (( 128*C*b)/(21*a^5) + (256*C*b^2*x^2)/(21*a^6))/(x*(a + b*x^2)^(1/2)) - (C/( 3*a^2) + (19*C*b*x^2)/(21*a^3))/(x^3*(a + b*x^2)^(5/2)) - ((167*A*b^2)/(35 *a^4) + (191*A*b^3*x^2)/(35*a^5))/(x^3*(a + b*x^2)^(5/2)) + ((1024*A*b^3)/ (35*a^7) + (2048*A*b^4*x^2)/(35*a^8))/(x*(a + b*x^2)^(1/2)) - ((512*B*b^2) /(35*a^6) + (1024*B*b^3*x^2)/(35*a^7))/(x*(a + b*x^2)^(1/2)) - (A*(a + b*x ^2)^(1/2))/(7*a^5*x^7) - (B*(a + b*x^2)^(1/2))/(5*a^5*x^5) - ((a/(b*x^2) + 1)^(9/2)*D*hypergeom([9/2, 5], 6, -a/(b*x^2)))/(10*x*(a + b*x^2)^(9/2)) + (34*A*b*(a + b*x^2)^(1/2))/(35*a^6*x^5) - (B*b)/(7*a^2*x^3*(a + b*x^2)^(7 /2)) - (32*C*b)/(21*a^4*x*(a + b*x^2)^(3/2)) + (C*b^2*x)/(7*a^3*(a + b*x^2 )^(7/2)) - (58*A*b^3)/(7*a^6*x*(a + b*x^2)^(3/2)) + (A*b^2)/(7*a^3*x^3*(a + b*x^2)^(7/2)) + (27*B*b^2)/(7*a^5*x*(a + b*x^2)^(3/2))