Integrand size = 32, antiderivative size = 392 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^{10} \left (a+b x^2\right )^{9/2}} \, dx=-\frac {A}{9 a x^9 \left (a+b x^2\right )^{7/2}}+\frac {16 A b-9 a B}{63 a^2 x^7 \left (a+b x^2\right )^{7/2}}-\frac {32 A b^2-9 a (2 b B-a C)}{45 a^3 x^5 \left (a+b x^2\right )^{7/2}}+\frac {128 A b^3-3 a \left (24 b^2 B-12 a b C+5 a^2 D\right )}{45 a^4 x^3 \left (a+b x^2\right )^{7/2}}-\frac {2 b \left (128 A b^3-3 a \left (24 b^2 B-12 a b C+5 a^2 D\right )\right )}{9 a^5 x \left (a+b x^2\right )^{7/2}}-\frac {16 b^2 \left (128 A b^3-3 a \left (24 b^2 B-12 a b C+5 a^2 D\right )\right ) x}{63 a^6 \left (a+b x^2\right )^{7/2}}-\frac {32 b^2 \left (128 A b^3-3 a \left (24 b^2 B-12 a b C+5 a^2 D\right )\right ) x}{105 a^7 \left (a+b x^2\right )^{5/2}}-\frac {128 b^2 \left (128 A b^3-3 a \left (24 b^2 B-12 a b C+5 a^2 D\right )\right ) x}{315 a^8 \left (a+b x^2\right )^{3/2}}-\frac {256 b^2 \left (128 A b^3-3 a \left (24 b^2 B-12 a b C+5 a^2 D\right )\right ) x}{315 a^9 \sqrt {a+b x^2}} \]
-1/9*A/a/x^9/(b*x^2+a)^(7/2)+1/63*(16*A*b-9*B*a)/a^2/x^7/(b*x^2+a)^(7/2)+1 /45*(-32*A*b^2+9*a*(2*B*b-C*a))/a^3/x^5/(b*x^2+a)^(7/2)+1/45*(128*A*b^3-3* a*(24*B*b^2-12*C*a*b+5*D*a^2))/a^4/x^3/(b*x^2+a)^(7/2)-2/9*b*(128*A*b^3-3* a*(24*B*b^2-12*C*a*b+5*D*a^2))/a^5/x/(b*x^2+a)^(7/2)-16/63*b^2*(128*A*b^3- 3*a*(24*B*b^2-12*C*a*b+5*D*a^2))*x/a^6/(b*x^2+a)^(7/2)-32/105*b^2*(128*A*b ^3-3*a*(24*B*b^2-12*C*a*b+5*D*a^2))*x/a^7/(b*x^2+a)^(5/2)-128/315*b^2*(128 *A*b^3-3*a*(24*B*b^2-12*C*a*b+5*D*a^2))*x/a^8/(b*x^2+a)^(3/2)-256/315*b^2* (128*A*b^3-3*a*(24*B*b^2-12*C*a*b+5*D*a^2))*x/a^9/(b*x^2+a)^(1/2)
Time = 0.72 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.69 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^{10} \left (a+b x^2\right )^{9/2}} \, dx=\frac {-32768 A b^8 x^{16}+2048 a b^7 x^{14} \left (-56 A+9 B x^2\right )-1024 a^2 b^6 x^{12} \left (140 A-63 B x^2+9 C x^4\right )-56 a^6 b^2 x^4 \left (4 A+9 B x^2+45 C x^4-150 D x^6\right )+4480 a^4 b^4 x^8 \left (-2 A+9 B x^2-9 C x^4+3 D x^6\right )+256 a^3 b^5 x^{10} \left (-280 A+315 B x^2-126 C x^4+15 D x^6\right )-a^8 \left (35 A+45 B x^2+63 C x^4+105 D x^6\right )+112 a^5 b^3 x^6 \left (8 A+45 B x^2-180 C x^4+150 D x^6\right )+2 a^7 b x^2 \left (40 A+21 \left (3 B x^2+6 C x^4+25 D x^6\right )\right )}{315 a^9 x^9 \left (a+b x^2\right )^{7/2}} \]
(-32768*A*b^8*x^16 + 2048*a*b^7*x^14*(-56*A + 9*B*x^2) - 1024*a^2*b^6*x^12 *(140*A - 63*B*x^2 + 9*C*x^4) - 56*a^6*b^2*x^4*(4*A + 9*B*x^2 + 45*C*x^4 - 150*D*x^6) + 4480*a^4*b^4*x^8*(-2*A + 9*B*x^2 - 9*C*x^4 + 3*D*x^6) + 256* a^3*b^5*x^10*(-280*A + 315*B*x^2 - 126*C*x^4 + 15*D*x^6) - a^8*(35*A + 45* B*x^2 + 63*C*x^4 + 105*D*x^6) + 112*a^5*b^3*x^6*(8*A + 45*B*x^2 - 180*C*x^ 4 + 150*D*x^6) + 2*a^7*b*x^2*(40*A + 21*(3*B*x^2 + 6*C*x^4 + 25*D*x^6)))/( 315*a^9*x^9*(a + b*x^2)^(7/2))
Time = 0.57 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.74, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.344, Rules used = {2334, 2089, 1588, 27, 359, 245, 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.
| \(\displaystyle \int \frac {A+B x^2+C x^4+D x^6}{x^{10} \left (a+b x^2\right )^{9/2}} \, dx\) |
| \(\Big \downarrow \) 2334 |
| \(\displaystyle -\frac {\int \frac {16 A b-9 a \left (D x^4+C x^2+B\right )}{x^8 \left (b x^2+a\right )^{9/2}}dx}{9 a}-\frac {A}{9 a x^9 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 2089 |
| \(\displaystyle -\frac {\int \frac {-9 a D x^4-9 a C x^2+16 A b-9 a B}{x^8 \left (b x^2+a\right )^{9/2}}dx}{9 a}-\frac {A}{9 a x^9 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 1588 |
| \(\displaystyle -\frac {-\frac {\int \frac {7 \left (32 A b^2+9 a^2 D x^2-9 a (2 b B-a C)\right )}{x^6 \left (b x^2+a\right )^{9/2}}dx}{7 a}-\frac {16 A b-9 a B}{7 a x^7 \left (a+b x^2\right )^{7/2}}}{9 a}-\frac {A}{9 a x^9 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\int \frac {32 A b^2+9 a^2 D x^2-9 a (2 b B-a C)}{x^6 \left (b x^2+a\right )^{9/2}}dx}{a}-\frac {16 A b-9 a B}{7 a x^7 \left (a+b x^2\right )^{7/2}}}{9 a}-\frac {A}{9 a x^9 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 359 |
| \(\displaystyle -\frac {-\frac {-\frac {3 \left (-15 a^3 D-36 a b (2 b B-a C)+128 A b^3\right ) \int \frac {1}{x^4 \left (b x^2+a\right )^{9/2}}dx}{5 a}-\frac {32 A b^2-9 a (2 b B-a C)}{5 a x^5 \left (a+b x^2\right )^{7/2}}}{a}-\frac {16 A b-9 a B}{7 a x^7 \left (a+b x^2\right )^{7/2}}}{9 a}-\frac {A}{9 a x^9 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle -\frac {-\frac {-\frac {3 \left (-15 a^3 D-36 a b (2 b B-a C)+128 A b^3\right ) \left (-\frac {10 b \int \frac {1}{x^2 \left (b x^2+a\right )^{9/2}}dx}{3 a}-\frac {1}{3 a x^3 \left (a+b x^2\right )^{7/2}}\right )}{5 a}-\frac {32 A b^2-9 a (2 b B-a C)}{5 a x^5 \left (a+b x^2\right )^{7/2}}}{a}-\frac {16 A b-9 a B}{7 a x^7 \left (a+b x^2\right )^{7/2}}}{9 a}-\frac {A}{9 a x^9 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle -\frac {-\frac {-\frac {3 \left (-15 a^3 D-36 a b (2 b B-a C)+128 A b^3\right ) \left (-\frac {10 b \left (-\frac {8 b \int \frac {1}{\left (b x^2+a\right )^{9/2}}dx}{a}-\frac {1}{a x \left (a+b x^2\right )^{7/2}}\right )}{3 a}-\frac {1}{3 a x^3 \left (a+b x^2\right )^{7/2}}\right )}{5 a}-\frac {32 A b^2-9 a (2 b B-a C)}{5 a x^5 \left (a+b x^2\right )^{7/2}}}{a}-\frac {16 A b-9 a B}{7 a x^7 \left (a+b x^2\right )^{7/2}}}{9 a}-\frac {A}{9 a x^9 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle -\frac {-\frac {-\frac {3 \left (-15 a^3 D-36 a b (2 b B-a C)+128 A b^3\right ) \left (-\frac {10 b \left (-\frac {8 b \left (\frac {6 \int \frac {1}{\left (b x^2+a\right )^{7/2}}dx}{7 a}+\frac {x}{7 a \left (a+b x^2\right )^{7/2}}\right )}{a}-\frac {1}{a x \left (a+b x^2\right )^{7/2}}\right )}{3 a}-\frac {1}{3 a x^3 \left (a+b x^2\right )^{7/2}}\right )}{5 a}-\frac {32 A b^2-9 a (2 b B-a C)}{5 a x^5 \left (a+b x^2\right )^{7/2}}}{a}-\frac {16 A b-9 a B}{7 a x^7 \left (a+b x^2\right )^{7/2}}}{9 a}-\frac {A}{9 a x^9 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle -\frac {-\frac {-\frac {3 \left (-15 a^3 D-36 a b (2 b B-a C)+128 A b^3\right ) \left (-\frac {10 b \left (-\frac {8 b \left (\frac {6 \left (\frac {4 \int \frac {1}{\left (b x^2+a\right )^{5/2}}dx}{5 a}+\frac {x}{5 a \left (a+b x^2\right )^{5/2}}\right )}{7 a}+\frac {x}{7 a \left (a+b x^2\right )^{7/2}}\right )}{a}-\frac {1}{a x \left (a+b x^2\right )^{7/2}}\right )}{3 a}-\frac {1}{3 a x^3 \left (a+b x^2\right )^{7/2}}\right )}{5 a}-\frac {32 A b^2-9 a (2 b B-a C)}{5 a x^5 \left (a+b x^2\right )^{7/2}}}{a}-\frac {16 A b-9 a B}{7 a x^7 \left (a+b x^2\right )^{7/2}}}{9 a}-\frac {A}{9 a x^9 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle -\frac {-\frac {-\frac {3 \left (-15 a^3 D-36 a b (2 b B-a C)+128 A b^3\right ) \left (-\frac {10 b \left (-\frac {8 b \left (\frac {6 \left (\frac {4 \left (\frac {2 \int \frac {1}{\left (b x^2+a\right )^{3/2}}dx}{3 a}+\frac {x}{3 a \left (a+b x^2\right )^{3/2}}\right )}{5 a}+\frac {x}{5 a \left (a+b x^2\right )^{5/2}}\right )}{7 a}+\frac {x}{7 a \left (a+b x^2\right )^{7/2}}\right )}{a}-\frac {1}{a x \left (a+b x^2\right )^{7/2}}\right )}{3 a}-\frac {1}{3 a x^3 \left (a+b x^2\right )^{7/2}}\right )}{5 a}-\frac {32 A b^2-9 a (2 b B-a C)}{5 a x^5 \left (a+b x^2\right )^{7/2}}}{a}-\frac {16 A b-9 a B}{7 a x^7 \left (a+b x^2\right )^{7/2}}}{9 a}-\frac {A}{9 a x^9 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 208 |
| \(\displaystyle -\frac {-\frac {-\frac {3 \left (-\frac {10 b \left (-\frac {8 b \left (\frac {6 \left (\frac {4 \left (\frac {2 x}{3 a^2 \sqrt {a+b x^2}}+\frac {x}{3 a \left (a+b x^2\right )^{3/2}}\right )}{5 a}+\frac {x}{5 a \left (a+b x^2\right )^{5/2}}\right )}{7 a}+\frac {x}{7 a \left (a+b x^2\right )^{7/2}}\right )}{a}-\frac {1}{a x \left (a+b x^2\right )^{7/2}}\right )}{3 a}-\frac {1}{3 a x^3 \left (a+b x^2\right )^{7/2}}\right ) \left (-15 a^3 D-36 a b (2 b B-a C)+128 A b^3\right )}{5 a}-\frac {32 A b^2-9 a (2 b B-a C)}{5 a x^5 \left (a+b x^2\right )^{7/2}}}{a}-\frac {16 A b-9 a B}{7 a x^7 \left (a+b x^2\right )^{7/2}}}{9 a}-\frac {A}{9 a x^9 \left (a+b x^2\right )^{7/2}}\) |
-1/9*A/(a*x^9*(a + b*x^2)^(7/2)) - (-1/7*(16*A*b - 9*a*B)/(a*x^7*(a + b*x^ 2)^(7/2)) - (-1/5*(32*A*b^2 - 9*a*(2*b*B - a*C))/(a*x^5*(a + b*x^2)^(7/2)) - (3*(128*A*b^3 - 36*a*b*(2*b*B - a*C) - 15*a^3*D)*(-1/3*1/(a*x^3*(a + b* x^2)^(7/2)) - (10*b*(-(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)/(9 *a)
3.2.68.3.1 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 = 3.65 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.64
| method | result | size |
| pseudoelliptic | \(\frac {\left (-105 D x^{6}-63 C \,x^{4}-45 x^{2} B -35 A \right ) a^{8}+80 b \,x^{2} \left (\frac {105}{8} D x^{6}+\frac {63}{20} C \,x^{4}+\frac {63}{40} x^{2} B +A \right ) a^{7}-224 \left (-\frac {75}{2} D x^{6}+\frac {45}{4} C \,x^{4}+\frac {9}{4} x^{2} B +A \right ) b^{2} x^{4} a^{6}+896 \left (\frac {75}{4} D x^{6}-\frac {45}{2} C \,x^{4}+\frac {45}{8} x^{2} B +A \right ) b^{3} x^{6} a^{5}-8960 b^{4} x^{8} \left (-\frac {3}{2} D x^{6}+\frac {9}{2} C \,x^{4}-\frac {9}{2} x^{2} B +A \right ) a^{4}-71680 \left (-\frac {3}{56} D x^{6}+\frac {9}{20} C \,x^{4}-\frac {9}{8} x^{2} B +A \right ) b^{5} x^{10} a^{3}-143360 \left (\frac {9}{140} C \,x^{4}-\frac {9}{20} x^{2} B +A \right ) b^{6} x^{12} a^{2}-114688 \left (-\frac {9 x^{2} B}{56}+A \right ) b^{7} x^{14} a -32768 A \,b^{8} x^{16}}{315 \left (b \,x^{2}+a \right )^{\frac {7}{2}} x^{9} a^{9}}\) | \(249\) |
| gosper | \(-\frac {32768 A \,b^{8} x^{16}-18432 B a \,b^{7} x^{16}+9216 C \,a^{2} b^{6} x^{16}-3840 D a^{3} b^{5} x^{16}+114688 A a \,b^{7} x^{14}-64512 B \,a^{2} b^{6} x^{14}+32256 C \,a^{3} b^{5} x^{14}-13440 D a^{4} b^{4} x^{14}+143360 A \,a^{2} b^{6} x^{12}-80640 B \,a^{3} b^{5} x^{12}+40320 C \,a^{4} b^{4} x^{12}-16800 D a^{5} b^{3} x^{12}+71680 A \,a^{3} b^{5} x^{10}-40320 B \,a^{4} b^{4} x^{10}+20160 C \,a^{5} b^{3} x^{10}-8400 D a^{6} b^{2} x^{10}+8960 A \,a^{4} b^{4} x^{8}-5040 B \,a^{5} b^{3} x^{8}+2520 C \,a^{6} b^{2} x^{8}-1050 D a^{7} b \,x^{8}-896 A \,a^{5} b^{3} x^{6}+504 B \,a^{6} b^{2} x^{6}-252 C \,a^{7} b \,x^{6}+105 D a^{8} x^{6}+224 A \,a^{6} b^{2} x^{4}-126 B \,a^{7} b \,x^{4}+63 C \,a^{8} x^{4}-80 A \,a^{7} b \,x^{2}+45 B \,a^{8} x^{2}+35 A \,a^{8}}{315 x^{9} \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{9}}\) | \(349\) |
| trager | \(-\frac {32768 A \,b^{8} x^{16}-18432 B a \,b^{7} x^{16}+9216 C \,a^{2} b^{6} x^{16}-3840 D a^{3} b^{5} x^{16}+114688 A a \,b^{7} x^{14}-64512 B \,a^{2} b^{6} x^{14}+32256 C \,a^{3} b^{5} x^{14}-13440 D a^{4} b^{4} x^{14}+143360 A \,a^{2} b^{6} x^{12}-80640 B \,a^{3} b^{5} x^{12}+40320 C \,a^{4} b^{4} x^{12}-16800 D a^{5} b^{3} x^{12}+71680 A \,a^{3} b^{5} x^{10}-40320 B \,a^{4} b^{4} x^{10}+20160 C \,a^{5} b^{3} x^{10}-8400 D a^{6} b^{2} x^{10}+8960 A \,a^{4} b^{4} x^{8}-5040 B \,a^{5} b^{3} x^{8}+2520 C \,a^{6} b^{2} x^{8}-1050 D a^{7} b \,x^{8}-896 A \,a^{5} b^{3} x^{6}+504 B \,a^{6} b^{2} x^{6}-252 C \,a^{7} b \,x^{6}+105 D a^{8} x^{6}+224 A \,a^{6} b^{2} x^{4}-126 B \,a^{7} b \,x^{4}+63 C \,a^{8} x^{4}-80 A \,a^{7} b \,x^{2}+45 B \,a^{8} x^{2}+35 A \,a^{8}}{315 x^{9} \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{9}}\) | \(349\) |
| default | \(B \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 )+D \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}{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 \left (-\frac {1}{9 a \,x^{9} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {16 b \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 )}{9 a}\right )\) | \(638\) |
1/315*((-105*D*x^6-63*C*x^4-45*B*x^2-35*A)*a^8+80*b*x^2*(105/8*D*x^6+63/20 *C*x^4+63/40*x^2*B+A)*a^7-224*(-75/2*D*x^6+45/4*C*x^4+9/4*x^2*B+A)*b^2*x^4 *a^6+896*(75/4*D*x^6-45/2*C*x^4+45/8*x^2*B+A)*b^3*x^6*a^5-8960*b^4*x^8*(-3 /2*D*x^6+9/2*C*x^4-9/2*x^2*B+A)*a^4-71680*(-3/56*D*x^6+9/20*C*x^4-9/8*x^2* B+A)*b^5*x^10*a^3-143360*(9/140*C*x^4-9/20*x^2*B+A)*b^6*x^12*a^2-114688*(- 9/56*x^2*B+A)*b^7*x^14*a-32768*A*b^8*x^16)/(b*x^2+a)^(7/2)/x^9/a^9
Time = 0.97 (sec) , antiderivative size = 354, normalized size of antiderivative = 0.90 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^{10} \left (a+b x^2\right )^{9/2}} \, dx=\frac {{\left (256 \, {\left (15 \, D a^{3} b^{5} - 36 \, C a^{2} b^{6} + 72 \, B a b^{7} - 128 \, A b^{8}\right )} x^{16} + 896 \, {\left (15 \, D a^{4} b^{4} - 36 \, C a^{3} b^{5} + 72 \, B a^{2} b^{6} - 128 \, A a b^{7}\right )} x^{14} + 1120 \, {\left (15 \, D a^{5} b^{3} - 36 \, C a^{4} b^{4} + 72 \, B a^{3} b^{5} - 128 \, A a^{2} b^{6}\right )} x^{12} + 560 \, {\left (15 \, D a^{6} b^{2} - 36 \, C a^{5} b^{3} + 72 \, B a^{4} b^{4} - 128 \, A a^{3} b^{5}\right )} x^{10} - 35 \, A a^{8} + 70 \, {\left (15 \, D a^{7} b - 36 \, C a^{6} b^{2} + 72 \, B a^{5} b^{3} - 128 \, A a^{4} b^{4}\right )} x^{8} - 7 \, {\left (15 \, D a^{8} - 36 \, C a^{7} b + 72 \, B a^{6} b^{2} - 128 \, A a^{5} b^{3}\right )} x^{6} - 7 \, {\left (9 \, C a^{8} - 18 \, B a^{7} b + 32 \, A a^{6} b^{2}\right )} x^{4} - 5 \, {\left (9 \, B a^{8} - 16 \, A a^{7} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{315 \, {\left (a^{9} b^{4} x^{17} + 4 \, a^{10} b^{3} x^{15} + 6 \, a^{11} b^{2} x^{13} + 4 \, a^{12} b x^{11} + a^{13} x^{9}\right )}} \]
1/315*(256*(15*D*a^3*b^5 - 36*C*a^2*b^6 + 72*B*a*b^7 - 128*A*b^8)*x^16 + 8 96*(15*D*a^4*b^4 - 36*C*a^3*b^5 + 72*B*a^2*b^6 - 128*A*a*b^7)*x^14 + 1120* (15*D*a^5*b^3 - 36*C*a^4*b^4 + 72*B*a^3*b^5 - 128*A*a^2*b^6)*x^12 + 560*(1 5*D*a^6*b^2 - 36*C*a^5*b^3 + 72*B*a^4*b^4 - 128*A*a^3*b^5)*x^10 - 35*A*a^8 + 70*(15*D*a^7*b - 36*C*a^6*b^2 + 72*B*a^5*b^3 - 128*A*a^4*b^4)*x^8 - 7*( 15*D*a^8 - 36*C*a^7*b + 72*B*a^6*b^2 - 128*A*a^5*b^3)*x^6 - 7*(9*C*a^8 - 1 8*B*a^7*b + 32*A*a^6*b^2)*x^4 - 5*(9*B*a^8 - 16*A*a^7*b)*x^2)*sqrt(b*x^2 + a)/(a^9*b^4*x^17 + 4*a^10*b^3*x^15 + 6*a^11*b^2*x^13 + 4*a^12*b*x^11 + a^ 13*x^9)
Timed out. \[ \int \frac {A+B x^2+C x^4+D x^6}{x^{10} \left (a+b x^2\right )^{9/2}} \, dx=\text {Timed out} \]
Time = 0.21 (sec) , antiderivative size = 579, normalized size of antiderivative = 1.48 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^{10} \left (a+b x^2\right )^{9/2}} \, dx=\frac {256 \, D b^{2} x}{21 \, \sqrt {b x^{2} + a} a^{6}} + \frac {128 \, D b^{2} x}{21 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{5}} + \frac {32 \, D b^{2} x}{7 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{4}} + \frac {80 \, D b^{2} x}{21 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{3}} - \frac {1024 \, C b^{3} x}{35 \, \sqrt {b x^{2} + a} a^{7}} - \frac {512 \, C b^{3} x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{6}} - \frac {384 \, C b^{3} x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{5}} - \frac {64 \, C b^{3} x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{4}} + \frac {2048 \, B b^{4} x}{35 \, \sqrt {b x^{2} + a} a^{8}} + \frac {1024 \, B b^{4} x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{7}} + \frac {768 \, B b^{4} x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{6}} + \frac {128 \, B b^{4} x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{5}} - \frac {32768 \, A b^{5} x}{315 \, \sqrt {b x^{2} + a} a^{9}} - \frac {16384 \, A b^{5} x}{315 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{8}} - \frac {4096 \, A b^{5} x}{105 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{7}} - \frac {2048 \, A b^{5} x}{63 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{6}} + \frac {10 \, D b}{3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2} x} - \frac {8 \, C b^{2}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{3} x} + \frac {16 \, B b^{3}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{4} x} - \frac {256 \, A b^{4}}{9 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{5} x} - \frac {D}{3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a x^{3}} + \frac {4 \, C b}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2} x^{3}} - \frac {8 \, B b^{2}}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{3} x^{3}} + \frac {128 \, A b^{3}}{45 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{4} x^{3}} - \frac {C}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a x^{5}} + \frac {2 \, B b}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2} x^{5}} - \frac {32 \, A b^{2}}{45 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{3} x^{5}} - \frac {B}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a x^{7}} + \frac {16 \, A b}{63 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2} x^{7}} - \frac {A}{9 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a x^{9}} \]
256/21*D*b^2*x/(sqrt(b*x^2 + a)*a^6) + 128/21*D*b^2*x/((b*x^2 + a)^(3/2)*a ^5) + 32/7*D*b^2*x/((b*x^2 + a)^(5/2)*a^4) + 80/21*D*b^2*x/((b*x^2 + a)^(7 /2)*a^3) - 1024/35*C*b^3*x/(sqrt(b*x^2 + a)*a^7) - 512/35*C*b^3*x/((b*x^2 + a)^(3/2)*a^6) - 384/35*C*b^3*x/((b*x^2 + a)^(5/2)*a^5) - 64/7*C*b^3*x/(( b*x^2 + a)^(7/2)*a^4) + 2048/35*B*b^4*x/(sqrt(b*x^2 + a)*a^8) + 1024/35*B* b^4*x/((b*x^2 + a)^(3/2)*a^7) + 768/35*B*b^4*x/((b*x^2 + a)^(5/2)*a^6) + 1 28/7*B*b^4*x/((b*x^2 + a)^(7/2)*a^5) - 32768/315*A*b^5*x/(sqrt(b*x^2 + a)* a^9) - 16384/315*A*b^5*x/((b*x^2 + a)^(3/2)*a^8) - 4096/105*A*b^5*x/((b*x^ 2 + a)^(5/2)*a^7) - 2048/63*A*b^5*x/((b*x^2 + a)^(7/2)*a^6) + 10/3*D*b/((b *x^2 + a)^(7/2)*a^2*x) - 8*C*b^2/((b*x^2 + a)^(7/2)*a^3*x) + 16*B*b^3/((b* x^2 + a)^(7/2)*a^4*x) - 256/9*A*b^4/((b*x^2 + a)^(7/2)*a^5*x) - 1/3*D/((b* x^2 + a)^(7/2)*a*x^3) + 4/5*C*b/((b*x^2 + a)^(7/2)*a^2*x^3) - 8/5*B*b^2/(( b*x^2 + a)^(7/2)*a^3*x^3) + 128/45*A*b^3/((b*x^2 + a)^(7/2)*a^4*x^3) - 1/5 *C/((b*x^2 + a)^(7/2)*a*x^5) + 2/5*B*b/((b*x^2 + a)^(7/2)*a^2*x^5) - 32/45 *A*b^2/((b*x^2 + a)^(7/2)*a^3*x^5) - 1/7*B/((b*x^2 + a)^(7/2)*a*x^7) + 16/ 63*A*b/((b*x^2 + a)^(7/2)*a^2*x^7) - 1/9*A/((b*x^2 + a)^(7/2)*a*x^9)
Leaf count of result is larger than twice the leaf count of optimal. 1162 vs. \(2 (355) = 710\).
Time = 0.31 (sec) , antiderivative size = 1162, normalized size of antiderivative = 2.96 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^{10} \left (a+b x^2\right )^{9/2}} \, dx=\text {Too large to display} \]
1/105*((x^2*((790*D*a^24*b^8 - 1686*C*a^23*b^9 + 3072*B*a^22*b^10 - 5053*A *a^21*b^11)*x^2/(a^30*b^3) + 7*(365*D*a^25*b^7 - 768*C*a^24*b^8 + 1386*B*a ^23*b^9 - 2264*A*a^22*b^10)/(a^30*b^3)) + 35*(80*D*a^26*b^6 - 165*C*a^25*b ^7 + 294*B*a^24*b^8 - 476*A*a^23*b^9)/(a^30*b^3))*x^2 + 105*(10*D*a^27*b^5 - 20*C*a^26*b^6 + 35*B*a^25*b^7 - 56*A*a^24*b^8)/(a^30*b^3))*x/(b*x^2 + a )^(7/2) - 2/315*(1260*(sqrt(b)*x - sqrt(b*x^2 + a))^16*D*a^3*b^(3/2) - 315 0*(sqrt(b)*x - sqrt(b*x^2 + a))^16*C*a^2*b^(5/2) + 6300*(sqrt(b)*x - sqrt( b*x^2 + a))^16*B*a*b^(7/2) - 11025*(sqrt(b)*x - sqrt(b*x^2 + a))^16*A*b^(9 /2) - 10710*(sqrt(b)*x - sqrt(b*x^2 + a))^14*D*a^4*b^(3/2) + 27720*(sqrt(b )*x - sqrt(b*x^2 + a))^14*C*a^3*b^(5/2) - 56700*(sqrt(b)*x - sqrt(b*x^2 + a))^14*B*a^2*b^(7/2) + 100800*(sqrt(b)*x - sqrt(b*x^2 + a))^14*A*a*b^(9/2) + 39270*(sqrt(b)*x - sqrt(b*x^2 + a))^12*D*a^5*b^(3/2) - 105840*(sqrt(b)* x - sqrt(b*x^2 + a))^12*C*a^4*b^(5/2) + 223020*(sqrt(b)*x - sqrt(b*x^2 + a ))^12*B*a^3*b^(7/2) - 405300*(sqrt(b)*x - sqrt(b*x^2 + a))^12*A*a^2*b^(9/2 ) - 81270*(sqrt(b)*x - sqrt(b*x^2 + a))^10*D*a^6*b^(3/2) + 226800*(sqrt(b) *x - sqrt(b*x^2 + a))^10*C*a^5*b^(5/2) - 495180*(sqrt(b)*x - sqrt(b*x^2 + a))^10*B*a^4*b^(7/2) + 927360*(sqrt(b)*x - sqrt(b*x^2 + a))^10*A*a^3*b^(9/ 2) + 103950*(sqrt(b)*x - sqrt(b*x^2 + a))^8*D*a^7*b^(3/2) - 297108*(sqrt(b )*x - sqrt(b*x^2 + a))^8*C*a^6*b^(5/2) + 666036*(sqrt(b)*x - sqrt(b*x^2 + a))^8*B*a^5*b^(7/2) - 1291374*(sqrt(b)*x - sqrt(b*x^2 + a))^8*A*a^4*b^(...
Timed out. \[ \int \frac {A+B x^2+C x^4+D x^6}{x^{10} \left (a+b x^2\right )^{9/2}} \, dx=\int \frac {A+B\,x^2+C\,x^4+x^6\,D}{x^{10}\,{\left (b\,x^2+a\right )}^{9/2}} \,d x \]