Integrand size = 32, antiderivative size = 330 \[ \int \frac {A+B x^2+C x^4+B x^6}{x^8 \left (a+b x^2\right )^{9/2}} \, dx=\frac {b \left (A b^3-a \left (a^2 B+b^2 B-a b C\right )\right ) x}{7 a^5 \left (a+b x^2\right )^{7/2}}+\frac {b \left (34 A b^3-a \left (13 a^2 B+27 b^2 B-20 a b C\right )\right ) x}{35 a^6 \left (a+b x^2\right )^{5/2}}+\frac {b \left (486 A b^3-a \left (87 a^2 B+318 b^2 B-185 a b C\right )\right ) x}{105 a^7 \left (a+b x^2\right )^{3/2}}+\frac {b \left (3072 A b^3-a \left (279 a^2 B+1686 b^2 B-790 a b C\right )\right ) x}{105 a^8 \sqrt {a+b x^2}}-\frac {A \sqrt {a+b x^2}}{7 a^5 x^7}+\frac {(34 A b-7 a B) \sqrt {a+b x^2}}{35 a^6 x^5}-\frac {\left (486 A b^2-168 a b B+35 a^2 C\right ) \sqrt {a+b x^2}}{105 a^7 x^3}+\frac {\left (3072 A b^3-7 a \left (15 a^2 B+198 b^2 B-70 a b C\right )\right ) \sqrt {a+b x^2}}{105 a^8 x} \] Output:
1/7*b*(A*b^3-a*(B*a^2+B*b^2-C*a*b))*x/a^5/(b*x^2+a)^(7/2)+1/35*b*(34*A*b^3 -a*(13*B*a^2+27*B*b^2-20*C*a*b))*x/a^6/(b*x^2+a)^(5/2)+1/105*b*(486*A*b^3- a*(87*B*a^2+318*B*b^2-185*C*a*b))*x/a^7/(b*x^2+a)^(3/2)+1/105*b*(3072*A*b^ 3-a*(279*B*a^2+1686*B*b^2-790*C*a*b))*x/a^8/(b*x^2+a)^(1/2)-1/7*A*(b*x^2+a )^(1/2)/a^5/x^7+1/35*(34*A*b-7*B*a)*(b*x^2+a)^(1/2)/a^6/x^5-1/105*(486*A*b ^2-168*B*a*b+35*C*a^2)*(b*x^2+a)^(1/2)/a^7/x^3+1/105*(3072*A*b^3-7*a*(15*B *a^2+198*B*b^2-70*C*a*b))*(b*x^2+a)^(1/2)/a^8/x
Time = 0.86 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.70 \[ \int \frac {A+B x^2+C x^4+B x^6}{x^8 \left (a+b x^2\right )^{9/2}} \, dx=\frac {6144 A b^7 x^{14}-3072 a b^6 x^{12} \left (-7 A+B x^2\right )+256 a^2 b^5 x^{10} \left (105 A-42 B x^2+5 C x^4\right )+14 a^6 b x^2 \left (3 A+6 B x^2+25 C x^4-60 B x^6\right )+112 a^4 b^3 x^6 \left (15 A+50 C x^4-12 B x^2 \left (5+x^4\right )\right )+128 a^3 b^4 x^8 \left (105 A+35 C x^4-3 B x^2 \left (35+x^4\right )\right )-56 a^5 b^2 x^4 \left (3 A-50 C x^4+15 B \left (x^2+2 x^6\right )\right )-a^7 \left (15 A+35 C x^4+21 B \left (x^2+5 x^6\right )\right )}{105 a^8 x^7 \left (a+b x^2\right )^{7/2}} \] Input:
Integrate[(A + B*x^2 + C*x^4 + B*x^6)/(x^8*(a + b*x^2)^(9/2)),x]
Output:
(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*B*x^ 6) + 112*a^4*b^3*x^6*(15*A + 50*C*x^4 - 12*B*x^2*(5 + x^4)) + 128*a^3*b^4* x^8*(105*A + 35*C*x^4 - 3*B*x^2*(35 + x^4)) - 56*a^5*b^2*x^4*(3*A - 50*C*x ^4 + 15*B*(x^2 + 2*x^6)) - a^7*(15*A + 35*C*x^4 + 21*B*(x^2 + 5*x^6)))/(10 5*a^8*x^7*(a + b*x^2)^(7/2))
Time = 0.59 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.79, 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.
| \(\displaystyle \int \frac {A+B x^6+B x^2+C x^4}{x^8 \left (a+b x^2\right )^{9/2}} \, dx\) |
| \(\Big \downarrow \) 2334 |
| \(\displaystyle -\frac {\int \frac {7 \left (2 A b-a \left (B x^4+C x^2+B\right )\right )}{x^6 \left (b x^2+a\right )^{9/2}}dx}{7 a}-\frac {A}{7 a x^7 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {2 A b-a \left (B x^4+C x^2+B\right )}{x^6 \left (b x^2+a\right )^{9/2}}dx}{a}-\frac {A}{7 a x^7 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 2089 |
| \(\displaystyle -\frac {\int \frac {-a B x^4-a C x^2+2 A b-a B}{x^6 \left (b x^2+a\right )^{9/2}}dx}{a}-\frac {A}{7 a x^7 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 1588 |
| \(\displaystyle -\frac {-\frac {\int \frac {24 A b^2+5 a^2 B x^2-a (12 b B-5 a C)}{x^4 \left (b x^2+a\right )^{9/2}}dx}{5 a}-\frac {2 A b-a B}{5 a x^5 \left (a+b x^2\right )^{7/2}}}{a}-\frac {A}{7 a x^7 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 359 |
| \(\displaystyle -\frac {-\frac {-\frac {5 \left (48 A b^3-a \left (3 a^2 B-10 a b C+24 b^2 B\right )\right ) \int \frac {1}{x^2 \left (b x^2+a\right )^{9/2}}dx}{3 a}-\frac {24 A b^2-a (12 b B-5 a C)}{3 a x^3 \left (a+b x^2\right )^{7/2}}}{5 a}-\frac {2 A b-a B}{5 a x^5 \left (a+b x^2\right )^{7/2}}}{a}-\frac {A}{7 a x^7 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle -\frac {-\frac {-\frac {5 \left (48 A b^3-a \left (3 a^2 B-10 a b C+24 b^2 B\right )\right ) \left (-\frac {8 b \int \frac {1}{\left (b x^2+a\right )^{9/2}}dx}{a}-\frac {1}{a x \left (a+b x^2\right )^{7/2}}\right )}{3 a}-\frac {24 A b^2-a (12 b B-5 a C)}{3 a x^3 \left (a+b x^2\right )^{7/2}}}{5 a}-\frac {2 A b-a B}{5 a x^5 \left (a+b x^2\right )^{7/2}}}{a}-\frac {A}{7 a x^7 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle -\frac {-\frac {-\frac {5 \left (48 A b^3-a \left (3 a^2 B-10 a b C+24 b^2 B\right )\right ) \left (-\frac {8 b \left (\frac {6 \int \frac {1}{\left (b x^2+a\right )^{7/2}}dx}{7 a}+\frac {x}{7 a \left (a+b x^2\right )^{7/2}}\right )}{a}-\frac {1}{a x \left (a+b x^2\right )^{7/2}}\right )}{3 a}-\frac {24 A b^2-a (12 b B-5 a C)}{3 a x^3 \left (a+b x^2\right )^{7/2}}}{5 a}-\frac {2 A b-a B}{5 a x^5 \left (a+b x^2\right )^{7/2}}}{a}-\frac {A}{7 a x^7 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle -\frac {-\frac {-\frac {5 \left (48 A b^3-a \left (3 a^2 B-10 a b C+24 b^2 B\right )\right ) \left (-\frac {8 b \left (\frac {6 \left (\frac {4 \int \frac {1}{\left (b x^2+a\right )^{5/2}}dx}{5 a}+\frac {x}{5 a \left (a+b x^2\right )^{5/2}}\right )}{7 a}+\frac {x}{7 a \left (a+b x^2\right )^{7/2}}\right )}{a}-\frac {1}{a x \left (a+b x^2\right )^{7/2}}\right )}{3 a}-\frac {24 A b^2-a (12 b B-5 a C)}{3 a x^3 \left (a+b x^2\right )^{7/2}}}{5 a}-\frac {2 A b-a B}{5 a x^5 \left (a+b x^2\right )^{7/2}}}{a}-\frac {A}{7 a x^7 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle -\frac {-\frac {-\frac {5 \left (48 A b^3-a \left (3 a^2 B-10 a b C+24 b^2 B\right )\right ) \left (-\frac {8 b \left (\frac {6 \left (\frac {4 \left (\frac {2 \int \frac {1}{\left (b x^2+a\right )^{3/2}}dx}{3 a}+\frac {x}{3 a \left (a+b x^2\right )^{3/2}}\right )}{5 a}+\frac {x}{5 a \left (a+b x^2\right )^{5/2}}\right )}{7 a}+\frac {x}{7 a \left (a+b x^2\right )^{7/2}}\right )}{a}-\frac {1}{a x \left (a+b x^2\right )^{7/2}}\right )}{3 a}-\frac {24 A b^2-a (12 b B-5 a C)}{3 a x^3 \left (a+b x^2\right )^{7/2}}}{5 a}-\frac {2 A b-a B}{5 a x^5 \left (a+b x^2\right )^{7/2}}}{a}-\frac {A}{7 a x^7 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 208 |
| \(\displaystyle -\frac {-\frac {-\frac {5 \left (-\frac {8 b \left (\frac {6 \left (\frac {4 \left (\frac {2 x}{3 a^2 \sqrt {a+b x^2}}+\frac {x}{3 a \left (a+b x^2\right )^{3/2}}\right )}{5 a}+\frac {x}{5 a \left (a+b x^2\right )^{5/2}}\right )}{7 a}+\frac {x}{7 a \left (a+b x^2\right )^{7/2}}\right )}{a}-\frac {1}{a x \left (a+b x^2\right )^{7/2}}\right ) \left (48 A b^3-a \left (3 a^2 B-10 a b C+24 b^2 B\right )\right )}{3 a}-\frac {24 A b^2-a (12 b B-5 a C)}{3 a x^3 \left (a+b x^2\right )^{7/2}}}{5 a}-\frac {2 A b-a B}{5 a x^5 \left (a+b x^2\right )^{7/2}}}{a}-\frac {A}{7 a x^7 \left (a+b x^2\right )^{7/2}}\) |
Input:
Int[(A + B*x^2 + C*x^4 + B*x^6)/(x^8*(a + b*x^2)^(9/2)),x]
Output:
-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*(3*a^2*B + 24*b^2*B - 10*a*b*C))*(-(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
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 = 0.94 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.66
| method | result | size |
| pseudoelliptic | \(\frac {\left (-105 B \,x^{6}-35 C \,x^{4}-21 x^{2} B -15 A \right ) a^{7}+42 \left (-20 B \,x^{6}+\frac {25}{3} C \,x^{4}+2 x^{2} B +A \right ) x^{2} b \,a^{6}-168 \left (10 B \,x^{6}-\frac {50}{3} C \,x^{4}+5 x^{2} B +A \right ) x^{4} b^{2} a^{5}+1680 \left (-\frac {4}{5} B \,x^{6}+\frac {10}{3} C \,x^{4}-4 x^{2} B +A \right ) x^{6} b^{3} a^{4}+13440 \left (-\frac {1}{35} B \,x^{6}+\frac {1}{3} C \,x^{4}-x^{2} B +A \right ) x^{8} b^{4} a^{3}+26880 \left (\frac {1}{21} C \,x^{4}-\frac {2}{5} x^{2} B +A \right ) x^{10} b^{5} a^{2}+21504 \left (-\frac {x^{2} B}{7}+A \right ) x^{12} b^{6} a +6144 A \,b^{7} x^{14}}{105 \left (b \,x^{2}+a \right )^{\frac {7}{2}} x^{7} a^{8}}\) | \(218\) |
| gosper | \(-\frac {-6144 A \,b^{7} x^{14}+384 B \,a^{3} b^{4} x^{14}+3072 B a \,b^{6} x^{14}-1280 C \,a^{2} b^{5} x^{14}-21504 A a \,b^{6} x^{12}+1344 B \,a^{4} b^{3} x^{12}+10752 B \,a^{2} b^{5} x^{12}-4480 C \,a^{3} b^{4} x^{12}-26880 A \,a^{2} b^{5} x^{10}+1680 B \,a^{5} b^{2} x^{10}+13440 B \,a^{3} b^{4} x^{10}-5600 C \,a^{4} b^{3} x^{10}-13440 A \,a^{3} b^{4} x^{8}+840 B \,a^{6} b \,x^{8}+6720 B \,a^{4} b^{3} x^{8}-2800 C \,a^{5} b^{2} x^{8}-1680 A \,a^{4} b^{3} x^{6}+105 B \,a^{7} x^{6}+840 B \,a^{5} b^{2} x^{6}-350 C \,a^{6} b \,x^{6}+168 A \,a^{5} b^{2} x^{4}-84 B \,a^{6} b \,x^{4}+35 C \,a^{7} x^{4}-42 A \,a^{6} b \,x^{2}+21 B \,a^{7} x^{2}+15 A \,a^{7}}{105 x^{7} \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{8}}\) | \(301\) |
| trager | \(-\frac {-6144 A \,b^{7} x^{14}+384 B \,a^{3} b^{4} x^{14}+3072 B a \,b^{6} x^{14}-1280 C \,a^{2} b^{5} x^{14}-21504 A a \,b^{6} x^{12}+1344 B \,a^{4} b^{3} x^{12}+10752 B \,a^{2} b^{5} x^{12}-4480 C \,a^{3} b^{4} x^{12}-26880 A \,a^{2} b^{5} x^{10}+1680 B \,a^{5} b^{2} x^{10}+13440 B \,a^{3} b^{4} x^{10}-5600 C \,a^{4} b^{3} x^{10}-13440 A \,a^{3} b^{4} x^{8}+840 B \,a^{6} b \,x^{8}+6720 B \,a^{4} b^{3} x^{8}-2800 C \,a^{5} b^{2} x^{8}-1680 A \,a^{4} b^{3} x^{6}+105 B \,a^{7} x^{6}+840 B \,a^{5} b^{2} x^{6}-350 C \,a^{6} b \,x^{6}+168 A \,a^{5} b^{2} x^{4}-84 B \,a^{6} b \,x^{4}+35 C \,a^{7} x^{4}-42 A \,a^{6} b \,x^{2}+21 B \,a^{7} x^{2}+15 A \,a^{7}}{105 x^{7} \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{8}}\) | \(301\) |
| orering | \(-\frac {-6144 A \,b^{7} x^{14}+384 B \,a^{3} b^{4} x^{14}+3072 B a \,b^{6} x^{14}-1280 C \,a^{2} b^{5} x^{14}-21504 A a \,b^{6} x^{12}+1344 B \,a^{4} b^{3} x^{12}+10752 B \,a^{2} b^{5} x^{12}-4480 C \,a^{3} b^{4} x^{12}-26880 A \,a^{2} b^{5} x^{10}+1680 B \,a^{5} b^{2} x^{10}+13440 B \,a^{3} b^{4} x^{10}-5600 C \,a^{4} b^{3} x^{10}-13440 A \,a^{3} b^{4} x^{8}+840 B \,a^{6} b \,x^{8}+6720 B \,a^{4} b^{3} x^{8}-2800 C \,a^{5} b^{2} x^{8}-1680 A \,a^{4} b^{3} x^{6}+105 B \,a^{7} x^{6}+840 B \,a^{5} b^{2} x^{6}-350 C \,a^{6} b \,x^{6}+168 A \,a^{5} b^{2} x^{4}-84 B \,a^{6} b \,x^{4}+35 C \,a^{7} x^{4}-42 A \,a^{6} b \,x^{2}+21 B \,a^{7} x^{2}+15 A \,a^{7}}{105 x^{7} \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{8}}\) | \(301\) |
| risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (-3072 A \,b^{3} x^{6}+105 B \,a^{3} x^{6}+1386 B a \,b^{2} x^{6}-490 C \,a^{2} b \,x^{6}+486 a A \,b^{2} x^{4}-168 B \,a^{2} b \,x^{4}+35 C \,a^{3} x^{4}-102 a^{2} A b \,x^{2}+21 B \,a^{3} x^{2}+15 a^{3} A \right )}{105 a^{8} x^{7}}+\frac {\sqrt {b \,x^{2}+a}\, x \left (3072 A \,b^{6} x^{6}-279 B \,a^{3} b^{3} x^{6}-1686 B a \,b^{5} x^{6}+790 C \,b^{4} a^{2} x^{6}+9702 A a \,b^{5} x^{4}-924 B \,a^{4} b^{2} x^{4}-5376 B \,a^{2} b^{4} x^{4}+2555 C \,a^{3} b^{3} x^{4}+10290 a^{2} x^{2} A \,b^{4}-1050 B \,a^{5} b \,x^{2}-5775 a^{3} x^{2} B \,b^{3}+2800 C \,a^{4} b^{2} x^{2}+3675 A \,a^{3} b^{3}-420 B \,a^{6}-2100 B \,a^{4} b^{2}+1050 C b \,a^{5}\right ) b}{105 a^{8} \left (b^{4} x^{8}+4 a \,b^{3} x^{6}+6 a^{2} b^{2} x^{4}+4 a^{3} b \,x^{2}+a^{4}\right )}\) | \(337\) |
| 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 )+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 )+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 )+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 )\) | \(542\) |
Input:
int((B*x^6+C*x^4+B*x^2+A)/x^8/(b*x^2+a)^(9/2),x,method=_RETURNVERBOSE)
Output:
1/105*((-105*B*x^6-35*C*x^4-21*B*x^2-15*A)*a^7+42*(-20*B*x^6+25/3*C*x^4+2* x^2*B+A)*x^2*b*a^6-168*(10*B*x^6-50/3*C*x^4+5*x^2*B+A)*x^4*b^2*a^5+1680*(- 4/5*B*x^6+10/3*C*x^4-4*x^2*B+A)*x^6*b^3*a^4+13440*(-1/35*B*x^6+1/3*C*x^4-x ^2*B+A)*x^8*b^4*a^3+26880*(1/21*C*x^4-2/5*x^2*B+A)*x^10*b^5*a^2+21504*(-1/ 7*x^2*B+A)*x^12*b^6*a+6144*A*b^7*x^14)/(b*x^2+a)^(7/2)/x^7/a^8
Time = 0.33 (sec) , antiderivative size = 311, normalized size of antiderivative = 0.94 \[ \int \frac {A+B x^2+C x^4+B x^6}{x^8 \left (a+b x^2\right )^{9/2}} \, dx=-\frac {{\left (128 \, {\left (3 \, B a^{3} b^{4} - 10 \, C a^{2} b^{5} + 24 \, B a b^{6} - 48 \, A b^{7}\right )} x^{14} + 448 \, {\left (3 \, B a^{4} b^{3} - 10 \, C a^{3} b^{4} + 24 \, B a^{2} b^{5} - 48 \, A a b^{6}\right )} x^{12} + 560 \, {\left (3 \, B a^{5} b^{2} - 10 \, C a^{4} b^{3} + 24 \, B a^{3} b^{4} - 48 \, A a^{2} b^{5}\right )} x^{10} + 280 \, {\left (3 \, B a^{6} b - 10 \, C a^{5} b^{2} + 24 \, B a^{4} b^{3} - 48 \, A a^{3} b^{4}\right )} x^{8} + 15 \, A a^{7} + 35 \, {\left (3 \, B a^{7} - 10 \, C a^{6} b + 24 \, B a^{5} b^{2} - 48 \, A a^{4} b^{3}\right )} x^{6} + 7 \, {\left (5 \, C a^{7} - 12 \, B a^{6} b + 24 \, A a^{5} b^{2}\right )} x^{4} + 21 \, {\left (B a^{7} - 2 \, A a^{6} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{105 \, {\left (a^{8} b^{4} x^{15} + 4 \, a^{9} b^{3} x^{13} + 6 \, a^{10} b^{2} x^{11} + 4 \, a^{11} b x^{9} + a^{12} x^{7}\right )}} \] Input:
integrate((B*x^6+C*x^4+B*x^2+A)/x^8/(b*x^2+a)^(9/2),x, algorithm="fricas")
Output:
-1/105*(128*(3*B*a^3*b^4 - 10*C*a^2*b^5 + 24*B*a*b^6 - 48*A*b^7)*x^14 + 44 8*(3*B*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*B *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*B*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*B* 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. \[ \int \frac {A+B x^2+C x^4+B x^6}{x^8 \left (a+b x^2\right )^{9/2}} \, dx=\text {Timed out} \] Input:
integrate((B*x**6+C*x**4+B*x**2+A)/x**8/(b*x**2+a)**(9/2),x)
Output:
Timed out
Time = 0.05 (sec) , antiderivative size = 489, normalized size of antiderivative = 1.48 \[ \int \frac {A+B x^2+C x^4+B x^6}{x^8 \left (a+b x^2\right )^{9/2}} \, dx=-\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 \, 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 {B}{{\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}} \] Input:
integrate((B*x^6+C*x^4+B*x^2+A)/x^8/(b*x^2+a)^(9/2),x, algorithm="maxima")
Output:
-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*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) - B/((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. \(2 (298) = 596\).
Time = 0.15 (sec) , antiderivative size = 938, normalized size of antiderivative = 2.84 \[ \int \frac {A+B x^2+C x^4+B x^6}{x^8 \left (a+b x^2\right )^{9/2}} \, dx =\text {Too large to display} \] Input:
integrate((B*x^6+C*x^4+B*x^2+A)/x^8/(b*x^2+a)^(9/2),x, algorithm="giac")
Output:
-1/105*((x^2*((279*B*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*B*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*B*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*B*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*B*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*B*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*B*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*B*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*B*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 = 2.49 (sec) , antiderivative size = 522, normalized size of antiderivative = 1.58 \[ \int \frac {A+B x^2+C x^4+B x^6}{x^8 \left (a+b x^2\right )^{9/2}} \, dx=\frac {16\,B}{35\,a^3\,x\,{\left (b\,x^2+a\right )}^{3/2}}-\frac {64\,B}{35\,a^4\,x\,\sqrt {b\,x^2+a}}-\frac {A\,\sqrt {b\,x^2+a}}{7\,a^5\,x^7}+\frac {8\,B}{35\,a^2\,x\,{\left (b\,x^2+a\right )}^{5/2}}-\frac {B\,\sqrt {b\,x^2+a}}{5\,a^5\,x^5}-\frac {10\,C}{21\,a^2\,x^3\,{\left (b\,x^2+a\right )}^{5/2}}-\frac {B}{7\,b\,x^3\,{\left (b\,x^2+a\right )}^{7/2}}+\frac {C}{7\,a\,x^3\,{\left (b\,x^2+a\right )}^{7/2}}+\frac {34\,A\,b\,\sqrt {b\,x^2+a}}{35\,a^6\,x^5}+\frac {2048\,A\,b^4\,x}{35\,a^8\,\sqrt {b\,x^2+a}}+\frac {61\,B\,b}{35\,a^3\,x^3\,{\left (b\,x^2+a\right )}^{5/2}}-\frac {1024\,B\,b^3\,x}{35\,a^7\,\sqrt {b\,x^2+a}}-\frac {B\,b}{7\,a^2\,x^3\,{\left (b\,x^2+a\right )}^{7/2}}+\frac {128\,C\,b}{21\,a^5\,x\,\sqrt {b\,x^2+a}}-\frac {32\,C\,b}{21\,a^4\,x\,{\left (b\,x^2+a\right )}^{3/2}}-\frac {16\,C\,b}{21\,a^3\,x\,{\left (b\,x^2+a\right )}^{5/2}}+\frac {256\,C\,b^2\,x}{21\,a^6\,\sqrt {b\,x^2+a}}+\frac {1024\,A\,b^3}{35\,a^7\,x\,\sqrt {b\,x^2+a}}-\frac {58\,A\,b^3}{7\,a^6\,x\,{\left (b\,x^2+a\right )}^{3/2}}-\frac {167\,A\,b^2}{35\,a^4\,x^3\,{\left (b\,x^2+a\right )}^{5/2}}-\frac {191\,A\,b^3}{35\,a^5\,x\,{\left (b\,x^2+a\right )}^{5/2}}+\frac {A\,b^2}{7\,a^3\,x^3\,{\left (b\,x^2+a\right )}^{7/2}}+\frac {B}{7\,a\,b\,x^3\,{\left (b\,x^2+a\right )}^{5/2}}-\frac {512\,B\,b^2}{35\,a^6\,x\,\sqrt {b\,x^2+a}}+\frac {27\,B\,b^2}{7\,a^5\,x\,{\left (b\,x^2+a\right )}^{3/2}}+\frac {78\,B\,b^2}{35\,a^4\,x\,{\left (b\,x^2+a\right )}^{5/2}}-\frac {128\,B\,b\,x}{35\,a^5\,\sqrt {b\,x^2+a}} \] Input:
int((A + B*x^2 + B*x^6 + C*x^4)/(x^8*(a + b*x^2)^(9/2)),x)
Output:
(16*B)/(35*a^3*x*(a + b*x^2)^(3/2)) - (64*B)/(35*a^4*x*(a + b*x^2)^(1/2)) - (A*(a + b*x^2)^(1/2))/(7*a^5*x^7) + (8*B)/(35*a^2*x*(a + b*x^2)^(5/2)) - (B*(a + b*x^2)^(1/2))/(5*a^5*x^5) - (10*C)/(21*a^2*x^3*(a + b*x^2)^(5/2)) - B/(7*b*x^3*(a + b*x^2)^(7/2)) + C/(7*a*x^3*(a + b*x^2)^(7/2)) + (34*A*b *(a + b*x^2)^(1/2))/(35*a^6*x^5) + (2048*A*b^4*x)/(35*a^8*(a + b*x^2)^(1/2 )) + (61*B*b)/(35*a^3*x^3*(a + b*x^2)^(5/2)) - (1024*B*b^3*x)/(35*a^7*(a + b*x^2)^(1/2)) - (B*b)/(7*a^2*x^3*(a + b*x^2)^(7/2)) + (128*C*b)/(21*a^5*x *(a + b*x^2)^(1/2)) - (32*C*b)/(21*a^4*x*(a + b*x^2)^(3/2)) - (16*C*b)/(21 *a^3*x*(a + b*x^2)^(5/2)) + (256*C*b^2*x)/(21*a^6*(a + b*x^2)^(1/2)) + (10 24*A*b^3)/(35*a^7*x*(a + b*x^2)^(1/2)) - (58*A*b^3)/(7*a^6*x*(a + b*x^2)^( 3/2)) - (167*A*b^2)/(35*a^4*x^3*(a + b*x^2)^(5/2)) - (191*A*b^3)/(35*a^5*x *(a + b*x^2)^(5/2)) + (A*b^2)/(7*a^3*x^3*(a + b*x^2)^(7/2)) + B/(7*a*b*x^3 *(a + b*x^2)^(5/2)) - (512*B*b^2)/(35*a^6*x*(a + b*x^2)^(1/2)) + (27*B*b^2 )/(7*a^5*x*(a + b*x^2)^(3/2)) + (78*B*b^2)/(35*a^4*x*(a + b*x^2)^(5/2)) - (128*B*b*x)/(35*a^5*(a + b*x^2)^(1/2))
Time = 5.34 (sec) , antiderivative size = 585, normalized size of antiderivative = 1.77 \[ \int \frac {A+B x^2+C x^4+B x^6}{x^8 \left (a+b x^2\right )^{9/2}} \, dx=\frac {21 \sqrt {b \,x^{2}+a}\, a^{6} b \,x^{2}-35 \sqrt {b \,x^{2}+a}\, a^{6} c \,x^{4}-84 \sqrt {b \,x^{2}+a}\, a^{5} b^{2} x^{4}+840 \sqrt {b \,x^{2}+a}\, a^{4} b^{3} x^{6}+6720 \sqrt {b \,x^{2}+a}\, a^{3} b^{4} x^{8}+13440 \sqrt {b \,x^{2}+a}\, a^{2} b^{5} x^{10}+10752 \sqrt {b \,x^{2}+a}\, a \,b^{6} x^{12}-1280 \sqrt {b}\, a \,b^{5} c \,x^{15}-1280 \sqrt {b}\, a^{5} b c \,x^{7}-5120 \sqrt {b}\, a^{4} b^{2} c \,x^{9}-7680 \sqrt {b}\, a^{3} b^{3} c \,x^{11}-5120 \sqrt {b}\, a^{2} b^{4} c \,x^{13}-105 \sqrt {b \,x^{2}+a}\, a^{6} b \,x^{6}-840 \sqrt {b \,x^{2}+a}\, a^{5} b^{2} x^{8}-1680 \sqrt {b \,x^{2}+a}\, a^{4} b^{3} x^{10}-1344 \sqrt {b \,x^{2}+a}\, a^{3} b^{4} x^{12}-384 \sqrt {b \,x^{2}+a}\, a^{2} b^{5} x^{14}+384 \sqrt {b}\, a^{6} b \,x^{7}+1536 \sqrt {b}\, a^{5} b^{2} x^{9}+2304 \sqrt {b}\, a^{4} b^{3} x^{11}-3072 \sqrt {b}\, a^{4} b^{3} x^{7}+1536 \sqrt {b}\, a^{3} b^{4} x^{13}-12288 \sqrt {b}\, a^{3} b^{4} x^{9}+384 \sqrt {b}\, a^{2} b^{5} x^{15}-18432 \sqrt {b}\, a^{2} b^{5} x^{11}-12288 \sqrt {b}\, a \,b^{6} x^{13}+3072 \sqrt {b \,x^{2}+a}\, b^{7} x^{14}-3072 \sqrt {b}\, b^{7} x^{15}+350 \sqrt {b \,x^{2}+a}\, a^{5} b c \,x^{6}+2800 \sqrt {b \,x^{2}+a}\, a^{4} b^{2} c \,x^{8}+5600 \sqrt {b \,x^{2}+a}\, a^{3} b^{3} c \,x^{10}+4480 \sqrt {b \,x^{2}+a}\, a^{2} b^{4} c \,x^{12}+1280 \sqrt {b \,x^{2}+a}\, a \,b^{5} c \,x^{14}-15 \sqrt {b \,x^{2}+a}\, a^{7}}{105 a^{7} x^{7} \left (b^{4} x^{8}+4 a \,b^{3} x^{6}+6 a^{2} b^{2} x^{4}+4 a^{3} b \,x^{2}+a^{4}\right )} \] Input:
int((B*x^6+C*x^4+B*x^2+A)/x^8/(b*x^2+a)^(9/2),x)
Output:
( - 15*sqrt(a + b*x**2)*a**7 - 105*sqrt(a + b*x**2)*a**6*b*x**6 + 21*sqrt( a + b*x**2)*a**6*b*x**2 - 35*sqrt(a + b*x**2)*a**6*c*x**4 - 840*sqrt(a + b *x**2)*a**5*b**2*x**8 - 84*sqrt(a + b*x**2)*a**5*b**2*x**4 + 350*sqrt(a + b*x**2)*a**5*b*c*x**6 - 1680*sqrt(a + b*x**2)*a**4*b**3*x**10 + 840*sqrt(a + b*x**2)*a**4*b**3*x**6 + 2800*sqrt(a + b*x**2)*a**4*b**2*c*x**8 - 1344* sqrt(a + b*x**2)*a**3*b**4*x**12 + 6720*sqrt(a + b*x**2)*a**3*b**4*x**8 + 5600*sqrt(a + b*x**2)*a**3*b**3*c*x**10 - 384*sqrt(a + b*x**2)*a**2*b**5*x **14 + 13440*sqrt(a + b*x**2)*a**2*b**5*x**10 + 4480*sqrt(a + b*x**2)*a**2 *b**4*c*x**12 + 10752*sqrt(a + b*x**2)*a*b**6*x**12 + 1280*sqrt(a + b*x**2 )*a*b**5*c*x**14 + 3072*sqrt(a + b*x**2)*b**7*x**14 + 384*sqrt(b)*a**6*b*x **7 + 1536*sqrt(b)*a**5*b**2*x**9 - 1280*sqrt(b)*a**5*b*c*x**7 + 2304*sqrt (b)*a**4*b**3*x**11 - 3072*sqrt(b)*a**4*b**3*x**7 - 5120*sqrt(b)*a**4*b**2 *c*x**9 + 1536*sqrt(b)*a**3*b**4*x**13 - 12288*sqrt(b)*a**3*b**4*x**9 - 76 80*sqrt(b)*a**3*b**3*c*x**11 + 384*sqrt(b)*a**2*b**5*x**15 - 18432*sqrt(b) *a**2*b**5*x**11 - 5120*sqrt(b)*a**2*b**4*c*x**13 - 12288*sqrt(b)*a*b**6*x **13 - 1280*sqrt(b)*a*b**5*c*x**15 - 3072*sqrt(b)*b**7*x**15)/(105*a**7*x* *7*(a**4 + 4*a**3*b*x**2 + 6*a**2*b**2*x**4 + 4*a*b**3*x**6 + b**4*x**8))