Integrand size = 25, antiderivative size = 188 \[ \int \frac {A+B x+C x^2}{x^2 \left (a+b x^2\right )^{9/2}} \, dx=\frac {B-\left (\frac {A b}{a}-C\right ) x}{7 a \left (a+b x^2\right )^{7/2}}+\frac {7 B-\left (\frac {13 A b}{a}-6 C\right ) x}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac {35 B-3 \left (\frac {29 A b}{a}-8 C\right ) x}{105 a^3 \left (a+b x^2\right )^{3/2}}+\frac {35 B-\left (\frac {93 A b}{a}-16 C\right ) x}{35 a^4 \sqrt {a+b x^2}}-\frac {A \sqrt {a+b x^2}}{a^5 x}-\frac {B \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{9/2}} \]
1/7*(B-(A*b/a-C)*x)/a/(b*x^2+a)^(7/2)+1/35*(7*B-(13*A*b/a-6*C)*x)/a^2/(b*x ^2+a)^(5/2)+1/105*(35*B-3*(29*A*b/a-8*C)*x)/a^3/(b*x^2+a)^(3/2)-B*arctanh( (b*x^2+a)^(1/2)/a^(1/2))/a^(9/2)+1/35*(35*B-(93*A*b/a-16*C)*x)/a^4/(b*x^2+ a)^(1/2)-A*(b*x^2+a)^(1/2)/a^5/x
Time = 0.94 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.89 \[ \int \frac {A+B x+C x^2}{x^2 \left (a+b x^2\right )^{9/2}} \, dx=\frac {-384 A b^4 x^8+14 a^2 b^2 x^4 (-120 A+x (25 B+12 C x))+14 a^3 b x^2 (-60 A+x (29 B+15 C x))+3 a b^3 x^6 (-448 A+x (35 B+16 C x))+a^4 (-105 A+x (176 B+105 C x))+210 \sqrt {a} B x \left (a+b x^2\right )^{7/2} \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{105 a^5 x \left (a+b x^2\right )^{7/2}} \]
(-384*A*b^4*x^8 + 14*a^2*b^2*x^4*(-120*A + x*(25*B + 12*C*x)) + 14*a^3*b*x ^2*(-60*A + x*(29*B + 15*C*x)) + 3*a*b^3*x^6*(-448*A + x*(35*B + 16*C*x)) + a^4*(-105*A + x*(176*B + 105*C*x)) + 210*Sqrt[a]*B*x*(a + b*x^2)^(7/2)*A rcTanh[(Sqrt[b]*x - Sqrt[a + b*x^2])/Sqrt[a]])/(105*a^5*x*(a + b*x^2)^(7/2 ))
Time = 0.76 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.13, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {2336, 25, 2336, 25, 2336, 27, 2336, 27, 534, 243, 73, 221}
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.
| \(\displaystyle \int \frac {A+B x+C x^2}{x^2 \left (a+b x^2\right )^{9/2}} \, dx\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle \frac {B-x \left (\frac {A b}{a}-C\right )}{7 a \left (a+b x^2\right )^{7/2}}-\frac {\int -\frac {-6 \left (\frac {A b}{a}-C\right ) x^2+7 B x+7 A}{x^2 \left (b x^2+a\right )^{7/2}}dx}{7 a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {-6 \left (\frac {A b}{a}-C\right ) x^2+7 B x+7 A}{x^2 \left (b x^2+a\right )^{7/2}}dx}{7 a}+\frac {B-x \left (\frac {A b}{a}-C\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle \frac {\frac {7 B-x \left (\frac {13 A b}{a}-6 C\right )}{5 a \left (a+b x^2\right )^{5/2}}-\frac {\int -\frac {-4 \left (\frac {13 A b}{a}-6 C\right ) x^2+35 B x+35 A}{x^2 \left (b x^2+a\right )^{5/2}}dx}{5 a}}{7 a}+\frac {B-x \left (\frac {A b}{a}-C\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {-4 \left (\frac {13 A b}{a}-6 C\right ) x^2+35 B x+35 A}{x^2 \left (b x^2+a\right )^{5/2}}dx}{5 a}+\frac {7 B-x \left (\frac {13 A b}{a}-6 C\right )}{5 a \left (a+b x^2\right )^{5/2}}}{7 a}+\frac {B-x \left (\frac {A b}{a}-C\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle \frac {\frac {\frac {35 B-3 x \left (\frac {29 A b}{a}-8 C\right )}{3 a \left (a+b x^2\right )^{3/2}}-\frac {\int -\frac {3 \left (-2 \left (\frac {29 A b}{a}-8 C\right ) x^2+35 B x+35 A\right )}{x^2 \left (b x^2+a\right )^{3/2}}dx}{3 a}}{5 a}+\frac {7 B-x \left (\frac {13 A b}{a}-6 C\right )}{5 a \left (a+b x^2\right )^{5/2}}}{7 a}+\frac {B-x \left (\frac {A b}{a}-C\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {-2 \left (\frac {29 A b}{a}-8 C\right ) x^2+35 B x+35 A}{x^2 \left (b x^2+a\right )^{3/2}}dx}{a}+\frac {35 B-3 x \left (\frac {29 A b}{a}-8 C\right )}{3 a \left (a+b x^2\right )^{3/2}}}{5 a}+\frac {7 B-x \left (\frac {13 A b}{a}-6 C\right )}{5 a \left (a+b x^2\right )^{5/2}}}{7 a}+\frac {B-x \left (\frac {A b}{a}-C\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle \frac {\frac {\frac {\frac {35 B-x \left (\frac {93 A b}{a}-16 C\right )}{a \sqrt {a+b x^2}}-\frac {\int -\frac {35 (A+B x)}{x^2 \sqrt {b x^2+a}}dx}{a}}{a}+\frac {35 B-3 x \left (\frac {29 A b}{a}-8 C\right )}{3 a \left (a+b x^2\right )^{3/2}}}{5 a}+\frac {7 B-x \left (\frac {13 A b}{a}-6 C\right )}{5 a \left (a+b x^2\right )^{5/2}}}{7 a}+\frac {B-x \left (\frac {A b}{a}-C\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\frac {35 \int \frac {A+B x}{x^2 \sqrt {b x^2+a}}dx}{a}+\frac {35 B-x \left (\frac {93 A b}{a}-16 C\right )}{a \sqrt {a+b x^2}}}{a}+\frac {35 B-3 x \left (\frac {29 A b}{a}-8 C\right )}{3 a \left (a+b x^2\right )^{3/2}}}{5 a}+\frac {7 B-x \left (\frac {13 A b}{a}-6 C\right )}{5 a \left (a+b x^2\right )^{5/2}}}{7 a}+\frac {B-x \left (\frac {A b}{a}-C\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 534 |
| \(\displaystyle \frac {\frac {\frac {\frac {35 \left (B \int \frac {1}{x \sqrt {b x^2+a}}dx-\frac {A \sqrt {a+b x^2}}{a x}\right )}{a}+\frac {35 B-x \left (\frac {93 A b}{a}-16 C\right )}{a \sqrt {a+b x^2}}}{a}+\frac {35 B-3 x \left (\frac {29 A b}{a}-8 C\right )}{3 a \left (a+b x^2\right )^{3/2}}}{5 a}+\frac {7 B-x \left (\frac {13 A b}{a}-6 C\right )}{5 a \left (a+b x^2\right )^{5/2}}}{7 a}+\frac {B-x \left (\frac {A b}{a}-C\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {\frac {\frac {\frac {35 \left (\frac {1}{2} B \int \frac {1}{x^2 \sqrt {b x^2+a}}dx^2-\frac {A \sqrt {a+b x^2}}{a x}\right )}{a}+\frac {35 B-x \left (\frac {93 A b}{a}-16 C\right )}{a \sqrt {a+b x^2}}}{a}+\frac {35 B-3 x \left (\frac {29 A b}{a}-8 C\right )}{3 a \left (a+b x^2\right )^{3/2}}}{5 a}+\frac {7 B-x \left (\frac {13 A b}{a}-6 C\right )}{5 a \left (a+b x^2\right )^{5/2}}}{7 a}+\frac {B-x \left (\frac {A b}{a}-C\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {\frac {\frac {35 \left (\frac {B \int \frac {1}{\frac {x^4}{b}-\frac {a}{b}}d\sqrt {b x^2+a}}{b}-\frac {A \sqrt {a+b x^2}}{a x}\right )}{a}+\frac {35 B-x \left (\frac {93 A b}{a}-16 C\right )}{a \sqrt {a+b x^2}}}{a}+\frac {35 B-3 x \left (\frac {29 A b}{a}-8 C\right )}{3 a \left (a+b x^2\right )^{3/2}}}{5 a}+\frac {7 B-x \left (\frac {13 A b}{a}-6 C\right )}{5 a \left (a+b x^2\right )^{5/2}}}{7 a}+\frac {B-x \left (\frac {A b}{a}-C\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\frac {\frac {35 \left (-\frac {A \sqrt {a+b x^2}}{a x}-\frac {B \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {a}}\right )}{a}+\frac {35 B-x \left (\frac {93 A b}{a}-16 C\right )}{a \sqrt {a+b x^2}}}{a}+\frac {35 B-3 x \left (\frac {29 A b}{a}-8 C\right )}{3 a \left (a+b x^2\right )^{3/2}}}{5 a}+\frac {7 B-x \left (\frac {13 A b}{a}-6 C\right )}{5 a \left (a+b x^2\right )^{5/2}}}{7 a}+\frac {B-x \left (\frac {A b}{a}-C\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
(B - ((A*b)/a - C)*x)/(7*a*(a + b*x^2)^(7/2)) + ((7*B - ((13*A*b)/a - 6*C) *x)/(5*a*(a + b*x^2)^(5/2)) + ((35*B - 3*((29*A*b)/a - 8*C)*x)/(3*a*(a + b *x^2)^(3/2)) + ((35*B - ((93*A*b)/a - 16*C)*x)/(a*Sqrt[a + b*x^2]) + (35*( -((A*Sqrt[a + b*x^2])/(a*x)) - (B*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/Sqrt[a ]))/a)/a)/(5*a))/(7*a)
3.1.56.3.1 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-c)*x^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1))), x] + Simp[d Int[ x^(m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && ILtQ[m, 0] && GtQ[p, -1] && EqQ[m + 2*p + 3, 0]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ {Q = PolynomialQuotient[(c*x)^m*Pq, a + b*x^2, x], f = Coeff[PolynomialRema inder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[(c*x) ^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b*f*x)*((a + b*x^2)^(p + 1)/(2*a* b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1)*Ex pandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x], x]] /; F reeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]
Time = 3.48 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.47
| method | result | size |
| default | \(C \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )+B \left (\frac {1}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {1}{5 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {\frac {1}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}}{a}}{a}}{a}\right )+A \left (-\frac {1}{a x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {8 b \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{a}\right )\) | \(277\) |
| risch | \(\text {Expression too large to display}\) | \(2044\) |
C*(1/7*x/a/(b*x^2+a)^(7/2)+6/7/a*(1/5*x/a/(b*x^2+a)^(5/2)+4/5/a*(1/3*x/a/( b*x^2+a)^(3/2)+2/3*x/a^2/(b*x^2+a)^(1/2))))+B*(1/7/a/(b*x^2+a)^(7/2)+1/a*( 1/5/a/(b*x^2+a)^(5/2)+1/a*(1/3/a/(b*x^2+a)^(3/2)+1/a*(1/a/(b*x^2+a)^(1/2)- 1/a^(3/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)))))+A*(-1/a/x/(b*x^2+a)^(7 /2)-8*b/a*(1/7*x/a/(b*x^2+a)^(7/2)+6/7/a*(1/5*x/a/(b*x^2+a)^(5/2)+4/5/a*(1 /3*x/a/(b*x^2+a)^(3/2)+2/3*x/a^2/(b*x^2+a)^(1/2)))))
Time = 0.35 (sec) , antiderivative size = 525, normalized size of antiderivative = 2.79 \[ \int \frac {A+B x+C x^2}{x^2 \left (a+b x^2\right )^{9/2}} \, dx=\left [\frac {105 \, {\left (B b^{4} x^{9} + 4 \, B a b^{3} x^{7} + 6 \, B a^{2} b^{2} x^{5} + 4 \, B a^{3} b x^{3} + B a^{4} x\right )} \sqrt {a} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (105 \, B a b^{3} x^{7} + 350 \, B a^{2} b^{2} x^{5} + 48 \, {\left (C a b^{3} - 8 \, A b^{4}\right )} x^{8} + 406 \, B a^{3} b x^{3} + 168 \, {\left (C a^{2} b^{2} - 8 \, A a b^{3}\right )} x^{6} + 176 \, B a^{4} x - 105 \, A a^{4} + 210 \, {\left (C a^{3} b - 8 \, A a^{2} b^{2}\right )} x^{4} + 105 \, {\left (C a^{4} - 8 \, A a^{3} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{210 \, {\left (a^{5} b^{4} x^{9} + 4 \, a^{6} b^{3} x^{7} + 6 \, a^{7} b^{2} x^{5} + 4 \, a^{8} b x^{3} + a^{9} x\right )}}, \frac {105 \, {\left (B b^{4} x^{9} + 4 \, B a b^{3} x^{7} + 6 \, B a^{2} b^{2} x^{5} + 4 \, B a^{3} b x^{3} + B a^{4} x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + {\left (105 \, B a b^{3} x^{7} + 350 \, B a^{2} b^{2} x^{5} + 48 \, {\left (C a b^{3} - 8 \, A b^{4}\right )} x^{8} + 406 \, B a^{3} b x^{3} + 168 \, {\left (C a^{2} b^{2} - 8 \, A a b^{3}\right )} x^{6} + 176 \, B a^{4} x - 105 \, A a^{4} + 210 \, {\left (C a^{3} b - 8 \, A a^{2} b^{2}\right )} x^{4} + 105 \, {\left (C a^{4} - 8 \, A a^{3} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{105 \, {\left (a^{5} b^{4} x^{9} + 4 \, a^{6} b^{3} x^{7} + 6 \, a^{7} b^{2} x^{5} + 4 \, a^{8} b x^{3} + a^{9} x\right )}}\right ] \]
[1/210*(105*(B*b^4*x^9 + 4*B*a*b^3*x^7 + 6*B*a^2*b^2*x^5 + 4*B*a^3*b*x^3 + B*a^4*x)*sqrt(a)*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2* (105*B*a*b^3*x^7 + 350*B*a^2*b^2*x^5 + 48*(C*a*b^3 - 8*A*b^4)*x^8 + 406*B* a^3*b*x^3 + 168*(C*a^2*b^2 - 8*A*a*b^3)*x^6 + 176*B*a^4*x - 105*A*a^4 + 21 0*(C*a^3*b - 8*A*a^2*b^2)*x^4 + 105*(C*a^4 - 8*A*a^3*b)*x^2)*sqrt(b*x^2 + a))/(a^5*b^4*x^9 + 4*a^6*b^3*x^7 + 6*a^7*b^2*x^5 + 4*a^8*b*x^3 + a^9*x), 1 /105*(105*(B*b^4*x^9 + 4*B*a*b^3*x^7 + 6*B*a^2*b^2*x^5 + 4*B*a^3*b*x^3 + B *a^4*x)*sqrt(-a)*arctan(sqrt(-a)/sqrt(b*x^2 + a)) + (105*B*a*b^3*x^7 + 350 *B*a^2*b^2*x^5 + 48*(C*a*b^3 - 8*A*b^4)*x^8 + 406*B*a^3*b*x^3 + 168*(C*a^2 *b^2 - 8*A*a*b^3)*x^6 + 176*B*a^4*x - 105*A*a^4 + 210*(C*a^3*b - 8*A*a^2*b ^2)*x^4 + 105*(C*a^4 - 8*A*a^3*b)*x^2)*sqrt(b*x^2 + a))/(a^5*b^4*x^9 + 4*a ^6*b^3*x^7 + 6*a^7*b^2*x^5 + 4*a^8*b*x^3 + a^9*x)]
Leaf count of result is larger than twice the leaf count of optimal. 6922 vs. \(2 (155) = 310\).
Time = 42.51 (sec) , antiderivative size = 6922, normalized size of antiderivative = 36.82 \[ \int \frac {A+B x+C x^2}{x^2 \left (a+b x^2\right )^{9/2}} \, dx=\text {Too large to display} \]
A*(-35*a**4*b**(33/2)*sqrt(a/(b*x**2) + 1)/(35*a**9*b**16 + 140*a**8*b**17 *x**2 + 210*a**7*b**18*x**4 + 140*a**6*b**19*x**6 + 35*a**5*b**20*x**8) - 280*a**3*b**(35/2)*x**2*sqrt(a/(b*x**2) + 1)/(35*a**9*b**16 + 140*a**8*b** 17*x**2 + 210*a**7*b**18*x**4 + 140*a**6*b**19*x**6 + 35*a**5*b**20*x**8) - 560*a**2*b**(37/2)*x**4*sqrt(a/(b*x**2) + 1)/(35*a**9*b**16 + 140*a**8*b **17*x**2 + 210*a**7*b**18*x**4 + 140*a**6*b**19*x**6 + 35*a**5*b**20*x**8 ) - 448*a*b**(39/2)*x**6*sqrt(a/(b*x**2) + 1)/(35*a**9*b**16 + 140*a**8*b* *17*x**2 + 210*a**7*b**18*x**4 + 140*a**6*b**19*x**6 + 35*a**5*b**20*x**8) - 128*b**(41/2)*x**8*sqrt(a/(b*x**2) + 1)/(35*a**9*b**16 + 140*a**8*b**17 *x**2 + 210*a**7*b**18*x**4 + 140*a**6*b**19*x**6 + 35*a**5*b**20*x**8)) + B*(352*a**32*sqrt(1 + b*x**2/a)/(210*a**(73/2) + 2100*a**(71/2)*b*x**2 + 9450*a**(69/2)*b**2*x**4 + 25200*a**(67/2)*b**3*x**6 + 44100*a**(65/2)*b** 4*x**8 + 52920*a**(63/2)*b**5*x**10 + 44100*a**(61/2)*b**6*x**12 + 25200*a **(59/2)*b**7*x**14 + 9450*a**(57/2)*b**8*x**16 + 2100*a**(55/2)*b**9*x**1 8 + 210*a**(53/2)*b**10*x**20) + 105*a**32*log(b*x**2/a)/(210*a**(73/2) + 2100*a**(71/2)*b*x**2 + 9450*a**(69/2)*b**2*x**4 + 25200*a**(67/2)*b**3*x* *6 + 44100*a**(65/2)*b**4*x**8 + 52920*a**(63/2)*b**5*x**10 + 44100*a**(61 /2)*b**6*x**12 + 25200*a**(59/2)*b**7*x**14 + 9450*a**(57/2)*b**8*x**16 + 2100*a**(55/2)*b**9*x**18 + 210*a**(53/2)*b**10*x**20) - 210*a**32*log(sqr t(1 + b*x**2/a) + 1)/(210*a**(73/2) + 2100*a**(71/2)*b*x**2 + 9450*a**(...
Time = 0.22 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.21 \[ \int \frac {A+B x+C x^2}{x^2 \left (a+b x^2\right )^{9/2}} \, dx=\frac {16 \, C x}{35 \, \sqrt {b x^{2} + a} a^{4}} + \frac {8 \, C x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3}} + \frac {6 \, C x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2}} + \frac {C x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a} - \frac {128 \, A b x}{35 \, \sqrt {b x^{2} + a} a^{5}} - \frac {64 \, A b x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{4}} - \frac {48 \, A b x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{3}} - \frac {8 \, A b x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2}} - \frac {B \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{a^{\frac {9}{2}}} + \frac {B}{\sqrt {b x^{2} + a} a^{4}} + \frac {B}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3}} + \frac {B}{5 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2}} + \frac {B}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a} - \frac {A}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} a x} \]
16/35*C*x/(sqrt(b*x^2 + a)*a^4) + 8/35*C*x/((b*x^2 + a)^(3/2)*a^3) + 6/35* C*x/((b*x^2 + a)^(5/2)*a^2) + 1/7*C*x/((b*x^2 + a)^(7/2)*a) - 128/35*A*b*x /(sqrt(b*x^2 + a)*a^5) - 64/35*A*b*x/((b*x^2 + a)^(3/2)*a^4) - 48/35*A*b*x /((b*x^2 + a)^(5/2)*a^3) - 8/7*A*b*x/((b*x^2 + a)^(7/2)*a^2) - B*arcsinh(a /(sqrt(a*b)*abs(x)))/a^(9/2) + B/(sqrt(b*x^2 + a)*a^4) + 1/3*B/((b*x^2 + a )^(3/2)*a^3) + 1/5*B/((b*x^2 + a)^(5/2)*a^2) + 1/7*B/((b*x^2 + a)^(7/2)*a) - A/((b*x^2 + a)^(7/2)*a*x)
Time = 0.31 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.27 \[ \int \frac {A+B x+C x^2}{x^2 \left (a+b x^2\right )^{9/2}} \, dx=\frac {{\left ({\left ({\left ({\left (3 \, {\left (x {\left (\frac {35 \, B b^{3}}{a^{4}} + \frac {{\left (16 \, C a^{20} b^{6} - 93 \, A a^{19} b^{7}\right )} x}{a^{24} b^{3}}\right )} + \frac {28 \, {\left (2 \, C a^{21} b^{5} - 11 \, A a^{20} b^{6}\right )}}{a^{24} b^{3}}\right )} x + \frac {350 \, B b^{2}}{a^{3}}\right )} x + \frac {210 \, {\left (C a^{22} b^{4} - 5 \, A a^{21} b^{5}\right )}}{a^{24} b^{3}}\right )} x + \frac {406 \, B b}{a^{2}}\right )} x + \frac {105 \, {\left (C a^{23} b^{3} - 4 \, A a^{22} b^{4}\right )}}{a^{24} b^{3}}\right )} x + \frac {176 \, B}{a}}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} + \frac {2 \, B \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{4}} + \frac {2 \, A \sqrt {b}}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )} a^{4}} \]
1/105*(((((3*(x*(35*B*b^3/a^4 + (16*C*a^20*b^6 - 93*A*a^19*b^7)*x/(a^24*b^ 3)) + 28*(2*C*a^21*b^5 - 11*A*a^20*b^6)/(a^24*b^3))*x + 350*B*b^2/a^3)*x + 210*(C*a^22*b^4 - 5*A*a^21*b^5)/(a^24*b^3))*x + 406*B*b/a^2)*x + 105*(C*a ^23*b^3 - 4*A*a^22*b^4)/(a^24*b^3))*x + 176*B/a)/(b*x^2 + a)^(7/2) + 2*B*a rctan(-(sqrt(b)*x - sqrt(b*x^2 + a))/sqrt(-a))/(sqrt(-a)*a^4) + 2*A*sqrt(b )/(((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)*a^4)
Time = 6.75 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.20 \[ \int \frac {A+B x+C x^2}{x^2 \left (a+b x^2\right )^{9/2}} \, dx=\frac {\frac {B}{7\,a}+\frac {B\,{\left (b\,x^2+a\right )}^2}{3\,a^3}+\frac {B\,{\left (b\,x^2+a\right )}^3}{a^4}+\frac {B\,\left (b\,x^2+a\right )}{5\,a^2}}{{\left (b\,x^2+a\right )}^{7/2}}-\frac {\frac {A}{a^4}+\frac {128\,A\,b\,x^2}{35\,a^5}}{x\,\sqrt {b\,x^2+a}}-\frac {B\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{a^{9/2}}+\frac {16\,C\,x}{35\,a^4\,\sqrt {b\,x^2+a}}+\frac {8\,C\,x}{35\,a^3\,{\left (b\,x^2+a\right )}^{3/2}}+\frac {6\,C\,x}{35\,a^2\,{\left (b\,x^2+a\right )}^{5/2}}+\frac {C\,x}{7\,a\,{\left (b\,x^2+a\right )}^{7/2}}-\frac {29\,A\,b\,x}{35\,a^4\,{\left (b\,x^2+a\right )}^{3/2}}-\frac {13\,A\,b\,x}{35\,a^3\,{\left (b\,x^2+a\right )}^{5/2}}-\frac {A\,b\,x}{7\,a^2\,{\left (b\,x^2+a\right )}^{7/2}} \]
(B/(7*a) + (B*(a + b*x^2)^2)/(3*a^3) + (B*(a + b*x^2)^3)/a^4 + (B*(a + b*x ^2))/(5*a^2))/(a + b*x^2)^(7/2) - (A/a^4 + (128*A*b*x^2)/(35*a^5))/(x*(a + b*x^2)^(1/2)) - (B*atanh((a + b*x^2)^(1/2)/a^(1/2)))/a^(9/2) + (16*C*x)/( 35*a^4*(a + b*x^2)^(1/2)) + (8*C*x)/(35*a^3*(a + b*x^2)^(3/2)) + (6*C*x)/( 35*a^2*(a + b*x^2)^(5/2)) + (C*x)/(7*a*(a + b*x^2)^(7/2)) - (29*A*b*x)/(35 *a^4*(a + b*x^2)^(3/2)) - (13*A*b*x)/(35*a^3*(a + b*x^2)^(5/2)) - (A*b*x)/ (7*a^2*(a + b*x^2)^(7/2))