Integrand size = 25, antiderivative size = 219 \[ \int \frac {A+B x+C x^2}{x^3 \left (a+b x^2\right )^{9/2}} \, dx=-\frac {a \left (\frac {A b}{a}-C\right )+b B x}{7 a^2 \left (a+b x^2\right )^{7/2}}-\frac {7 (2 A b-a C)+13 b B x}{35 a^3 \left (a+b x^2\right )^{5/2}}-\frac {35 (3 A b-a C)+87 b B x}{105 a^4 \left (a+b x^2\right )^{3/2}}-\frac {35 (4 A b-a C)+93 b B x}{35 a^5 \sqrt {a+b x^2}}-\frac {A \sqrt {a+b x^2}}{2 a^5 x^2}-\frac {B \sqrt {a+b x^2}}{a^5 x}+\frac {(9 A b-2 a C) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{11/2}} \]
1/7*(-a*(A*b/a-C)-B*b*x)/a^2/(b*x^2+a)^(7/2)+1/35*(-13*B*b*x-14*A*b+7*C*a) /a^3/(b*x^2+a)^(5/2)+1/105*(-87*B*b*x-105*A*b+35*C*a)/a^4/(b*x^2+a)^(3/2)+ 1/2*(9*A*b-2*C*a)*arctanh((b*x^2+a)^(1/2)/a^(1/2))/a^(11/2)+1/35*(-93*B*b* x-140*A*b+35*C*a)/a^5/(b*x^2+a)^(1/2)-1/2*A*(b*x^2+a)^(1/2)/a^5/x^2-B*(b*x ^2+a)^(1/2)/a^5/x
Time = 1.14 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.79 \[ \int \frac {A+B x+C x^2}{x^3 \left (a+b x^2\right )^{9/2}} \, dx=\frac {-3 b^4 x^8 (315 A+256 B x)+a^4 \left (-105 A-210 B x+352 C x^2\right )-4 a^3 b x^2 (396 A+7 x (60 B-29 C x))+42 a b^3 x^6 (-75 A+x (-64 B+5 C x))+14 a^2 b^2 x^4 (-261 A+10 x (-24 B+5 C x))}{210 a^5 x^2 \left (a+b x^2\right )^{7/2}}+\frac {(-9 A b+2 a C) \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{11/2}} \]
(-3*b^4*x^8*(315*A + 256*B*x) + a^4*(-105*A - 210*B*x + 352*C*x^2) - 4*a^3 *b*x^2*(396*A + 7*x*(60*B - 29*C*x)) + 42*a*b^3*x^6*(-75*A + x*(-64*B + 5* C*x)) + 14*a^2*b^2*x^4*(-261*A + 10*x*(-24*B + 5*C*x)))/(210*a^5*x^2*(a + b*x^2)^(7/2)) + ((-9*A*b + 2*a*C)*ArcTanh[(Sqrt[b]*x - Sqrt[a + b*x^2])/Sq rt[a]])/a^(11/2)
Time = 0.99 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.12, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.560, Rules used = {2336, 25, 2336, 25, 2336, 27, 2336, 27, 2338, 25, 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.
| \(\displaystyle \int \frac {A+B x+C x^2}{x^3 \left (a+b x^2\right )^{9/2}} \, dx\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle -\frac {\int -\frac {-\frac {6 b B x^3}{a}-7 \left (\frac {A b}{a}-C\right ) x^2+7 B x+7 A}{x^3 \left (b x^2+a\right )^{7/2}}dx}{7 a}-\frac {a \left (\frac {A b}{a}-C\right )+b B x}{7 a^2 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {-\frac {6 b B x^3}{a}-7 \left (\frac {A b}{a}-C\right ) x^2+7 B x+7 A}{x^3 \left (b x^2+a\right )^{7/2}}dx}{7 a}-\frac {a \left (\frac {A b}{a}-C\right )+b B x}{7 a^2 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle \frac {-\frac {\int -\frac {-\frac {52 b B x^3}{a}-35 \left (\frac {2 A b}{a}-C\right ) x^2+35 B x+35 A}{x^3 \left (b x^2+a\right )^{5/2}}dx}{5 a}-\frac {7 (2 A b-a C)+13 b B x}{5 a^2 \left (a+b x^2\right )^{5/2}}}{7 a}-\frac {a \left (\frac {A b}{a}-C\right )+b B x}{7 a^2 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {-\frac {52 b B x^3}{a}-35 \left (\frac {2 A b}{a}-C\right ) x^2+35 B x+35 A}{x^3 \left (b x^2+a\right )^{5/2}}dx}{5 a}-\frac {7 (2 A b-a C)+13 b B x}{5 a^2 \left (a+b x^2\right )^{5/2}}}{7 a}-\frac {a \left (\frac {A b}{a}-C\right )+b B x}{7 a^2 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle \frac {\frac {-\frac {\int -\frac {3 \left (-\frac {58 b B x^3}{a}-35 \left (\frac {3 A b}{a}-C\right ) x^2+35 B x+35 A\right )}{x^3 \left (b x^2+a\right )^{3/2}}dx}{3 a}-\frac {35 (3 A b-a C)+87 b B x}{3 a^2 \left (a+b x^2\right )^{3/2}}}{5 a}-\frac {7 (2 A b-a C)+13 b B x}{5 a^2 \left (a+b x^2\right )^{5/2}}}{7 a}-\frac {a \left (\frac {A b}{a}-C\right )+b B x}{7 a^2 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {-\frac {58 b B x^3}{a}-35 \left (\frac {3 A b}{a}-C\right ) x^2+35 B x+35 A}{x^3 \left (b x^2+a\right )^{3/2}}dx}{a}-\frac {35 (3 A b-a C)+87 b B x}{3 a^2 \left (a+b x^2\right )^{3/2}}}{5 a}-\frac {7 (2 A b-a C)+13 b B x}{5 a^2 \left (a+b x^2\right )^{5/2}}}{7 a}-\frac {a \left (\frac {A b}{a}-C\right )+b B x}{7 a^2 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\int -\frac {35 \left (-\left (\left (\frac {4 A b}{a}-C\right ) x^2\right )+B x+A\right )}{x^3 \sqrt {b x^2+a}}dx}{a}-\frac {35 (4 A b-a C)+93 b B x}{a^2 \sqrt {a+b x^2}}}{a}-\frac {35 (3 A b-a C)+87 b B x}{3 a^2 \left (a+b x^2\right )^{3/2}}}{5 a}-\frac {7 (2 A b-a C)+13 b B x}{5 a^2 \left (a+b x^2\right )^{5/2}}}{7 a}-\frac {a \left (\frac {A b}{a}-C\right )+b B x}{7 a^2 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\frac {35 \int \frac {-\left (\left (\frac {4 A b}{a}-C\right ) x^2\right )+B x+A}{x^3 \sqrt {b x^2+a}}dx}{a}-\frac {35 (4 A b-a C)+93 b B x}{a^2 \sqrt {a+b x^2}}}{a}-\frac {35 (3 A b-a C)+87 b B x}{3 a^2 \left (a+b x^2\right )^{3/2}}}{5 a}-\frac {7 (2 A b-a C)+13 b B x}{5 a^2 \left (a+b x^2\right )^{5/2}}}{7 a}-\frac {a \left (\frac {A b}{a}-C\right )+b B x}{7 a^2 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle \frac {\frac {\frac {\frac {35 \left (-\frac {\int -\frac {2 a B-(9 A b-2 a C) x}{x^2 \sqrt {b x^2+a}}dx}{2 a}-\frac {A \sqrt {a+b x^2}}{2 a x^2}\right )}{a}-\frac {35 (4 A b-a C)+93 b B x}{a^2 \sqrt {a+b x^2}}}{a}-\frac {35 (3 A b-a C)+87 b B x}{3 a^2 \left (a+b x^2\right )^{3/2}}}{5 a}-\frac {7 (2 A b-a C)+13 b B x}{5 a^2 \left (a+b x^2\right )^{5/2}}}{7 a}-\frac {a \left (\frac {A b}{a}-C\right )+b B x}{7 a^2 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\frac {35 \left (\frac {\int \frac {2 a B-(9 A b-2 a C) x}{x^2 \sqrt {b x^2+a}}dx}{2 a}-\frac {A \sqrt {a+b x^2}}{2 a x^2}\right )}{a}-\frac {35 (4 A b-a C)+93 b B x}{a^2 \sqrt {a+b x^2}}}{a}-\frac {35 (3 A b-a C)+87 b B x}{3 a^2 \left (a+b x^2\right )^{3/2}}}{5 a}-\frac {7 (2 A b-a C)+13 b B x}{5 a^2 \left (a+b x^2\right )^{5/2}}}{7 a}-\frac {a \left (\frac {A b}{a}-C\right )+b B x}{7 a^2 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 534 |
| \(\displaystyle \frac {\frac {\frac {\frac {35 \left (\frac {-(9 A b-2 a C) \int \frac {1}{x \sqrt {b x^2+a}}dx-\frac {2 B \sqrt {a+b x^2}}{x}}{2 a}-\frac {A \sqrt {a+b x^2}}{2 a x^2}\right )}{a}-\frac {35 (4 A b-a C)+93 b B x}{a^2 \sqrt {a+b x^2}}}{a}-\frac {35 (3 A b-a C)+87 b B x}{3 a^2 \left (a+b x^2\right )^{3/2}}}{5 a}-\frac {7 (2 A b-a C)+13 b B x}{5 a^2 \left (a+b x^2\right )^{5/2}}}{7 a}-\frac {a \left (\frac {A b}{a}-C\right )+b B x}{7 a^2 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {\frac {\frac {\frac {35 \left (\frac {-\frac {1}{2} (9 A b-2 a C) \int \frac {1}{x^2 \sqrt {b x^2+a}}dx^2-\frac {2 B \sqrt {a+b x^2}}{x}}{2 a}-\frac {A \sqrt {a+b x^2}}{2 a x^2}\right )}{a}-\frac {35 (4 A b-a C)+93 b B x}{a^2 \sqrt {a+b x^2}}}{a}-\frac {35 (3 A b-a C)+87 b B x}{3 a^2 \left (a+b x^2\right )^{3/2}}}{5 a}-\frac {7 (2 A b-a C)+13 b B x}{5 a^2 \left (a+b x^2\right )^{5/2}}}{7 a}-\frac {a \left (\frac {A b}{a}-C\right )+b B x}{7 a^2 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {\frac {\frac {35 \left (\frac {-\frac {(9 A b-2 a C) \int \frac {1}{\frac {x^4}{b}-\frac {a}{b}}d\sqrt {b x^2+a}}{b}-\frac {2 B \sqrt {a+b x^2}}{x}}{2 a}-\frac {A \sqrt {a+b x^2}}{2 a x^2}\right )}{a}-\frac {35 (4 A b-a C)+93 b B x}{a^2 \sqrt {a+b x^2}}}{a}-\frac {35 (3 A b-a C)+87 b B x}{3 a^2 \left (a+b x^2\right )^{3/2}}}{5 a}-\frac {7 (2 A b-a C)+13 b B x}{5 a^2 \left (a+b x^2\right )^{5/2}}}{7 a}-\frac {a \left (\frac {A b}{a}-C\right )+b B x}{7 a^2 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\frac {\frac {35 \left (\frac {\frac {(9 A b-2 a C) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {2 B \sqrt {a+b x^2}}{x}}{2 a}-\frac {A \sqrt {a+b x^2}}{2 a x^2}\right )}{a}-\frac {35 (4 A b-a C)+93 b B x}{a^2 \sqrt {a+b x^2}}}{a}-\frac {35 (3 A b-a C)+87 b B x}{3 a^2 \left (a+b x^2\right )^{3/2}}}{5 a}-\frac {7 (2 A b-a C)+13 b B x}{5 a^2 \left (a+b x^2\right )^{5/2}}}{7 a}-\frac {a \left (\frac {A b}{a}-C\right )+b B x}{7 a^2 \left (a+b x^2\right )^{7/2}}\) |
-1/7*(a*((A*b)/a - C) + b*B*x)/(a^2*(a + b*x^2)^(7/2)) + (-1/5*(7*(2*A*b - a*C) + 13*b*B*x)/(a^2*(a + b*x^2)^(5/2)) + (-1/3*(35*(3*A*b - a*C) + 87*b *B*x)/(a^2*(a + b*x^2)^(3/2)) + (-((35*(4*A*b - a*C) + 93*b*B*x)/(a^2*Sqrt [a + b*x^2])) + (35*(-1/2*(A*Sqrt[a + b*x^2])/(a*x^2) + ((-2*B*Sqrt[a + b* x^2])/x + ((9*A*b - 2*a*C)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/Sqrt[a])/(2*a )))/a)/a)/(5*a))/(7*a)
3.1.57.3.1 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]
Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{ Q = PolynomialQuotient[Pq, c*x, x], R = PolynomialRemainder[Pq, c*x, x]}, S imp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Simp[1/(a*c*( m + 1)) Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*( m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && Lt Q[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])
Time = 3.48 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.49
| method | result | size |
| default | \(C \left (\frac {1}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {1}{5 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {\frac {1}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}}{a}}{a}}{a}\right )+B \left (-\frac {1}{a x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {8 b \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{a}\right )+A \left (-\frac {1}{2 a \,x^{2} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {9 b \left (\frac {1}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {1}{5 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {\frac {1}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}}{a}}{a}}{a}\right )}{2 a}\right )\) | \(327\) |
| risch | \(\text {Expression too large to display}\) | \(2057\) |
C*(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)))))+B*(-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)))))+A*(-1/2/a/x^2/(b*x^2+a)^(7/2)-9/2*b/a*(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))))))
Time = 0.34 (sec) , antiderivative size = 688, normalized size of antiderivative = 3.14 \[ \int \frac {A+B x+C x^2}{x^3 \left (a+b x^2\right )^{9/2}} \, dx=\left [-\frac {105 \, {\left ({\left (2 \, C a b^{4} - 9 \, A b^{5}\right )} x^{10} + 4 \, {\left (2 \, C a^{2} b^{3} - 9 \, A a b^{4}\right )} x^{8} + 6 \, {\left (2 \, C a^{3} b^{2} - 9 \, A a^{2} b^{3}\right )} x^{6} + 4 \, {\left (2 \, C a^{4} b - 9 \, A a^{3} b^{2}\right )} x^{4} + {\left (2 \, C a^{5} - 9 \, A a^{4} b\right )} x^{2}\right )} \sqrt {a} \log \left (-\frac {b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (768 \, B a b^{4} x^{9} + 2688 \, B a^{2} b^{3} x^{7} + 3360 \, B a^{3} b^{2} x^{5} + 1680 \, B a^{4} b x^{3} - 105 \, {\left (2 \, C a^{2} b^{3} - 9 \, A a b^{4}\right )} x^{8} + 210 \, B a^{5} x - 350 \, {\left (2 \, C a^{3} b^{2} - 9 \, A a^{2} b^{3}\right )} x^{6} + 105 \, A a^{5} - 406 \, {\left (2 \, C a^{4} b - 9 \, A a^{3} b^{2}\right )} x^{4} - 176 \, {\left (2 \, C a^{5} - 9 \, A a^{4} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{420 \, {\left (a^{6} b^{4} x^{10} + 4 \, a^{7} b^{3} x^{8} + 6 \, a^{8} b^{2} x^{6} + 4 \, a^{9} b x^{4} + a^{10} x^{2}\right )}}, \frac {105 \, {\left ({\left (2 \, C a b^{4} - 9 \, A b^{5}\right )} x^{10} + 4 \, {\left (2 \, C a^{2} b^{3} - 9 \, A a b^{4}\right )} x^{8} + 6 \, {\left (2 \, C a^{3} b^{2} - 9 \, A a^{2} b^{3}\right )} x^{6} + 4 \, {\left (2 \, C a^{4} b - 9 \, A a^{3} b^{2}\right )} x^{4} + {\left (2 \, C a^{5} - 9 \, A a^{4} b\right )} x^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) - {\left (768 \, B a b^{4} x^{9} + 2688 \, B a^{2} b^{3} x^{7} + 3360 \, B a^{3} b^{2} x^{5} + 1680 \, B a^{4} b x^{3} - 105 \, {\left (2 \, C a^{2} b^{3} - 9 \, A a b^{4}\right )} x^{8} + 210 \, B a^{5} x - 350 \, {\left (2 \, C a^{3} b^{2} - 9 \, A a^{2} b^{3}\right )} x^{6} + 105 \, A a^{5} - 406 \, {\left (2 \, C a^{4} b - 9 \, A a^{3} b^{2}\right )} x^{4} - 176 \, {\left (2 \, C a^{5} - 9 \, A a^{4} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{210 \, {\left (a^{6} b^{4} x^{10} + 4 \, a^{7} b^{3} x^{8} + 6 \, a^{8} b^{2} x^{6} + 4 \, a^{9} b x^{4} + a^{10} x^{2}\right )}}\right ] \]
[-1/420*(105*((2*C*a*b^4 - 9*A*b^5)*x^10 + 4*(2*C*a^2*b^3 - 9*A*a*b^4)*x^8 + 6*(2*C*a^3*b^2 - 9*A*a^2*b^3)*x^6 + 4*(2*C*a^4*b - 9*A*a^3*b^2)*x^4 + ( 2*C*a^5 - 9*A*a^4*b)*x^2)*sqrt(a)*log(-(b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(768*B*a*b^4*x^9 + 2688*B*a^2*b^3*x^7 + 3360*B*a^3*b^2*x^5 + 1680*B*a^4*b*x^3 - 105*(2*C*a^2*b^3 - 9*A*a*b^4)*x^8 + 210*B*a^5*x - 35 0*(2*C*a^3*b^2 - 9*A*a^2*b^3)*x^6 + 105*A*a^5 - 406*(2*C*a^4*b - 9*A*a^3*b ^2)*x^4 - 176*(2*C*a^5 - 9*A*a^4*b)*x^2)*sqrt(b*x^2 + a))/(a^6*b^4*x^10 + 4*a^7*b^3*x^8 + 6*a^8*b^2*x^6 + 4*a^9*b*x^4 + a^10*x^2), 1/210*(105*((2*C* a*b^4 - 9*A*b^5)*x^10 + 4*(2*C*a^2*b^3 - 9*A*a*b^4)*x^8 + 6*(2*C*a^3*b^2 - 9*A*a^2*b^3)*x^6 + 4*(2*C*a^4*b - 9*A*a^3*b^2)*x^4 + (2*C*a^5 - 9*A*a^4*b )*x^2)*sqrt(-a)*arctan(sqrt(-a)/sqrt(b*x^2 + a)) - (768*B*a*b^4*x^9 + 2688 *B*a^2*b^3*x^7 + 3360*B*a^3*b^2*x^5 + 1680*B*a^4*b*x^3 - 105*(2*C*a^2*b^3 - 9*A*a*b^4)*x^8 + 210*B*a^5*x - 350*(2*C*a^3*b^2 - 9*A*a^2*b^3)*x^6 + 105 *A*a^5 - 406*(2*C*a^4*b - 9*A*a^3*b^2)*x^4 - 176*(2*C*a^5 - 9*A*a^4*b)*x^2 )*sqrt(b*x^2 + a))/(a^6*b^4*x^10 + 4*a^7*b^3*x^8 + 6*a^8*b^2*x^6 + 4*a^9*b *x^4 + a^10*x^2)]
Leaf count of result is larger than twice the leaf count of optimal. 11198 vs. \(2 (196) = 392\).
Time = 62.49 (sec) , antiderivative size = 11198, normalized size of antiderivative = 51.13 \[ \int \frac {A+B x+C x^2}{x^3 \left (a+b x^2\right )^{9/2}} \, dx=\text {Too large to display} \]
A*(-70*a**49*sqrt(1 + b*x**2/a)/(140*a**(107/2)*x**2 + 1400*a**(105/2)*b*x **4 + 6300*a**(103/2)*b**2*x**6 + 16800*a**(101/2)*b**3*x**8 + 29400*a**(9 9/2)*b**4*x**10 + 35280*a**(97/2)*b**5*x**12 + 29400*a**(95/2)*b**6*x**14 + 16800*a**(93/2)*b**7*x**16 + 6300*a**(91/2)*b**8*x**18 + 1400*a**(89/2)* b**9*x**20 + 140*a**(87/2)*b**10*x**22) - 1476*a**48*b*x**2*sqrt(1 + b*x** 2/a)/(140*a**(107/2)*x**2 + 1400*a**(105/2)*b*x**4 + 6300*a**(103/2)*b**2* x**6 + 16800*a**(101/2)*b**3*x**8 + 29400*a**(99/2)*b**4*x**10 + 35280*a** (97/2)*b**5*x**12 + 29400*a**(95/2)*b**6*x**14 + 16800*a**(93/2)*b**7*x**1 6 + 6300*a**(91/2)*b**8*x**18 + 1400*a**(89/2)*b**9*x**20 + 140*a**(87/2)* b**10*x**22) - 315*a**48*b*x**2*log(b*x**2/a)/(140*a**(107/2)*x**2 + 1400* a**(105/2)*b*x**4 + 6300*a**(103/2)*b**2*x**6 + 16800*a**(101/2)*b**3*x**8 + 29400*a**(99/2)*b**4*x**10 + 35280*a**(97/2)*b**5*x**12 + 29400*a**(95/ 2)*b**6*x**14 + 16800*a**(93/2)*b**7*x**16 + 6300*a**(91/2)*b**8*x**18 + 1 400*a**(89/2)*b**9*x**20 + 140*a**(87/2)*b**10*x**22) + 630*a**48*b*x**2*l og(sqrt(1 + b*x**2/a) + 1)/(140*a**(107/2)*x**2 + 1400*a**(105/2)*b*x**4 + 6300*a**(103/2)*b**2*x**6 + 16800*a**(101/2)*b**3*x**8 + 29400*a**(99/2)* b**4*x**10 + 35280*a**(97/2)*b**5*x**12 + 29400*a**(95/2)*b**6*x**14 + 168 00*a**(93/2)*b**7*x**16 + 6300*a**(91/2)*b**8*x**18 + 1400*a**(89/2)*b**9* x**20 + 140*a**(87/2)*b**10*x**22) - 9822*a**47*b**2*x**4*sqrt(1 + b*x**2/ a)/(140*a**(107/2)*x**2 + 1400*a**(105/2)*b*x**4 + 6300*a**(103/2)*b**2...
Time = 0.21 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.21 \[ \int \frac {A+B x+C x^2}{x^3 \left (a+b x^2\right )^{9/2}} \, dx=-\frac {128 \, B b x}{35 \, \sqrt {b x^{2} + a} a^{5}} - \frac {64 \, B b x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{4}} - \frac {48 \, B b x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{3}} - \frac {8 \, B b x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2}} - \frac {C \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{a^{\frac {9}{2}}} + \frac {9 \, A b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{2 \, a^{\frac {11}{2}}} + \frac {C}{\sqrt {b x^{2} + a} a^{4}} + \frac {C}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3}} + \frac {C}{5 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2}} + \frac {C}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a} - \frac {9 \, A b}{2 \, \sqrt {b x^{2} + a} a^{5}} - \frac {3 \, A b}{2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{4}} - \frac {9 \, A b}{10 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{3}} - \frac {9 \, A b}{14 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2}} - \frac {B}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} a x} - \frac {A}{2 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a x^{2}} \]
-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) - C*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(9/2) + 9/2*A*b*arcsinh(a/(sqrt(a*b)*a bs(x)))/a^(11/2) + C/(sqrt(b*x^2 + a)*a^4) + 1/3*C/((b*x^2 + a)^(3/2)*a^3) + 1/5*C/((b*x^2 + a)^(5/2)*a^2) + 1/7*C/((b*x^2 + a)^(7/2)*a) - 9/2*A*b/( sqrt(b*x^2 + a)*a^5) - 3/2*A*b/((b*x^2 + a)^(3/2)*a^4) - 9/10*A*b/((b*x^2 + a)^(5/2)*a^3) - 9/14*A*b/((b*x^2 + a)^(7/2)*a^2) - B/((b*x^2 + a)^(7/2)* a*x) - 1/2*A/((b*x^2 + a)^(7/2)*a*x^2)
Time = 0.31 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.48 \[ \int \frac {A+B x+C x^2}{x^3 \left (a+b x^2\right )^{9/2}} \, dx=-\frac {{\left ({\left ({\left ({\left (3 \, {\left ({\left (\frac {93 \, B b^{4} x}{a^{5}} - \frac {35 \, {\left (C a^{24} b^{6} - 4 \, A a^{23} b^{7}\right )}}{a^{28} b^{3}}\right )} x + \frac {308 \, B b^{3}}{a^{4}}\right )} x - \frac {35 \, {\left (10 \, C a^{25} b^{5} - 39 \, A a^{24} b^{6}\right )}}{a^{28} b^{3}}\right )} x + \frac {1050 \, B b^{2}}{a^{3}}\right )} x - \frac {14 \, {\left (29 \, C a^{26} b^{4} - 108 \, A a^{25} b^{5}\right )}}{a^{28} b^{3}}\right )} x + \frac {420 \, B b}{a^{2}}\right )} x - \frac {2 \, {\left (88 \, C a^{27} b^{3} - 291 \, A a^{26} b^{4}\right )}}{a^{28} b^{3}}}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} + \frac {{\left (2 \, C a - 9 \, A b\right )} \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{5}} + \frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} A b + 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a \sqrt {b} + {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} A a b - 2 \, B a^{2} \sqrt {b}}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{2} a^{5}} \]
-1/105*(((((3*((93*B*b^4*x/a^5 - 35*(C*a^24*b^6 - 4*A*a^23*b^7)/(a^28*b^3) )*x + 308*B*b^3/a^4)*x - 35*(10*C*a^25*b^5 - 39*A*a^24*b^6)/(a^28*b^3))*x + 1050*B*b^2/a^3)*x - 14*(29*C*a^26*b^4 - 108*A*a^25*b^5)/(a^28*b^3))*x + 420*B*b/a^2)*x - 2*(88*C*a^27*b^3 - 291*A*a^26*b^4)/(a^28*b^3))/(b*x^2 + a )^(7/2) + (2*C*a - 9*A*b)*arctan(-(sqrt(b)*x - sqrt(b*x^2 + a))/sqrt(-a))/ (sqrt(-a)*a^5) + ((sqrt(b)*x - sqrt(b*x^2 + a))^3*A*b + 2*(sqrt(b)*x - sqr t(b*x^2 + a))^2*B*a*sqrt(b) + (sqrt(b)*x - sqrt(b*x^2 + a))*A*a*b - 2*B*a^ 2*sqrt(b))/(((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^2*a^5)
Time = 7.46 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.27 \[ \int \frac {A+B x+C x^2}{x^3 \left (a+b x^2\right )^{9/2}} \, dx=\frac {\frac {C}{7\,a}+\frac {C\,{\left (b\,x^2+a\right )}^2}{3\,a^3}+\frac {C\,{\left (b\,x^2+a\right )}^3}{a^4}+\frac {C\,\left (b\,x^2+a\right )}{5\,a^2}}{{\left (b\,x^2+a\right )}^{7/2}}-\frac {\frac {A\,b}{7\,a}+\frac {9\,A\,b\,\left (b\,x^2+a\right )}{35\,a^2}+\frac {3\,A\,b\,{\left (b\,x^2+a\right )}^2}{5\,a^3}+\frac {3\,A\,b\,{\left (b\,x^2+a\right )}^3}{a^4}-\frac {9\,A\,b\,{\left (b\,x^2+a\right )}^4}{2\,a^5}}{a\,{\left (b\,x^2+a\right )}^{7/2}-{\left (b\,x^2+a\right )}^{9/2}}-\frac {\frac {B}{a^4}+\frac {128\,B\,b\,x^2}{35\,a^5}}{x\,\sqrt {b\,x^2+a}}-\frac {C\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{a^{9/2}}+\frac {9\,A\,b\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{2\,a^{11/2}}-\frac {29\,B\,b\,x}{35\,a^4\,{\left (b\,x^2+a\right )}^{3/2}}-\frac {13\,B\,b\,x}{35\,a^3\,{\left (b\,x^2+a\right )}^{5/2}}-\frac {B\,b\,x}{7\,a^2\,{\left (b\,x^2+a\right )}^{7/2}} \]
(C/(7*a) + (C*(a + b*x^2)^2)/(3*a^3) + (C*(a + b*x^2)^3)/a^4 + (C*(a + b*x ^2))/(5*a^2))/(a + b*x^2)^(7/2) - ((A*b)/(7*a) + (9*A*b*(a + b*x^2))/(35*a ^2) + (3*A*b*(a + b*x^2)^2)/(5*a^3) + (3*A*b*(a + b*x^2)^3)/a^4 - (9*A*b*( a + b*x^2)^4)/(2*a^5))/(a*(a + b*x^2)^(7/2) - (a + b*x^2)^(9/2)) - (B/a^4 + (128*B*b*x^2)/(35*a^5))/(x*(a + b*x^2)^(1/2)) - (C*atanh((a + b*x^2)^(1/ 2)/a^(1/2)))/a^(9/2) + (9*A*b*atanh((a + b*x^2)^(1/2)/a^(1/2)))/(2*a^(11/2 )) - (29*B*b*x)/(35*a^4*(a + b*x^2)^(3/2)) - (13*B*b*x)/(35*a^3*(a + b*x^2 )^(5/2)) - (B*b*x)/(7*a^2*(a + b*x^2)^(7/2))