Integrand size = 37, antiderivative size = 261 \[ \int \frac {x^2 \left (A+B x^2+C x^4+D x^6+F x^8\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {\left (A b^4-a \left (b^3 B-a b^2 C+a^2 b D-a^3 F\right )\right ) x^3}{7 a b^4 \left (a+b x^2\right )^{7/2}}+\frac {\left (4 A b^4+a \left (3 b^3 B-10 a b^2 C+17 a^2 b D-24 a^3 F\right )\right ) x^3}{35 a^2 b^4 \left (a+b x^2\right )^{5/2}}+\frac {\left (8 A b^4+a \left (6 b^3 B+15 a b^2 C-71 a^2 b D+162 a^3 F\right )\right ) x^3}{105 a^3 b^4 \left (a+b x^2\right )^{3/2}}-\frac {(b D-4 a F) x}{b^5 \sqrt {a+b x^2}}+\frac {F x \sqrt {a+b x^2}}{2 b^5}+\frac {(2 b D-9 a F) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{11/2}} \]
1/7*(A*b^4-a*(B*b^3-C*a*b^2+D*a^2*b-F*a^3))*x^3/a/b^4/(b*x^2+a)^(7/2)+1/35 *(4*A*b^4+a*(3*B*b^3-10*C*a*b^2+17*D*a^2*b-24*F*a^3))*x^3/a^2/b^4/(b*x^2+a )^(5/2)+1/105*(8*A*b^4+a*(6*B*b^3+15*C*a*b^2-71*D*a^2*b+162*F*a^3))*x^3/a^ 3/b^4/(b*x^2+a)^(3/2)+1/2*(2*D*b-9*F*a)*arctanh(x*b^(1/2)/(b*x^2+a)^(1/2)) /b^(11/2)-(D*b-4*F*a)*x/b^5/(b*x^2+a)^(1/2)+1/2*F*x*(b*x^2+a)^(1/2)/b^5
Time = 1.18 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.79 \[ \int \frac {x^2 \left (A+B x^2+C x^4+D x^6+F x^8\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {x \left (945 a^7 F+16 A b^7 x^6+4 a b^6 x^4 \left (14 A+3 B x^2\right )-210 a^6 b \left (D-15 F x^2\right )+a^3 b^4 x^6 \left (-352 D+105 F x^2\right )+14 a^5 b^2 x^2 \left (-50 D+261 F x^2\right )+4 a^4 b^3 x^4 \left (-203 D+396 F x^2\right )+2 a^2 b^5 x^2 \left (35 A+21 B x^2+15 C x^4\right )\right )}{210 a^3 b^5 \left (a+b x^2\right )^{7/2}}+\frac {(2 b D-9 a F) \text {arctanh}\left (\frac {\sqrt {b} x}{-\sqrt {a}+\sqrt {a+b x^2}}\right )}{b^{11/2}} \]
(x*(945*a^7*F + 16*A*b^7*x^6 + 4*a*b^6*x^4*(14*A + 3*B*x^2) - 210*a^6*b*(D - 15*F*x^2) + a^3*b^4*x^6*(-352*D + 105*F*x^2) + 14*a^5*b^2*x^2*(-50*D + 261*F*x^2) + 4*a^4*b^3*x^4*(-203*D + 396*F*x^2) + 2*a^2*b^5*x^2*(35*A + 21 *B*x^2 + 15*C*x^4)))/(210*a^3*b^5*(a + b*x^2)^(7/2)) + ((2*b*D - 9*a*F)*Ar cTanh[(Sqrt[b]*x)/(-Sqrt[a] + Sqrt[a + b*x^2])])/b^(11/2)
Time = 1.10 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.09, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.405, Rules used = {2335, 9, 25, 2335, 9, 25, 1586, 9, 27, 360, 25, 27, 299, 224, 219}
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.
| \(\displaystyle \int \frac {x^2 \left (A+B x^2+C x^4+D x^6+F x^8\right )}{\left (a+b x^2\right )^{9/2}} \, dx\) |
| \(\Big \downarrow \) 2335 |
| \(\displaystyle \frac {x^3 \left (\frac {A}{a}-\frac {a^3 (-F)+a^2 b D-a b^2 C+b^3 B}{b^4}\right )}{7 \left (a+b x^2\right )^{7/2}}-\frac {\int -\frac {x \left (7 a F x^7+7 a \left (D-\frac {a F}{b}\right ) x^5+\frac {7 a \left (F a^2-b D a+b^2 C\right ) x^3}{b^2}+\left (4 A b+\frac {3 a \left (-F a^3+b D a^2-b^2 C a+b^3 B\right )}{b^3}\right ) x\right )}{\left (b x^2+a\right )^{7/2}}dx}{7 a b}\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle \frac {x^3 \left (\frac {A}{a}-\frac {a^3 (-F)+a^2 b D-a b^2 C+b^3 B}{b^4}\right )}{7 \left (a+b x^2\right )^{7/2}}-\frac {\int -\frac {x^2 \left (7 a F x^6+7 a \left (D-\frac {a F}{b}\right ) x^4+7 a \left (C-\frac {a (b D-a F)}{b^2}\right ) x^2+4 A b+\frac {3 a \left (-F a^3+b D a^2-b^2 C a+b^3 B\right )}{b^3}\right )}{\left (b x^2+a\right )^{7/2}}dx}{7 a b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {x^2 \left (7 a F x^6+7 a \left (D-\frac {a F}{b}\right ) x^4+7 a \left (C-\frac {a (b D-a F)}{b^2}\right ) x^2+4 A b+\frac {3 a \left (-F a^3+b D a^2-b^2 C a+b^3 B\right )}{b^3}\right )}{\left (b x^2+a\right )^{7/2}}dx}{7 a b}+\frac {x^3 \left (\frac {A}{a}-\frac {a^3 (-F)+a^2 b D-a b^2 C+b^3 B}{b^4}\right )}{7 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 2335 |
| \(\displaystyle \frac {\frac {x^3 \left (\frac {a \left (-24 a^3 F+17 a^2 b D-10 a b^2 C+3 b^3 B\right )}{b^3}+4 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}-\frac {\int -\frac {x \left (35 a^2 F x^5+35 a^2 \left (D-\frac {2 a F}{b}\right ) x^3+\left (8 A b^2+3 a \left (\frac {19 F a^3}{b^2}-\frac {12 D a^2}{b}+5 C a+2 b B\right )\right ) x\right )}{\left (b x^2+a\right )^{5/2}}dx}{5 a b}}{7 a b}+\frac {x^3 \left (\frac {A}{a}-\frac {a^3 (-F)+a^2 b D-a b^2 C+b^3 B}{b^4}\right )}{7 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle \frac {\frac {x^3 \left (\frac {a \left (-24 a^3 F+17 a^2 b D-10 a b^2 C+3 b^3 B\right )}{b^3}+4 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}-\frac {\int -\frac {x^2 \left (35 a^2 F x^4+35 a^2 \left (D-\frac {2 a F}{b}\right ) x^2+8 A b^2+3 a \left (\frac {19 F a^3}{b^2}-\frac {12 D a^2}{b}+5 C a+2 b B\right )\right )}{\left (b x^2+a\right )^{5/2}}dx}{5 a b}}{7 a b}+\frac {x^3 \left (\frac {A}{a}-\frac {a^3 (-F)+a^2 b D-a b^2 C+b^3 B}{b^4}\right )}{7 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {x^2 \left (35 a^2 F x^4+35 a^2 \left (D-\frac {2 a F}{b}\right ) x^2+8 A b^2+3 a \left (\frac {19 F a^3}{b^2}-\frac {12 D a^2}{b}+5 C a+2 b B\right )\right )}{\left (b x^2+a\right )^{5/2}}dx}{5 a b}+\frac {x^3 \left (\frac {a \left (-24 a^3 F+17 a^2 b D-10 a b^2 C+3 b^3 B\right )}{b^3}+4 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}}{7 a b}+\frac {x^3 \left (\frac {A}{a}-\frac {a^3 (-F)+a^2 b D-a b^2 C+b^3 B}{b^4}\right )}{7 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 1586 |
| \(\displaystyle \frac {\frac {\frac {x^3 \left (a \left (\frac {162 a^3 F}{b^2}-\frac {71 a^2 D}{b}+15 a C+6 b B\right )+8 A b^2\right )}{3 a \left (a+b x^2\right )^{3/2}}-\frac {\int -\frac {105 x \left (\frac {F x^3 a^3}{b}+\frac {(b D-3 a F) x a^3}{b^2}\right )}{\left (b x^2+a\right )^{3/2}}dx}{3 a}}{5 a b}+\frac {x^3 \left (\frac {a \left (-24 a^3 F+17 a^2 b D-10 a b^2 C+3 b^3 B\right )}{b^3}+4 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}}{7 a b}+\frac {x^3 \left (\frac {A}{a}-\frac {a^3 (-F)+a^2 b D-a b^2 C+b^3 B}{b^4}\right )}{7 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle \frac {\frac {\frac {x^3 \left (a \left (\frac {162 a^3 F}{b^2}-\frac {71 a^2 D}{b}+15 a C+6 b B\right )+8 A b^2\right )}{3 a \left (a+b x^2\right )^{3/2}}-\frac {\int -\frac {105 a^3 x^2 \left (b F x^2+b D-3 a F\right )}{b^2 \left (b x^2+a\right )^{3/2}}dx}{3 a}}{5 a b}+\frac {x^3 \left (\frac {a \left (-24 a^3 F+17 a^2 b D-10 a b^2 C+3 b^3 B\right )}{b^3}+4 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}}{7 a b}+\frac {x^3 \left (\frac {A}{a}-\frac {a^3 (-F)+a^2 b D-a b^2 C+b^3 B}{b^4}\right )}{7 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {35 a^2 \int \frac {x^2 \left (b F x^2+b D-3 a F\right )}{\left (b x^2+a\right )^{3/2}}dx}{b^2}+\frac {x^3 \left (a \left (\frac {162 a^3 F}{b^2}-\frac {71 a^2 D}{b}+15 a C+6 b B\right )+8 A b^2\right )}{3 a \left (a+b x^2\right )^{3/2}}}{5 a b}+\frac {x^3 \left (\frac {a \left (-24 a^3 F+17 a^2 b D-10 a b^2 C+3 b^3 B\right )}{b^3}+4 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}}{7 a b}+\frac {x^3 \left (\frac {A}{a}-\frac {a^3 (-F)+a^2 b D-a b^2 C+b^3 B}{b^4}\right )}{7 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 360 |
| \(\displaystyle \frac {\frac {\frac {35 a^2 \left (-\frac {\int -\frac {b \left (b F x^2+b D-4 a F\right )}{\sqrt {b x^2+a}}dx}{b^2}-\frac {x (b D-4 a F)}{b \sqrt {a+b x^2}}\right )}{b^2}+\frac {x^3 \left (a \left (\frac {162 a^3 F}{b^2}-\frac {71 a^2 D}{b}+15 a C+6 b B\right )+8 A b^2\right )}{3 a \left (a+b x^2\right )^{3/2}}}{5 a b}+\frac {x^3 \left (\frac {a \left (-24 a^3 F+17 a^2 b D-10 a b^2 C+3 b^3 B\right )}{b^3}+4 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}}{7 a b}+\frac {x^3 \left (\frac {A}{a}-\frac {a^3 (-F)+a^2 b D-a b^2 C+b^3 B}{b^4}\right )}{7 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {35 a^2 \left (\frac {\int \frac {b \left (b F x^2+b D-4 a F\right )}{\sqrt {b x^2+a}}dx}{b^2}-\frac {x (b D-4 a F)}{b \sqrt {a+b x^2}}\right )}{b^2}+\frac {x^3 \left (a \left (\frac {162 a^3 F}{b^2}-\frac {71 a^2 D}{b}+15 a C+6 b B\right )+8 A b^2\right )}{3 a \left (a+b x^2\right )^{3/2}}}{5 a b}+\frac {x^3 \left (\frac {a \left (-24 a^3 F+17 a^2 b D-10 a b^2 C+3 b^3 B\right )}{b^3}+4 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}}{7 a b}+\frac {x^3 \left (\frac {A}{a}-\frac {a^3 (-F)+a^2 b D-a b^2 C+b^3 B}{b^4}\right )}{7 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {35 a^2 \left (\frac {\int \frac {b F x^2+b D-4 a F}{\sqrt {b x^2+a}}dx}{b}-\frac {x (b D-4 a F)}{b \sqrt {a+b x^2}}\right )}{b^2}+\frac {x^3 \left (a \left (\frac {162 a^3 F}{b^2}-\frac {71 a^2 D}{b}+15 a C+6 b B\right )+8 A b^2\right )}{3 a \left (a+b x^2\right )^{3/2}}}{5 a b}+\frac {x^3 \left (\frac {a \left (-24 a^3 F+17 a^2 b D-10 a b^2 C+3 b^3 B\right )}{b^3}+4 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}}{7 a b}+\frac {x^3 \left (\frac {A}{a}-\frac {a^3 (-F)+a^2 b D-a b^2 C+b^3 B}{b^4}\right )}{7 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle \frac {\frac {\frac {35 a^2 \left (\frac {\frac {1}{2} (2 b D-9 a F) \int \frac {1}{\sqrt {b x^2+a}}dx+\frac {1}{2} F x \sqrt {a+b x^2}}{b}-\frac {x (b D-4 a F)}{b \sqrt {a+b x^2}}\right )}{b^2}+\frac {x^3 \left (a \left (\frac {162 a^3 F}{b^2}-\frac {71 a^2 D}{b}+15 a C+6 b B\right )+8 A b^2\right )}{3 a \left (a+b x^2\right )^{3/2}}}{5 a b}+\frac {x^3 \left (\frac {a \left (-24 a^3 F+17 a^2 b D-10 a b^2 C+3 b^3 B\right )}{b^3}+4 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}}{7 a b}+\frac {x^3 \left (\frac {A}{a}-\frac {a^3 (-F)+a^2 b D-a b^2 C+b^3 B}{b^4}\right )}{7 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\frac {\frac {35 a^2 \left (\frac {\frac {1}{2} (2 b D-9 a F) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}+\frac {1}{2} F x \sqrt {a+b x^2}}{b}-\frac {x (b D-4 a F)}{b \sqrt {a+b x^2}}\right )}{b^2}+\frac {x^3 \left (a \left (\frac {162 a^3 F}{b^2}-\frac {71 a^2 D}{b}+15 a C+6 b B\right )+8 A b^2\right )}{3 a \left (a+b x^2\right )^{3/2}}}{5 a b}+\frac {x^3 \left (\frac {a \left (-24 a^3 F+17 a^2 b D-10 a b^2 C+3 b^3 B\right )}{b^3}+4 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}}{7 a b}+\frac {x^3 \left (\frac {A}{a}-\frac {a^3 (-F)+a^2 b D-a b^2 C+b^3 B}{b^4}\right )}{7 \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {35 a^2 \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b D-9 a F)}{2 \sqrt {b}}+\frac {1}{2} F x \sqrt {a+b x^2}}{b}-\frac {x (b D-4 a F)}{b \sqrt {a+b x^2}}\right )}{b^2}+\frac {x^3 \left (a \left (\frac {162 a^3 F}{b^2}-\frac {71 a^2 D}{b}+15 a C+6 b B\right )+8 A b^2\right )}{3 a \left (a+b x^2\right )^{3/2}}}{5 a b}+\frac {x^3 \left (\frac {a \left (-24 a^3 F+17 a^2 b D-10 a b^2 C+3 b^3 B\right )}{b^3}+4 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}}{7 a b}+\frac {x^3 \left (\frac {A}{a}-\frac {a^3 (-F)+a^2 b D-a b^2 C+b^3 B}{b^4}\right )}{7 \left (a+b x^2\right )^{7/2}}\) |
((A/a - (b^3*B - a*b^2*C + a^2*b*D - a^3*F)/b^4)*x^3)/(7*(a + b*x^2)^(7/2) ) + (((4*A*b + (a*(3*b^3*B - 10*a*b^2*C + 17*a^2*b*D - 24*a^3*F))/b^3)*x^3 )/(5*a*(a + b*x^2)^(5/2)) + (((8*A*b^2 + a*(6*b*B + 15*a*C - (71*a^2*D)/b + (162*a^3*F)/b^2))*x^3)/(3*a*(a + b*x^2)^(3/2)) + (35*a^2*(-(((b*D - 4*a* F)*x)/(b*Sqrt[a + b*x^2])) + ((F*x*Sqrt[a + b*x^2])/2 + ((2*b*D - 9*a*F)*A rcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*Sqrt[b]))/b))/b^2)/(5*a*b))/(7*a*b )
3.2.72.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[1/e^(p*r) Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && !MonomialQ[Px, x]
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)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 0]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] : > Simp[(-a)^(m/2 - 1)*(b*c - a*d)*x*((a + b*x^2)^(p + 1)/(2*b^(m/2 + 1)*(p + 1))), x] + Simp[1/(2*b^(m/2 + 1)*(p + 1)) Int[(a + b*x^2)^(p + 1)*Expan dToSum[2*b*(p + 1)*x^2*Together[(b^(m/2)*x^(m - 2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d))/(a + b*x^2)] - (-a)^(m/2 - 1)*(b*c - a*d), x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IGtQ[m/2, 0] & & (IntegerQ[p] || EqQ[m + 2*p + 1, 0])
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, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^ p, d + e*x^2, x], x, 0]}, Simp[(-R)*(f*x)^(m + 1)*((d + e*x^2)^(q + 1)/(2*d *f*(q + 1))), x] + Simp[f/(2*d*(q + 1)) Int[(f*x)^(m - 1)*(d + e*x^2)^(q + 1)*ExpandToSum[2*d*(q + 1)*x*Qx + R*(m + 2*q + 3)*x, x], x], x]] /; FreeQ [{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && LtQ[q, -1] && GtQ[m, 0]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ {Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq , a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(c*x)^m*(a + b*x^2)^(p + 1)*((a*g - b*f*x)/(2*a*b*(p + 1))), x] + Simp[c/(2*a*b*(p + 1)) Int[(c*x)^(m - 1)*(a + b*x^2)^(p + 1)*ExpandToSu m[2*a*b*(p + 1)*x*Q - a*g*m + b*f*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && GtQ[m, 0]
Time = 3.64 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.74
| method | result | size |
| pseudoelliptic | \(\frac {3 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{3} \left (D b -\frac {9 F a}{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )+\left (x^{2} a^{2} \left (\frac {3}{7} C \,x^{4}+\frac {3}{5} x^{2} B +A \right ) b^{\frac {11}{2}}+\frac {4 a \left (\frac {3 x^{2} B}{14}+A \right ) x^{4} b^{\frac {13}{2}}}{5}-3 a^{6} \left (-15 F \,x^{2}+D\right ) b^{\frac {3}{2}}-10 a^{5} \left (-\frac {261 F \,x^{2}}{50}+D\right ) x^{2} b^{\frac {5}{2}}-\frac {58 a^{4} \left (-\frac {396 F \,x^{2}}{203}+D\right ) x^{4} b^{\frac {7}{2}}}{5}-\frac {176 \left (-\frac {105 F \,x^{2}}{352}+D\right ) a^{3} x^{6} b^{\frac {9}{2}}}{35}+\frac {8 A \,b^{\frac {15}{2}} x^{6}}{35}+\frac {27 F \sqrt {b}\, a^{7}}{2}\right ) x}{3 b^{\frac {11}{2}} \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{3}}\) | \(193\) |
| default | \(D \left (-\frac {x^{7}}{7 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {-\frac {x^{5}}{5 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}}{b}}{b}\right )+C \left (-\frac {x^{5}}{2 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {5 a \left (-\frac {x^{3}}{4 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {3 a \left (-\frac {x}{6 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {a \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 )}{6 b}\right )}{4 b}\right )}{2 b}\right )+B \left (-\frac {x^{3}}{4 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {3 a \left (-\frac {x}{6 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {a \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 )}{6 b}\right )}{4 b}\right )+A \left (-\frac {x}{6 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {a \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 )}{6 b}\right )+F \left (\frac {x^{9}}{2 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {9 a \left (-\frac {x^{7}}{7 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {-\frac {x^{5}}{5 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}}{b}}{b}\right )}{2 b}\right )\) | \(597\) |
1/3/b^(11/2)*(3*(b*x^2+a)^(7/2)*a^3*(D*b-9/2*F*a)*arctanh((b*x^2+a)^(1/2)/ x/b^(1/2))+(x^2*a^2*(3/7*C*x^4+3/5*x^2*B+A)*b^(11/2)+4/5*a*(3/14*x^2*B+A)* x^4*b^(13/2)-3*a^6*(-15*F*x^2+D)*b^(3/2)-10*a^5*(-261/50*F*x^2+D)*x^2*b^(5 /2)-58/5*a^4*(-396/203*F*x^2+D)*x^4*b^(7/2)-176/35*(-105/352*F*x^2+D)*a^3* x^6*b^(9/2)+8/35*A*b^(15/2)*x^6+27/2*F*b^(1/2)*a^7)*x)/(b*x^2+a)^(7/2)/a^3
Time = 0.43 (sec) , antiderivative size = 705, normalized size of antiderivative = 2.70 \[ \int \frac {x^2 \left (A+B x^2+C x^4+D x^6+F x^8\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\left [-\frac {105 \, {\left (9 \, F a^{8} - 2 \, D a^{7} b + {\left (9 \, F a^{4} b^{4} - 2 \, D a^{3} b^{5}\right )} x^{8} + 4 \, {\left (9 \, F a^{5} b^{3} - 2 \, D a^{4} b^{4}\right )} x^{6} + 6 \, {\left (9 \, F a^{6} b^{2} - 2 \, D a^{5} b^{3}\right )} x^{4} + 4 \, {\left (9 \, F a^{7} b - 2 \, D a^{6} b^{2}\right )} x^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (105 \, F a^{3} b^{5} x^{9} + 2 \, {\left (792 \, F a^{4} b^{4} - 176 \, D a^{3} b^{5} + 15 \, C a^{2} b^{6} + 6 \, B a b^{7} + 8 \, A b^{8}\right )} x^{7} + 14 \, {\left (261 \, F a^{5} b^{3} - 58 \, D a^{4} b^{4} + 3 \, B a^{2} b^{6} + 4 \, A a b^{7}\right )} x^{5} + 70 \, {\left (45 \, F a^{6} b^{2} - 10 \, D a^{5} b^{3} + A a^{2} b^{6}\right )} x^{3} + 105 \, {\left (9 \, F a^{7} b - 2 \, D a^{6} b^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{420 \, {\left (a^{3} b^{10} x^{8} + 4 \, a^{4} b^{9} x^{6} + 6 \, a^{5} b^{8} x^{4} + 4 \, a^{6} b^{7} x^{2} + a^{7} b^{6}\right )}}, \frac {105 \, {\left (9 \, F a^{8} - 2 \, D a^{7} b + {\left (9 \, F a^{4} b^{4} - 2 \, D a^{3} b^{5}\right )} x^{8} + 4 \, {\left (9 \, F a^{5} b^{3} - 2 \, D a^{4} b^{4}\right )} x^{6} + 6 \, {\left (9 \, F a^{6} b^{2} - 2 \, D a^{5} b^{3}\right )} x^{4} + 4 \, {\left (9 \, F a^{7} b - 2 \, D a^{6} b^{2}\right )} x^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (105 \, F a^{3} b^{5} x^{9} + 2 \, {\left (792 \, F a^{4} b^{4} - 176 \, D a^{3} b^{5} + 15 \, C a^{2} b^{6} + 6 \, B a b^{7} + 8 \, A b^{8}\right )} x^{7} + 14 \, {\left (261 \, F a^{5} b^{3} - 58 \, D a^{4} b^{4} + 3 \, B a^{2} b^{6} + 4 \, A a b^{7}\right )} x^{5} + 70 \, {\left (45 \, F a^{6} b^{2} - 10 \, D a^{5} b^{3} + A a^{2} b^{6}\right )} x^{3} + 105 \, {\left (9 \, F a^{7} b - 2 \, D a^{6} b^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{210 \, {\left (a^{3} b^{10} x^{8} + 4 \, a^{4} b^{9} x^{6} + 6 \, a^{5} b^{8} x^{4} + 4 \, a^{6} b^{7} x^{2} + a^{7} b^{6}\right )}}\right ] \]
[-1/420*(105*(9*F*a^8 - 2*D*a^7*b + (9*F*a^4*b^4 - 2*D*a^3*b^5)*x^8 + 4*(9 *F*a^5*b^3 - 2*D*a^4*b^4)*x^6 + 6*(9*F*a^6*b^2 - 2*D*a^5*b^3)*x^4 + 4*(9*F *a^7*b - 2*D*a^6*b^2)*x^2)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b )*x - a) - 2*(105*F*a^3*b^5*x^9 + 2*(792*F*a^4*b^4 - 176*D*a^3*b^5 + 15*C* a^2*b^6 + 6*B*a*b^7 + 8*A*b^8)*x^7 + 14*(261*F*a^5*b^3 - 58*D*a^4*b^4 + 3* B*a^2*b^6 + 4*A*a*b^7)*x^5 + 70*(45*F*a^6*b^2 - 10*D*a^5*b^3 + A*a^2*b^6)* x^3 + 105*(9*F*a^7*b - 2*D*a^6*b^2)*x)*sqrt(b*x^2 + a))/(a^3*b^10*x^8 + 4* a^4*b^9*x^6 + 6*a^5*b^8*x^4 + 4*a^6*b^7*x^2 + a^7*b^6), 1/210*(105*(9*F*a^ 8 - 2*D*a^7*b + (9*F*a^4*b^4 - 2*D*a^3*b^5)*x^8 + 4*(9*F*a^5*b^3 - 2*D*a^4 *b^4)*x^6 + 6*(9*F*a^6*b^2 - 2*D*a^5*b^3)*x^4 + 4*(9*F*a^7*b - 2*D*a^6*b^2 )*x^2)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) + (105*F*a^3*b^5*x^9 + 2*(792*F*a^4*b^4 - 176*D*a^3*b^5 + 15*C*a^2*b^6 + 6*B*a*b^7 + 8*A*b^8)*x^7 + 14*(261*F*a^5*b^3 - 58*D*a^4*b^4 + 3*B*a^2*b^6 + 4*A*a*b^7)*x^5 + 70*(4 5*F*a^6*b^2 - 10*D*a^5*b^3 + A*a^2*b^6)*x^3 + 105*(9*F*a^7*b - 2*D*a^6*b^2 )*x)*sqrt(b*x^2 + a))/(a^3*b^10*x^8 + 4*a^4*b^9*x^6 + 6*a^5*b^8*x^4 + 4*a^ 6*b^7*x^2 + a^7*b^6)]
Leaf count of result is larger than twice the leaf count of optimal. 6987 vs. \(2 (253) = 506\).
Time = 108.77 (sec) , antiderivative size = 6987, normalized size of antiderivative = 26.77 \[ \int \frac {x^2 \left (A+B x^2+C x^4+D x^6+F x^8\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\text {Too large to display} \]
A*(35*a**5*x**3/(105*a**(19/2)*sqrt(1 + b*x**2/a) + 420*a**(17/2)*b*x**2*s qrt(1 + b*x**2/a) + 630*a**(15/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 420*a**(1 3/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 105*a**(11/2)*b**4*x**8*sqrt(1 + b*x** 2/a)) + 63*a**4*b*x**5/(105*a**(19/2)*sqrt(1 + b*x**2/a) + 420*a**(17/2)*b *x**2*sqrt(1 + b*x**2/a) + 630*a**(15/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 42 0*a**(13/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 105*a**(11/2)*b**4*x**8*sqrt(1 + b*x**2/a)) + 36*a**3*b**2*x**7/(105*a**(19/2)*sqrt(1 + b*x**2/a) + 420*a **(17/2)*b*x**2*sqrt(1 + b*x**2/a) + 630*a**(15/2)*b**2*x**4*sqrt(1 + b*x* *2/a) + 420*a**(13/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 105*a**(11/2)*b**4*x* *8*sqrt(1 + b*x**2/a)) + 8*a**2*b**3*x**9/(105*a**(19/2)*sqrt(1 + b*x**2/a ) + 420*a**(17/2)*b*x**2*sqrt(1 + b*x**2/a) + 630*a**(15/2)*b**2*x**4*sqrt (1 + b*x**2/a) + 420*a**(13/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 105*a**(11/2 )*b**4*x**8*sqrt(1 + b*x**2/a))) + B*(7*a*x**5/(35*a**(11/2)*sqrt(1 + b*x* *2/a) + 105*a**(9/2)*b*x**2*sqrt(1 + b*x**2/a) + 105*a**(7/2)*b**2*x**4*sq rt(1 + b*x**2/a) + 35*a**(5/2)*b**3*x**6*sqrt(1 + b*x**2/a)) + 2*b*x**7/(3 5*a**(11/2)*sqrt(1 + b*x**2/a) + 105*a**(9/2)*b*x**2*sqrt(1 + b*x**2/a) + 105*a**(7/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 35*a**(5/2)*b**3*x**6*sqrt(1 + b*x**2/a))) + C*x**7/(7*a**(9/2)*sqrt(1 + b*x**2/a) + 21*a**(7/2)*b*x**2* sqrt(1 + b*x**2/a) + 21*a**(5/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 7*a**(3/2) *b**3*x**6*sqrt(1 + b*x**2/a)) + D*(105*a**(205/2)*b**45*sqrt(1 + b*x**...
Leaf count of result is larger than twice the leaf count of optimal. 826 vs. \(2 (235) = 470\).
Time = 0.23 (sec) , antiderivative size = 826, normalized size of antiderivative = 3.16 \[ \int \frac {x^2 \left (A+B x^2+C x^4+D x^6+F x^8\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\text {Too large to display} \]
1/2*F*x^9/((b*x^2 + a)^(7/2)*b) - 1/35*(35*x^6/((b*x^2 + a)^(7/2)*b) + 70* a*x^4/((b*x^2 + a)^(7/2)*b^2) + 56*a^2*x^2/((b*x^2 + a)^(7/2)*b^3) + 16*a^ 3/((b*x^2 + a)^(7/2)*b^4))*D*x + 9/70*(35*x^6/((b*x^2 + a)^(7/2)*b) + 70*a *x^4/((b*x^2 + a)^(7/2)*b^2) + 56*a^2*x^2/((b*x^2 + a)^(7/2)*b^3) + 16*a^3 /((b*x^2 + a)^(7/2)*b^4))*F*a*x/b + 3/10*F*a*x*(15*x^4/((b*x^2 + a)^(5/2)* b) + 20*a*x^2/((b*x^2 + a)^(5/2)*b^2) + 8*a^2/((b*x^2 + a)^(5/2)*b^3))/b^2 - 1/15*D*x*(15*x^4/((b*x^2 + a)^(5/2)*b) + 20*a*x^2/((b*x^2 + a)^(5/2)*b^ 2) + 8*a^2/((b*x^2 + a)^(5/2)*b^3))/b - 1/2*C*x^5/((b*x^2 + a)^(7/2)*b) + 3/2*F*a*x*(3*x^2/((b*x^2 + a)^(3/2)*b) + 2*a/((b*x^2 + a)^(3/2)*b^2))/b^3 - 1/3*D*x*(3*x^2/((b*x^2 + a)^(3/2)*b) + 2*a/((b*x^2 + a)^(3/2)*b^2))/b^2 + 9/2*F*a^2*x^3/((b*x^2 + a)^(5/2)*b^4) - D*a*x^3/((b*x^2 + a)^(5/2)*b^3) - 5/8*C*a*x^3/((b*x^2 + a)^(7/2)*b^2) - 1/4*B*x^3/((b*x^2 + a)^(7/2)*b) - 417/70*F*a*x/(sqrt(b*x^2 + a)*b^5) - 51/70*F*a^2*x/((b*x^2 + a)^(3/2)*b^5) + 261/70*F*a^3*x/((b*x^2 + a)^(5/2)*b^5) + 139/105*D*x/(sqrt(b*x^2 + a)*b ^4) + 17/105*D*a*x/((b*x^2 + a)^(3/2)*b^4) - 29/35*D*a^2*x/((b*x^2 + a)^(5 /2)*b^4) + 1/14*C*x/((b*x^2 + a)^(3/2)*b^3) + 1/7*C*x/(sqrt(b*x^2 + a)*a*b ^3) + 3/56*C*a*x/((b*x^2 + a)^(5/2)*b^3) - 15/56*C*a^2*x/((b*x^2 + a)^(7/2 )*b^3) + 3/140*B*x/((b*x^2 + a)^(5/2)*b^2) + 2/35*B*x/(sqrt(b*x^2 + a)*a^2 *b^2) + 1/35*B*x/((b*x^2 + a)^(3/2)*a*b^2) - 3/28*B*a*x/((b*x^2 + a)^(7/2) *b^2) - 1/7*A*x/((b*x^2 + a)^(7/2)*b) + 8/105*A*x/(sqrt(b*x^2 + a)*a^3*...
Time = 0.30 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.86 \[ \int \frac {x^2 \left (A+B x^2+C x^4+D x^6+F x^8\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {{\left ({\left ({\left ({\left (\frac {105 \, F x^{2}}{b} + \frac {2 \, {\left (792 \, F a^{4} b^{7} - 176 \, D a^{3} b^{8} + 15 \, C a^{2} b^{9} + 6 \, B a b^{10} + 8 \, A b^{11}\right )}}{a^{3} b^{9}}\right )} x^{2} + \frac {14 \, {\left (261 \, F a^{5} b^{6} - 58 \, D a^{4} b^{7} + 3 \, B a^{2} b^{9} + 4 \, A a b^{10}\right )}}{a^{3} b^{9}}\right )} x^{2} + \frac {70 \, {\left (45 \, F a^{6} b^{5} - 10 \, D a^{5} b^{6} + A a^{2} b^{9}\right )}}{a^{3} b^{9}}\right )} x^{2} + \frac {105 \, {\left (9 \, F a^{7} b^{4} - 2 \, D a^{6} b^{5}\right )}}{a^{3} b^{9}}\right )} x}{210 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} + \frac {{\left (9 \, F a - 2 \, D b\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{2 \, b^{\frac {11}{2}}} \]
1/210*((((105*F*x^2/b + 2*(792*F*a^4*b^7 - 176*D*a^3*b^8 + 15*C*a^2*b^9 + 6*B*a*b^10 + 8*A*b^11)/(a^3*b^9))*x^2 + 14*(261*F*a^5*b^6 - 58*D*a^4*b^7 + 3*B*a^2*b^9 + 4*A*a*b^10)/(a^3*b^9))*x^2 + 70*(45*F*a^6*b^5 - 10*D*a^5*b^ 6 + A*a^2*b^9)/(a^3*b^9))*x^2 + 105*(9*F*a^7*b^4 - 2*D*a^6*b^5)/(a^3*b^9)) *x/(b*x^2 + a)^(7/2) + 1/2*(9*F*a - 2*D*b)*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(11/2)
Timed out. \[ \int \frac {x^2 \left (A+B x^2+C x^4+D x^6+F x^8\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\int \frac {x^2\,\left (A+B\,x^2+C\,x^4+F\,x^8+x^6\,D\right )}{{\left (b\,x^2+a\right )}^{9/2}} \,d x \]