Integrand size = 32, antiderivative size = 179 \[ \int \frac {x^2 \left (A+B x^2+C x^4+D x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx=-\frac {a^3 D x}{b^4 \left (a+b x^2\right )^{7/2}}+\frac {\left (A b^3-10 a^3 D\right ) x^3}{3 a b^3 \left (a+b x^2\right )^{7/2}}+\frac {\left (4 A b^3+3 a b^2 B-58 a^3 D\right ) x^5}{15 a^2 b^2 \left (a+b x^2\right )^{7/2}}+\frac {\left (8 A b^3+6 a b^2 B+15 a^2 b C-176 a^3 D\right ) x^7}{105 a^3 b \left (a+b x^2\right )^{7/2}}+\frac {D \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{9/2}} \]
-a^3*D*x/b^4/(b*x^2+a)^(7/2)+1/3*(A*b^3-10*D*a^3)*x^3/a/b^3/(b*x^2+a)^(7/2 )+1/15*(4*A*b^3+3*B*a*b^2-58*D*a^3)*x^5/a^2/b^2/(b*x^2+a)^(7/2)+1/105*(8*A *b^3+6*B*a*b^2+15*C*a^2*b-176*D*a^3)*x^7/a^3/b/(b*x^2+a)^(7/2)+D*arctanh(x *b^(1/2)/(b*x^2+a)^(1/2))/b^(9/2)
Time = 0.59 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.82 \[ \int \frac {x^2 \left (A+B x^2+C x^4+D x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {-105 a^6 D x-350 a^5 b D x^3-406 a^4 b^2 D x^5+8 A b^6 x^7-176 a^3 b^3 D x^7+2 a b^5 x^5 \left (14 A+3 B x^2\right )+a^2 b^4 x^3 \left (35 A+21 B x^2+15 C x^4\right )}{105 a^3 b^4 \left (a+b x^2\right )^{7/2}}-\frac {D \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{b^{9/2}} \]
(-105*a^6*D*x - 350*a^5*b*D*x^3 - 406*a^4*b^2*D*x^5 + 8*A*b^6*x^7 - 176*a^ 3*b^3*D*x^7 + 2*a*b^5*x^5*(14*A + 3*B*x^2) + a^2*b^4*x^3*(35*A + 21*B*x^2 + 15*C*x^4))/(105*a^3*b^4*(a + b*x^2)^(7/2)) - (D*Log[-(Sqrt[b]*x) + Sqrt[ a + b*x^2]])/b^(9/2)
Time = 0.56 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.23, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.344, Rules used = {2335, 9, 25, 1586, 9, 25, 27, 357, 252, 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.
| \(\displaystyle \int \frac {x^2 \left (A+B x^2+C x^4+D x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx\) |
| \(\Big \downarrow \) 2335 |
| \(\displaystyle \frac {x^3 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}-\frac {\int -\frac {x \left (7 a D x^5+7 a \left (C-\frac {a D}{b}\right ) x^3+\left (4 A b+\frac {3 a \left (D a^2-b C a+b^2 B\right )}{b^2}\right ) x\right )}{\left (b x^2+a\right )^{7/2}}dx}{7 a b}\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle \frac {x^3 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}-\frac {\int -\frac {x^2 \left (7 a D x^4+7 a \left (C-\frac {a D}{b}\right ) x^2+4 A b+\frac {3 a \left (D a^2-b C a+b^2 B\right )}{b^2}\right )}{\left (b x^2+a\right )^{7/2}}dx}{7 a b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {x^2 \left (7 a D x^4+7 a \left (C-\frac {a D}{b}\right ) x^2+4 A b+\frac {3 a \left (D a^2-b C a+b^2 B\right )}{b^2}\right )}{\left (b x^2+a\right )^{7/2}}dx}{7 a b}+\frac {x^3 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 1586 |
| \(\displaystyle \frac {\frac {x^3 \left (\frac {a \left (17 a^2 D-10 a b C+3 b^2 B\right )}{b^2}+4 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}-\frac {\int -\frac {x \left (\frac {35 a^2 D x^3}{b}+\left (8 A b+\frac {3 a \left (-12 D a^2+5 b C a+2 b^2 B\right )}{b^2}\right ) x\right )}{\left (b x^2+a\right )^{5/2}}dx}{5 a}}{7 a b}+\frac {x^3 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle \frac {\frac {x^3 \left (\frac {a \left (17 a^2 D-10 a b C+3 b^2 B\right )}{b^2}+4 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}-\frac {\int -\frac {x^2 \left (35 a^2 D x^2+b \left (8 A b+\frac {3 a \left (-12 D a^2+5 b C a+2 b^2 B\right )}{b^2}\right )\right )}{b \left (b x^2+a\right )^{5/2}}dx}{5 a}}{7 a b}+\frac {x^3 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {x^2 \left (8 A b^2+35 a^2 D x^2+3 a \left (-\frac {12 D a^2}{b}+5 C a+2 b B\right )\right )}{b \left (b x^2+a\right )^{5/2}}dx}{5 a}+\frac {x^3 \left (\frac {a \left (17 a^2 D-10 a b C+3 b^2 B\right )}{b^2}+4 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}}{7 a b}+\frac {x^3 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {x^2 \left (8 A b^2+35 a^2 D x^2+3 a \left (-\frac {12 D a^2}{b}+5 C a+2 b B\right )\right )}{\left (b x^2+a\right )^{5/2}}dx}{5 a b}+\frac {x^3 \left (\frac {a \left (17 a^2 D-10 a b C+3 b^2 B\right )}{b^2}+4 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}}{7 a b}+\frac {x^3 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 357 |
| \(\displaystyle \frac {\frac {\frac {35 a^2 D \int \frac {x^2}{\left (b x^2+a\right )^{3/2}}dx}{b}+\frac {x^3 \left (a \left (-71 a^2 D+15 a b C+6 b^2 B\right )+8 A b^3\right )}{3 a b \left (a+b x^2\right )^{3/2}}}{5 a b}+\frac {x^3 \left (\frac {a \left (17 a^2 D-10 a b C+3 b^2 B\right )}{b^2}+4 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}}{7 a b}+\frac {x^3 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {\frac {\frac {35 a^2 D \left (\frac {\int \frac {1}{\sqrt {b x^2+a}}dx}{b}-\frac {x}{b \sqrt {a+b x^2}}\right )}{b}+\frac {x^3 \left (a \left (-71 a^2 D+15 a b C+6 b^2 B\right )+8 A b^3\right )}{3 a b \left (a+b x^2\right )^{3/2}}}{5 a b}+\frac {x^3 \left (\frac {a \left (17 a^2 D-10 a b C+3 b^2 B\right )}{b^2}+4 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}}{7 a b}+\frac {x^3 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\frac {\frac {35 a^2 D \left (\frac {\int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{b}-\frac {x}{b \sqrt {a+b x^2}}\right )}{b}+\frac {x^3 \left (a \left (-71 a^2 D+15 a b C+6 b^2 B\right )+8 A b^3\right )}{3 a b \left (a+b x^2\right )^{3/2}}}{5 a b}+\frac {x^3 \left (\frac {a \left (17 a^2 D-10 a b C+3 b^2 B\right )}{b^2}+4 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}}{7 a b}+\frac {x^3 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {x^3 \left (a \left (-71 a^2 D+15 a b C+6 b^2 B\right )+8 A b^3\right )}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {35 a^2 D \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{3/2}}-\frac {x}{b \sqrt {a+b x^2}}\right )}{b}}{5 a b}+\frac {x^3 \left (\frac {a \left (17 a^2 D-10 a b C+3 b^2 B\right )}{b^2}+4 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}}{7 a b}+\frac {x^3 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
((A - (a*(b^2*B - a*b*C + a^2*D))/b^3)*x^3)/(7*a*(a + b*x^2)^(7/2)) + (((4 *A*b + (a*(3*b^2*B - 10*a*b*C + 17*a^2*D))/b^2)*x^3)/(5*a*(a + b*x^2)^(5/2 )) + (((8*A*b^3 + a*(6*b^2*B + 15*a*b*C - 71*a^2*D))*x^3)/(3*a*b*(a + b*x^ 2)^(3/2)) + (35*a^2*D*(-(x/(b*Sqrt[a + b*x^2])) + ArcTanh[(Sqrt[b]*x)/Sqrt [a + b*x^2]]/b^(3/2)))/b)/(5*a*b))/(7*a*b)
3.2.62.3.1 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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_ Symbol] :> Simp[(b*c - a*d)*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*b*e*(m + 1))), x] + Simp[d/b Int[(e*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b*c - a*d, 0] && EqQ[m + 2*p + 3, 0] && LtQ[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, 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 = 3.58 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.78
| method | result | size |
| pseudoelliptic | \(\frac {\frac {a^{2} x^{3} \left (\frac {3}{7} C \,x^{4}+\frac {3}{5} x^{2} B +A \right ) b^{\frac {9}{2}}}{3}+\frac {4 a \left (\frac {3 x^{2} B}{14}+A \right ) x^{5} b^{\frac {11}{2}}}{15}+\frac {8 A \,b^{\frac {13}{2}} x^{7}}{105}+a^{3} \left (\left (b \,x^{2}+a \right )^{\frac {7}{2}} \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )-\frac {176 x^{7} b^{\frac {7}{2}}}{105}-\frac {58 b^{\frac {5}{2}} a \,x^{5}}{15}-\frac {10 b^{\frac {3}{2}} a^{2} x^{3}}{3}-\sqrt {b}\, a^{3} x \right ) D}{\left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{\frac {9}{2}} a^{3}}\) | \(139\) |
| default | \(D \left (-\frac {x^{7}}{7 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {-\frac {x^{5}}{5 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}}{b}}{b}\right )+C \left (-\frac {x^{5}}{2 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {5 a \left (-\frac {x^{3}}{4 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {3 a \left (-\frac {x}{6 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {a \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{6 b}\right )}{4 b}\right )}{2 b}\right )+B \left (-\frac {x^{3}}{4 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {3 a \left (-\frac {x}{6 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {a \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{6 b}\right )}{4 b}\right )+A \left (-\frac {x}{6 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {a \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{6 b}\right )\) | \(469\) |
1/(b*x^2+a)^(7/2)*(1/3*a^2*x^3*(3/7*C*x^4+3/5*x^2*B+A)*b^(9/2)+4/15*a*(3/1 4*x^2*B+A)*x^5*b^(11/2)+8/105*A*b^(13/2)*x^7+a^3*((b*x^2+a)^(7/2)*arctanh( (b*x^2+a)^(1/2)/x/b^(1/2))-176/105*x^7*b^(7/2)-58/15*b^(5/2)*a*x^5-10/3*b^ (3/2)*a^2*x^3-b^(1/2)*a^3*x)*D)/b^(9/2)/a^3
Time = 0.38 (sec) , antiderivative size = 491, normalized size of antiderivative = 2.74 \[ \int \frac {x^2 \left (A+B x^2+C x^4+D x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\left [\frac {105 \, {\left (D a^{3} b^{4} x^{8} + 4 \, D a^{4} b^{3} x^{6} + 6 \, D a^{5} b^{2} x^{4} + 4 \, D a^{6} b x^{2} + D a^{7}\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (105 \, D a^{6} b x + {\left (176 \, D a^{3} b^{4} - 15 \, C a^{2} b^{5} - 6 \, B a b^{6} - 8 \, A b^{7}\right )} x^{7} + 7 \, {\left (58 \, D a^{4} b^{3} - 3 \, B a^{2} b^{5} - 4 \, A a b^{6}\right )} x^{5} + 35 \, {\left (10 \, D a^{5} b^{2} - A a^{2} b^{5}\right )} x^{3}\right )} \sqrt {b x^{2} + a}}{210 \, {\left (a^{3} b^{9} x^{8} + 4 \, a^{4} b^{8} x^{6} + 6 \, a^{5} b^{7} x^{4} + 4 \, a^{6} b^{6} x^{2} + a^{7} b^{5}\right )}}, -\frac {105 \, {\left (D a^{3} b^{4} x^{8} + 4 \, D a^{4} b^{3} x^{6} + 6 \, D a^{5} b^{2} x^{4} + 4 \, D a^{6} b x^{2} + D a^{7}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (105 \, D a^{6} b x + {\left (176 \, D a^{3} b^{4} - 15 \, C a^{2} b^{5} - 6 \, B a b^{6} - 8 \, A b^{7}\right )} x^{7} + 7 \, {\left (58 \, D a^{4} b^{3} - 3 \, B a^{2} b^{5} - 4 \, A a b^{6}\right )} x^{5} + 35 \, {\left (10 \, D a^{5} b^{2} - A a^{2} b^{5}\right )} x^{3}\right )} \sqrt {b x^{2} + a}}{105 \, {\left (a^{3} b^{9} x^{8} + 4 \, a^{4} b^{8} x^{6} + 6 \, a^{5} b^{7} x^{4} + 4 \, a^{6} b^{6} x^{2} + a^{7} b^{5}\right )}}\right ] \]
[1/210*(105*(D*a^3*b^4*x^8 + 4*D*a^4*b^3*x^6 + 6*D*a^5*b^2*x^4 + 4*D*a^6*b *x^2 + D*a^7)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 2* (105*D*a^6*b*x + (176*D*a^3*b^4 - 15*C*a^2*b^5 - 6*B*a*b^6 - 8*A*b^7)*x^7 + 7*(58*D*a^4*b^3 - 3*B*a^2*b^5 - 4*A*a*b^6)*x^5 + 35*(10*D*a^5*b^2 - A*a^ 2*b^5)*x^3)*sqrt(b*x^2 + a))/(a^3*b^9*x^8 + 4*a^4*b^8*x^6 + 6*a^5*b^7*x^4 + 4*a^6*b^6*x^2 + a^7*b^5), -1/105*(105*(D*a^3*b^4*x^8 + 4*D*a^4*b^3*x^6 + 6*D*a^5*b^2*x^4 + 4*D*a^6*b*x^2 + D*a^7)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt( b*x^2 + a)) + (105*D*a^6*b*x + (176*D*a^3*b^4 - 15*C*a^2*b^5 - 6*B*a*b^6 - 8*A*b^7)*x^7 + 7*(58*D*a^4*b^3 - 3*B*a^2*b^5 - 4*A*a*b^6)*x^5 + 35*(10*D* a^5*b^2 - A*a^2*b^5)*x^3)*sqrt(b*x^2 + a))/(a^3*b^9*x^8 + 4*a^4*b^8*x^6 + 6*a^5*b^7*x^4 + 4*a^6*b^6*x^2 + a^7*b^5)]
Leaf count of result is larger than twice the leaf count of optimal. 3803 vs. \(2 (178) = 356\).
Time = 66.25 (sec) , antiderivative size = 3803, normalized size of antiderivative = 21.25 \[ \int \frac {x^2 \left (A+B x^2+C x^4+D x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\text {Too large to display} \]
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. 533 vs. \(2 (160) = 320\).
Time = 0.25 (sec) , antiderivative size = 533, normalized size of antiderivative = 2.98 \[ \int \frac {x^2 \left (A+B x^2+C x^4+D x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx=-\frac {1}{35} \, {\left (\frac {35 \, x^{6}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b} + \frac {70 \, a x^{4}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} + \frac {56 \, a^{2} x^{2}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}} + \frac {16 \, a^{3}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{4}}\right )} D x - \frac {D x {\left (\frac {15 \, x^{4}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b} + \frac {20 \, a x^{2}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{2}} + \frac {8 \, a^{2}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{3}}\right )}}{15 \, b} - \frac {C x^{5}}{2 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} - \frac {D x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, a}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}\right )}}{3 \, b^{2}} - \frac {D a x^{3}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{3}} - \frac {5 \, C a x^{3}}{8 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} - \frac {B x^{3}}{4 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} + \frac {139 \, D x}{105 \, \sqrt {b x^{2} + a} b^{4}} + \frac {17 \, D a x}{105 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{4}} - \frac {29 \, D a^{2} x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{4}} + \frac {C x}{14 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{3}} + \frac {C x}{7 \, \sqrt {b x^{2} + a} a b^{3}} + \frac {3 \, C a x}{56 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{3}} - \frac {15 \, C a^{2} x}{56 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}} + \frac {3 \, B x}{140 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{2}} + \frac {2 \, B x}{35 \, \sqrt {b x^{2} + a} a^{2} b^{2}} + \frac {B x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a b^{2}} - \frac {3 \, B a x}{28 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} - \frac {A x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} + \frac {8 \, A x}{105 \, \sqrt {b x^{2} + a} a^{3} b} + \frac {4 \, A x}{105 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} b} + \frac {A x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a b} + \frac {D \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {9}{2}}} \]
-1/35*(35*x^6/((b*x^2 + a)^(7/2)*b) + 70*a*x^4/((b*x^2 + a)^(7/2)*b^2) + 5 6*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 - 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) - 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 - 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) + 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*b) + 4/105*A*x/((b *x^2 + a)^(3/2)*a^2*b) + 1/35*A*x/((b*x^2 + a)^(5/2)*a*b) + D*arcsinh(b*x/ sqrt(a*b))/b^(9/2)
Time = 0.29 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.89 \[ \int \frac {x^2 \left (A+B x^2+C x^4+D x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx=-\frac {{\left ({\left (x^{2} {\left (\frac {{\left (176 \, D a^{3} b^{6} - 15 \, C a^{2} b^{7} - 6 \, B a b^{8} - 8 \, A b^{9}\right )} x^{2}}{a^{3} b^{7}} + \frac {7 \, {\left (58 \, D a^{4} b^{5} - 3 \, B a^{2} b^{7} - 4 \, A a b^{8}\right )}}{a^{3} b^{7}}\right )} + \frac {35 \, {\left (10 \, D a^{5} b^{4} - A a^{2} b^{7}\right )}}{a^{3} b^{7}}\right )} x^{2} + \frac {105 \, D a^{3}}{b^{4}}\right )} x}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} - \frac {D \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{b^{\frac {9}{2}}} \]
-1/105*((x^2*((176*D*a^3*b^6 - 15*C*a^2*b^7 - 6*B*a*b^8 - 8*A*b^9)*x^2/(a^ 3*b^7) + 7*(58*D*a^4*b^5 - 3*B*a^2*b^7 - 4*A*a*b^8)/(a^3*b^7)) + 35*(10*D* a^5*b^4 - A*a^2*b^7)/(a^3*b^7))*x^2 + 105*D*a^3/b^4)*x/(b*x^2 + a)^(7/2) - D*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(9/2)
Timed out. \[ \int \frac {x^2 \left (A+B x^2+C x^4+D x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\int \frac {x^2\,\left (A+B\,x^2+C\,x^4+x^6\,D\right )}{{\left (b\,x^2+a\right )}^{9/2}} \,d x \]