Integrand size = 32, antiderivative size = 279 \[ \int \frac {x^6 \left (A+B x^2+C x^4+D x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {\left (A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) x^7}{7 a \left (a+b x^2\right )^{7/2}}+\frac {\left (b^2 B-2 a b C+3 a^2 D\right ) x^7}{5 a b^3 \left (a+b x^2\right )^{5/2}}+\frac {\left (8 b^2 B-36 a b C+99 a^2 D\right ) x^5}{60 a b^4 \left (a+b x^2\right )^{3/2}}+\frac {D x^7}{4 b^3 \left (a+b x^2\right )^{3/2}}+\frac {\left (8 b^2 B-36 a b C+99 a^2 D\right ) x^3}{12 a b^5 \sqrt {a+b x^2}}-\frac {\left (8 b^2 B-36 a b C+99 a^2 D\right ) x \sqrt {a+b x^2}}{8 a b^6}+\frac {\left (8 b^2 B-36 a b C+99 a^2 D\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{13/2}} \]
1/7*(A-a*(B*b^2-C*a*b+D*a^2)/b^3)*x^7/a/(b*x^2+a)^(7/2)+1/5*(B*b^2-2*C*a*b +3*D*a^2)*x^7/a/b^3/(b*x^2+a)^(5/2)+1/60*(8*B*b^2-36*C*a*b+99*D*a^2)*x^5/a /b^4/(b*x^2+a)^(3/2)+1/4*D*x^7/b^3/(b*x^2+a)^(3/2)+1/8*(8*B*b^2-36*C*a*b+9 9*D*a^2)*arctanh(x*b^(1/2)/(b*x^2+a)^(1/2))/b^(13/2)+1/12*(8*B*b^2-36*C*a* b+99*D*a^2)*x^3/a/b^5/(b*x^2+a)^(1/2)-1/8*(8*B*b^2-36*C*a*b+99*D*a^2)*x*(b *x^2+a)^(1/2)/a/b^6
Time = 1.01 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.77 \[ \int \frac {x^6 \left (A+B x^2+C x^4+D x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {x \left (-10395 a^6 D+120 A b^6 x^6+630 a^5 b \left (6 C-55 D x^2\right )+a^2 b^4 x^4 \left (-3248 B+6336 C x^2-1155 D x^4\right )-42 a^4 b^2 \left (20 B-300 C x^2+957 D x^4\right )-8 a^3 b^3 x^2 \left (350 B-1827 C x^2+2178 D x^4\right )+2 a b^5 x^6 \left (-704 B+105 \left (2 C x^2+D x^4\right )\right )\right )}{840 a b^6 \left (a+b x^2\right )^{7/2}}+\frac {\left (8 b^2 B-36 a b C+99 a^2 D\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{-\sqrt {a}+\sqrt {a+b x^2}}\right )}{4 b^{13/2}} \]
(x*(-10395*a^6*D + 120*A*b^6*x^6 + 630*a^5*b*(6*C - 55*D*x^2) + a^2*b^4*x^ 4*(-3248*B + 6336*C*x^2 - 1155*D*x^4) - 42*a^4*b^2*(20*B - 300*C*x^2 + 957 *D*x^4) - 8*a^3*b^3*x^2*(350*B - 1827*C*x^2 + 2178*D*x^4) + 2*a*b^5*x^6*(- 704*B + 105*(2*C*x^2 + D*x^4))))/(840*a*b^6*(a + b*x^2)^(7/2)) + ((8*b^2*B - 36*a*b*C + 99*a^2*D)*ArcTanh[(Sqrt[b]*x)/(-Sqrt[a] + Sqrt[a + b*x^2])]) /(4*b^(13/2))
Time = 0.55 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.88, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2335, 9, 27, 1586, 9, 27, 363, 252, 252, 262, 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.
| \(\displaystyle \int \frac {x^6 \left (A+B x^2+C x^4+D x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx\) |
| \(\Big \downarrow \) 2335 |
| \(\displaystyle \frac {x^7 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}-\frac {\int -\frac {7 x^5 \left (a D x^5+a \left (C-\frac {a D}{b}\right ) x^3+a \left (B-\frac {a (b C-a D)}{b^2}\right ) x\right )}{\left (b x^2+a\right )^{7/2}}dx}{7 a b}\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle \frac {x^7 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}-\frac {\int -\frac {7 x^6 \left (a D x^4+a \left (C-\frac {a D}{b}\right ) x^2+\frac {a \left (D a^2-b C a+b^2 B\right )}{b^2}\right )}{\left (b x^2+a\right )^{7/2}}dx}{7 a b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {x^6 \left (a D x^4+a \left (C-\frac {a D}{b}\right ) x^2+\frac {a \left (D a^2-b C a+b^2 B\right )}{b^2}\right )}{\left (b x^2+a\right )^{7/2}}dx}{a b}+\frac {x^7 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 1586 |
| \(\displaystyle \frac {\frac {x^7 \left (3 a^2 D-2 a b C+b^2 B\right )}{5 b^2 \left (a+b x^2\right )^{5/2}}-\frac {\int \frac {x^5 \left (\frac {a \left (16 D a^2-9 b C a+2 b^2 B\right ) x}{b^2}-\frac {5 a^2 D x^3}{b}\right )}{\left (b x^2+a\right )^{5/2}}dx}{5 a}}{a b}+\frac {x^7 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle \frac {\frac {x^7 \left (3 a^2 D-2 a b C+b^2 B\right )}{5 b^2 \left (a+b x^2\right )^{5/2}}-\frac {\int \frac {a x^6 \left (16 D a^2-5 b D x^2 a-9 b C a+2 b^2 B\right )}{b^2 \left (b x^2+a\right )^{5/2}}dx}{5 a}}{a b}+\frac {x^7 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {x^7 \left (3 a^2 D-2 a b C+b^2 B\right )}{5 b^2 \left (a+b x^2\right )^{5/2}}-\frac {\int \frac {x^6 \left (16 D a^2-5 b D x^2 a-9 b C a+2 b^2 B\right )}{\left (b x^2+a\right )^{5/2}}dx}{5 b^2}}{a b}+\frac {x^7 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 363 |
| \(\displaystyle \frac {\frac {x^7 \left (3 a^2 D-2 a b C+b^2 B\right )}{5 b^2 \left (a+b x^2\right )^{5/2}}-\frac {\frac {1}{4} \left (99 a^2 D-36 a b C+8 b^2 B\right ) \int \frac {x^6}{\left (b x^2+a\right )^{5/2}}dx-\frac {5 a D x^7}{4 \left (a+b x^2\right )^{3/2}}}{5 b^2}}{a b}+\frac {x^7 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {\frac {x^7 \left (3 a^2 D-2 a b C+b^2 B\right )}{5 b^2 \left (a+b x^2\right )^{5/2}}-\frac {\frac {1}{4} \left (99 a^2 D-36 a b C+8 b^2 B\right ) \left (\frac {5 \int \frac {x^4}{\left (b x^2+a\right )^{3/2}}dx}{3 b}-\frac {x^5}{3 b \left (a+b x^2\right )^{3/2}}\right )-\frac {5 a D x^7}{4 \left (a+b x^2\right )^{3/2}}}{5 b^2}}{a b}+\frac {x^7 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {\frac {x^7 \left (3 a^2 D-2 a b C+b^2 B\right )}{5 b^2 \left (a+b x^2\right )^{5/2}}-\frac {\frac {1}{4} \left (99 a^2 D-36 a b C+8 b^2 B\right ) \left (\frac {5 \left (\frac {3 \int \frac {x^2}{\sqrt {b x^2+a}}dx}{b}-\frac {x^3}{b \sqrt {a+b x^2}}\right )}{3 b}-\frac {x^5}{3 b \left (a+b x^2\right )^{3/2}}\right )-\frac {5 a D x^7}{4 \left (a+b x^2\right )^{3/2}}}{5 b^2}}{a b}+\frac {x^7 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {\frac {x^7 \left (3 a^2 D-2 a b C+b^2 B\right )}{5 b^2 \left (a+b x^2\right )^{5/2}}-\frac {\frac {1}{4} \left (99 a^2 D-36 a b C+8 b^2 B\right ) \left (\frac {5 \left (\frac {3 \left (\frac {x \sqrt {a+b x^2}}{2 b}-\frac {a \int \frac {1}{\sqrt {b x^2+a}}dx}{2 b}\right )}{b}-\frac {x^3}{b \sqrt {a+b x^2}}\right )}{3 b}-\frac {x^5}{3 b \left (a+b x^2\right )^{3/2}}\right )-\frac {5 a D x^7}{4 \left (a+b x^2\right )^{3/2}}}{5 b^2}}{a b}+\frac {x^7 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\frac {x^7 \left (3 a^2 D-2 a b C+b^2 B\right )}{5 b^2 \left (a+b x^2\right )^{5/2}}-\frac {\frac {1}{4} \left (99 a^2 D-36 a b C+8 b^2 B\right ) \left (\frac {5 \left (\frac {3 \left (\frac {x \sqrt {a+b x^2}}{2 b}-\frac {a \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 b}\right )}{b}-\frac {x^3}{b \sqrt {a+b x^2}}\right )}{3 b}-\frac {x^5}{3 b \left (a+b x^2\right )^{3/2}}\right )-\frac {5 a D x^7}{4 \left (a+b x^2\right )^{3/2}}}{5 b^2}}{a b}+\frac {x^7 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {x^7 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}+\frac {\frac {x^7 \left (3 a^2 D-2 a b C+b^2 B\right )}{5 b^2 \left (a+b x^2\right )^{5/2}}-\frac {\frac {1}{4} \left (\frac {5 \left (\frac {3 \left (\frac {x \sqrt {a+b x^2}}{2 b}-\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{3/2}}\right )}{b}-\frac {x^3}{b \sqrt {a+b x^2}}\right )}{3 b}-\frac {x^5}{3 b \left (a+b x^2\right )^{3/2}}\right ) \left (99 a^2 D-36 a b C+8 b^2 B\right )-\frac {5 a D x^7}{4 \left (a+b x^2\right )^{3/2}}}{5 b^2}}{a b}\) |
((A - (a*(b^2*B - a*b*C + a^2*D))/b^3)*x^7)/(7*a*(a + b*x^2)^(7/2)) + (((b ^2*B - 2*a*b*C + 3*a^2*D)*x^7)/(5*b^2*(a + b*x^2)^(5/2)) - ((-5*a*D*x^7)/( 4*(a + b*x^2)^(3/2)) + ((8*b^2*B - 36*a*b*C + 99*a^2*D)*(-1/3*x^5/(b*(a + b*x^2)^(3/2)) + (5*(-(x^3/(b*Sqrt[a + b*x^2])) + (3*((x*Sqrt[a + b*x^2])/( 2*b) - (a*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*b^(3/2))))/b))/(3*b)))/ 4)/(5*b^2))/(a*b)
3.2.60.3.1 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[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3)) Int[(e*x)^ m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d , 0] && NeQ[m + 2*p + 3, 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 = 3.67 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.70
| method | result | size |
| pseudoelliptic | \(\frac {7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}} \left (B \,b^{2}-\frac {9}{2} C a b +\frac {99}{8} D a^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )+x \left (-\frac {176 a \left (-\frac {105}{704} D x^{4}-\frac {105}{352} C \,x^{2}+B \right ) x^{6} b^{\frac {11}{2}}}{15}-7 a^{4} \left (\frac {957}{20} D x^{4}-15 C \,x^{2}+B \right ) b^{\frac {5}{2}}-\frac {70 a^{3} x^{2} \left (\frac {1089}{175} D x^{4}-\frac {261}{50} C \,x^{2}+B \right ) b^{\frac {7}{2}}}{3}-\frac {406 a^{2} x^{4} \left (\frac {165}{464} D x^{4}-\frac {396}{203} C \,x^{2}+B \right ) b^{\frac {9}{2}}}{15}+\frac {63 a^{5} \left (-\frac {55 D x^{2}}{6}+C \right ) b^{\frac {3}{2}}}{2}+A \,b^{\frac {13}{2}} x^{6}-\frac {693 D \sqrt {b}\, a^{6}}{8}\right )}{7 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{\frac {13}{2}} a}\) | \(196\) |
| default | \(B \left (-\frac {x^{7}}{7 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {-\frac {x^{5}}{5 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}}{b}}{b}\right )+A \left (-\frac {x^{5}}{2 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {5 a \left (-\frac {x^{3}}{4 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {3 a \left (-\frac {x}{6 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {a \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{6 b}\right )}{4 b}\right )}{2 b}\right )+D \left (\frac {x^{11}}{4 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {11 a \left (\frac {x^{9}}{2 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {9 a \left (-\frac {x^{7}}{7 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {-\frac {x^{5}}{5 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}}{b}}{b}\right )}{2 b}\right )}{4 b}\right )+C \left (\frac {x^{9}}{2 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {9 a \left (-\frac {x^{7}}{7 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {-\frac {x^{5}}{5 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}}{b}}{b}\right )}{2 b}\right )\) | \(531\) |
1/7/(b*x^2+a)^(7/2)*(7*a*(b*x^2+a)^(7/2)*(B*b^2-9/2*C*a*b+99/8*D*a^2)*arct anh((b*x^2+a)^(1/2)/x/b^(1/2))+x*(-176/15*a*(-105/704*D*x^4-105/352*C*x^2+ B)*x^6*b^(11/2)-7*a^4*(957/20*D*x^4-15*C*x^2+B)*b^(5/2)-70/3*a^3*x^2*(1089 /175*D*x^4-261/50*C*x^2+B)*b^(7/2)-406/15*a^2*x^4*(165/464*D*x^4-396/203*C *x^2+B)*b^(9/2)+63/2*a^5*(-55/6*D*x^2+C)*b^(3/2)+A*b^(13/2)*x^6-693/8*D*b^ (1/2)*a^6))/b^(13/2)/a
Time = 0.43 (sec) , antiderivative size = 816, normalized size of antiderivative = 2.92 \[ \int \frac {x^6 \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 ({\left (99 \, D a^{3} b^{4} - 36 \, C a^{2} b^{5} + 8 \, B a b^{6}\right )} x^{8} + 99 \, D a^{7} - 36 \, C a^{6} b + 8 \, B a^{5} b^{2} + 4 \, {\left (99 \, D a^{4} b^{3} - 36 \, C a^{3} b^{4} + 8 \, B a^{2} b^{5}\right )} x^{6} + 6 \, {\left (99 \, D a^{5} b^{2} - 36 \, C a^{4} b^{3} + 8 \, B a^{3} b^{4}\right )} x^{4} + 4 \, {\left (99 \, D a^{6} b - 36 \, C a^{5} b^{2} + 8 \, B a^{4} b^{3}\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 (210 \, D a b^{6} x^{11} - 105 \, {\left (11 \, D a^{2} b^{5} - 4 \, C a b^{6}\right )} x^{9} - 8 \, {\left (2178 \, D a^{3} b^{4} - 792 \, C a^{2} b^{5} + 176 \, B a b^{6} - 15 \, A b^{7}\right )} x^{7} - 406 \, {\left (99 \, D a^{4} b^{3} - 36 \, C a^{3} b^{4} + 8 \, B a^{2} b^{5}\right )} x^{5} - 350 \, {\left (99 \, D a^{5} b^{2} - 36 \, C a^{4} b^{3} + 8 \, B a^{3} b^{4}\right )} x^{3} - 105 \, {\left (99 \, D a^{6} b - 36 \, C a^{5} b^{2} + 8 \, B a^{4} b^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{1680 \, {\left (a b^{11} x^{8} + 4 \, a^{2} b^{10} x^{6} + 6 \, a^{3} b^{9} x^{4} + 4 \, a^{4} b^{8} x^{2} + a^{5} b^{7}\right )}}, -\frac {105 \, {\left ({\left (99 \, D a^{3} b^{4} - 36 \, C a^{2} b^{5} + 8 \, B a b^{6}\right )} x^{8} + 99 \, D a^{7} - 36 \, C a^{6} b + 8 \, B a^{5} b^{2} + 4 \, {\left (99 \, D a^{4} b^{3} - 36 \, C a^{3} b^{4} + 8 \, B a^{2} b^{5}\right )} x^{6} + 6 \, {\left (99 \, D a^{5} b^{2} - 36 \, C a^{4} b^{3} + 8 \, B a^{3} b^{4}\right )} x^{4} + 4 \, {\left (99 \, D a^{6} b - 36 \, C a^{5} b^{2} + 8 \, B a^{4} b^{3}\right )} x^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (210 \, D a b^{6} x^{11} - 105 \, {\left (11 \, D a^{2} b^{5} - 4 \, C a b^{6}\right )} x^{9} - 8 \, {\left (2178 \, D a^{3} b^{4} - 792 \, C a^{2} b^{5} + 176 \, B a b^{6} - 15 \, A b^{7}\right )} x^{7} - 406 \, {\left (99 \, D a^{4} b^{3} - 36 \, C a^{3} b^{4} + 8 \, B a^{2} b^{5}\right )} x^{5} - 350 \, {\left (99 \, D a^{5} b^{2} - 36 \, C a^{4} b^{3} + 8 \, B a^{3} b^{4}\right )} x^{3} - 105 \, {\left (99 \, D a^{6} b - 36 \, C a^{5} b^{2} + 8 \, B a^{4} b^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{840 \, {\left (a b^{11} x^{8} + 4 \, a^{2} b^{10} x^{6} + 6 \, a^{3} b^{9} x^{4} + 4 \, a^{4} b^{8} x^{2} + a^{5} b^{7}\right )}}\right ] \]
[1/1680*(105*((99*D*a^3*b^4 - 36*C*a^2*b^5 + 8*B*a*b^6)*x^8 + 99*D*a^7 - 3 6*C*a^6*b + 8*B*a^5*b^2 + 4*(99*D*a^4*b^3 - 36*C*a^3*b^4 + 8*B*a^2*b^5)*x^ 6 + 6*(99*D*a^5*b^2 - 36*C*a^4*b^3 + 8*B*a^3*b^4)*x^4 + 4*(99*D*a^6*b - 36 *C*a^5*b^2 + 8*B*a^4*b^3)*x^2)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sq rt(b)*x - a) + 2*(210*D*a*b^6*x^11 - 105*(11*D*a^2*b^5 - 4*C*a*b^6)*x^9 - 8*(2178*D*a^3*b^4 - 792*C*a^2*b^5 + 176*B*a*b^6 - 15*A*b^7)*x^7 - 406*(99* D*a^4*b^3 - 36*C*a^3*b^4 + 8*B*a^2*b^5)*x^5 - 350*(99*D*a^5*b^2 - 36*C*a^4 *b^3 + 8*B*a^3*b^4)*x^3 - 105*(99*D*a^6*b - 36*C*a^5*b^2 + 8*B*a^4*b^3)*x) *sqrt(b*x^2 + a))/(a*b^11*x^8 + 4*a^2*b^10*x^6 + 6*a^3*b^9*x^4 + 4*a^4*b^8 *x^2 + a^5*b^7), -1/840*(105*((99*D*a^3*b^4 - 36*C*a^2*b^5 + 8*B*a*b^6)*x^ 8 + 99*D*a^7 - 36*C*a^6*b + 8*B*a^5*b^2 + 4*(99*D*a^4*b^3 - 36*C*a^3*b^4 + 8*B*a^2*b^5)*x^6 + 6*(99*D*a^5*b^2 - 36*C*a^4*b^3 + 8*B*a^3*b^4)*x^4 + 4* (99*D*a^6*b - 36*C*a^5*b^2 + 8*B*a^4*b^3)*x^2)*sqrt(-b)*arctan(sqrt(-b)*x/ sqrt(b*x^2 + a)) - (210*D*a*b^6*x^11 - 105*(11*D*a^2*b^5 - 4*C*a*b^6)*x^9 - 8*(2178*D*a^3*b^4 - 792*C*a^2*b^5 + 176*B*a*b^6 - 15*A*b^7)*x^7 - 406*(9 9*D*a^4*b^3 - 36*C*a^3*b^4 + 8*B*a^2*b^5)*x^5 - 350*(99*D*a^5*b^2 - 36*C*a ^4*b^3 + 8*B*a^3*b^4)*x^3 - 105*(99*D*a^6*b - 36*C*a^5*b^2 + 8*B*a^4*b^3)* x)*sqrt(b*x^2 + a))/(a*b^11*x^8 + 4*a^2*b^10*x^6 + 6*a^3*b^9*x^4 + 4*a^4*b ^8*x^2 + a^5*b^7)]
Timed out. \[ \int \frac {x^6 \left (A+B x^2+C x^4+D x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 986 vs. \(2 (247) = 494\).
Time = 0.23 (sec) , antiderivative size = 986, normalized size of antiderivative = 3.53 \[ \int \frac {x^6 \left (A+B x^2+C x^4+D x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\text {Too large to display} \]
1/4*D*x^11/((b*x^2 + a)^(7/2)*b) - 11/8*D*a*x^9/((b*x^2 + a)^(7/2)*b^2) + 1/2*C*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))*B*x - 99/280*(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*a^2*x/b^2 + 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))*C*a*x/b - 33/40*D*a^2*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^3 + 3/10*C*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*B*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*A*x^5/((b*x^2 + a)^(7/2)*b) - 33/8*D*a^2*x*(3*x^2/( (b*x^2 + a)^(3/2)*b) + 2*a/((b*x^2 + a)^(3/2)*b^2))/b^4 + 3/2*C*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*B*x*(3*x^2 /((b*x^2 + a)^(3/2)*b) + 2*a/((b*x^2 + a)^(3/2)*b^2))/b^2 - 99/8*D*a^3*x^3 /((b*x^2 + a)^(5/2)*b^5) + 9/2*C*a^2*x^3/((b*x^2 + a)^(5/2)*b^4) - B*a*x^3 /((b*x^2 + a)^(5/2)*b^3) - 5/8*A*a*x^3/((b*x^2 + a)^(7/2)*b^2) + 4587/280* D*a^2*x/(sqrt(b*x^2 + a)*b^6) + 561/280*D*a^3*x/((b*x^2 + a)^(3/2)*b^6) - 2871/280*D*a^4*x/((b*x^2 + a)^(5/2)*b^6) - 417/70*C*a*x/(sqrt(b*x^2 + a...
Time = 0.31 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.95 \[ \int \frac {x^6 \left (A+B x^2+C x^4+D x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {{\left ({\left ({\left ({\left (105 \, {\left (\frac {2 \, D x^{2}}{b} - \frac {11 \, D a^{4} b^{9} - 4 \, C a^{3} b^{10}}{a^{3} b^{11}}\right )} x^{2} - \frac {8 \, {\left (2178 \, D a^{5} b^{8} - 792 \, C a^{4} b^{9} + 176 \, B a^{3} b^{10} - 15 \, A a^{2} b^{11}\right )}}{a^{3} b^{11}}\right )} x^{2} - \frac {406 \, {\left (99 \, D a^{6} b^{7} - 36 \, C a^{5} b^{8} + 8 \, B a^{4} b^{9}\right )}}{a^{3} b^{11}}\right )} x^{2} - \frac {350 \, {\left (99 \, D a^{7} b^{6} - 36 \, C a^{6} b^{7} + 8 \, B a^{5} b^{8}\right )}}{a^{3} b^{11}}\right )} x^{2} - \frac {105 \, {\left (99 \, D a^{8} b^{5} - 36 \, C a^{7} b^{6} + 8 \, B a^{6} b^{7}\right )}}{a^{3} b^{11}}\right )} x}{840 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} - \frac {{\left (99 \, D a^{2} - 36 \, C a b + 8 \, B b^{2}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{8 \, b^{\frac {13}{2}}} \]
1/840*((((105*(2*D*x^2/b - (11*D*a^4*b^9 - 4*C*a^3*b^10)/(a^3*b^11))*x^2 - 8*(2178*D*a^5*b^8 - 792*C*a^4*b^9 + 176*B*a^3*b^10 - 15*A*a^2*b^11)/(a^3* b^11))*x^2 - 406*(99*D*a^6*b^7 - 36*C*a^5*b^8 + 8*B*a^4*b^9)/(a^3*b^11))*x ^2 - 350*(99*D*a^7*b^6 - 36*C*a^6*b^7 + 8*B*a^5*b^8)/(a^3*b^11))*x^2 - 105 *(99*D*a^8*b^5 - 36*C*a^7*b^6 + 8*B*a^6*b^7)/(a^3*b^11))*x/(b*x^2 + a)^(7/ 2) - 1/8*(99*D*a^2 - 36*C*a*b + 8*B*b^2)*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(13/2)
Timed out. \[ \int \frac {x^6 \left (A+B x^2+C x^4+D x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\int \frac {x^6\,\left (A+B\,x^2+C\,x^4+x^6\,D\right )}{{\left (b\,x^2+a\right )}^{9/2}} \,d x \]