Integrand size = 32, antiderivative size = 381 \[ \int \frac {x^8 \left (A+B x^2+C x^4+D x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {\left (A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) x^9}{7 a \left (a+b x^2\right )^{7/2}}-\frac {\left (2 A b^3-a \left (9 b^2 B-16 a b C+23 a^2 D\right )\right ) x^9}{35 a^2 b^3 \left (a+b x^2\right )^{5/2}}-\frac {\left (16 A b^3-3 a \left (24 b^2 B-66 a b C+143 a^2 D\right )\right ) x^7}{210 a^2 b^4 \left (a+b x^2\right )^{3/2}}+\frac {D x^9}{6 b^3 \left (a+b x^2\right )^{3/2}}-\frac {\left (16 A b^3-3 a \left (24 b^2 B-66 a b C+143 a^2 D\right )\right ) x^5}{30 a^2 b^5 \sqrt {a+b x^2}}-\frac {\left (16 A b^3-3 a \left (24 b^2 B-66 a b C+143 a^2 D\right )\right ) x \sqrt {a+b x^2}}{16 a b^7}+\frac {\left (16 A b^3-3 a \left (24 b^2 B-66 a b C+143 a^2 D\right )\right ) x^3 \sqrt {a+b x^2}}{24 a^2 b^6}+\frac {\left (16 A b^3-72 a b^2 B+198 a^2 b C-429 a^3 D\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{16 b^{15/2}} \]
1/7*(A-a*(B*b^2-C*a*b+D*a^2)/b^3)*x^9/a/(b*x^2+a)^(7/2)-1/35*(2*A*b^3-a*(9 *B*b^2-16*C*a*b+23*D*a^2))*x^9/a^2/b^3/(b*x^2+a)^(5/2)-1/210*(16*A*b^3-3*a *(24*B*b^2-66*C*a*b+143*D*a^2))*x^7/a^2/b^4/(b*x^2+a)^(3/2)+1/6*D*x^9/b^3/ (b*x^2+a)^(3/2)+1/16*(16*A*b^3-72*B*a*b^2+198*C*a^2*b-429*D*a^3)*arctanh(x *b^(1/2)/(b*x^2+a)^(1/2))/b^(15/2)-1/30*(16*A*b^3-3*a*(24*B*b^2-66*C*a*b+1 43*D*a^2))*x^5/a^2/b^5/(b*x^2+a)^(1/2)-1/16*(16*A*b^3-3*a*(24*B*b^2-66*C*a *b+143*D*a^2))*x*(b*x^2+a)^(1/2)/a/b^7+1/24*(16*A*b^3-3*a*(24*B*b^2-66*C*a *b+143*D*a^2))*x^3*(b*x^2+a)^(1/2)/a^2/b^6
Time = 1.26 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.68 \[ \int \frac {x^8 \left (A+B x^2+C x^4+D x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {x \left (45045 a^6 D-2310 a^5 b \left (9 C-65 D x^2\right )+42 a^4 b^2 \left (180 B-1650 C x^2+4147 D x^4\right )-12 a^3 b^3 \left (140 A-2100 B x^2+6699 C x^4-6292 D x^6\right )-2 a b^5 x^4 \left (3248 A-6336 B x^2+1155 C x^4+455 D x^6\right )+a^2 b^4 x^2 \left (-5600 A+29232 B x^2-34848 C x^4+5005 D x^6\right )+4 b^6 x^6 \left (-704 A+35 \left (6 B x^2+3 C x^4+2 D x^6\right )\right )\right )}{1680 b^7 \left (a+b x^2\right )^{7/2}}+\frac {\left (16 A b^3-3 a \left (24 b^2 B-66 a b C+143 a^2 D\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{-\sqrt {a}+\sqrt {a+b x^2}}\right )}{8 b^{15/2}} \]
(x*(45045*a^6*D - 2310*a^5*b*(9*C - 65*D*x^2) + 42*a^4*b^2*(180*B - 1650*C *x^2 + 4147*D*x^4) - 12*a^3*b^3*(140*A - 2100*B*x^2 + 6699*C*x^4 - 6292*D* x^6) - 2*a*b^5*x^4*(3248*A - 6336*B*x^2 + 1155*C*x^4 + 455*D*x^6) + a^2*b^ 4*x^2*(-5600*A + 29232*B*x^2 - 34848*C*x^4 + 5005*D*x^6) + 4*b^6*x^6*(-704 *A + 35*(6*B*x^2 + 3*C*x^4 + 2*D*x^6))))/(1680*b^7*(a + b*x^2)^(7/2)) + (( 16*A*b^3 - 3*a*(24*b^2*B - 66*a*b*C + 143*a^2*D))*ArcTanh[(Sqrt[b]*x)/(-Sq rt[a] + Sqrt[a + b*x^2])])/(8*b^(15/2))
Time = 0.65 (sec) , antiderivative size = 310, normalized size of antiderivative = 0.81, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2335, 9, 1586, 9, 27, 363, 252, 252, 262, 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.
| \(\displaystyle \int \frac {x^8 \left (A+B x^2+C x^4+D x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx\) |
| \(\Big \downarrow \) 2335 |
| \(\displaystyle \frac {x^9 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}-\frac {\int \frac {x^7 \left (-7 a D x^5-7 a \left (C-\frac {a D}{b}\right ) x^3+\left (2 A b-\frac {9 a \left (D a^2-b C a+b^2 B\right )}{b^2}\right ) x\right )}{\left (b x^2+a\right )^{7/2}}dx}{7 a b}\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle \frac {x^9 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}-\frac {\int \frac {x^8 \left (-7 a D x^4-7 a \left (C-\frac {a D}{b}\right ) x^2+2 A b-\frac {9 a \left (D a^2-b C a+b^2 B\right )}{b^2}\right )}{\left (b x^2+a\right )^{7/2}}dx}{7 a b}\) |
| \(\Big \downarrow \) 1586 |
| \(\displaystyle \frac {x^9 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}-\frac {\frac {x^9 \left (2 A b-\frac {a \left (23 a^2 D-16 a b C+9 b^2 B\right )}{b^2}\right )}{5 a \left (a+b x^2\right )^{5/2}}-\frac {\int \frac {x^7 \left (\frac {35 a^2 D x^3}{b}+\left (8 A b-\frac {9 a \left (18 D a^2-11 b C a+4 b^2 B\right )}{b^2}\right ) x\right )}{\left (b x^2+a\right )^{5/2}}dx}{5 a}}{7 a b}\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle \frac {x^9 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}-\frac {\frac {x^9 \left (2 A b-\frac {a \left (23 a^2 D-16 a b C+9 b^2 B\right )}{b^2}\right )}{5 a \left (a+b x^2\right )^{5/2}}-\frac {\int \frac {x^8 \left (35 a^2 D x^2+b \left (8 A b-\frac {9 a \left (18 D a^2-11 b C a+4 b^2 B\right )}{b^2}\right )\right )}{b \left (b x^2+a\right )^{5/2}}dx}{5 a}}{7 a b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^9 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}-\frac {\frac {x^9 \left (2 A b-\frac {a \left (23 a^2 D-16 a b C+9 b^2 B\right )}{b^2}\right )}{5 a \left (a+b x^2\right )^{5/2}}-\frac {\int \frac {x^8 \left (8 A b^2+35 a^2 D x^2-9 a \left (\frac {18 D a^2}{b}-11 C a+4 b B\right )\right )}{\left (b x^2+a\right )^{5/2}}dx}{5 a b}}{7 a b}\) |
| \(\Big \downarrow \) 363 |
| \(\displaystyle \frac {x^9 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}-\frac {\frac {x^9 \left (2 A b-\frac {a \left (23 a^2 D-16 a b C+9 b^2 B\right )}{b^2}\right )}{5 a \left (a+b x^2\right )^{5/2}}-\frac {\frac {\left (-429 a^3 D+198 a^2 b C-72 a b^2 B+16 A b^3\right ) \int \frac {x^8}{\left (b x^2+a\right )^{5/2}}dx}{2 b}+\frac {35 a^2 D x^9}{6 b \left (a+b x^2\right )^{3/2}}}{5 a b}}{7 a b}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {x^9 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}-\frac {\frac {x^9 \left (2 A b-\frac {a \left (23 a^2 D-16 a b C+9 b^2 B\right )}{b^2}\right )}{5 a \left (a+b x^2\right )^{5/2}}-\frac {\frac {\left (-429 a^3 D+198 a^2 b C-72 a b^2 B+16 A b^3\right ) \left (\frac {7 \int \frac {x^6}{\left (b x^2+a\right )^{3/2}}dx}{3 b}-\frac {x^7}{3 b \left (a+b x^2\right )^{3/2}}\right )}{2 b}+\frac {35 a^2 D x^9}{6 b \left (a+b x^2\right )^{3/2}}}{5 a b}}{7 a b}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {x^9 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}-\frac {\frac {x^9 \left (2 A b-\frac {a \left (23 a^2 D-16 a b C+9 b^2 B\right )}{b^2}\right )}{5 a \left (a+b x^2\right )^{5/2}}-\frac {\frac {\left (-429 a^3 D+198 a^2 b C-72 a b^2 B+16 A b^3\right ) \left (\frac {7 \left (\frac {5 \int \frac {x^4}{\sqrt {b x^2+a}}dx}{b}-\frac {x^5}{b \sqrt {a+b x^2}}\right )}{3 b}-\frac {x^7}{3 b \left (a+b x^2\right )^{3/2}}\right )}{2 b}+\frac {35 a^2 D x^9}{6 b \left (a+b x^2\right )^{3/2}}}{5 a b}}{7 a b}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {x^9 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}-\frac {\frac {x^9 \left (2 A b-\frac {a \left (23 a^2 D-16 a b C+9 b^2 B\right )}{b^2}\right )}{5 a \left (a+b x^2\right )^{5/2}}-\frac {\frac {\left (-429 a^3 D+198 a^2 b C-72 a b^2 B+16 A b^3\right ) \left (\frac {7 \left (\frac {5 \left (\frac {x^3 \sqrt {a+b x^2}}{4 b}-\frac {3 a \int \frac {x^2}{\sqrt {b x^2+a}}dx}{4 b}\right )}{b}-\frac {x^5}{b \sqrt {a+b x^2}}\right )}{3 b}-\frac {x^7}{3 b \left (a+b x^2\right )^{3/2}}\right )}{2 b}+\frac {35 a^2 D x^9}{6 b \left (a+b x^2\right )^{3/2}}}{5 a b}}{7 a b}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {x^9 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}-\frac {\frac {x^9 \left (2 A b-\frac {a \left (23 a^2 D-16 a b C+9 b^2 B\right )}{b^2}\right )}{5 a \left (a+b x^2\right )^{5/2}}-\frac {\frac {\left (-429 a^3 D+198 a^2 b C-72 a b^2 B+16 A b^3\right ) \left (\frac {7 \left (\frac {5 \left (\frac {x^3 \sqrt {a+b x^2}}{4 b}-\frac {3 a \left (\frac {x \sqrt {a+b x^2}}{2 b}-\frac {a \int \frac {1}{\sqrt {b x^2+a}}dx}{2 b}\right )}{4 b}\right )}{b}-\frac {x^5}{b \sqrt {a+b x^2}}\right )}{3 b}-\frac {x^7}{3 b \left (a+b x^2\right )^{3/2}}\right )}{2 b}+\frac {35 a^2 D x^9}{6 b \left (a+b x^2\right )^{3/2}}}{5 a b}}{7 a b}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {x^9 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}-\frac {\frac {x^9 \left (2 A b-\frac {a \left (23 a^2 D-16 a b C+9 b^2 B\right )}{b^2}\right )}{5 a \left (a+b x^2\right )^{5/2}}-\frac {\frac {\left (-429 a^3 D+198 a^2 b C-72 a b^2 B+16 A b^3\right ) \left (\frac {7 \left (\frac {5 \left (\frac {x^3 \sqrt {a+b x^2}}{4 b}-\frac {3 a \left (\frac {x \sqrt {a+b x^2}}{2 b}-\frac {a \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 b}\right )}{4 b}\right )}{b}-\frac {x^5}{b \sqrt {a+b x^2}}\right )}{3 b}-\frac {x^7}{3 b \left (a+b x^2\right )^{3/2}}\right )}{2 b}+\frac {35 a^2 D x^9}{6 b \left (a+b x^2\right )^{3/2}}}{5 a b}}{7 a b}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {x^9 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}-\frac {\frac {x^9 \left (2 A b-\frac {a \left (23 a^2 D-16 a b C+9 b^2 B\right )}{b^2}\right )}{5 a \left (a+b x^2\right )^{5/2}}-\frac {\frac {35 a^2 D x^9}{6 b \left (a+b x^2\right )^{3/2}}+\frac {\left (\frac {7 \left (\frac {5 \left (\frac {x^3 \sqrt {a+b x^2}}{4 b}-\frac {3 a \left (\frac {x \sqrt {a+b x^2}}{2 b}-\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{3/2}}\right )}{4 b}\right )}{b}-\frac {x^5}{b \sqrt {a+b x^2}}\right )}{3 b}-\frac {x^7}{3 b \left (a+b x^2\right )^{3/2}}\right ) \left (-429 a^3 D+198 a^2 b C-72 a b^2 B+16 A b^3\right )}{2 b}}{5 a b}}{7 a b}\) |
((A - (a*(b^2*B - a*b*C + a^2*D))/b^3)*x^9)/(7*a*(a + b*x^2)^(7/2)) - (((2 *A*b - (a*(9*b^2*B - 16*a*b*C + 23*a^2*D))/b^2)*x^9)/(5*a*(a + b*x^2)^(5/2 )) - ((35*a^2*D*x^9)/(6*b*(a + b*x^2)^(3/2)) + ((16*A*b^3 - 72*a*b^2*B + 1 98*a^2*b*C - 429*a^3*D)*(-1/3*x^7/(b*(a + b*x^2)^(3/2)) + (7*(-(x^5/(b*Sqr t[a + b*x^2])) + (5*((x^3*Sqrt[a + b*x^2])/(4*b) - (3*a*((x*Sqrt[a + b*x^2 ])/(2*b) - (a*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*b^(3/2))))/(4*b)))/ b))/(3*b)))/(2*b))/(5*a*b))/(7*a*b)
3.2.59.3.1 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 = 3.75 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.62
| method | result | size |
| pseudoelliptic | \(\frac {-\frac {58 a \,x^{5} \left (\frac {65}{464} D x^{6}+\frac {165}{464} C \,x^{4}-\frac {396}{203} x^{2} B +A \right ) b^{\frac {11}{2}}}{15}+\left (\frac {1}{6} D x^{13}+\frac {1}{4} C \,x^{11}+\frac {1}{2} B \,x^{9}-\frac {176}{105} A \,x^{7}\right ) b^{\frac {13}{2}}-\frac {99 \left (-\frac {65 D x^{2}}{9}+C \right ) a^{5} x \,b^{\frac {3}{2}}}{8}+\frac {9 a^{4} x \left (\frac {4147}{180} D x^{4}-\frac {55}{6} C \,x^{2}+B \right ) b^{\frac {5}{2}}}{2}-a^{3} \left (-\frac {1573}{35} D x^{6}+\frac {957}{20} C \,x^{4}-15 x^{2} B +A \right ) x \,b^{\frac {7}{2}}-\frac {10 a^{2} x^{3} \left (-\frac {143}{160} D x^{6}+\frac {1089}{175} C \,x^{4}-\frac {261}{50} x^{2} B +A \right ) b^{\frac {9}{2}}}{3}+\frac {429 D \sqrt {b}\, a^{6} x}{16}+\left (b^{3} A -\frac {9}{2} a \,b^{2} B +\frac {99}{8} C \,a^{2} b -\frac {429}{16} D a^{3}\right ) \left (b \,x^{2}+a \right )^{\frac {7}{2}} \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{\left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{\frac {15}{2}}}\) | \(236\) |
| default | \(A \left (-\frac {x^{7}}{7 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {-\frac {x^{5}}{5 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}}{b}}{b}\right )+D \left (\frac {x^{13}}{6 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {13 a \left (\frac {x^{11}}{4 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {11 a \left (\frac {x^{9}}{2 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {9 a \left (-\frac {x^{7}}{7 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {-\frac {x^{5}}{5 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}}{b}}{b}\right )}{2 b}\right )}{4 b}\right )}{6 b}\right )+C \left (\frac {x^{11}}{4 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {11 a \left (\frac {x^{9}}{2 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {9 a \left (-\frac {x^{7}}{7 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {-\frac {x^{5}}{5 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}}{b}}{b}\right )}{2 b}\right )}{4 b}\right )+B \left (\frac {x^{9}}{2 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {9 a \left (-\frac {x^{7}}{7 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {-\frac {x^{5}}{5 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}}{b}}{b}\right )}{2 b}\right )\) | \(562\) |
1/(b*x^2+a)^(7/2)/b^(15/2)*(-58/15*a*x^5*(65/464*D*x^6+165/464*C*x^4-396/2 03*x^2*B+A)*b^(11/2)+(1/6*D*x^13+1/4*C*x^11+1/2*B*x^9-176/105*A*x^7)*b^(13 /2)-99/8*(-65/9*D*x^2+C)*a^5*x*b^(3/2)+9/2*a^4*x*(4147/180*D*x^4-55/6*C*x^ 2+B)*b^(5/2)-a^3*(-1573/35*D*x^6+957/20*C*x^4-15*x^2*B+A)*x*b^(7/2)-10/3*a ^2*x^3*(-143/160*D*x^6+1089/175*C*x^4-261/50*x^2*B+A)*b^(9/2)+429/16*D*b^( 1/2)*a^6*x+(b^3*A-9/2*a*b^2*B+99/8*C*a^2*b-429/16*D*a^3)*(b*x^2+a)^(7/2)*a rctanh((b*x^2+a)^(1/2)/x/b^(1/2)))
Time = 0.53 (sec) , antiderivative size = 987, normalized size of antiderivative = 2.59 \[ \int \frac {x^8 \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 (429 \, D a^{3} b^{4} - 198 \, C a^{2} b^{5} + 72 \, B a b^{6} - 16 \, A b^{7}\right )} x^{8} + 429 \, D a^{7} - 198 \, C a^{6} b + 72 \, B a^{5} b^{2} - 16 \, A a^{4} b^{3} + 4 \, {\left (429 \, D a^{4} b^{3} - 198 \, C a^{3} b^{4} + 72 \, B a^{2} b^{5} - 16 \, A a b^{6}\right )} x^{6} + 6 \, {\left (429 \, D a^{5} b^{2} - 198 \, C a^{4} b^{3} + 72 \, B a^{3} b^{4} - 16 \, A a^{2} b^{5}\right )} x^{4} + 4 \, {\left (429 \, D a^{6} b - 198 \, C a^{5} b^{2} + 72 \, B a^{4} b^{3} - 16 \, A a^{3} b^{4}\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 (280 \, D b^{7} x^{13} - 70 \, {\left (13 \, D a b^{6} - 6 \, C b^{7}\right )} x^{11} + 35 \, {\left (143 \, D a^{2} b^{5} - 66 \, C a b^{6} + 24 \, B b^{7}\right )} x^{9} + 176 \, {\left (429 \, D a^{3} b^{4} - 198 \, C a^{2} b^{5} + 72 \, B a b^{6} - 16 \, A b^{7}\right )} x^{7} + 406 \, {\left (429 \, D a^{4} b^{3} - 198 \, C a^{3} b^{4} + 72 \, B a^{2} b^{5} - 16 \, A a b^{6}\right )} x^{5} + 350 \, {\left (429 \, D a^{5} b^{2} - 198 \, C a^{4} b^{3} + 72 \, B a^{3} b^{4} - 16 \, A a^{2} b^{5}\right )} x^{3} + 105 \, {\left (429 \, D a^{6} b - 198 \, C a^{5} b^{2} + 72 \, B a^{4} b^{3} - 16 \, A a^{3} b^{4}\right )} x\right )} \sqrt {b x^{2} + a}}{3360 \, {\left (b^{12} x^{8} + 4 \, a b^{11} x^{6} + 6 \, a^{2} b^{10} x^{4} + 4 \, a^{3} b^{9} x^{2} + a^{4} b^{8}\right )}}, \frac {105 \, {\left ({\left (429 \, D a^{3} b^{4} - 198 \, C a^{2} b^{5} + 72 \, B a b^{6} - 16 \, A b^{7}\right )} x^{8} + 429 \, D a^{7} - 198 \, C a^{6} b + 72 \, B a^{5} b^{2} - 16 \, A a^{4} b^{3} + 4 \, {\left (429 \, D a^{4} b^{3} - 198 \, C a^{3} b^{4} + 72 \, B a^{2} b^{5} - 16 \, A a b^{6}\right )} x^{6} + 6 \, {\left (429 \, D a^{5} b^{2} - 198 \, C a^{4} b^{3} + 72 \, B a^{3} b^{4} - 16 \, A a^{2} b^{5}\right )} x^{4} + 4 \, {\left (429 \, D a^{6} b - 198 \, C a^{5} b^{2} + 72 \, B a^{4} b^{3} - 16 \, A a^{3} b^{4}\right )} x^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (280 \, D b^{7} x^{13} - 70 \, {\left (13 \, D a b^{6} - 6 \, C b^{7}\right )} x^{11} + 35 \, {\left (143 \, D a^{2} b^{5} - 66 \, C a b^{6} + 24 \, B b^{7}\right )} x^{9} + 176 \, {\left (429 \, D a^{3} b^{4} - 198 \, C a^{2} b^{5} + 72 \, B a b^{6} - 16 \, A b^{7}\right )} x^{7} + 406 \, {\left (429 \, D a^{4} b^{3} - 198 \, C a^{3} b^{4} + 72 \, B a^{2} b^{5} - 16 \, A a b^{6}\right )} x^{5} + 350 \, {\left (429 \, D a^{5} b^{2} - 198 \, C a^{4} b^{3} + 72 \, B a^{3} b^{4} - 16 \, A a^{2} b^{5}\right )} x^{3} + 105 \, {\left (429 \, D a^{6} b - 198 \, C a^{5} b^{2} + 72 \, B a^{4} b^{3} - 16 \, A a^{3} b^{4}\right )} x\right )} \sqrt {b x^{2} + a}}{1680 \, {\left (b^{12} x^{8} + 4 \, a b^{11} x^{6} + 6 \, a^{2} b^{10} x^{4} + 4 \, a^{3} b^{9} x^{2} + a^{4} b^{8}\right )}}\right ] \]
[1/3360*(105*((429*D*a^3*b^4 - 198*C*a^2*b^5 + 72*B*a*b^6 - 16*A*b^7)*x^8 + 429*D*a^7 - 198*C*a^6*b + 72*B*a^5*b^2 - 16*A*a^4*b^3 + 4*(429*D*a^4*b^3 - 198*C*a^3*b^4 + 72*B*a^2*b^5 - 16*A*a*b^6)*x^6 + 6*(429*D*a^5*b^2 - 198 *C*a^4*b^3 + 72*B*a^3*b^4 - 16*A*a^2*b^5)*x^4 + 4*(429*D*a^6*b - 198*C*a^5 *b^2 + 72*B*a^4*b^3 - 16*A*a^3*b^4)*x^2)*sqrt(b)*log(-2*b*x^2 + 2*sqrt(b*x ^2 + a)*sqrt(b)*x - a) + 2*(280*D*b^7*x^13 - 70*(13*D*a*b^6 - 6*C*b^7)*x^1 1 + 35*(143*D*a^2*b^5 - 66*C*a*b^6 + 24*B*b^7)*x^9 + 176*(429*D*a^3*b^4 - 198*C*a^2*b^5 + 72*B*a*b^6 - 16*A*b^7)*x^7 + 406*(429*D*a^4*b^3 - 198*C*a^ 3*b^4 + 72*B*a^2*b^5 - 16*A*a*b^6)*x^5 + 350*(429*D*a^5*b^2 - 198*C*a^4*b^ 3 + 72*B*a^3*b^4 - 16*A*a^2*b^5)*x^3 + 105*(429*D*a^6*b - 198*C*a^5*b^2 + 72*B*a^4*b^3 - 16*A*a^3*b^4)*x)*sqrt(b*x^2 + a))/(b^12*x^8 + 4*a*b^11*x^6 + 6*a^2*b^10*x^4 + 4*a^3*b^9*x^2 + a^4*b^8), 1/1680*(105*((429*D*a^3*b^4 - 198*C*a^2*b^5 + 72*B*a*b^6 - 16*A*b^7)*x^8 + 429*D*a^7 - 198*C*a^6*b + 72 *B*a^5*b^2 - 16*A*a^4*b^3 + 4*(429*D*a^4*b^3 - 198*C*a^3*b^4 + 72*B*a^2*b^ 5 - 16*A*a*b^6)*x^6 + 6*(429*D*a^5*b^2 - 198*C*a^4*b^3 + 72*B*a^3*b^4 - 16 *A*a^2*b^5)*x^4 + 4*(429*D*a^6*b - 198*C*a^5*b^2 + 72*B*a^4*b^3 - 16*A*a^3 *b^4)*x^2)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) + (280*D*b^7*x^13 - 70*(13*D*a*b^6 - 6*C*b^7)*x^11 + 35*(143*D*a^2*b^5 - 66*C*a*b^6 + 24*B*b^ 7)*x^9 + 176*(429*D*a^3*b^4 - 198*C*a^2*b^5 + 72*B*a*b^6 - 16*A*b^7)*x^7 + 406*(429*D*a^4*b^3 - 198*C*a^3*b^4 + 72*B*a^2*b^5 - 16*A*a*b^6)*x^5 + ...
Timed out. \[ \int \frac {x^8 \left (A+B x^2+C x^4+D x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 1221 vs. \(2 (345) = 690\).
Time = 0.22 (sec) , antiderivative size = 1221, normalized size of antiderivative = 3.20 \[ \int \frac {x^8 \left (A+B x^2+C x^4+D x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\text {Too large to display} \]
1/6*D*x^13/((b*x^2 + a)^(7/2)*b) - 13/24*D*a*x^11/((b*x^2 + a)^(7/2)*b^2) + 1/4*C*x^11/((b*x^2 + a)^(7/2)*b) + 143/48*D*a^2*x^9/((b*x^2 + a)^(7/2)*b ^3) - 11/8*C*a*x^9/((b*x^2 + a)^(7/2)*b^2) + 1/2*B*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))*A* x + 429/560*(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^3*x/b^3 - 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))*C*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))*B*a*x/b + 143/80*D*a^3*x*(15*x^4/((b*x^2 + a)^(5/2)*b) + 2 0*a*x^2/((b*x^2 + a)^(5/2)*b^2) + 8*a^2/((b*x^2 + a)^(5/2)*b^3))/b^4 - 33/ 40*C*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*B*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*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 + 143/16*D*a^3*x*(3*x^2/((b*x^2 + a)^(3/2)*b) + 2*a/((b*x^2 + a)^(3/2)*b^2))/b^5 - 33/8*C*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*B*a*x*(3*x...
Time = 0.31 (sec) , antiderivative size = 342, normalized size of antiderivative = 0.90 \[ \int \frac {x^8 \left (A+B x^2+C x^4+D x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {{\left ({\left ({\left ({\left (35 \, {\left (2 \, {\left (\frac {4 \, D x^{2}}{b} - \frac {13 \, D a^{4} b^{11} - 6 \, C a^{3} b^{12}}{a^{3} b^{13}}\right )} x^{2} + \frac {143 \, D a^{5} b^{10} - 66 \, C a^{4} b^{11} + 24 \, B a^{3} b^{12}}{a^{3} b^{13}}\right )} x^{2} + \frac {176 \, {\left (429 \, D a^{6} b^{9} - 198 \, C a^{5} b^{10} + 72 \, B a^{4} b^{11} - 16 \, A a^{3} b^{12}\right )}}{a^{3} b^{13}}\right )} x^{2} + \frac {406 \, {\left (429 \, D a^{7} b^{8} - 198 \, C a^{6} b^{9} + 72 \, B a^{5} b^{10} - 16 \, A a^{4} b^{11}\right )}}{a^{3} b^{13}}\right )} x^{2} + \frac {350 \, {\left (429 \, D a^{8} b^{7} - 198 \, C a^{7} b^{8} + 72 \, B a^{6} b^{9} - 16 \, A a^{5} b^{10}\right )}}{a^{3} b^{13}}\right )} x^{2} + \frac {105 \, {\left (429 \, D a^{9} b^{6} - 198 \, C a^{8} b^{7} + 72 \, B a^{7} b^{8} - 16 \, A a^{6} b^{9}\right )}}{a^{3} b^{13}}\right )} x}{1680 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} + \frac {{\left (429 \, D a^{3} - 198 \, C a^{2} b + 72 \, B a b^{2} - 16 \, A b^{3}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{16 \, b^{\frac {15}{2}}} \]
1/1680*((((35*(2*(4*D*x^2/b - (13*D*a^4*b^11 - 6*C*a^3*b^12)/(a^3*b^13))*x ^2 + (143*D*a^5*b^10 - 66*C*a^4*b^11 + 24*B*a^3*b^12)/(a^3*b^13))*x^2 + 17 6*(429*D*a^6*b^9 - 198*C*a^5*b^10 + 72*B*a^4*b^11 - 16*A*a^3*b^12)/(a^3*b^ 13))*x^2 + 406*(429*D*a^7*b^8 - 198*C*a^6*b^9 + 72*B*a^5*b^10 - 16*A*a^4*b ^11)/(a^3*b^13))*x^2 + 350*(429*D*a^8*b^7 - 198*C*a^7*b^8 + 72*B*a^6*b^9 - 16*A*a^5*b^10)/(a^3*b^13))*x^2 + 105*(429*D*a^9*b^6 - 198*C*a^8*b^7 + 72* B*a^7*b^8 - 16*A*a^6*b^9)/(a^3*b^13))*x/(b*x^2 + a)^(7/2) + 1/16*(429*D*a^ 3 - 198*C*a^2*b + 72*B*a*b^2 - 16*A*b^3)*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(15/2)
Timed out. \[ \int \frac {x^8 \left (A+B x^2+C x^4+D x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\int \frac {x^8\,\left (A+B\,x^2+C\,x^4+x^6\,D\right )}{{\left (b\,x^2+a\right )}^{9/2}} \,d x \]