Integrand size = 15, antiderivative size = 165 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{13}} \, dx=-\frac {a^4 \sqrt {a+b x^2}}{12 x^{12}}-\frac {49 a^3 b \sqrt {a+b x^2}}{120 x^{10}}-\frac {253 a^2 b^2 \sqrt {a+b x^2}}{320 x^8}-\frac {1429 a b^3 \sqrt {a+b x^2}}{1920 x^6}-\frac {491 b^4 \sqrt {a+b x^2}}{1536 x^4}-\frac {21 b^5 \sqrt {a+b x^2}}{1024 a x^2}+\frac {21 b^6 \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{1024 a^{3/2}} \] Output:
-1/12*a^4*(b*x^2+a)^(1/2)/x^12-49/120*a^3*b*(b*x^2+a)^(1/2)/x^10-253/320*a ^2*b^2*(b*x^2+a)^(1/2)/x^8-1429/1920*a*b^3*(b*x^2+a)^(1/2)/x^6-491/1536*b^ 4*(b*x^2+a)^(1/2)/x^4-21/1024*b^5*(b*x^2+a)^(1/2)/a/x^2+21/1024*b^6*arctan h((b*x^2+a)^(1/2)/a^(1/2))/a^(3/2)
Time = 0.14 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.64 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{13}} \, dx=\frac {\sqrt {a+b x^2} \left (-1280 a^5-6272 a^4 b x^2-12144 a^3 b^2 x^4-11432 a^2 b^3 x^6-4910 a b^4 x^8-315 b^5 x^{10}\right )}{15360 a x^{12}}+\frac {21 b^6 \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{1024 a^{3/2}} \] Input:
Integrate[(a + b*x^2)^(9/2)/x^13,x]
Output:
(Sqrt[a + b*x^2]*(-1280*a^5 - 6272*a^4*b*x^2 - 12144*a^3*b^2*x^4 - 11432*a ^2*b^3*x^6 - 4910*a*b^4*x^8 - 315*b^5*x^10))/(15360*a*x^12) + (21*b^6*ArcT anh[Sqrt[a + b*x^2]/Sqrt[a]])/(1024*a^(3/2))
Time = 0.21 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.02, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {243, 51, 51, 51, 51, 51, 52, 73, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{9/2}}{x^{13}} \, dx\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {1}{2} \int \frac {\left (b x^2+a\right )^{9/2}}{x^{14}}dx^2\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {1}{2} \left (\frac {3}{4} b \int \frac {\left (b x^2+a\right )^{7/2}}{x^{12}}dx^2-\frac {\left (a+b x^2\right )^{9/2}}{6 x^{12}}\right )\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {1}{2} \left (\frac {3}{4} b \left (\frac {7}{10} b \int \frac {\left (b x^2+a\right )^{5/2}}{x^{10}}dx^2-\frac {\left (a+b x^2\right )^{7/2}}{5 x^{10}}\right )-\frac {\left (a+b x^2\right )^{9/2}}{6 x^{12}}\right )\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {1}{2} \left (\frac {3}{4} b \left (\frac {7}{10} b \left (\frac {5}{8} b \int \frac {\left (b x^2+a\right )^{3/2}}{x^8}dx^2-\frac {\left (a+b x^2\right )^{5/2}}{4 x^8}\right )-\frac {\left (a+b x^2\right )^{7/2}}{5 x^{10}}\right )-\frac {\left (a+b x^2\right )^{9/2}}{6 x^{12}}\right )\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {1}{2} \left (\frac {3}{4} b \left (\frac {7}{10} b \left (\frac {5}{8} b \left (\frac {1}{2} b \int \frac {\sqrt {b x^2+a}}{x^6}dx^2-\frac {\left (a+b x^2\right )^{3/2}}{3 x^6}\right )-\frac {\left (a+b x^2\right )^{5/2}}{4 x^8}\right )-\frac {\left (a+b x^2\right )^{7/2}}{5 x^{10}}\right )-\frac {\left (a+b x^2\right )^{9/2}}{6 x^{12}}\right )\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {1}{2} \left (\frac {3}{4} b \left (\frac {7}{10} b \left (\frac {5}{8} b \left (\frac {1}{2} b \left (\frac {1}{4} b \int \frac {1}{x^4 \sqrt {b x^2+a}}dx^2-\frac {\sqrt {a+b x^2}}{2 x^4}\right )-\frac {\left (a+b x^2\right )^{3/2}}{3 x^6}\right )-\frac {\left (a+b x^2\right )^{5/2}}{4 x^8}\right )-\frac {\left (a+b x^2\right )^{7/2}}{5 x^{10}}\right )-\frac {\left (a+b x^2\right )^{9/2}}{6 x^{12}}\right )\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{2} \left (\frac {3}{4} b \left (\frac {7}{10} b \left (\frac {5}{8} b \left (\frac {1}{2} b \left (\frac {1}{4} b \left (-\frac {b \int \frac {1}{x^2 \sqrt {b x^2+a}}dx^2}{2 a}-\frac {\sqrt {a+b x^2}}{a x^2}\right )-\frac {\sqrt {a+b x^2}}{2 x^4}\right )-\frac {\left (a+b x^2\right )^{3/2}}{3 x^6}\right )-\frac {\left (a+b x^2\right )^{5/2}}{4 x^8}\right )-\frac {\left (a+b x^2\right )^{7/2}}{5 x^{10}}\right )-\frac {\left (a+b x^2\right )^{9/2}}{6 x^{12}}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{2} \left (\frac {3}{4} b \left (\frac {7}{10} b \left (\frac {5}{8} b \left (\frac {1}{2} b \left (\frac {1}{4} b \left (-\frac {\int \frac {1}{\frac {x^4}{b}-\frac {a}{b}}d\sqrt {b x^2+a}}{a}-\frac {\sqrt {a+b x^2}}{a x^2}\right )-\frac {\sqrt {a+b x^2}}{2 x^4}\right )-\frac {\left (a+b x^2\right )^{3/2}}{3 x^6}\right )-\frac {\left (a+b x^2\right )^{5/2}}{4 x^8}\right )-\frac {\left (a+b x^2\right )^{7/2}}{5 x^{10}}\right )-\frac {\left (a+b x^2\right )^{9/2}}{6 x^{12}}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{2} \left (\frac {3}{4} b \left (\frac {7}{10} b \left (\frac {5}{8} b \left (\frac {1}{2} b \left (\frac {1}{4} b \left (\frac {b \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {\sqrt {a+b x^2}}{a x^2}\right )-\frac {\sqrt {a+b x^2}}{2 x^4}\right )-\frac {\left (a+b x^2\right )^{3/2}}{3 x^6}\right )-\frac {\left (a+b x^2\right )^{5/2}}{4 x^8}\right )-\frac {\left (a+b x^2\right )^{7/2}}{5 x^{10}}\right )-\frac {\left (a+b x^2\right )^{9/2}}{6 x^{12}}\right )\) |
Input:
Int[(a + b*x^2)^(9/2)/x^13,x]
Output:
(-1/6*(a + b*x^2)^(9/2)/x^12 + (3*b*(-1/5*(a + b*x^2)^(7/2)/x^10 + (7*b*(- 1/4*(a + b*x^2)^(5/2)/x^8 + (5*b*(-1/3*(a + b*x^2)^(3/2)/x^6 + (b*(-1/2*Sq rt[a + b*x^2]/x^4 + (b*(-(Sqrt[a + b*x^2]/(a*x^2)) + (b*ArcTanh[Sqrt[a + b *x^2]/Sqrt[a]])/a^(3/2)))/4))/2))/8))/10))/4)/2
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Time = 0.59 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.61
| method | result | size |
| pseudoelliptic | \(-\frac {49 \left (-\frac {45 \,\operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{\sqrt {a}}\right ) b^{6} x^{12}}{896}+\sqrt {b \,x^{2}+a}\, \left (\frac {45 \sqrt {a}\, b^{5} x^{10}}{896}+\frac {2455 a^{\frac {3}{2}} b^{4} x^{8}}{3136}+\frac {1429 a^{\frac {5}{2}} b^{3} x^{6}}{784}+\frac {759 a^{\frac {7}{2}} b^{2} x^{4}}{392}+a^{\frac {9}{2}} b \,x^{2}+\frac {10 a^{\frac {11}{2}}}{49}\right )\right )}{120 a^{\frac {3}{2}} x^{12}}\) | \(100\) |
| risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (315 b^{5} x^{10}+4910 a \,b^{4} x^{8}+11432 a^{2} b^{3} x^{6}+12144 a^{3} b^{2} x^{4}+6272 a^{4} b \,x^{2}+1280 a^{5}\right )}{15360 x^{12} a}+\frac {21 b^{6} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{1024 a^{\frac {3}{2}}}\) | \(104\) |
| default | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{12 a \,x^{12}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{10 a \,x^{10}}+\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{8 a \,x^{8}}+\frac {3 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{6 a \,x^{6}}+\frac {5 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{4 a \,x^{4}}+\frac {7 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{2 a \,x^{2}}+\frac {9 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {9}{2}}}{9}+a \left (\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{7}+a \left (\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{5}+a \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )\right )\right )\right )}{2 a}\right )}{4 a}\right )}{6 a}\right )}{8 a}\right )}{10 a}\right )}{12 a}\) | \(239\) |
Input:
int((b*x^2+a)^(9/2)/x^13,x,method=_RETURNVERBOSE)
Output:
-49/120*(-45/896*arctanh((b*x^2+a)^(1/2)/a^(1/2))*b^6*x^12+(b*x^2+a)^(1/2) *(45/896*a^(1/2)*b^5*x^10+2455/3136*a^(3/2)*b^4*x^8+1429/784*a^(5/2)*b^3*x ^6+759/392*a^(7/2)*b^2*x^4+a^(9/2)*b*x^2+10/49*a^(11/2)))/a^(3/2)/x^12
Time = 0.10 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.37 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{13}} \, dx=\left [\frac {315 \, \sqrt {a} b^{6} x^{12} \log \left (-\frac {b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (315 \, a b^{5} x^{10} + 4910 \, a^{2} b^{4} x^{8} + 11432 \, a^{3} b^{3} x^{6} + 12144 \, a^{4} b^{2} x^{4} + 6272 \, a^{5} b x^{2} + 1280 \, a^{6}\right )} \sqrt {b x^{2} + a}}{30720 \, a^{2} x^{12}}, -\frac {315 \, \sqrt {-a} b^{6} x^{12} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) + {\left (315 \, a b^{5} x^{10} + 4910 \, a^{2} b^{4} x^{8} + 11432 \, a^{3} b^{3} x^{6} + 12144 \, a^{4} b^{2} x^{4} + 6272 \, a^{5} b x^{2} + 1280 \, a^{6}\right )} \sqrt {b x^{2} + a}}{15360 \, a^{2} x^{12}}\right ] \] Input:
integrate((b*x^2+a)^(9/2)/x^13,x, algorithm="fricas")
Output:
[1/30720*(315*sqrt(a)*b^6*x^12*log(-(b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(a) + 2 *a)/x^2) - 2*(315*a*b^5*x^10 + 4910*a^2*b^4*x^8 + 11432*a^3*b^3*x^6 + 1214 4*a^4*b^2*x^4 + 6272*a^5*b*x^2 + 1280*a^6)*sqrt(b*x^2 + a))/(a^2*x^12), -1 /15360*(315*sqrt(-a)*b^6*x^12*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) + (315*a* b^5*x^10 + 4910*a^2*b^4*x^8 + 11432*a^3*b^3*x^6 + 12144*a^4*b^2*x^4 + 6272 *a^5*b*x^2 + 1280*a^6)*sqrt(b*x^2 + a))/(a^2*x^12)]
Time = 48.64 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.24 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{13}} \, dx=- \frac {a^{5}}{12 \sqrt {b} x^{13} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {59 a^{4} \sqrt {b}}{120 x^{11} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {1151 a^{3} b^{\frac {3}{2}}}{960 x^{9} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {2947 a^{2} b^{\frac {5}{2}}}{1920 x^{7} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {8171 a b^{\frac {7}{2}}}{7680 x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {1045 b^{\frac {9}{2}}}{3072 x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {21 b^{\frac {11}{2}}}{1024 a x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {21 b^{6} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{1024 a^{\frac {3}{2}}} \] Input:
integrate((b*x**2+a)**(9/2)/x**13,x)
Output:
-a**5/(12*sqrt(b)*x**13*sqrt(a/(b*x**2) + 1)) - 59*a**4*sqrt(b)/(120*x**11 *sqrt(a/(b*x**2) + 1)) - 1151*a**3*b**(3/2)/(960*x**9*sqrt(a/(b*x**2) + 1) ) - 2947*a**2*b**(5/2)/(1920*x**7*sqrt(a/(b*x**2) + 1)) - 8171*a*b**(7/2)/ (7680*x**5*sqrt(a/(b*x**2) + 1)) - 1045*b**(9/2)/(3072*x**3*sqrt(a/(b*x**2 ) + 1)) - 21*b**(11/2)/(1024*a*x*sqrt(a/(b*x**2) + 1)) + 21*b**6*asinh(sqr t(a)/(sqrt(b)*x))/(1024*a**(3/2))
Time = 0.04 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.34 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{13}} \, dx=\frac {21 \, b^{6} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{1024 \, a^{\frac {3}{2}}} - \frac {7 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} b^{6}}{3072 \, a^{6}} - \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{6}}{1024 \, a^{5}} - \frac {21 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{6}}{5120 \, a^{4}} - \frac {7 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{6}}{1024 \, a^{3}} - \frac {21 \, \sqrt {b x^{2} + a} b^{6}}{1024 \, a^{2}} + \frac {7 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{5}}{3072 \, a^{6} x^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{4}}{1536 \, a^{5} x^{4}} + \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{3}}{1920 \, a^{4} x^{6}} + \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{2}}{960 \, a^{3} x^{8}} + \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}} b}{120 \, a^{2} x^{10}} - \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}}}{12 \, a x^{12}} \] Input:
integrate((b*x^2+a)^(9/2)/x^13,x, algorithm="maxima")
Output:
21/1024*b^6*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(3/2) - 7/3072*(b*x^2 + a)^(9/ 2)*b^6/a^6 - 3/1024*(b*x^2 + a)^(7/2)*b^6/a^5 - 21/5120*(b*x^2 + a)^(5/2)* b^6/a^4 - 7/1024*(b*x^2 + a)^(3/2)*b^6/a^3 - 21/1024*sqrt(b*x^2 + a)*b^6/a ^2 + 7/3072*(b*x^2 + a)^(11/2)*b^5/(a^6*x^2) + 1/1536*(b*x^2 + a)^(11/2)*b ^4/(a^5*x^4) + 1/1920*(b*x^2 + a)^(11/2)*b^3/(a^4*x^6) + 1/960*(b*x^2 + a) ^(11/2)*b^2/(a^3*x^8) + 1/120*(b*x^2 + a)^(11/2)*b/(a^2*x^10) - 1/12*(b*x^ 2 + a)^(11/2)/(a*x^12)
Time = 0.12 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{13}} \, dx=-\frac {\frac {315 \, b^{7} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} + \frac {315 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{7} + 3335 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} a b^{7} - 5058 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2} b^{7} + 4158 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{3} b^{7} - 1785 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{4} b^{7} + 315 \, \sqrt {b x^{2} + a} a^{5} b^{7}}{a b^{6} x^{12}}}{15360 \, b} \] Input:
integrate((b*x^2+a)^(9/2)/x^13,x, algorithm="giac")
Output:
-1/15360*(315*b^7*arctan(sqrt(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a) + (315*(b* x^2 + a)^(11/2)*b^7 + 3335*(b*x^2 + a)^(9/2)*a*b^7 - 5058*(b*x^2 + a)^(7/2 )*a^2*b^7 + 4158*(b*x^2 + a)^(5/2)*a^3*b^7 - 1785*(b*x^2 + a)^(3/2)*a^4*b^ 7 + 315*sqrt(b*x^2 + a)*a^5*b^7)/(a*b^6*x^12))/b
Time = 2.66 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{13}} \, dx=\frac {843\,a\,{\left (b\,x^2+a\right )}^{7/2}}{2560\,x^{12}}-\frac {667\,{\left (b\,x^2+a\right )}^{9/2}}{3072\,x^{12}}-\frac {21\,a^4\,\sqrt {b\,x^2+a}}{1024\,x^{12}}+\frac {119\,a^3\,{\left (b\,x^2+a\right )}^{3/2}}{1024\,x^{12}}-\frac {693\,a^2\,{\left (b\,x^2+a\right )}^{5/2}}{2560\,x^{12}}-\frac {21\,{\left (b\,x^2+a\right )}^{11/2}}{1024\,a\,x^{12}}-\frac {b^6\,\mathrm {atan}\left (\frac {\sqrt {b\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,21{}\mathrm {i}}{1024\,a^{3/2}} \] Input:
int((a + b*x^2)^(9/2)/x^13,x)
Output:
(843*a*(a + b*x^2)^(7/2))/(2560*x^12) - (b^6*atan(((a + b*x^2)^(1/2)*1i)/a ^(1/2))*21i)/(1024*a^(3/2)) - (667*(a + b*x^2)^(9/2))/(3072*x^12) - (21*a^ 4*(a + b*x^2)^(1/2))/(1024*x^12) + (119*a^3*(a + b*x^2)^(3/2))/(1024*x^12) - (693*a^2*(a + b*x^2)^(5/2))/(2560*x^12) - (21*(a + b*x^2)^(11/2))/(1024 *a*x^12)
Time = 0.37 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.07 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{13}} \, dx=\frac {-1280 \sqrt {b \,x^{2}+a}\, a^{6}-6272 \sqrt {b \,x^{2}+a}\, a^{5} b \,x^{2}-12144 \sqrt {b \,x^{2}+a}\, a^{4} b^{2} x^{4}-11432 \sqrt {b \,x^{2}+a}\, a^{3} b^{3} x^{6}-4910 \sqrt {b \,x^{2}+a}\, a^{2} b^{4} x^{8}-315 \sqrt {b \,x^{2}+a}\, a \,b^{5} x^{10}-315 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{6} x^{12}+315 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{6} x^{12}}{15360 a^{2} x^{12}} \] Input:
int((b*x^2+a)^(9/2)/x^13,x)
Output:
( - 1280*sqrt(a + b*x**2)*a**6 - 6272*sqrt(a + b*x**2)*a**5*b*x**2 - 12144 *sqrt(a + b*x**2)*a**4*b**2*x**4 - 11432*sqrt(a + b*x**2)*a**3*b**3*x**6 - 4910*sqrt(a + b*x**2)*a**2*b**4*x**8 - 315*sqrt(a + b*x**2)*a*b**5*x**10 - 315*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*b**6*x **12 + 315*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*b **6*x**12)/(15360*a**2*x**12)