Integrand size = 15, antiderivative size = 189 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{15}} \, dx=-\frac {a^4 \sqrt {a+b x^2}}{14 x^{14}}-\frac {19 a^3 b \sqrt {a+b x^2}}{56 x^{12}}-\frac {351 a^2 b^2 \sqrt {a+b x^2}}{560 x^{10}}-\frac {2441 a b^3 \sqrt {a+b x^2}}{4480 x^8}-\frac {253 b^4 \sqrt {a+b x^2}}{1280 x^6}-\frac {3 b^5 \sqrt {a+b x^2}}{1024 a x^4}+\frac {9 b^6 \sqrt {a+b x^2}}{2048 a^2 x^2}-\frac {9 b^7 \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2048 a^{5/2}} \] Output:
-1/14*a^4*(b*x^2+a)^(1/2)/x^14-19/56*a^3*b*(b*x^2+a)^(1/2)/x^12-351/560*a^ 2*b^2*(b*x^2+a)^(1/2)/x^10-2441/4480*a*b^3*(b*x^2+a)^(1/2)/x^8-253/1280*b^ 4*(b*x^2+a)^(1/2)/x^6-3/1024*b^5*(b*x^2+a)^(1/2)/a/x^4+9/2048*b^6*(b*x^2+a )^(1/2)/a^2/x^2-9/2048*b^7*arctanh((b*x^2+a)^(1/2)/a^(1/2))/a^(5/2)
Time = 0.17 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.62 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{15}} \, dx=\frac {\sqrt {a+b x^2} \left (-5120 a^6-24320 a^5 b x^2-44928 a^4 b^2 x^4-39056 a^3 b^3 x^6-14168 a^2 b^4 x^8-210 a b^5 x^{10}+315 b^6 x^{12}\right )}{71680 a^2 x^{14}}-\frac {9 b^7 \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2048 a^{5/2}} \] Input:
Integrate[(a + b*x^2)^(9/2)/x^15,x]
Output:
(Sqrt[a + b*x^2]*(-5120*a^6 - 24320*a^5*b*x^2 - 44928*a^4*b^2*x^4 - 39056* a^3*b^3*x^6 - 14168*a^2*b^4*x^8 - 210*a*b^5*x^10 + 315*b^6*x^12))/(71680*a ^2*x^14) - (9*b^7*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(2048*a^(5/2))
Time = 0.23 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.05, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {243, 51, 51, 51, 51, 51, 52, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{9/2}}{x^{15}} \, dx\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {1}{2} \int \frac {\left (b x^2+a\right )^{9/2}}{x^{16}}dx^2\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {1}{2} \left (\frac {9}{14} b \int \frac {\left (b x^2+a\right )^{7/2}}{x^{14}}dx^2-\frac {\left (a+b x^2\right )^{9/2}}{7 x^{14}}\right )\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {1}{2} \left (\frac {9}{14} b \left (\frac {7}{12} b \int \frac {\left (b x^2+a\right )^{5/2}}{x^{12}}dx^2-\frac {\left (a+b x^2\right )^{7/2}}{6 x^{12}}\right )-\frac {\left (a+b x^2\right )^{9/2}}{7 x^{14}}\right )\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {1}{2} \left (\frac {9}{14} b \left (\frac {7}{12} b \left (\frac {1}{2} b \int \frac {\left (b x^2+a\right )^{3/2}}{x^{10}}dx^2-\frac {\left (a+b x^2\right )^{5/2}}{5 x^{10}}\right )-\frac {\left (a+b x^2\right )^{7/2}}{6 x^{12}}\right )-\frac {\left (a+b x^2\right )^{9/2}}{7 x^{14}}\right )\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {1}{2} \left (\frac {9}{14} b \left (\frac {7}{12} b \left (\frac {1}{2} b \left (\frac {3}{8} b \int \frac {\sqrt {b x^2+a}}{x^8}dx^2-\frac {\left (a+b x^2\right )^{3/2}}{4 x^8}\right )-\frac {\left (a+b x^2\right )^{5/2}}{5 x^{10}}\right )-\frac {\left (a+b x^2\right )^{7/2}}{6 x^{12}}\right )-\frac {\left (a+b x^2\right )^{9/2}}{7 x^{14}}\right )\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {1}{2} \left (\frac {9}{14} b \left (\frac {7}{12} b \left (\frac {1}{2} b \left (\frac {3}{8} b \left (\frac {1}{6} b \int \frac {1}{x^6 \sqrt {b x^2+a}}dx^2-\frac {\sqrt {a+b x^2}}{3 x^6}\right )-\frac {\left (a+b x^2\right )^{3/2}}{4 x^8}\right )-\frac {\left (a+b x^2\right )^{5/2}}{5 x^{10}}\right )-\frac {\left (a+b x^2\right )^{7/2}}{6 x^{12}}\right )-\frac {\left (a+b x^2\right )^{9/2}}{7 x^{14}}\right )\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{2} \left (\frac {9}{14} b \left (\frac {7}{12} b \left (\frac {1}{2} b \left (\frac {3}{8} b \left (\frac {1}{6} b \left (-\frac {3 b \int \frac {1}{x^4 \sqrt {b x^2+a}}dx^2}{4 a}-\frac {\sqrt {a+b x^2}}{2 a x^4}\right )-\frac {\sqrt {a+b x^2}}{3 x^6}\right )-\frac {\left (a+b x^2\right )^{3/2}}{4 x^8}\right )-\frac {\left (a+b x^2\right )^{5/2}}{5 x^{10}}\right )-\frac {\left (a+b x^2\right )^{7/2}}{6 x^{12}}\right )-\frac {\left (a+b x^2\right )^{9/2}}{7 x^{14}}\right )\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{2} \left (\frac {9}{14} b \left (\frac {7}{12} b \left (\frac {1}{2} b \left (\frac {3}{8} b \left (\frac {1}{6} b \left (-\frac {3 b \left (-\frac {b \int \frac {1}{x^2 \sqrt {b x^2+a}}dx^2}{2 a}-\frac {\sqrt {a+b x^2}}{a x^2}\right )}{4 a}-\frac {\sqrt {a+b x^2}}{2 a x^4}\right )-\frac {\sqrt {a+b x^2}}{3 x^6}\right )-\frac {\left (a+b x^2\right )^{3/2}}{4 x^8}\right )-\frac {\left (a+b x^2\right )^{5/2}}{5 x^{10}}\right )-\frac {\left (a+b x^2\right )^{7/2}}{6 x^{12}}\right )-\frac {\left (a+b x^2\right )^{9/2}}{7 x^{14}}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{2} \left (\frac {9}{14} b \left (\frac {7}{12} b \left (\frac {1}{2} b \left (\frac {3}{8} b \left (\frac {1}{6} b \left (-\frac {3 b \left (-\frac {\int \frac {1}{\frac {x^4}{b}-\frac {a}{b}}d\sqrt {b x^2+a}}{a}-\frac {\sqrt {a+b x^2}}{a x^2}\right )}{4 a}-\frac {\sqrt {a+b x^2}}{2 a x^4}\right )-\frac {\sqrt {a+b x^2}}{3 x^6}\right )-\frac {\left (a+b x^2\right )^{3/2}}{4 x^8}\right )-\frac {\left (a+b x^2\right )^{5/2}}{5 x^{10}}\right )-\frac {\left (a+b x^2\right )^{7/2}}{6 x^{12}}\right )-\frac {\left (a+b x^2\right )^{9/2}}{7 x^{14}}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{2} \left (\frac {9}{14} b \left (\frac {7}{12} b \left (\frac {1}{2} b \left (\frac {3}{8} b \left (\frac {1}{6} b \left (-\frac {3 b \left (\frac {b \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {\sqrt {a+b x^2}}{a x^2}\right )}{4 a}-\frac {\sqrt {a+b x^2}}{2 a x^4}\right )-\frac {\sqrt {a+b x^2}}{3 x^6}\right )-\frac {\left (a+b x^2\right )^{3/2}}{4 x^8}\right )-\frac {\left (a+b x^2\right )^{5/2}}{5 x^{10}}\right )-\frac {\left (a+b x^2\right )^{7/2}}{6 x^{12}}\right )-\frac {\left (a+b x^2\right )^{9/2}}{7 x^{14}}\right )\) |
Input:
Int[(a + b*x^2)^(9/2)/x^15,x]
Output:
(-1/7*(a + b*x^2)^(9/2)/x^14 + (9*b*(-1/6*(a + b*x^2)^(7/2)/x^12 + (7*b*(- 1/5*(a + b*x^2)^(5/2)/x^10 + (b*(-1/4*(a + b*x^2)^(3/2)/x^8 + (3*b*(-1/3*S qrt[a + b*x^2]/x^6 + (b*(-1/2*Sqrt[a + b*x^2]/(a*x^4) - (3*b*(-(Sqrt[a + b *x^2]/(a*x^2)) + (b*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/a^(3/2)))/(4*a)))/6) )/8))/2))/12))/14)/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 = 0.95 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.59
| method | result | size |
| pseudoelliptic | \(-\frac {351 \left (\frac {35 \,\operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{\sqrt {a}}\right ) x^{14} b^{7}}{4992}+\sqrt {b \,x^{2}+a}\, \left (-\frac {35 \sqrt {a}\, b^{6} x^{12}}{4992}+\frac {35 a^{\frac {3}{2}} b^{5} x^{10}}{7488}+\frac {1771 a^{\frac {5}{2}} b^{4} x^{8}}{5616}+\frac {2441 a^{\frac {7}{2}} b^{3} x^{6}}{2808}+a^{\frac {9}{2}} b^{2} x^{4}+\frac {190 a^{\frac {11}{2}} b \,x^{2}}{351}+\frac {40 a^{\frac {13}{2}}}{351}\right )\right )}{560 a^{\frac {5}{2}} x^{14}}\) | \(111\) |
| risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (-315 b^{6} x^{12}+210 a \,b^{5} x^{10}+14168 a^{2} b^{4} x^{8}+39056 a^{3} b^{3} x^{6}+44928 a^{4} b^{2} x^{4}+24320 a^{5} b \,x^{2}+5120 a^{6}\right )}{71680 x^{14} a^{2}}-\frac {9 b^{7} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{2048 a^{\frac {5}{2}}}\) | \(115\) |
| default | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{14 a \,x^{14}}-\frac {3 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{12 a \,x^{12}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{10 a \,x^{10}}+\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{8 a \,x^{8}}+\frac {3 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{6 a \,x^{6}}+\frac {5 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{4 a \,x^{4}}+\frac {7 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{2 a \,x^{2}}+\frac {9 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {9}{2}}}{9}+a \left (\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{7}+a \left (\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{5}+a \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )\right )\right )\right )}{2 a}\right )}{4 a}\right )}{6 a}\right )}{8 a}\right )}{10 a}\right )}{12 a}\right )}{14 a}\) | \(263\) |
Input:
int((b*x^2+a)^(9/2)/x^15,x,method=_RETURNVERBOSE)
Output:
-351/560/a^(5/2)*(35/4992*arctanh((b*x^2+a)^(1/2)/a^(1/2))*x^14*b^7+(b*x^2 +a)^(1/2)*(-35/4992*a^(1/2)*b^6*x^12+35/7488*a^(3/2)*b^5*x^10+1771/5616*a^ (5/2)*b^4*x^8+2441/2808*a^(7/2)*b^3*x^6+a^(9/2)*b^2*x^4+190/351*a^(11/2)*b *x^2+40/351*a^(13/2)))/x^14
Time = 0.13 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.31 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{15}} \, dx=\left [\frac {315 \, \sqrt {a} b^{7} x^{14} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (315 \, a b^{6} x^{12} - 210 \, a^{2} b^{5} x^{10} - 14168 \, a^{3} b^{4} x^{8} - 39056 \, a^{4} b^{3} x^{6} - 44928 \, a^{5} b^{2} x^{4} - 24320 \, a^{6} b x^{2} - 5120 \, a^{7}\right )} \sqrt {b x^{2} + a}}{143360 \, a^{3} x^{14}}, \frac {315 \, \sqrt {-a} b^{7} x^{14} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) + {\left (315 \, a b^{6} x^{12} - 210 \, a^{2} b^{5} x^{10} - 14168 \, a^{3} b^{4} x^{8} - 39056 \, a^{4} b^{3} x^{6} - 44928 \, a^{5} b^{2} x^{4} - 24320 \, a^{6} b x^{2} - 5120 \, a^{7}\right )} \sqrt {b x^{2} + a}}{71680 \, a^{3} x^{14}}\right ] \] Input:
integrate((b*x^2+a)^(9/2)/x^15,x, algorithm="fricas")
Output:
[1/143360*(315*sqrt(a)*b^7*x^14*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(315*a*b^6*x^12 - 210*a^2*b^5*x^10 - 14168*a^3*b^4*x^8 - 390 56*a^4*b^3*x^6 - 44928*a^5*b^2*x^4 - 24320*a^6*b*x^2 - 5120*a^7)*sqrt(b*x^ 2 + a))/(a^3*x^14), 1/71680*(315*sqrt(-a)*b^7*x^14*arctan(sqrt(b*x^2 + a)* sqrt(-a)/a) + (315*a*b^6*x^12 - 210*a^2*b^5*x^10 - 14168*a^3*b^4*x^8 - 390 56*a^4*b^3*x^6 - 44928*a^5*b^2*x^4 - 24320*a^6*b*x^2 - 5120*a^7)*sqrt(b*x^ 2 + a))/(a^3*x^14)]
Timed out. \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{15}} \, dx=\text {Timed out} \] Input:
integrate((b*x**2+a)**(9/2)/x**15,x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.28 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{15}} \, dx=-\frac {9 \, b^{7} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{2048 \, a^{\frac {5}{2}}} + \frac {{\left (b x^{2} + a\right )}^{\frac {9}{2}} b^{7}}{2048 \, a^{7}} + \frac {9 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{7}}{14336 \, a^{6}} + \frac {9 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{7}}{10240 \, a^{5}} + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{7}}{2048 \, a^{4}} + \frac {9 \, \sqrt {b x^{2} + a} b^{7}}{2048 \, a^{3}} - \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{6}}{2048 \, a^{7} x^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{5}}{7168 \, a^{6} x^{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{4}}{8960 \, a^{5} x^{6}} - \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{3}}{4480 \, a^{4} x^{8}} - \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{2}}{560 \, a^{3} x^{10}} + \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}} b}{56 \, a^{2} x^{12}} - \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}}}{14 \, a x^{14}} \] Input:
integrate((b*x^2+a)^(9/2)/x^15,x, algorithm="maxima")
Output:
-9/2048*b^7*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(5/2) + 1/2048*(b*x^2 + a)^(9/ 2)*b^7/a^7 + 9/14336*(b*x^2 + a)^(7/2)*b^7/a^6 + 9/10240*(b*x^2 + a)^(5/2) *b^7/a^5 + 3/2048*(b*x^2 + a)^(3/2)*b^7/a^4 + 9/2048*sqrt(b*x^2 + a)*b^7/a ^3 - 1/2048*(b*x^2 + a)^(11/2)*b^6/(a^7*x^2) - 1/7168*(b*x^2 + a)^(11/2)*b ^5/(a^6*x^4) - 1/8960*(b*x^2 + a)^(11/2)*b^4/(a^5*x^6) - 1/4480*(b*x^2 + a )^(11/2)*b^3/(a^4*x^8) - 1/560*(b*x^2 + a)^(11/2)*b^2/(a^3*x^10) + 1/56*(b *x^2 + a)^(11/2)*b/(a^2*x^12) - 1/14*(b*x^2 + a)^(11/2)/(a*x^14)
Time = 0.13 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.72 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{15}} \, dx=\frac {1}{71680} \, b^{7} {\left (\frac {315 \, \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} + \frac {315 \, {\left (b x^{2} + a\right )}^{\frac {13}{2}} - 2100 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} a - 8393 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} a^{2} + 9216 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{3} - 5943 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{4} + 2100 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{5} - 315 \, \sqrt {b x^{2} + a} a^{6}}{a^{2} b^{7} x^{14}}\right )} \] Input:
integrate((b*x^2+a)^(9/2)/x^15,x, algorithm="giac")
Output:
1/71680*b^7*(315*arctan(sqrt(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a^2) + (315*(b *x^2 + a)^(13/2) - 2100*(b*x^2 + a)^(11/2)*a - 8393*(b*x^2 + a)^(9/2)*a^2 + 9216*(b*x^2 + a)^(7/2)*a^3 - 5943*(b*x^2 + a)^(5/2)*a^4 + 2100*(b*x^2 + a)^(3/2)*a^5 - 315*sqrt(b*x^2 + a)*a^6)/(a^2*b^7*x^14))
Time = 3.26 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.74 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{15}} \, dx=\frac {9\,a\,{\left (b\,x^2+a\right )}^{7/2}}{70\,x^{14}}-\frac {1199\,{\left (b\,x^2+a\right )}^{9/2}}{10240\,x^{14}}-\frac {9\,a^4\,\sqrt {b\,x^2+a}}{2048\,x^{14}}+\frac {15\,a^3\,{\left (b\,x^2+a\right )}^{3/2}}{512\,x^{14}}-\frac {849\,a^2\,{\left (b\,x^2+a\right )}^{5/2}}{10240\,x^{14}}-\frac {15\,{\left (b\,x^2+a\right )}^{11/2}}{512\,a\,x^{14}}+\frac {9\,{\left (b\,x^2+a\right )}^{13/2}}{2048\,a^2\,x^{14}}+\frac {b^7\,\mathrm {atan}\left (\frac {\sqrt {b\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,9{}\mathrm {i}}{2048\,a^{5/2}} \] Input:
int((a + b*x^2)^(9/2)/x^15,x)
Output:
(b^7*atan(((a + b*x^2)^(1/2)*1i)/a^(1/2))*9i)/(2048*a^(5/2)) - (1199*(a + b*x^2)^(9/2))/(10240*x^14) + (9*a*(a + b*x^2)^(7/2))/(70*x^14) - (9*a^4*(a + b*x^2)^(1/2))/(2048*x^14) + (15*a^3*(a + b*x^2)^(3/2))/(512*x^14) - (84 9*a^2*(a + b*x^2)^(5/2))/(10240*x^14) - (15*(a + b*x^2)^(11/2))/(512*a*x^1 4) + (9*(a + b*x^2)^(13/2))/(2048*a^2*x^14)
Time = 0.36 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.04 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{15}} \, dx=\frac {-5120 \sqrt {b \,x^{2}+a}\, a^{7}-24320 \sqrt {b \,x^{2}+a}\, a^{6} b \,x^{2}-44928 \sqrt {b \,x^{2}+a}\, a^{5} b^{2} x^{4}-39056 \sqrt {b \,x^{2}+a}\, a^{4} b^{3} x^{6}-14168 \sqrt {b \,x^{2}+a}\, a^{3} b^{4} x^{8}-210 \sqrt {b \,x^{2}+a}\, a^{2} b^{5} x^{10}+315 \sqrt {b \,x^{2}+a}\, a \,b^{6} x^{12}+315 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{7} x^{14}-315 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{7} x^{14}}{71680 a^{3} x^{14}} \] Input:
int((b*x^2+a)^(9/2)/x^15,x)
Output:
( - 5120*sqrt(a + b*x**2)*a**7 - 24320*sqrt(a + b*x**2)*a**6*b*x**2 - 4492 8*sqrt(a + b*x**2)*a**5*b**2*x**4 - 39056*sqrt(a + b*x**2)*a**4*b**3*x**6 - 14168*sqrt(a + b*x**2)*a**3*b**4*x**8 - 210*sqrt(a + b*x**2)*a**2*b**5*x **10 + 315*sqrt(a + b*x**2)*a*b**6*x**12 + 315*sqrt(a)*log((sqrt(a + b*x** 2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*b**7*x**14 - 315*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*b**7*x**14)/(71680*a**3*x**14)