Integrand size = 15, antiderivative size = 141 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{11}} \, dx=-\frac {a^4 \sqrt {a+b x^2}}{10 x^{10}}-\frac {41 a^3 b \sqrt {a+b x^2}}{80 x^8}-\frac {171 a^2 b^2 \sqrt {a+b x^2}}{160 x^6}-\frac {149 a b^3 \sqrt {a+b x^2}}{128 x^4}-\frac {193 b^4 \sqrt {a+b x^2}}{256 x^2}-\frac {63 b^5 \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{256 \sqrt {a}} \] Output:
-1/10*a^4*(b*x^2+a)^(1/2)/x^10-41/80*a^3*b*(b*x^2+a)^(1/2)/x^8-171/160*a^2 *b^2*(b*x^2+a)^(1/2)/x^6-149/128*a*b^3*(b*x^2+a)^(1/2)/x^4-193/256*b^4*(b* x^2+a)^(1/2)/x^2-63/256*b^5*arctanh((b*x^2+a)^(1/2)/a^(1/2))/a^(1/2)
Time = 0.14 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.65 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{11}} \, dx=\frac {\sqrt {a+b x^2} \left (-128 a^4-656 a^3 b x^2-1368 a^2 b^2 x^4-1490 a b^3 x^6-965 b^4 x^8\right )}{1280 x^{10}}-\frac {63 b^5 \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{256 \sqrt {a}} \] Input:
Integrate[(a + b*x^2)^(9/2)/x^11,x]
Output:
(Sqrt[a + b*x^2]*(-128*a^4 - 656*a^3*b*x^2 - 1368*a^2*b^2*x^4 - 1490*a*b^3 *x^6 - 965*b^4*x^8))/(1280*x^10) - (63*b^5*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a] ])/(256*Sqrt[a])
Time = 0.20 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.01, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {243, 51, 51, 51, 51, 51, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{9/2}}{x^{11}} \, dx\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {1}{2} \int \frac {\left (b x^2+a\right )^{9/2}}{x^{12}}dx^2\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {1}{2} \left (\frac {9}{10} b \int \frac {\left (b x^2+a\right )^{7/2}}{x^{10}}dx^2-\frac {\left (a+b x^2\right )^{9/2}}{5 x^{10}}\right )\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {1}{2} \left (\frac {9}{10} b \left (\frac {7}{8} b \int \frac {\left (b x^2+a\right )^{5/2}}{x^8}dx^2-\frac {\left (a+b x^2\right )^{7/2}}{4 x^8}\right )-\frac {\left (a+b x^2\right )^{9/2}}{5 x^{10}}\right )\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {1}{2} \left (\frac {9}{10} b \left (\frac {7}{8} b \left (\frac {5}{6} b \int \frac {\left (b x^2+a\right )^{3/2}}{x^6}dx^2-\frac {\left (a+b x^2\right )^{5/2}}{3 x^6}\right )-\frac {\left (a+b x^2\right )^{7/2}}{4 x^8}\right )-\frac {\left (a+b x^2\right )^{9/2}}{5 x^{10}}\right )\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {1}{2} \left (\frac {9}{10} b \left (\frac {7}{8} b \left (\frac {5}{6} b \left (\frac {3}{4} b \int \frac {\sqrt {b x^2+a}}{x^4}dx^2-\frac {\left (a+b x^2\right )^{3/2}}{2 x^4}\right )-\frac {\left (a+b x^2\right )^{5/2}}{3 x^6}\right )-\frac {\left (a+b x^2\right )^{7/2}}{4 x^8}\right )-\frac {\left (a+b x^2\right )^{9/2}}{5 x^{10}}\right )\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {1}{2} \left (\frac {9}{10} b \left (\frac {7}{8} b \left (\frac {5}{6} b \left (\frac {3}{4} b \left (\frac {1}{2} b \int \frac {1}{x^2 \sqrt {b x^2+a}}dx^2-\frac {\sqrt {a+b x^2}}{x^2}\right )-\frac {\left (a+b x^2\right )^{3/2}}{2 x^4}\right )-\frac {\left (a+b x^2\right )^{5/2}}{3 x^6}\right )-\frac {\left (a+b x^2\right )^{7/2}}{4 x^8}\right )-\frac {\left (a+b x^2\right )^{9/2}}{5 x^{10}}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{2} \left (\frac {9}{10} b \left (\frac {7}{8} b \left (\frac {5}{6} b \left (\frac {3}{4} b \left (\int \frac {1}{\frac {x^4}{b}-\frac {a}{b}}d\sqrt {b x^2+a}-\frac {\sqrt {a+b x^2}}{x^2}\right )-\frac {\left (a+b x^2\right )^{3/2}}{2 x^4}\right )-\frac {\left (a+b x^2\right )^{5/2}}{3 x^6}\right )-\frac {\left (a+b x^2\right )^{7/2}}{4 x^8}\right )-\frac {\left (a+b x^2\right )^{9/2}}{5 x^{10}}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{2} \left (\frac {9}{10} b \left (\frac {7}{8} b \left (\frac {5}{6} b \left (\frac {3}{4} b \left (-\frac {b \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {\sqrt {a+b x^2}}{x^2}\right )-\frac {\left (a+b x^2\right )^{3/2}}{2 x^4}\right )-\frac {\left (a+b x^2\right )^{5/2}}{3 x^6}\right )-\frac {\left (a+b x^2\right )^{7/2}}{4 x^8}\right )-\frac {\left (a+b x^2\right )^{9/2}}{5 x^{10}}\right )\) |
Input:
Int[(a + b*x^2)^(9/2)/x^11,x]
Output:
(-1/5*(a + b*x^2)^(9/2)/x^10 + (9*b*(-1/4*(a + b*x^2)^(7/2)/x^8 + (7*b*(-1 /3*(a + b*x^2)^(5/2)/x^6 + (5*b*(-1/2*(a + b*x^2)^(3/2)/x^4 + (3*b*(-(Sqrt [a + b*x^2]/x^2) - (b*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/Sqrt[a]))/4))/6))/ 8))/10)/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] :> 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 = 0.48 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.64
| method | result | size |
| risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (965 b^{4} x^{8}+1490 a \,b^{3} x^{6}+1368 a^{2} b^{2} x^{4}+656 a^{3} b \,x^{2}+128 a^{4}\right )}{1280 x^{10}}-\frac {63 b^{5} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{256 \sqrt {a}}\) | \(90\) |
| pseudoelliptic | \(\frac {-315 \,\operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{\sqrt {a}}\right ) b^{5} x^{10}-965 b^{4} x^{8} \sqrt {a}\, \sqrt {b \,x^{2}+a}-1490 a^{\frac {3}{2}} b^{3} x^{6} \sqrt {b \,x^{2}+a}-1368 a^{\frac {5}{2}} b^{2} x^{4} \sqrt {b \,x^{2}+a}-656 a^{\frac {7}{2}} b \,x^{2} \sqrt {b \,x^{2}+a}-128 a^{\frac {9}{2}} \sqrt {b \,x^{2}+a}}{1280 x^{10} \sqrt {a}}\) | \(124\) |
| default | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{10 a \,x^{10}}+\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{8 a \,x^{8}}+\frac {3 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{6 a \,x^{6}}+\frac {5 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{4 a \,x^{4}}+\frac {7 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{2 a \,x^{2}}+\frac {9 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {9}{2}}}{9}+a \left (\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{7}+a \left (\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{5}+a \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )\right )\right )\right )}{2 a}\right )}{4 a}\right )}{6 a}\right )}{8 a}\right )}{10 a}\) | \(215\) |
Input:
int((b*x^2+a)^(9/2)/x^11,x,method=_RETURNVERBOSE)
Output:
-1/1280*(b*x^2+a)^(1/2)*(965*b^4*x^8+1490*a*b^3*x^6+1368*a^2*b^2*x^4+656*a ^3*b*x^2+128*a^4)/x^10-63/256*b^5/a^(1/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2 ))/x)
Time = 0.09 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.45 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{11}} \, dx=\left [\frac {315 \, \sqrt {a} b^{5} x^{10} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (965 \, a b^{4} x^{8} + 1490 \, a^{2} b^{3} x^{6} + 1368 \, a^{3} b^{2} x^{4} + 656 \, a^{4} b x^{2} + 128 \, a^{5}\right )} \sqrt {b x^{2} + a}}{2560 \, a x^{10}}, \frac {315 \, \sqrt {-a} b^{5} x^{10} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) - {\left (965 \, a b^{4} x^{8} + 1490 \, a^{2} b^{3} x^{6} + 1368 \, a^{3} b^{2} x^{4} + 656 \, a^{4} b x^{2} + 128 \, a^{5}\right )} \sqrt {b x^{2} + a}}{1280 \, a x^{10}}\right ] \] Input:
integrate((b*x^2+a)^(9/2)/x^11,x, algorithm="fricas")
Output:
[1/2560*(315*sqrt(a)*b^5*x^10*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2* a)/x^2) - 2*(965*a*b^4*x^8 + 1490*a^2*b^3*x^6 + 1368*a^3*b^2*x^4 + 656*a^4 *b*x^2 + 128*a^5)*sqrt(b*x^2 + a))/(a*x^10), 1/1280*(315*sqrt(-a)*b^5*x^10 *arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) - (965*a*b^4*x^8 + 1490*a^2*b^3*x^6 + 1368*a^3*b^2*x^4 + 656*a^4*b*x^2 + 128*a^5)*sqrt(b*x^2 + a))/(a*x^10)]
Time = 10.48 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{11}} \, dx=- \frac {a^{4} \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{10 x^{9}} - \frac {41 a^{3} b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{80 x^{7}} - \frac {171 a^{2} b^{\frac {5}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{160 x^{5}} - \frac {149 a b^{\frac {7}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{128 x^{3}} - \frac {193 b^{\frac {9}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{256 x} - \frac {63 b^{5} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{256 \sqrt {a}} \] Input:
integrate((b*x**2+a)**(9/2)/x**11,x)
Output:
-a**4*sqrt(b)*sqrt(a/(b*x**2) + 1)/(10*x**9) - 41*a**3*b**(3/2)*sqrt(a/(b* x**2) + 1)/(80*x**7) - 171*a**2*b**(5/2)*sqrt(a/(b*x**2) + 1)/(160*x**5) - 149*a*b**(7/2)*sqrt(a/(b*x**2) + 1)/(128*x**3) - 193*b**(9/2)*sqrt(a/(b*x **2) + 1)/(256*x) - 63*b**5*asinh(sqrt(a)/(sqrt(b)*x))/(256*sqrt(a))
Time = 0.04 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.43 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{11}} \, dx=-\frac {63 \, b^{5} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{256 \, \sqrt {a}} + \frac {7 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} b^{5}}{256 \, a^{5}} + \frac {9 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{5}}{256 \, a^{4}} + \frac {63 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{5}}{1280 \, a^{3}} + \frac {21 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{5}}{256 \, a^{2}} + \frac {63 \, \sqrt {b x^{2} + a} b^{5}}{256 \, a} - \frac {7 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{4}}{256 \, a^{5} x^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{3}}{128 \, a^{4} x^{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{2}}{160 \, a^{3} x^{6}} - \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}} b}{80 \, a^{2} x^{8}} - \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}}}{10 \, a x^{10}} \] Input:
integrate((b*x^2+a)^(9/2)/x^11,x, algorithm="maxima")
Output:
-63/256*b^5*arcsinh(a/(sqrt(a*b)*abs(x)))/sqrt(a) + 7/256*(b*x^2 + a)^(9/2 )*b^5/a^5 + 9/256*(b*x^2 + a)^(7/2)*b^5/a^4 + 63/1280*(b*x^2 + a)^(5/2)*b^ 5/a^3 + 21/256*(b*x^2 + a)^(3/2)*b^5/a^2 + 63/256*sqrt(b*x^2 + a)*b^5/a - 7/256*(b*x^2 + a)^(11/2)*b^4/(a^5*x^2) - 1/128*(b*x^2 + a)^(11/2)*b^3/(a^4 *x^4) - 1/160*(b*x^2 + a)^(11/2)*b^2/(a^3*x^6) - 1/80*(b*x^2 + a)^(11/2)*b /(a^2*x^8) - 1/10*(b*x^2 + a)^(11/2)/(a*x^10)
Time = 0.12 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.73 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{11}} \, dx=\frac {1}{1280} \, b^{5} {\left (\frac {315 \, \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {965 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} - 2370 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a + 2688 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2} - 1470 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x^{2} + a} a^{4}}{b^{5} x^{10}}\right )} \] Input:
integrate((b*x^2+a)^(9/2)/x^11,x, algorithm="giac")
Output:
1/1280*b^5*(315*arctan(sqrt(b*x^2 + a)/sqrt(-a))/sqrt(-a) - (965*(b*x^2 + a)^(9/2) - 2370*(b*x^2 + a)^(7/2)*a + 2688*(b*x^2 + a)^(5/2)*a^2 - 1470*(b *x^2 + a)^(3/2)*a^3 + 315*sqrt(b*x^2 + a)*a^4)/(b^5*x^10))
Time = 2.00 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{11}} \, dx=\frac {237\,a\,{\left (b\,x^2+a\right )}^{7/2}}{128\,x^{10}}-\frac {193\,{\left (b\,x^2+a\right )}^{9/2}}{256\,x^{10}}-\frac {63\,a^4\,\sqrt {b\,x^2+a}}{256\,x^{10}}+\frac {147\,a^3\,{\left (b\,x^2+a\right )}^{3/2}}{128\,x^{10}}-\frac {21\,a^2\,{\left (b\,x^2+a\right )}^{5/2}}{10\,x^{10}}+\frac {b^5\,\mathrm {atan}\left (\frac {\sqrt {b\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,63{}\mathrm {i}}{256\,\sqrt {a}} \] Input:
int((a + b*x^2)^(9/2)/x^11,x)
Output:
(b^5*atan(((a + b*x^2)^(1/2)*1i)/a^(1/2))*63i)/(256*a^(1/2)) - (193*(a + b *x^2)^(9/2))/(256*x^10) + (237*a*(a + b*x^2)^(7/2))/(128*x^10) - (63*a^4*( a + b*x^2)^(1/2))/(256*x^10) + (147*a^3*(a + b*x^2)^(3/2))/(128*x^10) - (2 1*a^2*(a + b*x^2)^(5/2))/(10*x^10)
Time = 0.29 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.12 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{11}} \, dx=\frac {-128 \sqrt {b \,x^{2}+a}\, a^{5}-656 \sqrt {b \,x^{2}+a}\, a^{4} b \,x^{2}-1368 \sqrt {b \,x^{2}+a}\, a^{3} b^{2} x^{4}-1490 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} x^{6}-965 \sqrt {b \,x^{2}+a}\, a \,b^{4} x^{8}+315 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{5} x^{10}-315 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{5} x^{10}}{1280 a \,x^{10}} \] Input:
int((b*x^2+a)^(9/2)/x^11,x)
Output:
( - 128*sqrt(a + b*x**2)*a**5 - 656*sqrt(a + b*x**2)*a**4*b*x**2 - 1368*sq rt(a + b*x**2)*a**3*b**2*x**4 - 1490*sqrt(a + b*x**2)*a**2*b**3*x**6 - 965 *sqrt(a + b*x**2)*a*b**4*x**8 + 315*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a ) + sqrt(b)*x)/sqrt(a))*b**5*x**10 - 315*sqrt(a)*log((sqrt(a + b*x**2) + s qrt(a) + sqrt(b)*x)/sqrt(a))*b**5*x**10)/(1280*a*x**10)