Integrand size = 15, antiderivative size = 124 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{10}} \, dx=-\frac {b^4 \sqrt {a+b x^2}}{x}-\frac {b^3 \left (a+b x^2\right )^{3/2}}{3 x^3}-\frac {b^2 \left (a+b x^2\right )^{5/2}}{5 x^5}-\frac {b \left (a+b x^2\right )^{7/2}}{7 x^7}-\frac {\left (a+b x^2\right )^{9/2}}{9 x^9}+b^{9/2} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \]
-1/3*b^3*(b*x^2+a)^(3/2)/x^3-1/5*b^2*(b*x^2+a)^(5/2)/x^5-1/7*b*(b*x^2+a)^( 7/2)/x^7-1/9*(b*x^2+a)^(9/2)/x^9+b^(9/2)*arctanh(x*b^(1/2)/(b*x^2+a)^(1/2) )-b^4*(b*x^2+a)^(1/2)/x
Time = 0.16 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.73 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{10}} \, dx=\frac {\sqrt {a+b x^2} \left (-35 a^4-185 a^3 b x^2-408 a^2 b^2 x^4-506 a b^3 x^6-563 b^4 x^8\right )}{315 x^9}-b^{9/2} \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right ) \]
(Sqrt[a + b*x^2]*(-35*a^4 - 185*a^3*b*x^2 - 408*a^2*b^2*x^4 - 506*a*b^3*x^ 6 - 563*b^4*x^8))/(315*x^9) - b^(9/2)*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]]
Time = 0.22 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {247, 247, 247, 247, 247, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{9/2}}{x^{10}} \, dx\) |
| \(\Big \downarrow \) 247 |
| \(\displaystyle b \int \frac {\left (b x^2+a\right )^{7/2}}{x^8}dx-\frac {\left (a+b x^2\right )^{9/2}}{9 x^9}\) |
| \(\Big \downarrow \) 247 |
| \(\displaystyle b \left (b \int \frac {\left (b x^2+a\right )^{5/2}}{x^6}dx-\frac {\left (a+b x^2\right )^{7/2}}{7 x^7}\right )-\frac {\left (a+b x^2\right )^{9/2}}{9 x^9}\) |
| \(\Big \downarrow \) 247 |
| \(\displaystyle b \left (b \left (b \int \frac {\left (b x^2+a\right )^{3/2}}{x^4}dx-\frac {\left (a+b x^2\right )^{5/2}}{5 x^5}\right )-\frac {\left (a+b x^2\right )^{7/2}}{7 x^7}\right )-\frac {\left (a+b x^2\right )^{9/2}}{9 x^9}\) |
| \(\Big \downarrow \) 247 |
| \(\displaystyle b \left (b \left (b \left (b \int \frac {\sqrt {b x^2+a}}{x^2}dx-\frac {\left (a+b x^2\right )^{3/2}}{3 x^3}\right )-\frac {\left (a+b x^2\right )^{5/2}}{5 x^5}\right )-\frac {\left (a+b x^2\right )^{7/2}}{7 x^7}\right )-\frac {\left (a+b x^2\right )^{9/2}}{9 x^9}\) |
| \(\Big \downarrow \) 247 |
| \(\displaystyle b \left (b \left (b \left (b \left (b \int \frac {1}{\sqrt {b x^2+a}}dx-\frac {\sqrt {a+b x^2}}{x}\right )-\frac {\left (a+b x^2\right )^{3/2}}{3 x^3}\right )-\frac {\left (a+b x^2\right )^{5/2}}{5 x^5}\right )-\frac {\left (a+b x^2\right )^{7/2}}{7 x^7}\right )-\frac {\left (a+b x^2\right )^{9/2}}{9 x^9}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle b \left (b \left (b \left (b \left (b \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}-\frac {\sqrt {a+b x^2}}{x}\right )-\frac {\left (a+b x^2\right )^{3/2}}{3 x^3}\right )-\frac {\left (a+b x^2\right )^{5/2}}{5 x^5}\right )-\frac {\left (a+b x^2\right )^{7/2}}{7 x^7}\right )-\frac {\left (a+b x^2\right )^{9/2}}{9 x^9}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle b \left (b \left (b \left (b \left (\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )-\frac {\sqrt {a+b x^2}}{x}\right )-\frac {\left (a+b x^2\right )^{3/2}}{3 x^3}\right )-\frac {\left (a+b x^2\right )^{5/2}}{5 x^5}\right )-\frac {\left (a+b x^2\right )^{7/2}}{7 x^7}\right )-\frac {\left (a+b x^2\right )^{9/2}}{9 x^9}\) |
-1/9*(a + b*x^2)^(9/2)/x^9 + b*(-1/7*(a + b*x^2)^(7/2)/x^7 + b*(-1/5*(a + b*x^2)^(5/2)/x^5 + b*(-1/3*(a + b*x^2)^(3/2)/x^3 + b*(-(Sqrt[a + b*x^2]/x) + Sqrt[b]*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]]))))
3.5.33.3.1 Defintions of rubi rules used
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*x)^ (m + 1)*((a + b*x^2)^p/(c*(m + 1))), x] - Simp[2*b*(p/(c^2*(m + 1))) Int[ (c*x)^(m + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomialQ[a, b, c, 2, m, p, x]
Time = 2.01 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.64
| method | result | size |
| risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (563 x^{8} b^{4}+506 a \,b^{3} x^{6}+408 a^{2} x^{4} b^{2}+185 a^{3} b \,x^{2}+35 a^{4}\right )}{315 x^{9}}+b^{\frac {9}{2}} \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )\) | \(79\) |
| pseudoelliptic | \(\frac {315 b^{\frac {9}{2}} \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right ) x^{9}-\sqrt {b \,x^{2}+a}\, \left (563 x^{8} b^{4}+506 a \,b^{3} x^{6}+408 a^{2} x^{4} b^{2}+185 a^{3} b \,x^{2}+35 a^{4}\right )}{315 x^{9}}\) | \(86\) |
| default | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{9 a \,x^{9}}+\frac {2 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{7 a \,x^{7}}+\frac {4 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{5 a \,x^{5}}+\frac {6 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{3 a \,x^{3}}+\frac {8 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{a x}+\frac {10 b \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {9}{2}}}{10}+\frac {9 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{8}+\frac {7 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )}{8}\right )}{10}\right )}{a}\right )}{3 a}\right )}{5 a}\right )}{7 a}\right )}{9 a}\) | \(220\) |
-1/315*(b*x^2+a)^(1/2)*(563*b^4*x^8+506*a*b^3*x^6+408*a^2*b^2*x^4+185*a^3* b*x^2+35*a^4)/x^9+b^(9/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))
Time = 0.27 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.48 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{10}} \, dx=\left [\frac {315 \, b^{\frac {9}{2}} x^{9} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (563 \, b^{4} x^{8} + 506 \, a b^{3} x^{6} + 408 \, a^{2} b^{2} x^{4} + 185 \, a^{3} b x^{2} + 35 \, a^{4}\right )} \sqrt {b x^{2} + a}}{630 \, x^{9}}, -\frac {315 \, \sqrt {-b} b^{4} x^{9} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (563 \, b^{4} x^{8} + 506 \, a b^{3} x^{6} + 408 \, a^{2} b^{2} x^{4} + 185 \, a^{3} b x^{2} + 35 \, a^{4}\right )} \sqrt {b x^{2} + a}}{315 \, x^{9}}\right ] \]
[1/630*(315*b^(9/2)*x^9*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 2*(563*b^4*x^8 + 506*a*b^3*x^6 + 408*a^2*b^2*x^4 + 185*a^3*b*x^2 + 35*a^4) *sqrt(b*x^2 + a))/x^9, -1/315*(315*sqrt(-b)*b^4*x^9*arctan(sqrt(-b)*x/sqrt (b*x^2 + a)) + (563*b^4*x^8 + 506*a*b^3*x^6 + 408*a^2*b^2*x^4 + 185*a^3*b* x^2 + 35*a^4)*sqrt(b*x^2 + a))/x^9]
Time = 11.16 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.29 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{10}} \, dx=- \frac {a^{4} \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{9 x^{8}} - \frac {37 a^{3} b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{63 x^{6}} - \frac {136 a^{2} b^{\frac {5}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{105 x^{4}} - \frac {506 a b^{\frac {7}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{315 x^{2}} - \frac {563 b^{\frac {9}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{315} - \frac {b^{\frac {9}{2}} \log {\left (\frac {a}{b x^{2}} \right )}}{2} + b^{\frac {9}{2}} \log {\left (\sqrt {\frac {a}{b x^{2}} + 1} + 1 \right )} \]
-a**4*sqrt(b)*sqrt(a/(b*x**2) + 1)/(9*x**8) - 37*a**3*b**(3/2)*sqrt(a/(b*x **2) + 1)/(63*x**6) - 136*a**2*b**(5/2)*sqrt(a/(b*x**2) + 1)/(105*x**4) - 506*a*b**(7/2)*sqrt(a/(b*x**2) + 1)/(315*x**2) - 563*b**(9/2)*sqrt(a/(b*x* *2) + 1)/315 - b**(9/2)*log(a/(b*x**2))/2 + b**(9/2)*log(sqrt(a/(b*x**2) + 1) + 1)
Time = 0.21 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.45 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{10}} \, dx=\frac {16 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{5} x}{35 \, a^{4}} + \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{5} x}{15 \, a^{3}} + \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{5} x}{3 \, a^{2}} + \frac {\sqrt {b x^{2} + a} b^{5} x}{a} + b^{\frac {9}{2}} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - \frac {128 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} b^{4}}{315 \, a^{4} x} - \frac {16 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{3}}{315 \, a^{4} x^{3}} - \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{2}}{315 \, a^{3} x^{5}} - \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b}{63 \, a^{2} x^{7}} - \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}}}{9 \, a x^{9}} \]
16/35*(b*x^2 + a)^(7/2)*b^5*x/a^4 + 8/15*(b*x^2 + a)^(5/2)*b^5*x/a^3 + 2/3 *(b*x^2 + a)^(3/2)*b^5*x/a^2 + sqrt(b*x^2 + a)*b^5*x/a + b^(9/2)*arcsinh(b *x/sqrt(a*b)) - 128/315*(b*x^2 + a)^(9/2)*b^4/(a^4*x) - 16/315*(b*x^2 + a) ^(11/2)*b^3/(a^4*x^3) - 8/315*(b*x^2 + a)^(11/2)*b^2/(a^3*x^5) - 2/63*(b*x ^2 + a)^(11/2)*b/(a^2*x^7) - 1/9*(b*x^2 + a)^(11/2)/(a*x^9)
Leaf count of result is larger than twice the leaf count of optimal. 276 vs. \(2 (100) = 200\).
Time = 0.30 (sec) , antiderivative size = 276, normalized size of antiderivative = 2.23 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{10}} \, dx=-\frac {1}{2} \, b^{\frac {9}{2}} \log \left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2}\right ) + \frac {2 \, {\left (1575 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{16} a b^{\frac {9}{2}} - 6300 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{14} a^{2} b^{\frac {9}{2}} + 21000 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} a^{3} b^{\frac {9}{2}} - 31500 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a^{4} b^{\frac {9}{2}} + 39438 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{5} b^{\frac {9}{2}} - 26292 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{6} b^{\frac {9}{2}} + 13968 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{7} b^{\frac {9}{2}} - 3492 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{8} b^{\frac {9}{2}} + 563 \, a^{9} b^{\frac {9}{2}}\right )}}{315 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{9}} \]
-1/2*b^(9/2)*log((sqrt(b)*x - sqrt(b*x^2 + a))^2) + 2/315*(1575*(sqrt(b)*x - sqrt(b*x^2 + a))^16*a*b^(9/2) - 6300*(sqrt(b)*x - sqrt(b*x^2 + a))^14*a ^2*b^(9/2) + 21000*(sqrt(b)*x - sqrt(b*x^2 + a))^12*a^3*b^(9/2) - 31500*(s qrt(b)*x - sqrt(b*x^2 + a))^10*a^4*b^(9/2) + 39438*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^5*b^(9/2) - 26292*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^6*b^(9/2) + 13968*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^7*b^(9/2) - 3492*(sqrt(b)*x - sqr t(b*x^2 + a))^2*a^8*b^(9/2) + 563*a^9*b^(9/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^9
Timed out. \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{10}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{9/2}}{x^{10}} \,d x \]