Integrand size = 15, antiderivative size = 129 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^6} \, dx=\frac {63}{8} a b^3 x \sqrt {a+b x^2}+\frac {21}{4} b^3 x \left (a+b x^2\right )^{3/2}-\frac {21 b^2 \left (a+b x^2\right )^{5/2}}{5 x}-\frac {3 b \left (a+b x^2\right )^{7/2}}{5 x^3}-\frac {\left (a+b x^2\right )^{9/2}}{5 x^5}+\frac {63}{8} a^2 b^{5/2} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \]
21/4*b^3*x*(b*x^2+a)^(3/2)-21/5*b^2*(b*x^2+a)^(5/2)/x-3/5*b*(b*x^2+a)^(7/2 )/x^3-1/5*(b*x^2+a)^(9/2)/x^5+63/8*a^2*b^(5/2)*arctanh(x*b^(1/2)/(b*x^2+a) ^(1/2))+63/8*a*b^3*x*(b*x^2+a)^(1/2)
Time = 0.15 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.74 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^6} \, dx=\frac {\sqrt {a+b x^2} \left (-8 a^4-56 a^3 b x^2-288 a^2 b^2 x^4+85 a b^3 x^6+10 b^4 x^8\right )}{40 x^5}-\frac {63}{8} a^2 b^{5/2} \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right ) \]
(Sqrt[a + b*x^2]*(-8*a^4 - 56*a^3*b*x^2 - 288*a^2*b^2*x^4 + 85*a*b^3*x^6 + 10*b^4*x^8))/(40*x^5) - (63*a^2*b^(5/2)*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2 ]])/8
Time = 0.21 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.05, 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, 211, 211, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{9/2}}{x^6} \, dx\) |
| \(\Big \downarrow \) 247 |
| \(\displaystyle \frac {9}{5} b \int \frac {\left (b x^2+a\right )^{7/2}}{x^4}dx-\frac {\left (a+b x^2\right )^{9/2}}{5 x^5}\) |
| \(\Big \downarrow \) 247 |
| \(\displaystyle \frac {9}{5} b \left (\frac {7}{3} b \int \frac {\left (b x^2+a\right )^{5/2}}{x^2}dx-\frac {\left (a+b x^2\right )^{7/2}}{3 x^3}\right )-\frac {\left (a+b x^2\right )^{9/2}}{5 x^5}\) |
| \(\Big \downarrow \) 247 |
| \(\displaystyle \frac {9}{5} b \left (\frac {7}{3} b \left (5 b \int \left (b x^2+a\right )^{3/2}dx-\frac {\left (a+b x^2\right )^{5/2}}{x}\right )-\frac {\left (a+b x^2\right )^{7/2}}{3 x^3}\right )-\frac {\left (a+b x^2\right )^{9/2}}{5 x^5}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {9}{5} b \left (\frac {7}{3} b \left (5 b \left (\frac {3}{4} a \int \sqrt {b x^2+a}dx+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )-\frac {\left (a+b x^2\right )^{5/2}}{x}\right )-\frac {\left (a+b x^2\right )^{7/2}}{3 x^3}\right )-\frac {\left (a+b x^2\right )^{9/2}}{5 x^5}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {9}{5} b \left (\frac {7}{3} b \left (5 b \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{\sqrt {b x^2+a}}dx+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )-\frac {\left (a+b x^2\right )^{5/2}}{x}\right )-\frac {\left (a+b x^2\right )^{7/2}}{3 x^3}\right )-\frac {\left (a+b x^2\right )^{9/2}}{5 x^5}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {9}{5} b \left (\frac {7}{3} b \left (5 b \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )-\frac {\left (a+b x^2\right )^{5/2}}{x}\right )-\frac {\left (a+b x^2\right )^{7/2}}{3 x^3}\right )-\frac {\left (a+b x^2\right )^{9/2}}{5 x^5}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {9}{5} b \left (\frac {7}{3} b \left (5 b \left (\frac {3}{4} a \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )-\frac {\left (a+b x^2\right )^{5/2}}{x}\right )-\frac {\left (a+b x^2\right )^{7/2}}{3 x^3}\right )-\frac {\left (a+b x^2\right )^{9/2}}{5 x^5}\) |
-1/5*(a + b*x^2)^(9/2)/x^5 + (9*b*(-1/3*(a + b*x^2)^(7/2)/x^3 + (7*b*(-((a + b*x^2)^(5/2)/x) + 5*b*((x*(a + b*x^2)^(3/2))/4 + (3*a*((x*Sqrt[a + b*x^ 2])/2 + (a*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*Sqrt[b])))/4)))/3))/5
3.5.31.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
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 = 1.94 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.64
| method | result | size |
| risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (-10 x^{8} b^{4}-85 a \,b^{3} x^{6}+288 a^{2} x^{4} b^{2}+56 a^{3} b \,x^{2}+8 a^{4}\right )}{40 x^{5}}+\frac {63 a^{2} b^{\frac {5}{2}} \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{8}\) | \(83\) |
| pseudoelliptic | \(-\frac {-\frac {315 \,\operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right ) a^{2} b^{3} x^{5}}{8}+\sqrt {b \,x^{2}+a}\, \left (-\frac {5 b^{\frac {9}{2}} x^{8}}{4}-\frac {85 a \,b^{\frac {7}{2}} x^{6}}{8}+36 a^{2} b^{\frac {5}{2}} x^{4}+7 a^{3} b^{\frac {3}{2}} x^{2}+a^{4} \sqrt {b}\right )}{5 \sqrt {b}\, x^{5}}\) | \(95\) |
| default | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{5 a \,x^{5}}+\frac {6 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{3 a \,x^{3}}+\frac {8 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{a x}+\frac {10 b \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {9}{2}}}{10}+\frac {9 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{8}+\frac {7 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )}{8}\right )}{10}\right )}{a}\right )}{3 a}\right )}{5 a}\) | \(172\) |
-1/40*(b*x^2+a)^(1/2)*(-10*b^4*x^8-85*a*b^3*x^6+288*a^2*b^2*x^4+56*a^3*b*x ^2+8*a^4)/x^5+63/8*a^2*b^(5/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))
Time = 0.29 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.48 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^6} \, dx=\left [\frac {315 \, a^{2} b^{\frac {5}{2}} x^{5} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (10 \, b^{4} x^{8} + 85 \, a b^{3} x^{6} - 288 \, a^{2} b^{2} x^{4} - 56 \, a^{3} b x^{2} - 8 \, a^{4}\right )} \sqrt {b x^{2} + a}}{80 \, x^{5}}, -\frac {315 \, a^{2} \sqrt {-b} b^{2} x^{5} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (10 \, b^{4} x^{8} + 85 \, a b^{3} x^{6} - 288 \, a^{2} b^{2} x^{4} - 56 \, a^{3} b x^{2} - 8 \, a^{4}\right )} \sqrt {b x^{2} + a}}{40 \, x^{5}}\right ] \]
[1/80*(315*a^2*b^(5/2)*x^5*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(10*b^4*x^8 + 85*a*b^3*x^6 - 288*a^2*b^2*x^4 - 56*a^3*b*x^2 - 8*a^4)* sqrt(b*x^2 + a))/x^5, -1/40*(315*a^2*sqrt(-b)*b^2*x^5*arctan(sqrt(-b)*x/sq rt(b*x^2 + a)) - (10*b^4*x^8 + 85*a*b^3*x^6 - 288*a^2*b^2*x^4 - 56*a^3*b*x ^2 - 8*a^4)*sqrt(b*x^2 + a))/x^5]
Time = 9.78 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.36 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^6} \, dx=- \frac {a^{\frac {9}{2}}}{5 x^{5} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {8 a^{\frac {7}{2}} b}{5 x^{3} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {43 a^{\frac {5}{2}} b^{2}}{5 x \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {203 a^{\frac {3}{2}} b^{3} x}{40 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {19 \sqrt {a} b^{4} x^{3}}{8 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {63 a^{2} b^{\frac {5}{2}} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{8} + \frac {b^{5} x^{5}}{4 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \]
-a**(9/2)/(5*x**5*sqrt(1 + b*x**2/a)) - 8*a**(7/2)*b/(5*x**3*sqrt(1 + b*x* *2/a)) - 43*a**(5/2)*b**2/(5*x*sqrt(1 + b*x**2/a)) - 203*a**(3/2)*b**3*x/( 40*sqrt(1 + b*x**2/a)) + 19*sqrt(a)*b**4*x**3/(8*sqrt(1 + b*x**2/a)) + 63* a**2*b**(5/2)*asinh(sqrt(b)*x/sqrt(a))/8 + b**5*x**5/(4*sqrt(a)*sqrt(1 + b *x**2/a))
Time = 0.21 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^6} \, dx=\frac {21}{4} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{3} x + \frac {18 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3} x}{5 \, a^{2}} + \frac {21 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{3} x}{5 \, a} + \frac {63}{8} \, \sqrt {b x^{2} + a} a b^{3} x + \frac {63}{8} \, a^{2} b^{\frac {5}{2}} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - \frac {16 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} b^{2}}{5 \, a^{2} x} - \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b}{5 \, a^{2} x^{3}} - \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}}}{5 \, a x^{5}} \]
21/4*(b*x^2 + a)^(3/2)*b^3*x + 18/5*(b*x^2 + a)^(7/2)*b^3*x/a^2 + 21/5*(b* x^2 + a)^(5/2)*b^3*x/a + 63/8*sqrt(b*x^2 + a)*a*b^3*x + 63/8*a^2*b^(5/2)*a rcsinh(b*x/sqrt(a*b)) - 16/5*(b*x^2 + a)^(9/2)*b^2/(a^2*x) - 2/5*(b*x^2 + a)^(11/2)*b/(a^2*x^3) - 1/5*(b*x^2 + a)^(11/2)/(a*x^5)
Time = 0.30 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.55 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^6} \, dx=-\frac {63}{16} \, a^{2} b^{\frac {5}{2}} \log \left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2}\right ) + \frac {1}{8} \, {\left (2 \, b^{4} x^{2} + 17 \, a b^{3}\right )} \sqrt {b x^{2} + a} x + \frac {4 \, {\left (25 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{3} b^{\frac {5}{2}} - 75 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{4} b^{\frac {5}{2}} + 105 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{5} b^{\frac {5}{2}} - 65 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{6} b^{\frac {5}{2}} + 18 \, a^{7} b^{\frac {5}{2}}\right )}}{5 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{5}} \]
-63/16*a^2*b^(5/2)*log((sqrt(b)*x - sqrt(b*x^2 + a))^2) + 1/8*(2*b^4*x^2 + 17*a*b^3)*sqrt(b*x^2 + a)*x + 4/5*(25*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^3 *b^(5/2) - 75*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^4*b^(5/2) + 105*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^5*b^(5/2) - 65*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^6 *b^(5/2) + 18*a^7*b^(5/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^5
Timed out. \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^6} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{9/2}}{x^6} \,d x \]