Integrand size = 15, antiderivative size = 128 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^4} \, dx=\frac {105}{16} a^2 b^2 x \sqrt {a+b x^2}+\frac {35}{8} a b^2 x \left (a+b x^2\right )^{3/2}+\frac {7}{2} b^2 x \left (a+b x^2\right )^{5/2}-\frac {3 b \left (a+b x^2\right )^{7/2}}{x}-\frac {\left (a+b x^2\right )^{9/2}}{3 x^3}+\frac {105}{16} a^3 b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \]
35/8*a*b^2*x*(b*x^2+a)^(3/2)+7/2*b^2*x*(b*x^2+a)^(5/2)-3*b*(b*x^2+a)^(7/2) /x-1/3*(b*x^2+a)^(9/2)/x^3+105/16*a^3*b^(3/2)*arctanh(x*b^(1/2)/(b*x^2+a)^ (1/2))+105/16*a^2*b^2*x*(b*x^2+a)^(1/2)
Time = 0.13 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.74 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^4} \, dx=\frac {\sqrt {a+b x^2} \left (-16 a^4-208 a^3 b x^2+165 a^2 b^2 x^4+50 a b^3 x^6+8 b^4 x^8\right )}{48 x^3}-\frac {105}{16} a^3 b^{3/2} \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right ) \]
(Sqrt[a + b*x^2]*(-16*a^4 - 208*a^3*b*x^2 + 165*a^2*b^2*x^4 + 50*a*b^3*x^6 + 8*b^4*x^8))/(48*x^3) - (105*a^3*b^(3/2)*Log[-(Sqrt[b]*x) + Sqrt[a + b*x ^2]])/16
Time = 0.21 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.03, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {247, 247, 211, 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.
\(\displaystyle \int \frac {\left (a+b x^2\right )^{9/2}}{x^4} \, dx\) |
\(\Big \downarrow \) 247 |
\(\displaystyle 3 b \int \frac {\left (b x^2+a\right )^{7/2}}{x^2}dx-\frac {\left (a+b x^2\right )^{9/2}}{3 x^3}\) |
\(\Big \downarrow \) 247 |
\(\displaystyle 3 b \left (7 b \int \left (b x^2+a\right )^{5/2}dx-\frac {\left (a+b x^2\right )^{7/2}}{x}\right )-\frac {\left (a+b x^2\right )^{9/2}}{3 x^3}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle 3 b \left (7 b \left (\frac {5}{6} a \int \left (b x^2+a\right )^{3/2}dx+\frac {1}{6} x \left (a+b x^2\right )^{5/2}\right )-\frac {\left (a+b x^2\right )^{7/2}}{x}\right )-\frac {\left (a+b x^2\right )^{9/2}}{3 x^3}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle 3 b \left (7 b \left (\frac {5}{6} a \left (\frac {3}{4} a \int \sqrt {b x^2+a}dx+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+b x^2\right )^{5/2}\right )-\frac {\left (a+b x^2\right )^{7/2}}{x}\right )-\frac {\left (a+b x^2\right )^{9/2}}{3 x^3}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle 3 b \left (7 b \left (\frac {5}{6} a \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{\sqrt {b x^2+a}}dx+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+b x^2\right )^{5/2}\right )-\frac {\left (a+b x^2\right )^{7/2}}{x}\right )-\frac {\left (a+b x^2\right )^{9/2}}{3 x^3}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle 3 b \left (7 b \left (\frac {5}{6} a \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+b x^2\right )^{5/2}\right )-\frac {\left (a+b x^2\right )^{7/2}}{x}\right )-\frac {\left (a+b x^2\right )^{9/2}}{3 x^3}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle 3 b \left (7 b \left (\frac {5}{6} a \left (\frac {3}{4} a \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+b x^2\right )^{5/2}\right )-\frac {\left (a+b x^2\right )^{7/2}}{x}\right )-\frac {\left (a+b x^2\right )^{9/2}}{3 x^3}\) |
-1/3*(a + b*x^2)^(9/2)/x^3 + 3*b*(-((a + b*x^2)^(7/2)/x) + 7*b*((x*(a + b* x^2)^(5/2))/6 + (5*a*((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))/6))
3.5.30.3.1 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 = 1.93 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.65
method | result | size |
risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (-8 x^{8} b^{4}-50 a \,b^{3} x^{6}-165 a^{2} x^{4} b^{2}+208 a^{3} b \,x^{2}+16 a^{4}\right )}{48 x^{3}}+\frac {105 a^{3} b^{\frac {3}{2}} \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{16}\) | \(83\) |
pseudoelliptic | \(-\frac {-\frac {315 \,\operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right ) a^{3} b^{2} x^{3}}{16}+\sqrt {b \,x^{2}+a}\, \left (-\frac {b^{\frac {9}{2}} x^{8}}{2}-\frac {25 a \,b^{\frac {7}{2}} x^{6}}{8}-\frac {165 a^{2} b^{\frac {5}{2}} x^{4}}{16}+13 a^{3} b^{\frac {3}{2}} x^{2}+a^{4} \sqrt {b}\right )}{3 \sqrt {b}\, x^{3}}\) | \(95\) |
default | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{3 a \,x^{3}}+\frac {8 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{a x}+\frac {10 b \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {9}{2}}}{10}+\frac {9 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{8}+\frac {7 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )}{8}\right )}{10}\right )}{a}\right )}{3 a}\) | \(148\) |
-1/48*(b*x^2+a)^(1/2)*(-8*b^4*x^8-50*a*b^3*x^6-165*a^2*b^2*x^4+208*a^3*b*x ^2+16*a^4)/x^3+105/16*a^3*b^(3/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))
Time = 0.32 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.48 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^4} \, dx=\left [\frac {315 \, a^{3} b^{\frac {3}{2}} x^{3} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (8 \, b^{4} x^{8} + 50 \, a b^{3} x^{6} + 165 \, a^{2} b^{2} x^{4} - 208 \, a^{3} b x^{2} - 16 \, a^{4}\right )} \sqrt {b x^{2} + a}}{96 \, x^{3}}, -\frac {315 \, a^{3} \sqrt {-b} b x^{3} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (8 \, b^{4} x^{8} + 50 \, a b^{3} x^{6} + 165 \, a^{2} b^{2} x^{4} - 208 \, a^{3} b x^{2} - 16 \, a^{4}\right )} \sqrt {b x^{2} + a}}{48 \, x^{3}}\right ] \]
[1/96*(315*a^3*b^(3/2)*x^3*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(8*b^4*x^8 + 50*a*b^3*x^6 + 165*a^2*b^2*x^4 - 208*a^3*b*x^2 - 16*a^4) *sqrt(b*x^2 + a))/x^3, -1/48*(315*a^3*sqrt(-b)*b*x^3*arctan(sqrt(-b)*x/sqr t(b*x^2 + a)) - (8*b^4*x^8 + 50*a*b^3*x^6 + 165*a^2*b^2*x^4 - 208*a^3*b*x^ 2 - 16*a^4)*sqrt(b*x^2 + a))/x^3]
Time = 9.54 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.37 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^4} \, dx=- \frac {a^{\frac {9}{2}}}{3 x^{3} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {14 a^{\frac {7}{2}} b}{3 x \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {43 a^{\frac {5}{2}} b^{2} x}{48 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {215 a^{\frac {3}{2}} b^{3} x^{3}}{48 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {29 \sqrt {a} b^{4} x^{5}}{24 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {105 a^{3} b^{\frac {3}{2}} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{16} + \frac {b^{5} x^{7}}{6 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \]
-a**(9/2)/(3*x**3*sqrt(1 + b*x**2/a)) - 14*a**(7/2)*b/(3*x*sqrt(1 + b*x**2 /a)) - 43*a**(5/2)*b**2*x/(48*sqrt(1 + b*x**2/a)) + 215*a**(3/2)*b**3*x**3 /(48*sqrt(1 + b*x**2/a)) + 29*sqrt(a)*b**4*x**5/(24*sqrt(1 + b*x**2/a)) + 105*a**3*b**(3/2)*asinh(sqrt(b)*x/sqrt(a))/16 + b**5*x**7/(6*sqrt(a)*sqrt( 1 + b*x**2/a))
Time = 0.21 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^4} \, dx=\frac {7}{2} \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{2} x + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2} x}{a} + \frac {35}{8} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a b^{2} x + \frac {105}{16} \, \sqrt {b x^{2} + a} a^{2} b^{2} x + \frac {105}{16} \, a^{3} b^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} b}{3 \, a x} - \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}}}{3 \, a x^{3}} \]
7/2*(b*x^2 + a)^(5/2)*b^2*x + 3*(b*x^2 + a)^(7/2)*b^2*x/a + 35/8*(b*x^2 + a)^(3/2)*a*b^2*x + 105/16*sqrt(b*x^2 + a)*a^2*b^2*x + 105/16*a^3*b^(3/2)*a rcsinh(b*x/sqrt(a*b)) - 8/3*(b*x^2 + a)^(9/2)*b/(a*x) - 1/3*(b*x^2 + a)^(1 1/2)/(a*x^3)
Time = 0.30 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.25 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^4} \, dx=-\frac {105}{32} \, a^{3} b^{\frac {3}{2}} \log \left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2}\right ) + \frac {1}{48} \, {\left (165 \, a^{2} b^{2} + 2 \, {\left (4 \, b^{4} x^{2} + 25 \, a b^{3}\right )} x^{2}\right )} \sqrt {b x^{2} + a} x + \frac {2 \, {\left (15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{4} b^{\frac {3}{2}} - 24 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{5} b^{\frac {3}{2}} + 13 \, a^{6} b^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{3}} \]
-105/32*a^3*b^(3/2)*log((sqrt(b)*x - sqrt(b*x^2 + a))^2) + 1/48*(165*a^2*b ^2 + 2*(4*b^4*x^2 + 25*a*b^3)*x^2)*sqrt(b*x^2 + a)*x + 2/3*(15*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^4*b^(3/2) - 24*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^5* b^(3/2) + 13*a^6*b^(3/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^3
Timed out. \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^4} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{9/2}}{x^4} \,d x \]