Integrand size = 15, antiderivative size = 116 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{x^4} \, dx=-\frac {5 a^2 \sqrt {a+\frac {b}{x^2}}}{64 x^3}-\frac {5 a \left (a+\frac {b}{x^2}\right )^{3/2}}{48 x^3}-\frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{8 x^3}-\frac {5 a^3 \sqrt {a+\frac {b}{x^2}}}{128 b x}+\frac {5 a^4 \text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x^2}} x}\right )}{128 b^{3/2}} \]
-5/48*a*(a+b/x^2)^(3/2)/x^3-1/8*(a+b/x^2)^(5/2)/x^3+5/128*a^4*arctanh(b^(1 /2)/x/(a+b/x^2)^(1/2))/b^(3/2)-5/64*a^2*(a+b/x^2)^(1/2)/x^3-5/128*a^3*(a+b /x^2)^(1/2)/b/x
Time = 0.23 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{x^4} \, dx=\frac {\sqrt {a+\frac {b}{x^2}} \left (-\sqrt {b} \left (48 b^3+136 a b^2 x^2+118 a^2 b x^4+15 a^3 x^6\right )+\frac {15 a^4 x^8 \text {arctanh}\left (\frac {\sqrt {b+a x^2}}{\sqrt {b}}\right )}{\sqrt {b+a x^2}}\right )}{384 b^{3/2} x^7} \]
(Sqrt[a + b/x^2]*(-(Sqrt[b]*(48*b^3 + 136*a*b^2*x^2 + 118*a^2*b*x^4 + 15*a ^3*x^6)) + (15*a^4*x^8*ArcTanh[Sqrt[b + a*x^2]/Sqrt[b]])/Sqrt[b + a*x^2])) /(384*b^(3/2)*x^7)
Time = 0.23 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.08, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {858, 248, 248, 248, 262, 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.
\(\displaystyle \int \frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{x^4} \, dx\) |
\(\Big \downarrow \) 858 |
\(\displaystyle -\int \frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{x^2}d\frac {1}{x}\) |
\(\Big \downarrow \) 248 |
\(\displaystyle -\frac {5}{8} a \int \frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{x^2}d\frac {1}{x}-\frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{8 x^3}\) |
\(\Big \downarrow \) 248 |
\(\displaystyle -\frac {5}{8} a \left (\frac {1}{2} a \int \frac {\sqrt {a+\frac {b}{x^2}}}{x^2}d\frac {1}{x}+\frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{6 x^3}\right )-\frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{8 x^3}\) |
\(\Big \downarrow \) 248 |
\(\displaystyle -\frac {5}{8} a \left (\frac {1}{2} a \left (\frac {1}{4} a \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^2}d\frac {1}{x}+\frac {\sqrt {a+\frac {b}{x^2}}}{4 x^3}\right )+\frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{6 x^3}\right )-\frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{8 x^3}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle -\frac {5}{8} a \left (\frac {1}{2} a \left (\frac {1}{4} a \left (\frac {\sqrt {a+\frac {b}{x^2}}}{2 b x}-\frac {a \int \frac {1}{\sqrt {a+\frac {b}{x^2}}}d\frac {1}{x}}{2 b}\right )+\frac {\sqrt {a+\frac {b}{x^2}}}{4 x^3}\right )+\frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{6 x^3}\right )-\frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{8 x^3}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle -\frac {5}{8} a \left (\frac {1}{2} a \left (\frac {1}{4} a \left (\frac {\sqrt {a+\frac {b}{x^2}}}{2 b x}-\frac {a \int \frac {1}{1-\frac {b}{x^2}}d\frac {1}{\sqrt {a+\frac {b}{x^2}} x}}{2 b}\right )+\frac {\sqrt {a+\frac {b}{x^2}}}{4 x^3}\right )+\frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{6 x^3}\right )-\frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{8 x^3}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {5}{8} a \left (\frac {1}{2} a \left (\frac {1}{4} a \left (\frac {\sqrt {a+\frac {b}{x^2}}}{2 b x}-\frac {a \text {arctanh}\left (\frac {\sqrt {b}}{x \sqrt {a+\frac {b}{x^2}}}\right )}{2 b^{3/2}}\right )+\frac {\sqrt {a+\frac {b}{x^2}}}{4 x^3}\right )+\frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{6 x^3}\right )-\frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{8 x^3}\) |
-1/8*(a + b/x^2)^(5/2)/x^3 - (5*a*((a + b/x^2)^(3/2)/(6*x^3) + (a*(Sqrt[a + b/x^2]/(4*x^3) + (a*(Sqrt[a + b/x^2]/(2*b*x) - (a*ArcTanh[Sqrt[b]/(Sqrt[ a + b/x^2]*x)])/(2*b^(3/2))))/4))/2))/8
3.20.14.3.1 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 + 2*p + 1))), x] + Simp[2*a*(p/(m + 2*p + 1)) Int[(c*x)^m*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && GtQ[ p, 0] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, p}, x] && ILtQ[n, 0] && Int egerQ[m]
Time = 0.06 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.94
method | result | size |
risch | \(-\frac {\left (15 x^{6} a^{3}+118 a^{2} b \,x^{4}+136 a \,b^{2} x^{2}+48 b^{3}\right ) \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}{384 x^{7} b}+\frac {5 a^{4} \ln \left (\frac {2 b +2 \sqrt {b}\, \sqrt {a \,x^{2}+b}}{x}\right ) \sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, x}{128 b^{\frac {3}{2}} \sqrt {a \,x^{2}+b}}\) | \(109\) |
default | \(\frac {\left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {5}{2}} \left (-3 \left (a \,x^{2}+b \right )^{\frac {5}{2}} a^{4} x^{8}+15 b^{\frac {5}{2}} \ln \left (\frac {2 b +2 \sqrt {b}\, \sqrt {a \,x^{2}+b}}{x}\right ) a^{4} x^{8}+3 \left (a \,x^{2}+b \right )^{\frac {7}{2}} a^{3} x^{6}-5 \left (a \,x^{2}+b \right )^{\frac {3}{2}} a^{4} b \,x^{8}-15 \sqrt {a \,x^{2}+b}\, a^{4} b^{2} x^{8}+2 \left (a \,x^{2}+b \right )^{\frac {7}{2}} a^{2} b \,x^{4}+8 \left (a \,x^{2}+b \right )^{\frac {7}{2}} a \,b^{2} x^{2}-48 \left (a \,x^{2}+b \right )^{\frac {7}{2}} b^{3}\right )}{384 x^{3} \left (a \,x^{2}+b \right )^{\frac {5}{2}} b^{4}}\) | \(186\) |
-1/384*(15*a^3*x^6+118*a^2*b*x^4+136*a*b^2*x^2+48*b^3)/x^7/b*((a*x^2+b)/x^ 2)^(1/2)+5/128/b^(3/2)*a^4*ln((2*b+2*b^(1/2)*(a*x^2+b)^(1/2))/x)*((a*x^2+b )/x^2)^(1/2)*x/(a*x^2+b)^(1/2)
Time = 0.30 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.78 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{x^4} \, dx=\left [\frac {15 \, a^{4} \sqrt {b} x^{7} \log \left (-\frac {a x^{2} + 2 \, \sqrt {b} x \sqrt {\frac {a x^{2} + b}{x^{2}}} + 2 \, b}{x^{2}}\right ) - 2 \, {\left (15 \, a^{3} b x^{6} + 118 \, a^{2} b^{2} x^{4} + 136 \, a b^{3} x^{2} + 48 \, b^{4}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{768 \, b^{2} x^{7}}, -\frac {15 \, a^{4} \sqrt {-b} x^{7} \arctan \left (\frac {\sqrt {-b} x \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right ) + {\left (15 \, a^{3} b x^{6} + 118 \, a^{2} b^{2} x^{4} + 136 \, a b^{3} x^{2} + 48 \, b^{4}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{384 \, b^{2} x^{7}}\right ] \]
[1/768*(15*a^4*sqrt(b)*x^7*log(-(a*x^2 + 2*sqrt(b)*x*sqrt((a*x^2 + b)/x^2) + 2*b)/x^2) - 2*(15*a^3*b*x^6 + 118*a^2*b^2*x^4 + 136*a*b^3*x^2 + 48*b^4) *sqrt((a*x^2 + b)/x^2))/(b^2*x^7), -1/384*(15*a^4*sqrt(-b)*x^7*arctan(sqrt (-b)*x*sqrt((a*x^2 + b)/x^2)/(a*x^2 + b)) + (15*a^3*b*x^6 + 118*a^2*b^2*x^ 4 + 136*a*b^3*x^2 + 48*b^4)*sqrt((a*x^2 + b)/x^2))/(b^2*x^7)]
Time = 6.77 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.29 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{x^4} \, dx=- \frac {5 a^{\frac {7}{2}}}{128 b x \sqrt {1 + \frac {b}{a x^{2}}}} - \frac {133 a^{\frac {5}{2}}}{384 x^{3} \sqrt {1 + \frac {b}{a x^{2}}}} - \frac {127 a^{\frac {3}{2}} b}{192 x^{5} \sqrt {1 + \frac {b}{a x^{2}}}} - \frac {23 \sqrt {a} b^{2}}{48 x^{7} \sqrt {1 + \frac {b}{a x^{2}}}} + \frac {5 a^{4} \operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} x} \right )}}{128 b^{\frac {3}{2}}} - \frac {b^{3}}{8 \sqrt {a} x^{9} \sqrt {1 + \frac {b}{a x^{2}}}} \]
-5*a**(7/2)/(128*b*x*sqrt(1 + b/(a*x**2))) - 133*a**(5/2)/(384*x**3*sqrt(1 + b/(a*x**2))) - 127*a**(3/2)*b/(192*x**5*sqrt(1 + b/(a*x**2))) - 23*sqrt (a)*b**2/(48*x**7*sqrt(1 + b/(a*x**2))) + 5*a**4*asinh(sqrt(b)/(sqrt(a)*x) )/(128*b**(3/2)) - b**3/(8*sqrt(a)*x**9*sqrt(1 + b/(a*x**2)))
Leaf count of result is larger than twice the leaf count of optimal. 190 vs. \(2 (92) = 184\).
Time = 0.27 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.64 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{x^4} \, dx=-\frac {5 \, a^{4} \log \left (\frac {\sqrt {a + \frac {b}{x^{2}}} x - \sqrt {b}}{\sqrt {a + \frac {b}{x^{2}}} x + \sqrt {b}}\right )}{256 \, b^{\frac {3}{2}}} - \frac {15 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {7}{2}} a^{4} x^{7} + 73 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {5}{2}} a^{4} b x^{5} - 55 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} a^{4} b^{2} x^{3} + 15 \, \sqrt {a + \frac {b}{x^{2}}} a^{4} b^{3} x}{384 \, {\left ({\left (a + \frac {b}{x^{2}}\right )}^{4} b x^{8} - 4 \, {\left (a + \frac {b}{x^{2}}\right )}^{3} b^{2} x^{6} + 6 \, {\left (a + \frac {b}{x^{2}}\right )}^{2} b^{3} x^{4} - 4 \, {\left (a + \frac {b}{x^{2}}\right )} b^{4} x^{2} + b^{5}\right )}} \]
-5/256*a^4*log((sqrt(a + b/x^2)*x - sqrt(b))/(sqrt(a + b/x^2)*x + sqrt(b)) )/b^(3/2) - 1/384*(15*(a + b/x^2)^(7/2)*a^4*x^7 + 73*(a + b/x^2)^(5/2)*a^4 *b*x^5 - 55*(a + b/x^2)^(3/2)*a^4*b^2*x^3 + 15*sqrt(a + b/x^2)*a^4*b^3*x)/ ((a + b/x^2)^4*b*x^8 - 4*(a + b/x^2)^3*b^2*x^6 + 6*(a + b/x^2)^2*b^3*x^4 - 4*(a + b/x^2)*b^4*x^2 + b^5)
Time = 0.31 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{x^4} \, dx=-\frac {\frac {15 \, a^{5} \arctan \left (\frac {\sqrt {a x^{2} + b}}{\sqrt {-b}}\right ) \mathrm {sgn}\left (x\right )}{\sqrt {-b} b} + \frac {15 \, {\left (a x^{2} + b\right )}^{\frac {7}{2}} a^{5} \mathrm {sgn}\left (x\right ) + 73 \, {\left (a x^{2} + b\right )}^{\frac {5}{2}} a^{5} b \mathrm {sgn}\left (x\right ) - 55 \, {\left (a x^{2} + b\right )}^{\frac {3}{2}} a^{5} b^{2} \mathrm {sgn}\left (x\right ) + 15 \, \sqrt {a x^{2} + b} a^{5} b^{3} \mathrm {sgn}\left (x\right )}{a^{4} b x^{8}}}{384 \, a} \]
-1/384*(15*a^5*arctan(sqrt(a*x^2 + b)/sqrt(-b))*sgn(x)/(sqrt(-b)*b) + (15* (a*x^2 + b)^(7/2)*a^5*sgn(x) + 73*(a*x^2 + b)^(5/2)*a^5*b*sgn(x) - 55*(a*x ^2 + b)^(3/2)*a^5*b^2*sgn(x) + 15*sqrt(a*x^2 + b)*a^5*b^3*sgn(x))/(a^4*b*x ^8))/a
Timed out. \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{x^4} \, dx=\int \frac {{\left (a+\frac {b}{x^2}\right )}^{5/2}}{x^4} \,d x \]