Optimal. Leaf size=72 \[ a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )-a^2 \sqrt {a+\frac {b}{x^2}}-\frac {1}{3} a \left (a+\frac {b}{x^2}\right )^{3/2}-\frac {1}{5} \left (a+\frac {b}{x^2}\right )^{5/2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {266, 50, 63, 208} \[ -a^2 \sqrt {a+\frac {b}{x^2}}+a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )-\frac {1}{3} a \left (a+\frac {b}{x^2}\right )^{3/2}-\frac {1}{5} \left (a+\frac {b}{x^2}\right )^{5/2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 50
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{x} \, dx &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^{5/2}}{x} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=-\frac {1}{5} \left (a+\frac {b}{x^2}\right )^{5/2}-\frac {1}{2} a \operatorname {Subst}\left (\int \frac {(a+b x)^{3/2}}{x} \, dx,x,\frac {1}{x^2}\right )\\ &=-\frac {1}{3} a \left (a+\frac {b}{x^2}\right )^{3/2}-\frac {1}{5} \left (a+\frac {b}{x^2}\right )^{5/2}-\frac {1}{2} a^2 \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,\frac {1}{x^2}\right )\\ &=-a^2 \sqrt {a+\frac {b}{x^2}}-\frac {1}{3} a \left (a+\frac {b}{x^2}\right )^{3/2}-\frac {1}{5} \left (a+\frac {b}{x^2}\right )^{5/2}-\frac {1}{2} a^3 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x^2}\right )\\ &=-a^2 \sqrt {a+\frac {b}{x^2}}-\frac {1}{3} a \left (a+\frac {b}{x^2}\right )^{3/2}-\frac {1}{5} \left (a+\frac {b}{x^2}\right )^{5/2}-\frac {a^3 \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x^2}}\right )}{b}\\ &=-a^2 \sqrt {a+\frac {b}{x^2}}-\frac {1}{3} a \left (a+\frac {b}{x^2}\right )^{3/2}-\frac {1}{5} \left (a+\frac {b}{x^2}\right )^{5/2}+a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.01, size = 54, normalized size = 0.75 \[ -\frac {b^2 \sqrt {a+\frac {b}{x^2}} \, _2F_1\left (-\frac {5}{2},-\frac {5}{2};-\frac {3}{2};-\frac {a x^2}{b}\right )}{5 x^4 \sqrt {\frac {a x^2}{b}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.90, size = 169, normalized size = 2.35 \[ \left [\frac {15 \, a^{\frac {5}{2}} x^{4} \log \left (-2 \, a x^{2} - 2 \, \sqrt {a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}} - b\right ) - 2 \, {\left (23 \, a^{2} x^{4} + 11 \, a b x^{2} + 3 \, b^{2}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{30 \, x^{4}}, -\frac {15 \, \sqrt {-a} a^{2} x^{4} \arctan \left (\frac {\sqrt {-a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right ) + {\left (23 \, a^{2} x^{4} + 11 \, a b x^{2} + 3 \, b^{2}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{15 \, x^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 1.06, size = 180, normalized size = 2.50 \[ -\frac {1}{2} \, a^{\frac {5}{2}} \log \left ({\left (\sqrt {a} x - \sqrt {a x^{2} + b}\right )}^{2}\right ) \mathrm {sgn}\relax (x) + \frac {2 \, {\left (45 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b}\right )}^{8} a^{\frac {5}{2}} b \mathrm {sgn}\relax (x) - 90 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b}\right )}^{6} a^{\frac {5}{2}} b^{2} \mathrm {sgn}\relax (x) + 140 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b}\right )}^{4} a^{\frac {5}{2}} b^{3} \mathrm {sgn}\relax (x) - 70 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b}\right )}^{2} a^{\frac {5}{2}} b^{4} \mathrm {sgn}\relax (x) + 23 \, a^{\frac {5}{2}} b^{5} \mathrm {sgn}\relax (x)\right )}}{15 \, {\left ({\left (\sqrt {a} x - \sqrt {a x^{2} + b}\right )}^{2} - b\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.01, size = 166, normalized size = 2.31 \[ \frac {\left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {5}{2}} \left (15 a^{3} b^{3} x^{5} \ln \left (\sqrt {a}\, x +\sqrt {a \,x^{2}+b}\right )+15 \sqrt {a \,x^{2}+b}\, a^{\frac {7}{2}} b^{2} x^{6}+10 \left (a \,x^{2}+b \right )^{\frac {3}{2}} a^{\frac {7}{2}} b \,x^{6}+8 \left (a \,x^{2}+b \right )^{\frac {5}{2}} a^{\frac {7}{2}} x^{6}-8 \left (a \,x^{2}+b \right )^{\frac {7}{2}} a^{\frac {5}{2}} x^{4}-2 \left (a \,x^{2}+b \right )^{\frac {7}{2}} a^{\frac {3}{2}} b \,x^{2}-3 \left (a \,x^{2}+b \right )^{\frac {7}{2}} \sqrt {a}\, b^{2}\right )}{15 \left (a \,x^{2}+b \right )^{\frac {5}{2}} \sqrt {a}\, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.83, size = 75, normalized size = 1.04 \[ -\frac {1}{2} \, a^{\frac {5}{2}} \log \left (\frac {\sqrt {a + \frac {b}{x^{2}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{2}}} + \sqrt {a}}\right ) - \frac {1}{5} \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {5}{2}} - \frac {1}{3} \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} a - \sqrt {a + \frac {b}{x^{2}}} a^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.65, size = 60, normalized size = 0.83 \[ -\frac {a\,{\left (a+\frac {b}{x^2}\right )}^{3/2}}{3}-\frac {{\left (a+\frac {b}{x^2}\right )}^{5/2}}{5}-a^2\,\sqrt {a+\frac {b}{x^2}}-a^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {a+\frac {b}{x^2}}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,1{}\mathrm {i} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 4.10, size = 105, normalized size = 1.46 \[ - \frac {23 a^{\frac {5}{2}} \sqrt {1 + \frac {b}{a x^{2}}}}{15} - \frac {a^{\frac {5}{2}} \log {\left (\frac {b}{a x^{2}} \right )}}{2} + a^{\frac {5}{2}} \log {\left (\sqrt {1 + \frac {b}{a x^{2}}} + 1 \right )} - \frac {11 a^{\frac {3}{2}} b \sqrt {1 + \frac {b}{a x^{2}}}}{15 x^{2}} - \frac {\sqrt {a} b^{2} \sqrt {1 + \frac {b}{a x^{2}}}}{5 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________