Optimal. Leaf size=71 \[ -\frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{8 a^{3/2}}+\frac {b x^2 \sqrt {a+\frac {b}{x^2}}}{8 a}+\frac {1}{4} x^4 \sqrt {a+\frac {b}{x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {266, 47, 51, 63, 208} \[ -\frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{8 a^{3/2}}+\frac {1}{4} x^4 \sqrt {a+\frac {b}{x^2}}+\frac {b x^2 \sqrt {a+\frac {b}{x^2}}}{8 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 47
Rule 51
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \sqrt {a+\frac {b}{x^2}} x^3 \, dx &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x^3} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=\frac {1}{4} \sqrt {a+\frac {b}{x^2}} x^4-\frac {1}{8} b \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,\frac {1}{x^2}\right )\\ &=\frac {b \sqrt {a+\frac {b}{x^2}} x^2}{8 a}+\frac {1}{4} \sqrt {a+\frac {b}{x^2}} x^4+\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x^2}\right )}{16 a}\\ &=\frac {b \sqrt {a+\frac {b}{x^2}} x^2}{8 a}+\frac {1}{4} \sqrt {a+\frac {b}{x^2}} x^4+\frac {b \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x^2}}\right )}{8 a}\\ &=\frac {b \sqrt {a+\frac {b}{x^2}} x^2}{8 a}+\frac {1}{4} \sqrt {a+\frac {b}{x^2}} x^4-\frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{8 a^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 88, normalized size = 1.24 \[ x \sqrt {a+\frac {b}{x^2}} \left (\frac {b x}{8 a}+\frac {x^3}{4}\right )-\frac {b^2 x \sqrt {a+\frac {b}{x^2}} \log \left (\sqrt {a} \sqrt {a x^2+b}+a x\right )}{8 a^{3/2} \sqrt {a x^2+b}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.89, size = 152, normalized size = 2.14 \[ \left [\frac {\sqrt {a} b^{2} \log \left (-2 \, a x^{2} + 2 \, \sqrt {a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}} - b\right ) + 2 \, {\left (2 \, a^{2} x^{4} + a b x^{2}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{16 \, a^{2}}, \frac {\sqrt {-a} b^{2} \arctan \left (\frac {\sqrt {-a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right ) + {\left (2 \, a^{2} x^{4} + a b x^{2}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{8 \, a^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 69, normalized size = 0.97 \[ \frac {1}{8} \, \sqrt {a x^{2} + b} {\left (2 \, x^{2} \mathrm {sgn}\relax (x) + \frac {b \mathrm {sgn}\relax (x)}{a}\right )} x + \frac {b^{2} \log \left ({\left | -\sqrt {a} x + \sqrt {a x^{2} + b} \right |}\right ) \mathrm {sgn}\relax (x)}{8 \, a^{\frac {3}{2}}} - \frac {b^{2} \log \left ({\left | b \right |}\right ) \mathrm {sgn}\relax (x)}{16 \, a^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 82, normalized size = 1.15 \[ \frac {\sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, \left (-b^{2} \ln \left (\sqrt {a}\, x +\sqrt {a \,x^{2}+b}\right )-\sqrt {a \,x^{2}+b}\, \sqrt {a}\, b x +2 \left (a \,x^{2}+b \right )^{\frac {3}{2}} \sqrt {a}\, x \right ) x}{8 \sqrt {a \,x^{2}+b}\, a^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.97, size = 100, normalized size = 1.41 \[ \frac {b^{2} \log \left (\frac {\sqrt {a + \frac {b}{x^{2}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{2}}} + \sqrt {a}}\right )}{16 \, a^{\frac {3}{2}}} + \frac {{\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} b^{2} + \sqrt {a + \frac {b}{x^{2}}} a b^{2}}{8 \, {\left ({\left (a + \frac {b}{x^{2}}\right )}^{2} a - 2 \, {\left (a + \frac {b}{x^{2}}\right )} a^{2} + a^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.61, size = 54, normalized size = 0.76 \[ \frac {x^4\,\sqrt {a+\frac {b}{x^2}}}{8}-\frac {b^2\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{8\,a^{3/2}}+\frac {x^4\,{\left (a+\frac {b}{x^2}\right )}^{3/2}}{8\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 3.93, size = 92, normalized size = 1.30 \[ \frac {a x^{5}}{4 \sqrt {b} \sqrt {\frac {a x^{2}}{b} + 1}} + \frac {3 \sqrt {b} x^{3}}{8 \sqrt {\frac {a x^{2}}{b} + 1}} + \frac {b^{\frac {3}{2}} x}{8 a \sqrt {\frac {a x^{2}}{b} + 1}} - \frac {b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a} x}{\sqrt {b}} \right )}}{8 a^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________