Optimal. Leaf size=63 \[ -\frac {\left (a+b x^2\right )^{3/2}}{2 x^2}+\frac {3}{2} b \sqrt {a+b x^2}-\frac {3}{2} \sqrt {a} b \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) \]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 63, 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, 50, 63, 208} \begin {gather*} -\frac {\left (a+b x^2\right )^{3/2}}{2 x^2}+\frac {3}{2} b \sqrt {a+b x^2}-\frac {3}{2} \sqrt {a} b \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 47
Rule 50
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{3/2}}{x^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^{3/2}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {\left (a+b x^2\right )^{3/2}}{2 x^2}+\frac {1}{4} (3 b) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,x^2\right )\\ &=\frac {3}{2} b \sqrt {a+b x^2}-\frac {\left (a+b x^2\right )^{3/2}}{2 x^2}+\frac {1}{4} (3 a b) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )\\ &=\frac {3}{2} b \sqrt {a+b x^2}-\frac {\left (a+b x^2\right )^{3/2}}{2 x^2}+\frac {1}{2} (3 a) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )\\ &=\frac {3}{2} b \sqrt {a+b x^2}-\frac {\left (a+b x^2\right )^{3/2}}{2 x^2}-\frac {3}{2} \sqrt {a} b \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.01, size = 37, normalized size = 0.59 \begin {gather*} \frac {b \left (a+b x^2\right )^{5/2} \, _2F_1\left (2,\frac {5}{2};\frac {7}{2};\frac {b x^2}{a}+1\right )}{5 a^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.08, size = 57, normalized size = 0.90 \begin {gather*} \frac {\sqrt {a+b x^2} \left (2 b x^2-a\right )}{2 x^2}-\frac {3}{2} \sqrt {a} b \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.62, size = 119, normalized size = 1.89 \begin {gather*} \left [\frac {3 \, \sqrt {a} b x^{2} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (2 \, b x^{2} - a\right )} \sqrt {b x^{2} + a}}{4 \, x^{2}}, \frac {3 \, \sqrt {-a} b x^{2} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + {\left (2 \, b x^{2} - a\right )} \sqrt {b x^{2} + a}}{2 \, x^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.68, size = 63, normalized size = 1.00 \begin {gather*} \frac {\frac {3 \, a b^{2} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + 2 \, \sqrt {b x^{2} + a} b^{2} - \frac {\sqrt {b x^{2} + a} a b}{x^{2}}}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 75, normalized size = 1.19 \begin {gather*} -\frac {3 \sqrt {a}\, b \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{2}+\frac {3 \sqrt {b \,x^{2}+a}\, b}{2}+\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} b}{2 a}-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.34, size = 63, normalized size = 1.00 \begin {gather*} -\frac {3}{2} \, \sqrt {a} b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right ) + \frac {3}{2} \, \sqrt {b x^{2} + a} b + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} b}{2 \, a} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{2 \, a x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.82, size = 47, normalized size = 0.75 \begin {gather*} b\,\sqrt {b\,x^2+a}-\frac {a\,\sqrt {b\,x^2+a}}{2\,x^2}-\frac {3\,\sqrt {a}\,b\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 2.33, size = 88, normalized size = 1.40 \begin {gather*} - \frac {3 \sqrt {a} b \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{2} - \frac {a^{2}}{2 \sqrt {b} x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {a \sqrt {b}}{2 x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {b^{\frac {3}{2}} x}{\sqrt {\frac {a}{b x^{2}} + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________