Optimal. Leaf size=31 \[ \frac {x^3 \sqrt {b-\frac {a}{x^2}}}{2 \sqrt {a-b x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {515, 23, 30} \[ \frac {x^3 \sqrt {b-\frac {a}{x^2}}}{2 \sqrt {a-b x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 23
Rule 30
Rule 515
Rubi steps
\begin {align*} \int \frac {\sqrt {b-\frac {a}{x^2}} x^2}{\sqrt {a-b x^2}} \, dx &=\frac {\left (\sqrt {b-\frac {a}{x^2}} x\right ) \int \frac {x \sqrt {-a+b x^2}}{\sqrt {a-b x^2}} \, dx}{\sqrt {-a+b x^2}}\\ &=\frac {\left (\sqrt {b-\frac {a}{x^2}} x\right ) \int x \, dx}{\sqrt {a-b x^2}}\\ &=\frac {\sqrt {b-\frac {a}{x^2}} x^3}{2 \sqrt {a-b x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 31, normalized size = 1.00 \[ \frac {x^3 \sqrt {b-\frac {a}{x^2}}}{2 \sqrt {a-b x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.42, size = 41, normalized size = 1.32 \[ -\frac {\sqrt {-b x^{2} + a} x^{3} \sqrt {\frac {b x^{2} - a}{x^{2}}}}{2 \, {\left (b x^{2} - a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [C] time = 0.33, size = 15, normalized size = 0.48 \[ -\frac {i \, b x^{2} - i \, a}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 31, normalized size = 1.00 \[ \frac {\sqrt {-\frac {-b \,x^{2}+a}{x^{2}}}\, x^{3}}{2 \sqrt {-b \,x^{2}+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [C] time = 1.02, size = 5, normalized size = 0.16 \[ -\frac {1}{2} i \, x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.14, size = 25, normalized size = 0.81 \[ \frac {x^3\,\sqrt {b-\frac {a}{x^2}}}{2\,\sqrt {a-b\,x^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \sqrt {- \frac {a}{x^{2}} + b}}{\sqrt {a - b x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________