Optimal. Leaf size=27 \[ \frac {a^2 x^2}{2}+2 a b \log (x)-\frac {b^2}{2 x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {263, 266, 43} \[ \frac {a^2 x^2}{2}+2 a b \log (x)-\frac {b^2}{2 x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 263
Rule 266
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x^2}\right )^2 x \, dx &=\int \frac {\left (b+a x^2\right )^2}{x^3} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(b+a x)^2}{x^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (a^2+\frac {b^2}{x^2}+\frac {2 a b}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac {b^2}{2 x^2}+\frac {a^2 x^2}{2}+2 a b \log (x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 27, normalized size = 1.00 \[ \frac {a^2 x^2}{2}+2 a b \log (x)-\frac {b^2}{2 x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.83, size = 27, normalized size = 1.00 \[ \frac {a^{2} x^{4} + 4 \, a b x^{2} \log \relax (x) - b^{2}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 24, normalized size = 0.89 \[ \frac {1}{2} \, a^{2} x^{2} + 2 \, a b \log \left ({\left | x \right |}\right ) - \frac {b^{2}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 24, normalized size = 0.89 \[ \frac {a^{2} x^{2}}{2}+2 a b \ln \relax (x )-\frac {b^{2}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.88, size = 24, normalized size = 0.89 \[ \frac {1}{2} \, a^{2} x^{2} + a b \log \left (x^{2}\right ) - \frac {b^{2}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.06, size = 23, normalized size = 0.85 \[ \frac {a^2\,x^2}{2}-\frac {b^2}{2\,x^2}+2\,a\,b\,\ln \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.14, size = 24, normalized size = 0.89 \[ \frac {a^{2} x^{2}}{2} + 2 a b \log {\relax (x )} - \frac {b^{2}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________