Optimal. Leaf size=27 \[ -\frac {a^2}{2 x^2}+2 a b \log (x)+\frac {b^2 x^2}{2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 43} \[ -\frac {a^2}{2 x^2}+2 a b \log (x)+\frac {b^2 x^2}{2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2}{x^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^2}{x^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (b^2+\frac {a^2}{x^2}+\frac {2 a b}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac {a^2}{2 x^2}+\frac {b^2 x^2}{2}+2 a b \log (x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 27, normalized size = 1.00 \[ -\frac {a^2}{2 x^2}+2 a b \log (x)+\frac {b^2 x^2}{2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.81, size = 27, normalized size = 1.00 \[ \frac {b^{2} x^{4} + 4 \, a b x^{2} \log \relax (x) - a^{2}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.14, size = 32, normalized size = 1.19 \[ \frac {1}{2} \, b^{2} x^{2} + a b \log \left (x^{2}\right ) - \frac {2 \, a b x^{2} + a^{2}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 24, normalized size = 0.89 \[ \frac {b^{2} x^{2}}{2}+2 a b \ln \relax (x )-\frac {a^{2}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.39, size = 24, normalized size = 0.89 \[ \frac {1}{2} \, b^{2} x^{2} + a b \log \left (x^{2}\right ) - \frac {a^{2}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.94, size = 23, normalized size = 0.85 \[ \frac {b^2\,x^2}{2}-\frac {a^2}{2\,x^2}+2\,a\,b\,\ln \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.13, size = 24, normalized size = 0.89 \[ - \frac {a^{2}}{2 x^{2}} + 2 a b \log {\relax (x )} + \frac {b^{2} x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________