Optimal. Leaf size=27 \[ \frac {x^2}{2 a}-\frac {b \log \left (b+a x^2\right )}{2 a^2} \]
[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 = {272, 45}
\begin {gather*} \frac {x^2}{2 a}-\frac {b \log \left (a x^2+b\right )}{2 a^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {x^3}{b+a x^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x}{b+a x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{a}-\frac {b}{a (b+a x)}\right ) \, dx,x,x^2\right )\\ &=\frac {x^2}{2 a}-\frac {b \log \left (b+a x^2\right )}{2 a^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.00, size = 27, normalized size = 1.00 \begin {gather*} \frac {x^2}{2 a}-\frac {b \log \left (b+a x^2\right )}{2 a^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.03, size = 24, normalized size = 0.89
method | result | size |
default | \(\frac {x^{2}}{2 a}-\frac {b \ln \left (a \,x^{2}+b \right )}{2 a^{2}}\) | \(24\) |
norman | \(\frac {x^{2}}{2 a}-\frac {b \ln \left (a \,x^{2}+b \right )}{2 a^{2}}\) | \(24\) |
risch | \(\frac {x^{2}}{2 a}-\frac {b \ln \left (a \,x^{2}+b \right )}{2 a^{2}}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 5.54, size = 23, normalized size = 0.85 \begin {gather*} \frac {x^{2}}{2 \, a} - \frac {b \log \left (a x^{2} + b\right )}{2 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.85, size = 22, normalized size = 0.81 \begin {gather*} \frac {a x^{2} - b \log \left (a x^{2} + b\right )}{2 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.05, size = 20, normalized size = 0.74 \begin {gather*} \frac {x^{2}}{2 a} - \frac {b \log {\left (a x^{2} + b \right )}}{2 a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.44, size = 24, normalized size = 0.89 \begin {gather*} \frac {x^{2}}{2 \, a} - \frac {b \log \left ({\left | a x^{2} + b \right |}\right )}{2 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.15, size = 22, normalized size = 0.81 \begin {gather*} -\frac {b\,\ln \left (a\,x^2+b\right )-a\,x^2}{2\,a^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________