Optimal. Leaf size=33 \[ \frac{a}{2 b^2 \left (a+b x^2\right )}+\frac{\log \left (a+b x^2\right )}{2 b^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0284405, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {28, 266, 43} \[ \frac{a}{2 b^2 \left (a+b x^2\right )}+\frac{\log \left (a+b x^2\right )}{2 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 28
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^3}{a^2+2 a b x^2+b^2 x^4} \, dx &=b^2 \int \frac{x^3}{\left (a b+b^2 x^2\right )^2} \, dx\\ &=\frac{1}{2} b^2 \operatorname{Subst}\left (\int \frac{x}{\left (a b+b^2 x\right )^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} b^2 \operatorname{Subst}\left (\int \left (-\frac{a}{b^3 (a+b x)^2}+\frac{1}{b^3 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac{a}{2 b^2 \left (a+b x^2\right )}+\frac{\log \left (a+b x^2\right )}{2 b^2}\\ \end{align*}
Mathematica [A] time = 0.0085283, size = 27, normalized size = 0.82 \[ \frac{\frac{a}{a+b x^2}+\log \left (a+b x^2\right )}{2 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.046, size = 30, normalized size = 0.9 \begin{align*}{\frac{a}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{\ln \left ( b{x}^{2}+a \right ) }{2\,{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.01066, size = 43, normalized size = 1.3 \begin{align*} \frac{a}{2 \,{\left (b^{3} x^{2} + a b^{2}\right )}} + \frac{\log \left (b x^{2} + a\right )}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.66032, size = 76, normalized size = 2.3 \begin{align*} \frac{{\left (b x^{2} + a\right )} \log \left (b x^{2} + a\right ) + a}{2 \,{\left (b^{3} x^{2} + a b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.339552, size = 29, normalized size = 0.88 \begin{align*} \frac{a}{2 a b^{2} + 2 b^{3} x^{2}} + \frac{\log{\left (a + b x^{2} \right )}}{2 b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.1272, size = 41, normalized size = 1.24 \begin{align*} \frac{\log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{2}} + \frac{a}{2 \,{\left (b x^{2} + a\right )} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]