Optimal. Leaf size=33 \[ \frac {a}{2 b^2 \left (a+b x^2\right )}+\frac {\log \left (a+b x^2\right )}{2 b^2} \]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 33, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {a}{2 b^2 \left (a+b x^2\right )}+\frac {\log \left (a+b x^2\right )}{2 b^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 28
Rule 43
Rule 266
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 = 0.01, size = 27, normalized size = 0.82 \begin {gather*} \frac {\frac {a}{a+b x^2}+\log \left (a+b x^2\right )}{2 b^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^3}{a^2+2 a b x^2+b^2 x^4} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.85, size = 35, normalized size = 1.06 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 30, normalized size = 0.91 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 30, normalized size = 0.91 \begin {gather*} \frac {a}{2 \left (b \,x^{2}+a \right ) b^{2}}+\frac {\ln \left (b \,x^{2}+a \right )}{2 b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.41, size = 32, normalized size = 0.97 \begin {gather*} \frac {a}{2 \, {\left (b^{3} x^{2} + a b^{2}\right )}} + \frac {\log \left (b x^{2} + a\right )}{2 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.05, size = 29, normalized size = 0.88 \begin {gather*} \frac {\ln \left (b\,x^2+a\right )}{2\,b^2}+\frac {a}{2\,b^2\,\left (b\,x^2+a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.21, size = 29, normalized size = 0.88 \begin {gather*} \frac {a}{2 a b^{2} + 2 b^{3} x^{2}} + \frac {\log {\left (a + b x^{2} \right )}}{2 b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________