Optimal. Leaf size=49 \[ -\frac {1}{2 a^2 x^2}-\frac {b}{2 a^2 \left (a+b x^2\right )}-\frac {2 b \log (x)}{a^3}+\frac {b \log \left (a+b x^2\right )}{a^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1598, 272, 46}
\begin {gather*} \frac {b \log \left (a+b x^2\right )}{a^3}-\frac {2 b \log (x)}{a^3}-\frac {b}{2 a^2 \left (a+b x^2\right )}-\frac {1}{2 a^2 x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 46
Rule 272
Rule 1598
Rubi steps
\begin {align*} \int \frac {1}{x \left (a x+b x^3\right )^2} \, dx &=\int \frac {1}{x^3 \left (a+b x^2\right )^2} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 (a+b x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{a^2 x^2}-\frac {2 b}{a^3 x}+\frac {b^2}{a^2 (a+b x)^2}+\frac {2 b^2}{a^3 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{2 a^2 x^2}-\frac {b}{2 a^2 \left (a+b x^2\right )}-\frac {2 b \log (x)}{a^3}+\frac {b \log \left (a+b x^2\right )}{a^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 41, normalized size = 0.84 \begin {gather*} -\frac {a \left (\frac {1}{x^2}+\frac {b}{a+b x^2}\right )+4 b \log (x)-2 b \log \left (a+b x^2\right )}{2 a^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.36, size = 55, normalized size = 1.12
method | result | size |
norman | \(\frac {\frac {b^{2} x^{4}}{a^{3}}-\frac {1}{2 a}}{x^{2} \left (b \,x^{2}+a \right )}+\frac {b \ln \left (b \,x^{2}+a \right )}{a^{3}}-\frac {2 b \ln \left (x \right )}{a^{3}}\) | \(52\) |
risch | \(\frac {-\frac {b \,x^{2}}{a^{2}}-\frac {1}{2 a}}{x^{2} \left (b \,x^{2}+a \right )}-\frac {2 b \ln \left (x \right )}{a^{3}}+\frac {b \ln \left (-b \,x^{2}-a \right )}{a^{3}}\) | \(54\) |
default | \(\frac {b^{2} \left (-\frac {a}{b \left (b \,x^{2}+a \right )}+\frac {2 \ln \left (b \,x^{2}+a \right )}{b}\right )}{2 a^{3}}-\frac {1}{2 a^{2} x^{2}}-\frac {2 b \ln \left (x \right )}{a^{3}}\) | \(55\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.29, size = 50, normalized size = 1.02 \begin {gather*} -\frac {2 \, b x^{2} + a}{2 \, {\left (a^{2} b x^{4} + a^{3} x^{2}\right )}} + \frac {b \log \left (b x^{2} + a\right )}{a^{3}} - \frac {2 \, b \log \left (x\right )}{a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.35, size = 73, normalized size = 1.49 \begin {gather*} -\frac {2 \, a b x^{2} + a^{2} - 2 \, {\left (b^{2} x^{4} + a b x^{2}\right )} \log \left (b x^{2} + a\right ) + 4 \, {\left (b^{2} x^{4} + a b x^{2}\right )} \log \left (x\right )}{2 \, {\left (a^{3} b x^{4} + a^{4} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.18, size = 51, normalized size = 1.04 \begin {gather*} \frac {- a - 2 b x^{2}}{2 a^{3} x^{2} + 2 a^{2} b x^{4}} - \frac {2 b \log {\left (x \right )}}{a^{3}} + \frac {b \log {\left (\frac {a}{b} + x^{2} \right )}}{a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.37, size = 51, normalized size = 1.04 \begin {gather*} -\frac {b \log \left (x^{2}\right )}{a^{3}} + \frac {b \log \left ({\left | b x^{2} + a \right |}\right )}{a^{3}} - \frac {2 \, b x^{2} + a}{2 \, {\left (b x^{4} + a x^{2}\right )} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.05, size = 51, normalized size = 1.04 \begin {gather*} \frac {b\,\ln \left (b\,x^2+a\right )}{a^3}-\frac {\frac {1}{2\,a}+\frac {b\,x^2}{a^2}}{b\,x^4+a\,x^2}-\frac {2\,b\,\ln \left (x\right )}{a^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________