Optimal. Leaf size=45 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 b^{3/2} \sqrt {c}}+\frac {x}{2 b \left (b+c x^2\right )} \]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1584, 199, 205} \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 b^{3/2} \sqrt {c}}+\frac {x}{2 b \left (b+c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 199
Rule 205
Rule 1584
Rubi steps
\begin {align*} \int \frac {x^4}{\left (b x^2+c x^4\right )^2} \, dx &=\int \frac {1}{\left (b+c x^2\right )^2} \, dx\\ &=\frac {x}{2 b \left (b+c x^2\right )}+\frac {\int \frac {1}{b+c x^2} \, dx}{2 b}\\ &=\frac {x}{2 b \left (b+c x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 b^{3/2} \sqrt {c}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 45, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 b^{3/2} \sqrt {c}}+\frac {x}{2 b \left (b+c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^4}{\left (b x^2+c x^4\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.71, size = 120, normalized size = 2.67 \begin {gather*} \left [\frac {2 \, b c x - {\left (c x^{2} + b\right )} \sqrt {-b c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-b c} x - b}{c x^{2} + b}\right )}{4 \, {\left (b^{2} c^{2} x^{2} + b^{3} c\right )}}, \frac {b c x + {\left (c x^{2} + b\right )} \sqrt {b c} \arctan \left (\frac {\sqrt {b c} x}{b}\right )}{2 \, {\left (b^{2} c^{2} x^{2} + b^{3} c\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.15, size = 35, normalized size = 0.78 \begin {gather*} \frac {\arctan \left (\frac {c x}{\sqrt {b c}}\right )}{2 \, \sqrt {b c} b} + \frac {x}{2 \, {\left (c x^{2} + b\right )} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 36, normalized size = 0.80 \begin {gather*} \frac {x}{2 \left (c \,x^{2}+b \right ) b}+\frac {\arctan \left (\frac {c x}{\sqrt {b c}}\right )}{2 \sqrt {b c}\, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 3.00, size = 35, normalized size = 0.78 \begin {gather*} \frac {x}{2 \, {\left (b c x^{2} + b^{2}\right )}} + \frac {\arctan \left (\frac {c x}{\sqrt {b c}}\right )}{2 \, \sqrt {b c} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.04, size = 33, normalized size = 0.73 \begin {gather*} \frac {x}{2\,b\,\left (c\,x^2+b\right )}+\frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {b}}\right )}{2\,b^{3/2}\,\sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.25, size = 78, normalized size = 1.73 \begin {gather*} \frac {x}{2 b^{2} + 2 b c x^{2}} - \frac {\sqrt {- \frac {1}{b^{3} c}} \log {\left (- b^{2} \sqrt {- \frac {1}{b^{3} c}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{b^{3} c}} \log {\left (b^{2} \sqrt {- \frac {1}{b^{3} c}} + x \right )}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________