Optimal. Leaf size=83 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (3 c d^2-e (a e+b d)\right )}{2 d^{3/2} e^{5/2}}+\frac {x \left (a+\frac {d (c d-b e)}{e^2}\right )}{2 d \left (d+e x^2\right )}+\frac {c x}{e^2} \]
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Rubi [A] time = 0.09, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1157, 388, 205} \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (3 c d^2-e (a e+b d)\right )}{2 d^{3/2} e^{5/2}}+\frac {x \left (a+\frac {d (c d-b e)}{e^2}\right )}{2 d \left (d+e x^2\right )}+\frac {c x}{e^2} \]
Antiderivative was successfully verified.
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Rule 205
Rule 388
Rule 1157
Rubi steps
\begin {align*} \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^2} \, dx &=\frac {\left (a+\frac {d (c d-b e)}{e^2}\right ) x}{2 d \left (d+e x^2\right )}-\frac {\int \frac {\frac {c d^2-e (b d+a e)}{e^2}-\frac {2 c d x^2}{e}}{d+e x^2} \, dx}{2 d}\\ &=\frac {c x}{e^2}+\frac {\left (a+\frac {d (c d-b e)}{e^2}\right ) x}{2 d \left (d+e x^2\right )}-\frac {\left (3 c d^2-e (b d+a e)\right ) \int \frac {1}{d+e x^2} \, dx}{2 d e^2}\\ &=\frac {c x}{e^2}+\frac {\left (a+\frac {d (c d-b e)}{e^2}\right ) x}{2 d \left (d+e x^2\right )}-\frac {\left (3 c d^2-e (b d+a e)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} e^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 88, normalized size = 1.06 \[ \frac {x \left (a e^2-b d e+c d^2\right )}{2 d e^2 \left (d+e x^2\right )}-\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (-a e^2-b d e+3 c d^2\right )}{2 d^{3/2} e^{5/2}}+\frac {c x}{e^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 268, normalized size = 3.23 \[ \left [\frac {4 \, c d^{2} e^{2} x^{3} + {\left (3 \, c d^{3} - b d^{2} e - a d e^{2} + {\left (3 \, c d^{2} e - b d e^{2} - a e^{3}\right )} x^{2}\right )} \sqrt {-d e} \log \left (\frac {e x^{2} - 2 \, \sqrt {-d e} x - d}{e x^{2} + d}\right ) + 2 \, {\left (3 \, c d^{3} e - b d^{2} e^{2} + a d e^{3}\right )} x}{4 \, {\left (d^{2} e^{4} x^{2} + d^{3} e^{3}\right )}}, \frac {2 \, c d^{2} e^{2} x^{3} - {\left (3 \, c d^{3} - b d^{2} e - a d e^{2} + {\left (3 \, c d^{2} e - b d e^{2} - a e^{3}\right )} x^{2}\right )} \sqrt {d e} \arctan \left (\frac {\sqrt {d e} x}{d}\right ) + {\left (3 \, c d^{3} e - b d^{2} e^{2} + a d e^{3}\right )} x}{2 \, {\left (d^{2} e^{4} x^{2} + d^{3} e^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 75, normalized size = 0.90 \[ c x e^{\left (-2\right )} - \frac {{\left (3 \, c d^{2} - b d e - a e^{2}\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {5}{2}\right )}}{2 \, d^{\frac {3}{2}}} + \frac {{\left (c d^{2} x - b d x e + a x e^{2}\right )} e^{\left (-2\right )}}{2 \, {\left (x^{2} e + d\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 118, normalized size = 1.42 \[ \frac {a x}{2 \left (e \,x^{2}+d \right ) d}+\frac {a \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 \sqrt {d e}\, d}-\frac {b x}{2 \left (e \,x^{2}+d \right ) e}+\frac {b \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 \sqrt {d e}\, e}+\frac {c d x}{2 \left (e \,x^{2}+d \right ) e^{2}}-\frac {3 c d \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 \sqrt {d e}\, e^{2}}+\frac {c x}{e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.25, size = 84, normalized size = 1.01 \[ \frac {{\left (c d^{2} - b d e + a e^{2}\right )} x}{2 \, {\left (d e^{3} x^{2} + d^{2} e^{2}\right )}} + \frac {c x}{e^{2}} - \frac {{\left (3 \, c d^{2} - b d e - a e^{2}\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 \, \sqrt {d e} d e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.67, size = 77, normalized size = 0.93 \[ \frac {c\,x}{e^2}+\frac {\mathrm {atan}\left (\frac {\sqrt {e}\,x}{\sqrt {d}}\right )\,\left (-3\,c\,d^2+b\,d\,e+a\,e^2\right )}{2\,d^{3/2}\,e^{5/2}}+\frac {x\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{2\,d\,\left (e^3\,x^2+d\,e^2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.23, size = 153, normalized size = 1.84 \[ \frac {c x}{e^{2}} + \frac {x \left (a e^{2} - b d e + c d^{2}\right )}{2 d^{2} e^{2} + 2 d e^{3} x^{2}} - \frac {\sqrt {- \frac {1}{d^{3} e^{5}}} \left (a e^{2} + b d e - 3 c d^{2}\right ) \log {\left (- d^{2} e^{2} \sqrt {- \frac {1}{d^{3} e^{5}}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{d^{3} e^{5}}} \left (a e^{2} + b d e - 3 c d^{2}\right ) \log {\left (d^{2} e^{2} \sqrt {- \frac {1}{d^{3} e^{5}}} + x \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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