Optimal. Leaf size=74 \[ \frac {x \left (a+\frac {c d^2}{e^2}\right )}{2 d \left (d+e x^2\right )}-\frac {\left (3 c d^2-a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} e^{5/2}}+\frac {c x}{e^2} \]
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Rubi [A] time = 0.05, antiderivative size = 74, 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 = {1158, 388, 205} \begin {gather*} \frac {x \left (a+\frac {c d^2}{e^2}\right )}{2 d \left (d+e x^2\right )}-\frac {\left (3 c d^2-a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} e^{5/2}}+\frac {c x}{e^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 388
Rule 1158
Rubi steps
\begin {align*} \int \frac {a+c x^4}{\left (d+e x^2\right )^2} \, dx &=\frac {\left (a+\frac {c d^2}{e^2}\right ) x}{2 d \left (d+e x^2\right )}-\frac {\int \frac {-a+\frac {c d^2}{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 {c d^2}{e^2}\right ) x}{2 d \left (d+e x^2\right )}+\frac {\left (a-\frac {3 c d^2}{e^2}\right ) \int \frac {1}{d+e x^2} \, dx}{2 d}\\ &=\frac {c x}{e^2}+\frac {\left (a+\frac {c d^2}{e^2}\right ) x}{2 d \left (d+e x^2\right )}+\frac {\left (a-\frac {3 c d^2}{e^2}\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {e}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 78, normalized size = 1.05 \begin {gather*} \frac {x \left (a e^2+c d^2\right )}{2 d e^2 \left (d+e x^2\right )}-\frac {\left (3 c d^2-a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} e^{5/2}}+\frac {c x}{e^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+c x^4}{\left (d+e x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.11, size = 222, normalized size = 3.00 \begin {gather*} \left [\frac {4 \, c d^{2} e^{2} x^{3} + {\left (3 \, c d^{3} - a d e^{2} + {\left (3 \, c d^{2} e - 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 + 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} - a d e^{2} + {\left (3 \, c d^{2} e - 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 + a d e^{3}\right )} x}{2 \, {\left (d^{2} e^{4} x^{2} + d^{3} e^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 62, normalized size = 0.84 \begin {gather*} c x e^{\left (-2\right )} - \frac {{\left (3 \, c d^{2} - 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 + a x e^{2}\right )} e^{\left (-2\right )}}{2 \, {\left (x^{2} e + d\right )} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 82, normalized size = 1.11 \begin {gather*} \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 {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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.24, size = 74, normalized size = 1.00 \begin {gather*} \frac {{\left (c d^{2} + 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} - a e^{2}\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 \, \sqrt {d e} d e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.44, size = 68, normalized size = 0.92 \begin {gather*} \frac {c\,x}{e^2}+\frac {\mathrm {atan}\left (\frac {\sqrt {e}\,x}{\sqrt {d}}\right )\,\left (a\,e^2-3\,c\,d^2\right )}{2\,d^{3/2}\,e^{5/2}}+\frac {x\,\left (c\,d^2+a\,e^2\right )}{2\,d\,\left (e^3\,x^2+d\,e^2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.51, size = 138, normalized size = 1.86 \begin {gather*} \frac {c x}{e^{2}} + \frac {x \left (a e^{2} + 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} - 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} - 3 c d^{2}\right ) \log {\left (d^{2} e^{2} \sqrt {- \frac {1}{d^{3} e^{5}}} + x \right )}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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