Optimal. Leaf size=134 \[ \frac {e^{3/2} \log \left (\sqrt {e} x \sqrt {2 c d-b e}+\sqrt {c} d+\sqrt {c} e x^2\right )}{2 \sqrt {c} \sqrt {2 c d-b e}}-\frac {e^{3/2} \log \left (-\sqrt {e} x \sqrt {2 c d-b e}+\sqrt {c} d+\sqrt {c} e x^2\right )}{2 \sqrt {c} \sqrt {2 c d-b e}} \]
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Rubi [A] time = 0.10, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1164, 628} \begin {gather*} \frac {e^{3/2} \log \left (\sqrt {e} x \sqrt {2 c d-b e}+\sqrt {c} d+\sqrt {c} e x^2\right )}{2 \sqrt {c} \sqrt {2 c d-b e}}-\frac {e^{3/2} \log \left (-\sqrt {e} x \sqrt {2 c d-b e}+\sqrt {c} d+\sqrt {c} e x^2\right )}{2 \sqrt {c} \sqrt {2 c d-b e}} \end {gather*}
Antiderivative was successfully verified.
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Rule 628
Rule 1164
Rubi steps
\begin {align*} \int \frac {d-e x^2}{\frac {c d^2}{e^2}+b x^2+c x^4} \, dx &=-\frac {e^{3/2} \int \frac {\frac {\sqrt {2 c d-b e}}{\sqrt {c} \sqrt {e}}+2 x}{-\frac {d}{e}-\frac {\sqrt {2 c d-b e} x}{\sqrt {c} \sqrt {e}}-x^2} \, dx}{2 \sqrt {c} \sqrt {2 c d-b e}}-\frac {e^{3/2} \int \frac {\frac {\sqrt {2 c d-b e}}{\sqrt {c} \sqrt {e}}-2 x}{-\frac {d}{e}+\frac {\sqrt {2 c d-b e} x}{\sqrt {c} \sqrt {e}}-x^2} \, dx}{2 \sqrt {c} \sqrt {2 c d-b e}}\\ &=-\frac {e^{3/2} \log \left (\sqrt {c} d-\sqrt {e} \sqrt {2 c d-b e} x+\sqrt {c} e x^2\right )}{2 \sqrt {c} \sqrt {2 c d-b e}}+\frac {e^{3/2} \log \left (\sqrt {c} d+\sqrt {e} \sqrt {2 c d-b e} x+\sqrt {c} e x^2\right )}{2 \sqrt {c} \sqrt {2 c d-b e}}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 250, normalized size = 1.87 \begin {gather*} \frac {e^{3/2} \left (-\frac {\left (\sqrt {b^2 e^2-4 c^2 d^2}-b e-2 c d\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {e} x}{\sqrt {b e-\sqrt {b^2 e^2-4 c^2 d^2}}}\right )}{\sqrt {b e-\sqrt {b^2 e^2-4 c^2 d^2}}}-\frac {\left (\sqrt {b^2 e^2-4 c^2 d^2}+b e+2 c d\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {e} x}{\sqrt {\sqrt {b^2 e^2-4 c^2 d^2}+b e}}\right )}{\sqrt {\sqrt {b^2 e^2-4 c^2 d^2}+b e}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b^2 e^2-4 c^2 d^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 {d-e x^2}{\frac {c d^2}{e^2}+b x^2+c x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.82, size = 244, normalized size = 1.82 \begin {gather*} \left [\frac {1}{2} \, e \sqrt {\frac {e}{2 \, c^{2} d - b c e}} \log \left (\frac {c e^{2} x^{4} + c d^{2} + {\left (4 \, c d e - b e^{2}\right )} x^{2} + 2 \, {\left ({\left (2 \, c^{2} d e - b c e^{2}\right )} x^{3} + {\left (2 \, c^{2} d^{2} - b c d e\right )} x\right )} \sqrt {\frac {e}{2 \, c^{2} d - b c e}}}{c e^{2} x^{4} + b e^{2} x^{2} + c d^{2}}\right ), -e \sqrt {-\frac {e}{2 \, c^{2} d - b c e}} \arctan \left (c x \sqrt {-\frac {e}{2 \, c^{2} d - b c e}}\right ) + e \sqrt {-\frac {e}{2 \, c^{2} d - b c e}} \arctan \left (\frac {{\left (c e x^{3} - {\left (c d - b e\right )} x\right )} \sqrt {-\frac {e}{2 \, c^{2} d - b c e}}}{d}\right )\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.37, size = 2202, normalized size = 16.43
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 582, normalized size = 4.34 \begin {gather*} -\frac {\sqrt {2}\, b \,e^{4} \arctanh \left (\frac {\sqrt {2}\, c e x}{\sqrt {\left (-b \,e^{2}+\sqrt {\left (b e -2 c d \right ) \left (b e +2 c d \right ) e^{2}}\right ) c}}\right )}{2 \sqrt {\left (b e -2 c d \right ) \left (b e +2 c d \right ) e^{2}}\, \sqrt {\left (-b \,e^{2}+\sqrt {\left (b e -2 c d \right ) \left (b e +2 c d \right ) e^{2}}\right ) c}}-\frac {\sqrt {2}\, b \,e^{4} \arctan \left (\frac {\sqrt {2}\, c e x}{\sqrt {\left (b \,e^{2}+\sqrt {\left (b e -2 c d \right ) \left (b e +2 c d \right ) e^{2}}\right ) c}}\right )}{2 \sqrt {\left (b e -2 c d \right ) \left (b e +2 c d \right ) e^{2}}\, \sqrt {\left (b \,e^{2}+\sqrt {\left (b e -2 c d \right ) \left (b e +2 c d \right ) e^{2}}\right ) c}}-\frac {\sqrt {2}\, c d \,e^{3} \arctanh \left (\frac {\sqrt {2}\, c e x}{\sqrt {\left (-b \,e^{2}+\sqrt {\left (b e -2 c d \right ) \left (b e +2 c d \right ) e^{2}}\right ) c}}\right )}{\sqrt {\left (b e -2 c d \right ) \left (b e +2 c d \right ) e^{2}}\, \sqrt {\left (-b \,e^{2}+\sqrt {\left (b e -2 c d \right ) \left (b e +2 c d \right ) e^{2}}\right ) c}}-\frac {\sqrt {2}\, c d \,e^{3} \arctan \left (\frac {\sqrt {2}\, c e x}{\sqrt {\left (b \,e^{2}+\sqrt {\left (b e -2 c d \right ) \left (b e +2 c d \right ) e^{2}}\right ) c}}\right )}{\sqrt {\left (b e -2 c d \right ) \left (b e +2 c d \right ) e^{2}}\, \sqrt {\left (b \,e^{2}+\sqrt {\left (b e -2 c d \right ) \left (b e +2 c d \right ) e^{2}}\right ) c}}+\frac {\sqrt {2}\, e^{2} \arctanh \left (\frac {\sqrt {2}\, c e x}{\sqrt {\left (-b \,e^{2}+\sqrt {\left (b e -2 c d \right ) \left (b e +2 c d \right ) e^{2}}\right ) c}}\right )}{2 \sqrt {\left (-b \,e^{2}+\sqrt {\left (b e -2 c d \right ) \left (b e +2 c d \right ) e^{2}}\right ) c}}-\frac {\sqrt {2}\, e^{2} \arctan \left (\frac {\sqrt {2}\, c e x}{\sqrt {\left (b \,e^{2}+\sqrt {\left (b e -2 c d \right ) \left (b e +2 c d \right ) e^{2}}\right ) c}}\right )}{2 \sqrt {\left (b \,e^{2}+\sqrt {\left (b e -2 c d \right ) \left (b e +2 c d \right ) e^{2}}\right ) c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {e x^{2} - d}{c x^{4} + b x^{2} + \frac {c d^{2}}{e^{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 129, normalized size = 0.96 \begin {gather*} -\frac {e^{3/2}\,\left (\mathrm {atan}\left (\frac {\sqrt {e}\,x\,\sqrt {b\,c\,e-2\,c^2\,d}}{b\,e-2\,c\,d}\right )+\mathrm {atan}\left (\frac {c\,e^{3/2}\,x^3\,\sqrt {b\,c\,e-2\,c^2\,d}+b\,e^{3/2}\,x\,\sqrt {b\,c\,e-2\,c^2\,d}-c\,d\,\sqrt {e}\,x\,\sqrt {b\,c\,e-2\,c^2\,d}}{d\,\left (2\,c^2\,d-b\,c\,e\right )}\right )\right )}{\sqrt {b\,c\,e-2\,c^2\,d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.86, size = 158, normalized size = 1.18 \begin {gather*} \frac {\sqrt {- \frac {e^{3}}{c \left (b e - 2 c d\right )}} \log {\left (\frac {d}{e} + x^{2} + \frac {x \left (- b e \sqrt {- \frac {e^{3}}{c \left (b e - 2 c d\right )}} + 2 c d \sqrt {- \frac {e^{3}}{c \left (b e - 2 c d\right )}}\right )}{e^{2}} \right )}}{2} - \frac {\sqrt {- \frac {e^{3}}{c \left (b e - 2 c d\right )}} \log {\left (\frac {d}{e} + x^{2} + \frac {x \left (b e \sqrt {- \frac {e^{3}}{c \left (b e - 2 c d\right )}} - 2 c d \sqrt {- \frac {e^{3}}{c \left (b e - 2 c d\right )}}\right )}{e^{2}} \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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