Optimal. Leaf size=130 \[ \frac {e^{3/2} \tan ^{-1}\left (\frac {\sqrt {2 c d-b e}+2 \sqrt {c} \sqrt {e} x}{\sqrt {b e+2 c d}}\right )}{\sqrt {c} \sqrt {b e+2 c d}}-\frac {e^{3/2} \tan ^{-1}\left (\frac {\sqrt {2 c d-b e}-2 \sqrt {c} \sqrt {e} x}{\sqrt {b e+2 c d}}\right )}{\sqrt {c} \sqrt {b e+2 c d}} \]
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Rubi [A] time = 0.17, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {1161, 618, 204} \begin {gather*} \frac {e^{3/2} \tan ^{-1}\left (\frac {\sqrt {2 c d-b e}+2 \sqrt {c} \sqrt {e} x}{\sqrt {b e+2 c d}}\right )}{\sqrt {c} \sqrt {b e+2 c d}}-\frac {e^{3/2} \tan ^{-1}\left (\frac {\sqrt {2 c d-b e}-2 \sqrt {c} \sqrt {e} x}{\sqrt {b e+2 c d}}\right )}{\sqrt {c} \sqrt {b e+2 c d}} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 1161
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 \int \frac {1}{\frac {d}{e}-\frac {\sqrt {2 c d-b e} x}{\sqrt {c} \sqrt {e}}+x^2} \, dx}{2 c}+\frac {e \int \frac {1}{\frac {d}{e}+\frac {\sqrt {2 c d-b e} x}{\sqrt {c} \sqrt {e}}+x^2} \, dx}{2 c}\\ &=-\frac {e \operatorname {Subst}\left (\int \frac {1}{-\frac {b}{c}-\frac {2 d}{e}-x^2} \, dx,x,-\frac {\sqrt {2 c d-b e}}{\sqrt {c} \sqrt {e}}+2 x\right )}{c}-\frac {e \operatorname {Subst}\left (\int \frac {1}{-\frac {b}{c}-\frac {2 d}{e}-x^2} \, dx,x,\frac {\sqrt {2 c d-b e}}{\sqrt {c} \sqrt {e}}+2 x\right )}{c}\\ &=-\frac {e^{3/2} \tan ^{-1}\left (\frac {\sqrt {2 c d-b e}-2 \sqrt {c} \sqrt {e} x}{\sqrt {2 c d+b e}}\right )}{\sqrt {c} \sqrt {2 c d+b e}}+\frac {e^{3/2} \tan ^{-1}\left (\frac {\sqrt {2 c d-b e}+2 \sqrt {c} \sqrt {e} x}{\sqrt {2 c d+b e}}\right )}{\sqrt {c} \sqrt {2 c d+b e}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 248, normalized size = 1.91 \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 = 0.65, size = 232, normalized size = 1.78 \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 = 1.40, size = 2202, normalized size = 16.94
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 582, normalized size = 4.48 \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 = 4.52, size = 232, normalized size = 1.78 \begin {gather*} \frac {e^{3/2}\,\left (\mathrm {atan}\left (\frac {c\,\sqrt {e}\,x}{\sqrt {c\,\left (b\,e+2\,c\,d\right )}}\right )-\mathrm {atan}\left (\frac {\left (2\,d\,c^2+b\,e\,c\right )\,\left (x\,\left (\frac {\sqrt {e}\,\left (c\,d\,e^7-\frac {4\,c^3\,d^2\,e^7}{2\,d\,c^2+b\,e\,c}\right )}{d\,\sqrt {c\,\left (b\,e+2\,c\,d\right )}\,\left (b\,e-2\,c\,d\right )}+\frac {e^{3/2}\,\left (2\,c^2\,d\,e^6-b\,c\,e^7\right )}{c\,d\,\sqrt {2\,d\,c^2+b\,e\,c}\,\left (b\,e-2\,c\,d\right )}\right )+\frac {\sqrt {e}\,x^3\,\left (c\,e^8-\frac {2\,b\,c^2\,e^9}{2\,d\,c^2+b\,e\,c}\right )}{d\,\sqrt {c\,\left (b\,e+2\,c\,d\right )}\,\left (b\,e-2\,c\,d\right )}\right )}{c\,e^7}\right )\right )}{\sqrt {2\,d\,c^2+b\,e\,c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.77, size = 160, normalized size = 1.23 \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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