Optimal. Leaf size=74 \[ \frac{x \left (a+\frac{c d^2}{e^2}\right )}{2 d \left (d+e x^2\right )}-\frac{\left (3 c d^2-a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} e^{5/2}}+\frac{c x}{e^2} \]
[Out]
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Rubi [A] time = 0.113144, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{x \left (a+\frac{c d^2}{e^2}\right )}{2 d \left (d+e x^2\right )}-\frac{\left (3 c d^2-a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} e^{5/2}}+\frac{c x}{e^2} \]
Antiderivative was successfully verified.
[In] Int[(a + c*x^4)/(d + e*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 23.8383, size = 65, normalized size = 0.88 \[ \frac{c x}{e^{2}} + \frac{x \left (\frac{a}{2 d} + \frac{c d}{2 e^{2}}\right )}{d + e x^{2}} + \frac{\left (a e^{2} - 3 c d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{2 d^{\frac{3}{2}} e^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**4+a)/(e*x**2+d)**2,x)
[Out]
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Mathematica [A] time = 0.0833572, size = 78, normalized size = 1.05 \[ \frac{x \left (a e^2+c d^2\right )}{2 d e^2 \left (d+e x^2\right )}-\frac{\left (3 c d^2-a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} e^{5/2}}+\frac{c x}{e^2} \]
Antiderivative was successfully verified.
[In] Integrate[(a + c*x^4)/(d + e*x^2)^2,x]
[Out]
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Maple [A] time = 0.012, size = 82, normalized size = 1.1 \[{\frac{cx}{{e}^{2}}}+{\frac{ax}{2\,d \left ( e{x}^{2}+d \right ) }}+{\frac{cdx}{2\,{e}^{2} \left ( e{x}^{2}+d \right ) }}+{\frac{a}{2\,d}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}-{\frac{3\,cd}{2\,{e}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^4+a)/(e*x^2+d)^2,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + a)/(e*x^2 + d)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.29123, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (3 \, c d^{3} - a d e^{2} +{\left (3 \, c d^{2} e - a e^{3}\right )} x^{2}\right )} \log \left (\frac{2 \, d e x +{\left (e x^{2} - d\right )} \sqrt{-d e}}{e x^{2} + d}\right ) - 2 \,{\left (2 \, c d e x^{3} +{\left (3 \, c d^{2} + a e^{2}\right )} x\right )} \sqrt{-d e}}{4 \,{\left (d e^{3} x^{2} + d^{2} e^{2}\right )} \sqrt{-d e}}, -\frac{{\left (3 \, c d^{3} - a d e^{2} +{\left (3 \, c d^{2} e - a e^{3}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{d e} x}{d}\right ) -{\left (2 \, c d e x^{3} +{\left (3 \, c d^{2} + a e^{2}\right )} x\right )} \sqrt{d e}}{2 \,{\left (d e^{3} x^{2} + d^{2} e^{2}\right )} \sqrt{d e}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + a)/(e*x^2 + d)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.29111, size = 138, normalized size = 1.86 \[ \frac{c x}{e^{2}} + \frac{x \left (a e^{2} + c d^{2}\right )}{2 d^{2} e^{2} + 2 d e^{3} x^{2}} - \frac{\sqrt{- \frac{1}{d^{3} e^{5}}} \left (a e^{2} - 3 c d^{2}\right ) \log{\left (- d^{2} e^{2} \sqrt{- \frac{1}{d^{3} e^{5}}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{d^{3} e^{5}}} \left (a e^{2} - 3 c d^{2}\right ) \log{\left (d^{2} e^{2} \sqrt{- \frac{1}{d^{3} e^{5}}} + x \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**4+a)/(e*x**2+d)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.270064, size = 84, normalized size = 1.14 \[ c x e^{\left (-2\right )} - \frac{{\left (3 \, c d^{2} - a e^{2}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{5}{2}\right )}}{2 \, d^{\frac{3}{2}}} + \frac{{\left (c d^{2} x + a x e^{2}\right )} e^{\left (-2\right )}}{2 \,{\left (x^{2} e + d\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + a)/(e*x^2 + d)^2,x, algorithm="giac")
[Out]