Optimal. Leaf size=93 \[ \frac{x \left (\frac{3 a}{d^2}-\frac{5 c}{e^2}\right )}{8 \left (d+e x^2\right )}+\frac{x \left (a+\frac{c d^2}{e^2}\right )}{4 d \left (d+e x^2\right )^2}+\frac{3 \left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{5/2} e^{5/2}} \]
[Out]
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Rubi [A] time = 0.143192, antiderivative size = 93, 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 \[ \frac{x \left (\frac{3 a}{d^2}-\frac{5 c}{e^2}\right )}{8 \left (d+e x^2\right )}+\frac{x \left (a+\frac{c d^2}{e^2}\right )}{4 d \left (d+e x^2\right )^2}+\frac{3 \left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{5/2} e^{5/2}} \]
Antiderivative was successfully verified.
[In] Int[(a + c*x^4)/(d + e*x^2)^3,x]
[Out]
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Rubi in Sympy [A] time = 24.203, size = 85, normalized size = 0.91 \[ \frac{x \left (\frac{3 a}{8 d^{2}} - \frac{5 c}{8 e^{2}}\right )}{d + e x^{2}} + \frac{x \left (\frac{a}{4 d} + \frac{c d}{4 e^{2}}\right )}{\left (d + e x^{2}\right )^{2}} + \frac{3 \left (a e^{2} + c d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{8 d^{\frac{5}{2}} e^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**4+a)/(e*x**2+d)**3,x)
[Out]
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Mathematica [A] time = 0.098541, size = 92, normalized size = 0.99 \[ \frac{a e^2 x \left (5 d+3 e x^2\right )-c d^2 x \left (3 d+5 e x^2\right )}{8 d^2 e^2 \left (d+e x^2\right )^2}+\frac{3 \left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{5/2} e^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + c*x^4)/(d + e*x^2)^3,x]
[Out]
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Maple [A] time = 0.012, size = 99, normalized size = 1.1 \[{\frac{1}{ \left ( e{x}^{2}+d \right ) ^{2}} \left ({\frac{ \left ( 3\,a{e}^{2}-5\,c{d}^{2} \right ){x}^{3}}{8\,{d}^{2}e}}+{\frac{ \left ( 5\,a{e}^{2}-3\,c{d}^{2} \right ) x}{8\,{e}^{2}d}} \right ) }+{\frac{3\,a}{8\,{d}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{3\,c}{8\,{e}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^4+a)/(e*x^2+d)^3,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)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.290071, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (c d^{4} + a d^{2} e^{2} +{\left (c d^{2} e^{2} + a e^{4}\right )} x^{4} + 2 \,{\left (c d^{3} e + a d e^{3}\right )} x^{2}\right )} \log \left (\frac{2 \, d e x +{\left (e x^{2} - d\right )} \sqrt{-d e}}{e x^{2} + d}\right ) - 2 \,{\left ({\left (5 \, c d^{2} e - 3 \, a e^{3}\right )} x^{3} +{\left (3 \, c d^{3} - 5 \, a d e^{2}\right )} x\right )} \sqrt{-d e}}{16 \,{\left (d^{2} e^{4} x^{4} + 2 \, d^{3} e^{3} x^{2} + d^{4} e^{2}\right )} \sqrt{-d e}}, \frac{3 \,{\left (c d^{4} + a d^{2} e^{2} +{\left (c d^{2} e^{2} + a e^{4}\right )} x^{4} + 2 \,{\left (c d^{3} e + a d e^{3}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{d e} x}{d}\right ) -{\left ({\left (5 \, c d^{2} e - 3 \, a e^{3}\right )} x^{3} +{\left (3 \, c d^{3} - 5 \, a d e^{2}\right )} x\right )} \sqrt{d e}}{8 \,{\left (d^{2} e^{4} x^{4} + 2 \, d^{3} e^{3} x^{2} + d^{4} e^{2}\right )} \sqrt{d e}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + a)/(e*x^2 + d)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.11724, size = 219, normalized size = 2.35 \[ - \frac{3 \sqrt{- \frac{1}{d^{5} e^{5}}} \left (a e^{2} + c d^{2}\right ) \log{\left (- \frac{3 d^{3} e^{2} \sqrt{- \frac{1}{d^{5} e^{5}}} \left (a e^{2} + c d^{2}\right )}{3 a e^{2} + 3 c d^{2}} + x \right )}}{16} + \frac{3 \sqrt{- \frac{1}{d^{5} e^{5}}} \left (a e^{2} + c d^{2}\right ) \log{\left (\frac{3 d^{3} e^{2} \sqrt{- \frac{1}{d^{5} e^{5}}} \left (a e^{2} + c d^{2}\right )}{3 a e^{2} + 3 c d^{2}} + x \right )}}{16} + \frac{x^{3} \left (3 a e^{3} - 5 c d^{2} e\right ) + x \left (5 a d e^{2} - 3 c d^{3}\right )}{8 d^{4} e^{2} + 16 d^{3} e^{3} x^{2} + 8 d^{2} e^{4} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**4+a)/(e*x**2+d)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.271349, size = 104, normalized size = 1.12 \[ \frac{3 \,{\left (c d^{2} + a e^{2}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{5}{2}\right )}}{8 \, d^{\frac{5}{2}}} - \frac{{\left (5 \, c d^{2} x^{3} e + 3 \, c d^{3} x - 3 \, a x^{3} e^{3} - 5 \, a d x e^{2}\right )} e^{\left (-2\right )}}{8 \,{\left (x^{2} e + d\right )}^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + a)/(e*x^2 + d)^3,x, algorithm="giac")
[Out]