Optimal. Leaf size=97 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right ) \left (8 a e^2-4 b d e+3 c d^2\right )}{8 e^{5/2}}-\frac{x \sqrt{d+e x^2} (3 c d-4 b e)}{8 e^2}+\frac{c x^3 \sqrt{d+e x^2}}{4 e} \]
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Rubi [A] time = 0.13211, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right ) \left (8 a e^2-4 b d e+3 c d^2\right )}{8 e^{5/2}}-\frac{x \sqrt{d+e x^2} (3 c d-4 b e)}{8 e^2}+\frac{c x^3 \sqrt{d+e x^2}}{4 e} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2 + c*x^4)/Sqrt[d + e*x^2],x]
[Out]
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Rubi in Sympy [A] time = 16.893, size = 90, normalized size = 0.93 \[ \frac{c x^{3} \sqrt{d + e x^{2}}}{4 e} + \frac{x \sqrt{d + e x^{2}} \left (4 b e - 3 c d\right )}{8 e^{2}} + \frac{\left (8 a e^{2} - 4 b d e + 3 c d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{e} x}{\sqrt{d + e x^{2}}} \right )}}{8 e^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**4+b*x**2+a)/(e*x**2+d)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0928843, size = 85, normalized size = 0.88 \[ \frac{\log \left (\sqrt{e} \sqrt{d+e x^2}+e x\right ) \left (8 a e^2-4 b d e+3 c d^2\right )+\sqrt{e} x \sqrt{d+e x^2} \left (4 b e-3 c d+2 c e x^2\right )}{8 e^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2 + c*x^4)/Sqrt[d + e*x^2],x]
[Out]
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Maple [A] time = 0.011, size = 122, normalized size = 1.3 \[{a\ln \left ( x\sqrt{e}+\sqrt{e{x}^{2}+d} \right ){\frac{1}{\sqrt{e}}}}+{\frac{bx}{2\,e}\sqrt{e{x}^{2}+d}}-{\frac{bd}{2}\ln \left ( x\sqrt{e}+\sqrt{e{x}^{2}+d} \right ){e}^{-{\frac{3}{2}}}}+{\frac{c{x}^{3}}{4\,e}\sqrt{e{x}^{2}+d}}-{\frac{3\,cdx}{8\,{e}^{2}}\sqrt{e{x}^{2}+d}}+{\frac{3\,c{d}^{2}}{8}\ln \left ( x\sqrt{e}+\sqrt{e{x}^{2}+d} \right ){e}^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^4+b*x^2+a)/(e*x^2+d)^(1/2),x)
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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 + b*x^2 + a)/sqrt(e*x^2 + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.2953, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (2 \, c e x^{3} -{\left (3 \, c d - 4 \, b e\right )} x\right )} \sqrt{e x^{2} + d} \sqrt{e} +{\left (3 \, c d^{2} - 4 \, b d e + 8 \, a e^{2}\right )} \log \left (-2 \, \sqrt{e x^{2} + d} e x -{\left (2 \, e x^{2} + d\right )} \sqrt{e}\right )}{16 \, e^{\frac{5}{2}}}, \frac{{\left (2 \, c e x^{3} -{\left (3 \, c d - 4 \, b e\right )} x\right )} \sqrt{e x^{2} + d} \sqrt{-e} +{\left (3 \, c d^{2} - 4 \, b d e + 8 \, a e^{2}\right )} \arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right )}{8 \, \sqrt{-e} e^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + b*x^2 + a)/sqrt(e*x^2 + d),x, algorithm="fricas")
[Out]
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Sympy [A] time = 19.5, size = 230, normalized size = 2.37 \[ a \left (\begin{cases} \frac{\sqrt{- \frac{d}{e}} \operatorname{asin}{\left (x \sqrt{- \frac{e}{d}} \right )}}{\sqrt{d}} & \text{for}\: d > 0 \wedge e < 0 \\\frac{\sqrt{\frac{d}{e}} \operatorname{asinh}{\left (x \sqrt{\frac{e}{d}} \right )}}{\sqrt{d}} & \text{for}\: d > 0 \wedge e > 0 \\\frac{\sqrt{- \frac{d}{e}} \operatorname{acosh}{\left (x \sqrt{- \frac{e}{d}} \right )}}{\sqrt{- d}} & \text{for}\: e > 0 \wedge d < 0 \end{cases}\right ) + \frac{b \sqrt{d} x \sqrt{1 + \frac{e x^{2}}{d}}}{2 e} - \frac{b d \operatorname{asinh}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{2 e^{\frac{3}{2}}} - \frac{3 c d^{\frac{3}{2}} x}{8 e^{2} \sqrt{1 + \frac{e x^{2}}{d}}} - \frac{c \sqrt{d} x^{3}}{8 e \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{3 c d^{2} \operatorname{asinh}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{8 e^{\frac{5}{2}}} + \frac{c x^{5}}{4 \sqrt{d} \sqrt{1 + \frac{e x^{2}}{d}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**4+b*x**2+a)/(e*x**2+d)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.268649, size = 107, normalized size = 1.1 \[ -\frac{1}{8} \,{\left (3 \, c d^{2} - 4 \, b d e + 8 \, a e^{2}\right )} e^{\left (-\frac{5}{2}\right )}{\rm ln}\left ({\left | -x e^{\frac{1}{2}} + \sqrt{x^{2} e + d} \right |}\right ) + \frac{1}{8} \,{\left (2 \, c x^{2} e^{\left (-1\right )} -{\left (3 \, c d e - 4 \, b e^{2}\right )} e^{\left (-3\right )}\right )} \sqrt{x^{2} e + d} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + b*x^2 + a)/sqrt(e*x^2 + d),x, algorithm="giac")
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