Optimal. Leaf size=86 \[ -\frac{(2 c d-b e)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{c d-b e}}\right )}{c^{5/2} \sqrt{e} \sqrt{c d-b e}}+\frac{x (3 c d-b e)}{c^2}+\frac{e x^3}{3 c} \]
[Out]
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Rubi [A] time = 0.183219, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{(2 c d-b e)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{c d-b e}}\right )}{c^{5/2} \sqrt{e} \sqrt{c d-b e}}+\frac{x (3 c d-b e)}{c^2}+\frac{e x^3}{3 c} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x^2)^3/(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \left (b e - 3 c d\right ) \int \frac{1}{c^{2}}\, dx + \frac{e x^{3}}{3 c} + \frac{\left (b e - 2 c d\right )^{2} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{b e - c d}} \right )}}{c^{\frac{5}{2}} \sqrt{e} \sqrt{b e - c d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x**2+d)**3/(c*e**2*x**4+b*e**2*x**2+b*d*e-c*d**2),x)
[Out]
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Mathematica [A] time = 0.073463, size = 84, normalized size = 0.98 \[ \frac{(b e-2 c d)^2 \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{b e-c d}}\right )}{c^{5/2} \sqrt{e} \sqrt{b e-c d}}-\frac{x (b e-3 c d)}{c^2}+\frac{e x^3}{3 c} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x^2)^3/(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4),x]
[Out]
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Maple [A] time = 0.004, size = 142, normalized size = 1.7 \[{\frac{e{x}^{3}}{3\,c}}-{\frac{bex}{{c}^{2}}}+3\,{\frac{dx}{c}}+{\frac{{b}^{2}{e}^{2}}{{c}^{2}}\arctan \left ({cex{\frac{1}{\sqrt{ \left ( be-cd \right ) ce}}}} \right ){\frac{1}{\sqrt{ \left ( be-cd \right ) ce}}}}-4\,{\frac{bde}{c\sqrt{ \left ( be-cd \right ) ce}}\arctan \left ({\frac{cex}{\sqrt{ \left ( be-cd \right ) ce}}} \right ) }+4\,{\frac{{d}^{2}}{\sqrt{ \left ( be-cd \right ) ce}}\arctan \left ({\frac{cex}{\sqrt{ \left ( be-cd \right ) ce}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x^2+d)^3/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*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((e*x^2 + d)^3/(c*e^2*x^4 + b*e^2*x^2 - c*d^2 + b*d*e),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.287878, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (4 \, c^{2} d^{2} - 4 \, b c d e + b^{2} e^{2}\right )} \log \left (-\frac{2 \,{\left (c^{2} d e - b c e^{2}\right )} x - \sqrt{c^{2} d e - b c e^{2}}{\left (c e x^{2} + c d - b e\right )}}{c e x^{2} - c d + b e}\right ) + 2 \,{\left (c e x^{3} + 3 \,{\left (3 \, c d - b e\right )} x\right )} \sqrt{c^{2} d e - b c e^{2}}}{6 \, \sqrt{c^{2} d e - b c e^{2}} c^{2}}, \frac{3 \,{\left (4 \, c^{2} d^{2} - 4 \, b c d e + b^{2} e^{2}\right )} \arctan \left (-\frac{\sqrt{-c^{2} d e + b c e^{2}} x}{c d - b e}\right ) +{\left (c e x^{3} + 3 \,{\left (3 \, c d - b e\right )} x\right )} \sqrt{-c^{2} d e + b c e^{2}}}{3 \, \sqrt{-c^{2} d e + b c e^{2}} c^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^2 + d)^3/(c*e^2*x^4 + b*e^2*x^2 - c*d^2 + b*d*e),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.57207, size = 275, normalized size = 3.2 \[ - \frac{\sqrt{- \frac{1}{c^{5} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{2} \log{\left (x + \frac{- b c^{2} e \sqrt{- \frac{1}{c^{5} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{2} + c^{3} d \sqrt{- \frac{1}{c^{5} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{2}}{b^{2} e^{2} - 4 b c d e + 4 c^{2} d^{2}} \right )}}{2} + \frac{\sqrt{- \frac{1}{c^{5} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{2} \log{\left (x + \frac{b c^{2} e \sqrt{- \frac{1}{c^{5} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{2} - c^{3} d \sqrt{- \frac{1}{c^{5} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{2}}{b^{2} e^{2} - 4 b c d e + 4 c^{2} d^{2}} \right )}}{2} + \frac{e x^{3}}{3 c} - \frac{x \left (b e - 3 c d\right )}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x**2+d)**3/(c*e**2*x**4+b*e**2*x**2+b*d*e-c*d**2),x)
[Out]
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GIAC/XCAS [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^2 + d)^3/(c*e^2*x^4 + b*e^2*x^2 - c*d^2 + b*d*e),x, algorithm="giac")
[Out]