Optimal. Leaf size=139 \[ -\frac{(2 c d-b e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{e} x \sqrt{2 c d-b e}}{\sqrt{d+e x^2} \sqrt{c d-b e}}\right )}{c^2 \sqrt{e} \sqrt{c d-b e}}+\frac{(5 c d-2 b e) \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{2 c^2 \sqrt{e}}+\frac{x \sqrt{d+e x^2}}{2 c} \]
[Out]
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Rubi [A] time = 0.551695, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171 \[ -\frac{(2 c d-b e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{e} x \sqrt{2 c d-b e}}{\sqrt{d+e x^2} \sqrt{c d-b e}}\right )}{c^2 \sqrt{e} \sqrt{c d-b e}}+\frac{(5 c d-2 b e) \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{2 c^2 \sqrt{e}}+\frac{x \sqrt{d+e x^2}}{2 c} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x^2)^(5/2)/(-(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 [A] time = 102.221, size = 124, normalized size = 0.89 \[ \frac{x \sqrt{d + e x^{2}}}{2 c} + \frac{\left (b e - 2 c d\right )^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{e} x \sqrt{b e - 2 c d}}{\sqrt{d + e x^{2}} \sqrt{b e - c d}} \right )}}{c^{2} \sqrt{e} \sqrt{b e - c d}} - \frac{\left (2 b e - 5 c d\right ) \operatorname{atanh}{\left (\frac{\sqrt{e} x}{\sqrt{d + e x^{2}}} \right )}}{2 c^{2} \sqrt{e}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x**2+d)**(5/2)/(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 = 0.392921, size = 134, normalized size = 0.96 \[ -\frac{\frac{(2 b e-5 c d) \log \left (\sqrt{e} \sqrt{d+e x^2}+e x\right )}{\sqrt{e}}-\frac{2 (b e-2 c d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{e} x \sqrt{b e-2 c d}}{\sqrt{d+e x^2} \sqrt{b e-c d}}\right )}{\sqrt{e} \sqrt{b e-c d}}-c x \sqrt{d+e x^2}}{2 c^2} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x^2)^(5/2)/(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4),x]
[Out]
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Maple [B] time = 0.069, size = 7043, normalized size = 50.7 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x^2+d)^(5/2)/(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)^(5/2)/(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 = 0.783323, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^2 + d)^(5/2)/(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 [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x^{2}\right )^{\frac{3}{2}}}{b e - c d + c e x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x**2+d)**(5/2)/(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)^(5/2)/(c*e^2*x^4 + b*e^2*x^2 - c*d^2 + b*d*e),x, algorithm="giac")
[Out]