Optimal. Leaf size=20 \[ \frac{d x}{b c \sqrt{c+d x^2}} \]
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Rubi [A] time = 0.0142504, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{d x}{b c \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
[In] Int[1/(((b*c)/d + b*x^2)*Sqrt[c + d*x^2]),x]
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Rubi in Sympy [A] time = 4.93944, size = 15, normalized size = 0.75 \[ \frac{d x}{b c \sqrt{c + d x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*c/d+b*x**2)/(d*x**2+c)**(1/2),x)
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Mathematica [A] time = 0.017386, size = 20, normalized size = 1. \[ \frac{d x}{b c \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(((b*c)/d + b*x^2)*Sqrt[c + d*x^2]),x]
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Maple [A] time = 0.004, size = 19, normalized size = 1. \[{\frac{dx}{bc}{\frac{1}{\sqrt{d{x}^{2}+c}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*c/d+b*x^2)/(d*x^2+c)^(1/2),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(1/((b*x^2 + b*c/d)*sqrt(d*x^2 + c)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.210956, size = 36, normalized size = 1.8 \[ \frac{\sqrt{d x^{2} + c} d x}{b c d x^{2} + b c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + b*c/d)*sqrt(d*x^2 + c)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{d \int \frac{1}{c \sqrt{c + d x^{2}} + d x^{2} \sqrt{c + d x^{2}}}\, dx}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*c/d+b*x**2)/(d*x**2+c)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.240655, size = 24, normalized size = 1.2 \[ \frac{d x}{\sqrt{d x^{2} + c} b c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + b*c/d)*sqrt(d*x^2 + c)),x, algorithm="giac")
[Out]