3.105 \(\int \frac{1}{\left (\frac{b c}{d}+b x^2\right ) \sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=20 \[ \frac{d x}{b c \sqrt{c+d x^2}} \]

[Out]

(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]

[Out]

(d*x)/(b*c*Sqrt[c + d*x^2])

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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)

[Out]

d*x/(b*c*sqrt(c + d*x**2))

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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]

[Out]

(d*x)/(b*c*Sqrt[c + d*x^2])

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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]

d*x/b/c/(d*x^2+c)^(1/2)

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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]

Exception raised: ValueError

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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]

sqrt(d*x^2 + c)*d*x/(b*c*d*x^2 + b*c^2)

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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]

d*Integral(1/(c*sqrt(c + d*x**2) + d*x**2*sqrt(c + d*x**2)), x)/b

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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]

d*x/(sqrt(d*x^2 + c)*b*c)