3.434 \(\int \frac{\sqrt{b-\frac{a}{x^2}} x^m}{\sqrt{a-b x^2}} \, dx\)

Optimal. Leaf size=33 \[ \frac{x^{m+1} \sqrt{b-\frac{a}{x^2}}}{m \sqrt{a-b x^2}} \]

[Out]

(Sqrt[b - a/x^2]*x^(1 + m))/(m*Sqrt[a - b*x^2])

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Rubi [A]  time = 0.111759, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ \frac{x^{m+1} \sqrt{b-\frac{a}{x^2}}}{m \sqrt{a-b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[b - a/x^2]*x^m)/Sqrt[a - b*x^2],x]

[Out]

(Sqrt[b - a/x^2]*x^(1 + m))/(m*Sqrt[a - b*x^2])

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Rubi in Sympy [A]  time = 6.00444, size = 27, normalized size = 0.82 \[ - \frac{x^{m} \sqrt{a - b x^{2}}}{m x \sqrt{- \frac{a}{x^{2}} + b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m*(b-a/x**2)**(1/2)/(-b*x**2+a)**(1/2),x)

[Out]

-x**m*sqrt(a - b*x**2)/(m*x*sqrt(-a/x**2 + b))

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Mathematica [A]  time = 0.0927068, size = 33, normalized size = 1. \[ \frac{x^{m+1} \sqrt{b-\frac{a}{x^2}}}{m \sqrt{a-b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[b - a/x^2]*x^m)/Sqrt[a - b*x^2],x]

[Out]

(Sqrt[b - a/x^2]*x^(1 + m))/(m*Sqrt[a - b*x^2])

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Maple [A]  time = 0.004, size = 35, normalized size = 1.1 \[{\frac{{x}^{1+m}}{m}\sqrt{-{\frac{-b{x}^{2}+a}{{x}^{2}}}}{\frac{1}{\sqrt{-b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(b-a/x^2)^(1/2)/(-b*x^2+a)^(1/2),x)

[Out]

x^(1+m)/m*(-(-b*x^2+a)/x^2)^(1/2)/(-b*x^2+a)^(1/2)

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Maxima [A]  time = 0.71332, size = 11, normalized size = 0.33 \[ -\frac{i \, x^{m}}{m} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b - a/x^2)*x^m/sqrt(-b*x^2 + a),x, algorithm="maxima")

[Out]

-I*x^m/m

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Fricas [A]  time = 0.279225, size = 59, normalized size = 1.79 \[ -\frac{\sqrt{-b x^{2} + a} x x^{m} \sqrt{\frac{b x^{2} - a}{x^{2}}}}{b m x^{2} - a m} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b - a/x^2)*x^m/sqrt(-b*x^2 + a),x, algorithm="fricas")

[Out]

-sqrt(-b*x^2 + a)*x*x^m*sqrt((b*x^2 - a)/x^2)/(b*m*x^2 - a*m)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m} \sqrt{- \frac{a}{x^{2}} + b}}{\sqrt{a - b x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**m*(b-a/x**2)**(1/2)/(-b*x**2+a)**(1/2),x)

[Out]

Integral(x**m*sqrt(-a/x**2 + b)/sqrt(a - b*x**2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b - \frac{a}{x^{2}}} x^{m}}{\sqrt{-b x^{2} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b - a/x^2)*x^m/sqrt(-b*x^2 + a),x, algorithm="giac")

[Out]

integrate(sqrt(b - a/x^2)*x^m/sqrt(-b*x^2 + a), x)