3.2685 \(\int \frac{x^{-1-\frac{3 n}{2}}}{\sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=58 \[ \frac{4 b x^{-n/2} \sqrt{a+b x^n}}{3 a^2 n}-\frac{2 x^{-3 n/2} \sqrt{a+b x^n}}{3 a n} \]

[Out]

(-2*Sqrt[a + b*x^n])/(3*a*n*x^((3*n)/2)) + (4*b*Sqrt[a + b*x^n])/(3*a^2*n*x^(n/2
))

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Rubi [A]  time = 0.056414, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{4 b x^{-n/2} \sqrt{a+b x^n}}{3 a^2 n}-\frac{2 x^{-3 n/2} \sqrt{a+b x^n}}{3 a n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - (3*n)/2)/Sqrt[a + b*x^n],x]

[Out]

(-2*Sqrt[a + b*x^n])/(3*a*n*x^((3*n)/2)) + (4*b*Sqrt[a + b*x^n])/(3*a^2*n*x^(n/2
))

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Rubi in Sympy [A]  time = 5.36357, size = 48, normalized size = 0.83 \[ - \frac{2 x^{- \frac{3 n}{2}} \sqrt{a + b x^{n}}}{3 a n} + \frac{4 b x^{- \frac{n}{2}} \sqrt{a + b x^{n}}}{3 a^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1-3/2*n)/(a+b*x**n)**(1/2),x)

[Out]

-2*x**(-3*n/2)*sqrt(a + b*x**n)/(3*a*n) + 4*b*x**(-n/2)*sqrt(a + b*x**n)/(3*a**2
*n)

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Mathematica [A]  time = 0.0474612, size = 36, normalized size = 0.62 \[ -\frac{2 x^{-3 n/2} \left (a-2 b x^n\right ) \sqrt{a+b x^n}}{3 a^2 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 - (3*n)/2)/Sqrt[a + b*x^n],x]

[Out]

(-2*(a - 2*b*x^n)*Sqrt[a + b*x^n])/(3*a^2*n*x^((3*n)/2))

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Maple [F]  time = 0.083, size = 0, normalized size = 0. \[ \int{1{x}^{-1-{\frac{3\,n}{2}}}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-3/2*n)/(a+b*x^n)^(1/2),x)

[Out]

int(x^(-1-3/2*n)/(a+b*x^n)^(1/2),x)

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Maxima [A]  time = 1.53043, size = 62, normalized size = 1.07 \[ \frac{2 \, \sqrt{b x^{n} + a} b x^{-\frac{1}{2} \, n}}{a^{2} n} - \frac{2 \,{\left (b x^{n} + a\right )}^{\frac{3}{2}} x^{-\frac{3}{2} \, n}}{3 \, a^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(-3/2*n - 1)/sqrt(b*x^n + a),x, algorithm="maxima")

[Out]

2*sqrt(b*x^n + a)*b*x^(-1/2*n)/(a^2*n) - 2/3*(b*x^n + a)^(3/2)*x^(-3/2*n)/(a^2*n
)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(-3/2*n - 1)/sqrt(b*x^n + a),x, algorithm="fricas")

[Out]

Exception raised: TypeError

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(-1-3/2*n)/(a+b*x**n)**(1/2),x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{-\frac{3}{2} \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(-3/2*n - 1)/sqrt(b*x^n + a),x, algorithm="giac")

[Out]

integrate(x^(-3/2*n - 1)/sqrt(b*x^n + a), x)