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

Optimal. Leaf size=19 \[ \frac{x^{n/2} \log (x)}{\sqrt{b x^n}} \]

[Out]

(x^(n/2)*Log[x])/Sqrt[b*x^n]

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Rubi [A]  time = 0.00991851, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{x^{n/2} \log (x)}{\sqrt{b x^n}} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + n/2)/Sqrt[b*x^n],x]

[Out]

(x^(n/2)*Log[x])/Sqrt[b*x^n]

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Rubi in Sympy [A]  time = 2.64999, size = 17, normalized size = 0.89 \[ \frac{x^{- \frac{n}{2}} \sqrt{b x^{n}} \log{\left (x \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**(-n/2)*sqrt(b*x**n)*log(x)/b

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Mathematica [A]  time = 0.00572642, size = 19, normalized size = 1. \[ \frac{x^{n/2} \log (x)}{\sqrt{b x^n}} \]

Antiderivative was successfully verified.

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

[Out]

(x^(n/2)*Log[x])/Sqrt[b*x^n]

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Maple [A]  time = 0.044, size = 20, normalized size = 1.1 \[{\ln \left ( x \right ){x}^{{\frac{n}{2}}}{\frac{1}{\sqrt{b \left ({x}^{{\frac{n}{2}}} \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(b*(x^(1/2*n))^2)^(1/2)*x^(1/2*n)*ln(x)

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Maxima [A]  time = 1.51275, size = 8, normalized size = 0.42 \[ \frac{\log \left (x\right )}{\sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x)/sqrt(b)

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Fricas [A]  time = 0.229689, size = 41, normalized size = 2.16 \[ \frac{\sqrt{b x^{2} x^{n - 2}} \log \left (x\right )}{b x x^{\frac{1}{2} \, n - 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(b*x^2*x^(n - 2))*log(x)/(b*x*x^(1/2*n - 1))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**(n/2 - 1)/sqrt(b*x**n), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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