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

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

[Out]

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

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Rubi [A]  time = 0.0093851, 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 \[ x^{-n/2} \log (x) \sqrt{b x^n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - n/2)*Sqrt[b*x^n],x]

[Out]

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

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Rubi in Sympy [A]  time = 2.32132, size = 15, normalized size = 0.79 \[ x^{- \frac{n}{2}} \sqrt{b x^{n}} \log{\left (x \right )} \]

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)

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Mathematica [A]  time = 0.00699099, size = 19, normalized size = 1. \[ x^{-n/2} \log (x) \sqrt{b x^n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.044, size = 22, normalized size = 1.2 \[{\ln \left ( x \right ) \sqrt{b \left ({x}^{{\frac{n}{2}}} \right ) ^{2}} \left ({x}^{{\frac{n}{2}}} \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.51002, size = 8, normalized size = 0.42 \[ \sqrt{b} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(b)*log(x)

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Fricas [A]  time = 0.226512, size = 36, normalized size = 1.89 \[ x x^{-\frac{1}{2} \, n - 1} \sqrt{\frac{b}{x^{2} x^{-n - 2}}} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.23396, size = 8, normalized size = 0.42 \[ \sqrt{b}{\rm ln}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(b)*ln(x)