3.171 \(\int \frac{x^{-1+\frac{3 n}{2}}}{\left (b x^n\right )^{3/2}} \, dx\)

Optimal. Leaf size=22 \[ \frac{x^{n/2} \log (x)}{b \sqrt{b x^n}} \]

[Out]

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

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Rubi [A]  time = 0.00877873, antiderivative size = 22, 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 \[ \frac{x^{n/2} \log (x)}{b \sqrt{b x^n}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 2.65139, size = 19, normalized size = 0.86 \[ \frac{x^{- \frac{n}{2}} \sqrt{b x^{n}} \log{\left (x \right )}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00662045, size = 19, normalized size = 0.86 \[ \frac{x^{3 n/2} \log (x)}{\left (b x^n\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(x^((3*n)/2)*Log[x])/(b*x^n)^(3/2)

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Maple [A]  time = 0.045, size = 23, normalized size = 1.1 \[{\frac{\ln \left ( x \right ) }{b}{x}^{{\frac{n}{2}}}{\frac{1}{\sqrt{b \left ({x}^{{\frac{n}{2}}} \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.53026, size = 8, normalized size = 0.36 \[ \frac{\log \left (x\right )}{b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x)/b^(3/2)

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Fricas [A]  time = 0.237995, size = 27, normalized size = 1.23 \[ \frac{\sqrt{b x^{n}} \log \left (x\right )}{b^{2} x^{\frac{1}{2} \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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+3/2*n)/(b*x**n)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{3}{2} \, n - 1}}{\left (b x^{n}\right )^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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