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

Optimal. Leaf size=20 \[ b x^{-n/2} \log (x) \sqrt{b x^n} \]

[Out]

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

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Rubi [A]  time = 0.0086117, antiderivative size = 20, 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 \[ b x^{-n/2} \log (x) \sqrt{b x^n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 2.41357, size = 17, normalized size = 0.85 \[ b x^{- \frac{n}{2}} \sqrt{b x^{n}} \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0079647, size = 19, normalized size = 0.95 \[ x^{-3 n/2} \log (x) \left (b x^n\right )^{3/2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.054, size = 23, normalized size = 1.2 \[{b\ln \left ( x \right ) \sqrt{b \left ({x}^{{\frac{n}{2}}} \right ) ^{2}} \left ({x}^{{\frac{n}{2}}} \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.49516, size = 8, normalized size = 0.4 \[ b^{\frac{3}{2}} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.239329, size = 8, normalized size = 0.4 \[ b^{\frac{3}{2}} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.247724, size = 8, normalized size = 0.4 \[ b^{\frac{3}{2}}{\rm ln}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b^(3/2)*ln(x)