3.138 \(\int \left (b x^n\right )^{3/2} \, dx\)

Optimal. Leaf size=24 \[ \frac{2 b x^{n+1} \sqrt{b x^n}}{3 n+2} \]

[Out]

(2*b*x^(1 + n)*Sqrt[b*x^n])/(2 + 3*n)

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Rubi [A]  time = 0.01689, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{2 b x^{n+1} \sqrt{b x^n}}{3 n+2} \]

Antiderivative was successfully verified.

[In]  Int[(b*x^n)^(3/2),x]

[Out]

(2*b*x^(1 + n)*Sqrt[b*x^n])/(2 + 3*n)

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Rubi in Sympy [A]  time = 2.28161, size = 29, normalized size = 1.21 \[ \frac{2 b x^{- \frac{n}{2}} x^{\frac{3 n}{2} + 1} \sqrt{b x^{n}}}{3 n + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00640926, size = 20, normalized size = 0.83 \[ \frac{x \left (b x^n\right )^{3/2}}{\frac{3 n}{2}+1} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x^n)^(3/2),x]

[Out]

(x*(b*x^n)^(3/2))/(1 + (3*n)/2)

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Maple [A]  time = 0.001, size = 18, normalized size = 0.8 \[ 2\,{\frac{x \left ( b{x}^{n} \right ) ^{3/2}}{2+3\,n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^n)^(3/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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

[Out]

Exception raised: TypeError

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GIAC/XCAS [A]  time = 0.227146, size = 26, normalized size = 1.08 \[ \frac{2 \, b^{\frac{3}{2}} x e^{\left (\frac{3}{2} \, n{\rm ln}\left (x\right )\right )}}{3 \, n + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b^(3/2)*x*e^(3/2*n*ln(x))/(3*n + 2)