3.131 \(\int x^2 \sqrt{b x^n} \, dx\)

Optimal. Leaf size=19 \[ \frac{2 x^3 \sqrt{b x^n}}{n+6} \]

[Out]

(2*x^3*Sqrt[b*x^n])/(6 + n)

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Rubi [A]  time = 0.015164, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{2 x^3 \sqrt{b x^n}}{n+6} \]

Antiderivative was successfully verified.

[In]  Int[x^2*Sqrt[b*x^n],x]

[Out]

(2*x^3*Sqrt[b*x^n])/(6 + n)

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Rubi in Sympy [A]  time = 3.01567, size = 24, normalized size = 1.26 \[ \frac{2 x^{- \frac{n}{2}} x^{\frac{n}{2} + 3} \sqrt{b x^{n}}}{n + 6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x**(-n/2)*x**(n/2 + 3)*sqrt(b*x**n)/(n + 6)

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Mathematica [A]  time = 0.00592225, size = 19, normalized size = 1. \[ \frac{2 x^3 \sqrt{b x^n}}{n+6} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*Sqrt[b*x^n],x]

[Out]

(2*x^3*Sqrt[b*x^n])/(6 + n)

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Maple [A]  time = 0.003, size = 18, normalized size = 1. \[ 2\,{\frac{{x}^{3}\sqrt{b{x}^{n}}}{6+n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*x^n)^(1/2),x)

[Out]

2*x^3*(b*x^n)^(1/2)/(6+n)

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

[Out]

Exception raised: TypeError

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GIAC/XCAS [A]  time = 0.226392, size = 26, normalized size = 1.37 \[ \frac{2 \, \sqrt{b} x^{3} e^{\left (\frac{1}{2} \, n{\rm ln}\left (x\right )\right )}}{n + 6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(b)*x^3*e^(1/2*n*ln(x))/(n + 6)