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

Optimal. Leaf size=24 \[ \frac{2 b x^{n+2} \sqrt{b x^n}}{3 n+4} \]

[Out]

(2*b*x^(2 + n)*Sqrt[b*x^n])/(4 + 3*n)

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Rubi [A]  time = 0.017304, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{2 b x^{n+2} \sqrt{b x^n}}{3 n+4} \]

Antiderivative was successfully verified.

[In]  Int[x*(b*x^n)^(3/2),x]

[Out]

(2*b*x^(2 + n)*Sqrt[b*x^n])/(4 + 3*n)

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Rubi in Sympy [A]  time = 2.95375, size = 29, normalized size = 1.21 \[ \frac{2 b x^{- \frac{n}{2}} x^{\frac{3 n}{2} + 2} \sqrt{b x^{n}}}{3 n + 4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b*x**(-n/2)*x**(3*n/2 + 2)*sqrt(b*x**n)/(3*n + 4)

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Mathematica [A]  time = 0.00963597, size = 22, normalized size = 0.92 \[ \frac{x^2 \left (b x^n\right )^{3/2}}{\frac{3 n}{2}+2} \]

Antiderivative was successfully verified.

[In]  Integrate[x*(b*x^n)^(3/2),x]

[Out]

(x^2*(b*x^n)^(3/2))/(2 + (3*n)/2)

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Maple [A]  time = 0.004, size = 20, normalized size = 0.8 \[ 2\,{\frac{{x}^{2} \left ( b{x}^{n} \right ) ^{3/2}}{4+3\,n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x^n)^(3/2),x)

[Out]

2*x^2/(4+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,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,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*(b*x**n)**(3/2),x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [A]  time = 0.229467, size = 28, normalized size = 1.17 \[ \frac{2 \, b^{\frac{3}{2}} x^{2} e^{\left (\frac{3}{2} \, n{\rm ln}\left (x\right )\right )}}{3 \, n + 4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b^(3/2)*x^2*e^(3/2*n*ln(x))/(3*n + 4)