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

Optimal. Leaf size=20 \[ \frac{2 b x^n \sqrt{b x^n}}{3 n} \]

[Out]

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

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Rubi [A]  time = 0.0114605, antiderivative size = 20, 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 \[ \frac{2 b x^n \sqrt{b x^n}}{3 n} \]

Antiderivative was successfully verified.

[In]  Int[(b*x^n)^(3/2)/x,x]

[Out]

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

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Rubi in Sympy [A]  time = 2.80194, size = 12, normalized size = 0.6 \[ \frac{2 \left (b x^{n}\right )^{\frac{3}{2}}}{3 n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0042129, size = 16, normalized size = 0.8 \[ \frac{2 \left (b x^n\right )^{3/2}}{3 n} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x^n)^(3/2)/x,x]

[Out]

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

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Maple [A]  time = 0.002, size = 13, normalized size = 0.7 \[{\frac{2}{3\,n} \left ( b{x}^{n} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^n)^(3/2)/x,x)

[Out]

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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.229261, size = 22, normalized size = 1.1 \[ \frac{2 \, \sqrt{b x^{n}} b x^{n}}{3 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 12.4628, size = 24, normalized size = 1.2 \[ \begin{cases} \frac{2 b^{\frac{3}{2}} \left (x^{n}\right )^{\frac{3}{2}}}{3 n} & \text{for}\: n \neq 0 \\b^{\frac{3}{2}} \log{\left (x \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**n)**(3/2)/x,x)

[Out]

Piecewise((2*b**(3/2)*(x**n)**(3/2)/(3*n), Ne(n, 0)), (b**(3/2)*log(x), True))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^n)^(3/2)/x, x)