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

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 = 0.0061238, 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, Rules used = {15, 30} \[ \frac{2 b x^{n+2} \sqrt{b x^n}}{3 n+4} \]

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)

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int x \left (b x^n\right )^{3/2} \, dx &=\left (b x^{-n/2} \sqrt{b x^n}\right ) \int x^{1+\frac{3 n}{2}} \, dx\\ &=\frac{2 b x^{2+n} \sqrt{b x^n}}{4+3 n}\\ \end{align*}

Mathematica [A]  time = 0.0053048, 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 = 0.001, size = 20, normalized size = 0.8 \begin{align*} 2\,{\frac{{x}^{2} \left ( b{x}^{n} \right ) ^{3/2}}{4+3\,n}} \end{align*}

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(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(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. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

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 [A]  time = 1.20751, size = 27, normalized size = 1.12 \begin{align*} \frac{2 \, b^{\frac{3}{2}} x^{2} x^{\frac{3}{2} \, n}}{3 \, n + 4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b^(3/2)*x^2*x^(3/2*n)/(3*n + 4)