3.2 \(\int x^3 \sqrt{b x^2} \, dx\)

Optimal. Leaf size=16 \[ \frac{1}{5} x^4 \sqrt{b x^2} \]

[Out]

(x^4*Sqrt[b*x^2])/5

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Rubi [A]  time = 0.0015588, antiderivative size = 16, 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, Rules used = {15, 30} \[ \frac{1}{5} x^4 \sqrt{b x^2} \]

Antiderivative was successfully verified.

[In]

Int[x^3*Sqrt[b*x^2],x]

[Out]

(x^4*Sqrt[b*x^2])/5

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^3 \sqrt{b x^2} \, dx &=\frac{\sqrt{b x^2} \int x^4 \, dx}{x}\\ &=\frac{1}{5} x^4 \sqrt{b x^2}\\ \end{align*}

Mathematica [A]  time = 0.0016718, size = 16, normalized size = 1. \[ \frac{1}{5} x^4 \sqrt{b x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*Sqrt[b*x^2],x]

[Out]

(x^4*Sqrt[b*x^2])/5

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Maple [A]  time = 0.003, size = 13, normalized size = 0.8 \begin{align*}{\frac{{x}^{4}}{5}\sqrt{b{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(b*x^2)^(1/2),x)

[Out]

1/5*x^4*(b*x^2)^(1/2)

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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^3*(b*x^2)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.15891, size = 28, normalized size = 1.75 \begin{align*} \frac{1}{5} \, \sqrt{b x^{2}} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x^2)^(1/2),x, algorithm="fricas")

[Out]

1/5*sqrt(b*x^2)*x^4

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Sympy [A]  time = 0.409797, size = 15, normalized size = 0.94 \begin{align*} \frac{\sqrt{b} x^{4} \sqrt{x^{2}}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(b*x**2)**(1/2),x)

[Out]

sqrt(b)*x**4*sqrt(x**2)/5

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Giac [A]  time = 1.13399, size = 14, normalized size = 0.88 \begin{align*} \frac{1}{5} \, \sqrt{b} x^{5} \mathrm{sgn}\left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x^2)^(1/2),x, algorithm="giac")

[Out]

1/5*sqrt(b)*x^5*sgn(x)