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

Optimal. Leaf size=16 \[ \frac{x^4}{3 \sqrt{b x^2}} \]

[Out]

x^4/(3*Sqrt[b*x^2])

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Rubi [A]  time = 0.0017573, 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} \[ \frac{x^4}{3 \sqrt{b x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x^4/(3*Sqrt[b*x^2])

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

Mathematica [A]  time = 0.0014684, size = 16, normalized size = 1. \[ \frac{x^4}{3 \sqrt{b x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x^4/(3*Sqrt[b*x^2])

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Maple [A]  time = 0.001, size = 13, normalized size = 0.8 \begin{align*}{\frac{{x}^{4}}{3}{\frac{1}{\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]

1/3*x^4/(b*x^2)^(1/2)

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Maxima [A]  time = 0.98179, size = 20, normalized size = 1.25 \begin{align*} \frac{\sqrt{b x^{2}} x^{2}}{3 \, b} \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]

1/3*sqrt(b*x^2)*x^2/b

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Fricas [A]  time = 1.1742, size = 31, normalized size = 1.94 \begin{align*} \frac{\sqrt{b x^{2}} x^{2}}{3 \, b} \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]

1/3*sqrt(b*x^2)*x^2/b

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Sympy [A]  time = 0.469438, size = 15, normalized size = 0.94 \begin{align*} \frac{x^{4}}{3 \sqrt{b} \sqrt{x^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**4/(3*sqrt(b)*sqrt(x**2))

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Giac [A]  time = 1.15481, size = 20, normalized size = 1.25 \begin{align*} \frac{\sqrt{b x^{2}} x^{2}}{3 \, b} \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]

1/3*sqrt(b*x^2)*x^2/b