3.42 \(\int \frac{x^5}{(b x^2)^{3/2}} \, dx\)

Optimal. Leaf size=19 \[ \frac{x^4}{3 b \sqrt{b x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0017836, antiderivative size = 19, 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} \[ \frac{x^4}{3 b \sqrt{b x^2}} \]

Antiderivative was successfully verified.

[In]

Int[x^5/(b*x^2)^(3/2),x]

[Out]

x^4/(3*b*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^5}{\left (b x^2\right )^{3/2}} \, dx &=\frac{x \int x^2 \, dx}{b \sqrt{b x^2}}\\ &=\frac{x^4}{3 b \sqrt{b x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0010452, size = 16, normalized size = 0.84 \[ \frac{x^6}{3 \left (b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(b*x^2)^(3/2),x]

[Out]

x^6/(3*(b*x^2)^(3/2))

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Maple [A]  time = 0.001, size = 13, normalized size = 0.7 \begin{align*}{\frac{{x}^{6}}{3} \left ( b{x}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(b*x^2)^(3/2),x)

[Out]

1/3*x^6/(b*x^2)^(3/2)

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Maxima [A]  time = 0.990119, size = 20, normalized size = 1.05 \begin{align*} \frac{x^{4}}{3 \, \sqrt{b x^{2}} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x^2)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.46354, size = 34, normalized size = 1.79 \begin{align*} \frac{\sqrt{b x^{2}} x^{2}}{3 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x^2)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.944193, size = 15, normalized size = 0.79 \begin{align*} \frac{x^{6}}{3 b^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(b*x**2)**(3/2),x)

[Out]

x**6/(3*b**(3/2)*(x**2)**(3/2))

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Giac [A]  time = 1.12634, size = 20, normalized size = 1.05 \begin{align*} \frac{\sqrt{b x^{2}} x^{2}}{3 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x^2)^(3/2),x, algorithm="giac")

[Out]

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