3.57 \(\int \frac{x^6}{(b x^2)^{5/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0018147, 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} \[ \frac{x^3}{2 b^2 \sqrt{b x^2}} \]

Antiderivative was successfully verified.

[In]

Int[x^6/(b*x^2)^(5/2),x]

[Out]

x^3/(2*b^2*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^6}{\left (b x^2\right )^{5/2}} \, dx &=\frac{x \int x \, dx}{b^2 \sqrt{b x^2}}\\ &=\frac{x^3}{2 b^2 \sqrt{b x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0015981, size = 16, normalized size = 0.84 \[ \frac{x^7}{2 \left (b x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6/(b*x^2)^(5/2),x]

[Out]

x^7/(2*(b*x^2)^(5/2))

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Maple [A]  time = 0.001, size = 13, normalized size = 0.7 \begin{align*}{\frac{{x}^{7}}{2} \left ( b{x}^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(b*x^2)^(5/2),x)

[Out]

1/2*x^7/(b*x^2)^(5/2)

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Maxima [A]  time = 0.992369, size = 20, normalized size = 1.05 \begin{align*} \frac{x^{5}}{2 \, \left (b x^{2}\right )^{\frac{3}{2}} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x^2)^(5/2),x, algorithm="maxima")

[Out]

1/2*x^5/((b*x^2)^(3/2)*b)

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Fricas [A]  time = 1.46111, size = 31, normalized size = 1.63 \begin{align*} \frac{\sqrt{b x^{2}} x}{2 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x^2)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.33395, size = 15, normalized size = 0.79 \begin{align*} \frac{x^{7}}{2 b^{\frac{5}{2}} \left (x^{2}\right )^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6/(b*x**2)**(5/2),x)

[Out]

x**7/(2*b**(5/2)*(x**2)**(5/2))

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Giac [A]  time = 1.22201, size = 18, normalized size = 0.95 \begin{align*} \frac{\sqrt{b x^{2}} x}{2 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x^2)^(5/2),x, algorithm="giac")

[Out]

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