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

Optimal. Leaf size=13 \[ b^2 \sqrt{b x^2} \]

[Out]

b^2*Sqrt[b*x^2]

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Rubi [A]  time = 0.0012499, antiderivative size = 13, 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, 8} \[ b^2 \sqrt{b x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{\left (b x^2\right )^{5/2}}{x^5} \, dx &=\frac{\left (b^2 \sqrt{b x^2}\right ) \int 1 \, dx}{x}\\ &=b^2 \sqrt{b x^2}\\ \end{align*}

Mathematica [A]  time = 0.0011131, size = 13, normalized size = 1. \[ \frac{\left (b x^2\right )^{5/2}}{x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0., size = 12, normalized size = 0.9 \begin{align*}{\frac{1}{{x}^{4}} \left ( b{x}^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 1.21152, size = 23, normalized size = 1.77 \begin{align*} \sqrt{b x^{2}} b^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(b*x^2)*b^2

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Sympy [A]  time = 1.20033, size = 14, normalized size = 1.08 \begin{align*} \frac{b^{\frac{5}{2}} \left (x^{2}\right )^{\frac{5}{2}}}{x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.53789, size = 9, normalized size = 0.69 \begin{align*} b^{\frac{5}{2}} x \mathrm{sgn}\left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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