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

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Rubi [A]  time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, 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 8

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

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]

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*}

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Mathematica [A]  time = 0.00, size = 13, normalized size = 1.00 \[ \frac {\left (b x^2\right )^{5/2}}{x^4} \]

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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fricas [A]  time = 0.52, size = 11, normalized size = 0.85 \[ \sqrt {b x^{2}} b^{2} \]

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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giac [A]  time = 0.17, size = 7, normalized size = 0.54 \[ b^{\frac {5}{2}} x \mathrm {sgn}\relax (x) \]

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)

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maple [A]  time = 0.00, size = 12, normalized size = 0.92 \[ \frac {\left (b \,x^{2}\right )^{\frac {5}{2}}}{x^{4}} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

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 >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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mupad [B]  time = 0.95, size = 6, normalized size = 0.46 \[ b^{5/2}\,\relax |x| \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 1.24, size = 14, normalized size = 1.08 \[ \frac {b^{\frac {5}{2}} \left (x^{2}\right )^{\frac {5}{2}}}{x^{4}} \]

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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