3.1.29 \(\int \frac {(b x^2)^{5/2}}{x^5} \, dx\) [29]

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} \begin {gather*} b^2 \sqrt {b x^2} \end {gather*}

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 \begin {gather*} \frac {\left (b x^2\right )^{5/2}}{x^4} \end {gather*}

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 = 0.02, size = 12, normalized size = 0.92

method	result	size
default	\(\frac {\left (b \,x^{2}\right )^{\frac {5}{2}}}{x^{4}}\)	\(12\)
risch	\(b^{2} \sqrt {b \,x^{2}}\)	\(12\)
trager	\(\frac {b^{2} \left (x -1\right ) \sqrt {b \,x^{2}}}{x}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2)^(5/2)/x^5,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

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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Fricas [A]
time = 0.33, size = 11, normalized size = 0.85 \begin {gather*} \sqrt {b x^{2}} b^{2} \end {gather*}

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 = 0.29, size = 10, normalized size = 0.77 \begin {gather*} \frac {\left (b x^{2}\right )^{\frac {5}{2}}}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.53, size = 7, normalized size = 0.54 \begin {gather*} b^{\frac {5}{2}} x \mathrm {sgn}\left (x\right ) \end {gather*}

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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Mupad [B]
time = 0.95, size = 6, normalized size = 0.46 \begin {gather*} b^{5/2}\,\left |x\right | \end {gather*}

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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