3.1.48 \(\int \frac {x^4}{(b x^2)^{3/2}} \, dx\) [48]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 19, 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, 30} \begin {gather*} \frac {x^3}{2 b \sqrt {b x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]
time = 0.00, size = 16, normalized size = 0.84 \begin {gather*} \frac {x^5}{2 \left (b x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.02, size = 13, normalized size = 0.68

method	result	size
gosper	\(\frac {x^{5}}{2 \left (b \,x^{2}\right )^{\frac {3}{2}}}\)	\(13\)
default	\(\frac {x^{5}}{2 \left (b \,x^{2}\right )^{\frac {3}{2}}}\)	\(13\)
risch	\(\frac {x^{3}}{2 b \sqrt {b \,x^{2}}}\)	\(16\)
trager	\(\frac {\left (x -1\right ) \left (x +1\right ) \sqrt {b \,x^{2}}}{2 b^{2} x}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(b*x^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 15, normalized size = 0.79 \begin {gather*} \frac {x^{3}}{2 \, \sqrt {b x^{2}} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^2)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.34, size = 13, normalized size = 0.68 \begin {gather*} \frac {\sqrt {b x^{2}} x}{2 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^2)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.21, size = 12, normalized size = 0.63 \begin {gather*} \frac {x^{5}}{2 \left (b x^{2}\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(b*x**2)**(3/2),x)

[Out]

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

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Giac [A]
time = 0.91, size = 12, normalized size = 0.63 \begin {gather*} \frac {x^{2}}{2 \, b^{\frac {3}{2}} \mathrm {sgn}\left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^2)^(3/2),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {x^4}{{\left (b\,x^2\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(b*x^2)^(3/2),x)

[Out]

int(x^4/(b*x^2)^(3/2), x)

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