3.1.85 \(\int (b x)^{3/2} \, dx\) [85]

Optimal. Leaf size=14 \[ \frac {2 (b x)^{5/2}}{5 b} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {32} \begin {gather*} \frac {2 (b x)^{5/2}}{5 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 12, normalized size = 0.86 \begin {gather*} \frac {2}{5} x (b x)^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.02, size = 11, normalized size = 0.79

method	result	size
gosper	\(\frac {2 x \left (b x \right )^{\frac {3}{2}}}{5}\)	\(9\)
derivativedivides	\(\frac {2 \left (b x \right )^{\frac {5}{2}}}{5 b}\)	\(11\)
default	\(\frac {2 \left (b x \right )^{\frac {5}{2}}}{5 b}\)	\(11\)
trager	\(\frac {2 b \,x^{2} \sqrt {b x}}{5}\)	\(12\)
risch	\(\frac {2 b^{2} x^{3}}{5 \sqrt {b x}}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 10, normalized size = 0.71 \begin {gather*} \frac {2 \, \left (b x\right )^{\frac {5}{2}}}{5 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.35, size = 11, normalized size = 0.79 \begin {gather*} \frac {2}{5} \, \sqrt {b x} b x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.01, size = 10, normalized size = 0.71 \begin {gather*} \frac {2 \left (b x\right )^{\frac {5}{2}}}{5 b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.48, size = 11, normalized size = 0.79 \begin {gather*} \frac {2}{5} \, \sqrt {b x} b x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.01, size = 10, normalized size = 0.71 \begin {gather*} \frac {2\,{\left (b\,x\right )}^{5/2}}{5\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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