3.1.83 \(\int (b x^3)^{3/2} \, dx\) [83]

Optimal. Leaf size=17 \[ \frac {2}{11} b x^4 \sqrt {b x^3} \]

[Out]

2/11*b*x^4*(b*x^3)^(1/2)

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Rubi [A]
time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {15, 30} \begin {gather*} \frac {2}{11} b x^4 \sqrt {b x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b*x^4*Sqrt[b*x^3])/11

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 \left (b x^3\right )^{3/2} \, dx &=\frac {\left (b \sqrt {b x^3}\right ) \int x^{9/2} \, dx}{x^{3/2}}\\ &=\frac {2}{11} b x^4 \sqrt {b x^3}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 14, normalized size = 0.82 \begin {gather*} \frac {2}{11} x \left (b x^3\right )^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*(b*x^3)^(3/2))/11

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Maple [A]
time = 0.03, size = 11, normalized size = 0.65

method	result	size
gosper	\(\frac {2 x \left (b \,x^{3}\right )^{\frac {3}{2}}}{11}\)	\(11\)
default	\(\frac {2 x \left (b \,x^{3}\right )^{\frac {3}{2}}}{11}\)	\(11\)
trager	\(\frac {2 b \,x^{4} \sqrt {b \,x^{3}}}{11}\)	\(14\)
risch	\(\frac {2 b \,x^{4} \sqrt {b \,x^{3}}}{11}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/11*x*(b*x^3)^(3/2)

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Maxima [A]
time = 0.28, size = 10, normalized size = 0.59 \begin {gather*} \frac {2}{11} \, \left (b x^{3}\right )^{\frac {3}{2}} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/11*(b*x^3)^(3/2)*x

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Fricas [A]
time = 0.37, size = 13, normalized size = 0.76 \begin {gather*} \frac {2}{11} \, \sqrt {b x^{3}} b x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/11*sqrt(b*x^3)*b*x^4

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Sympy [A]
time = 0.11, size = 12, normalized size = 0.71 \begin {gather*} \frac {2 x \left (b x^{3}\right )^{\frac {3}{2}}}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x*(b*x**3)**(3/2)/11

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Giac [A]
time = 1.11, size = 13, normalized size = 0.76 \begin {gather*} \frac {2}{11} \, \sqrt {b x} b x^{5} \mathrm {sgn}\left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/11*sqrt(b*x)*b*x^5*sgn(x)

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Mupad [B]
time = 0.95, size = 10, normalized size = 0.59 \begin {gather*} \frac {2\,x\,{\left (b\,x^3\right )}^{3/2}}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*x*(b*x^3)^(3/2))/11

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