3.5.24 \(\int \frac {x^2}{(a+b x^3)^{3/2}} \, dx\) [424]

Optimal. Leaf size=18 \[ -\frac {2}{3 b \sqrt {a+b x^3}} \]

[Out]

-2/3/b/(b*x^3+a)^(1/2)

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Rubi [A]
time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {267} \begin {gather*} -\frac {2}{3 b \sqrt {a+b x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*x^3)^(3/2),x]

[Out]

-2/(3*b*Sqrt[a + b*x^3])

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 18, normalized size = 1.00 \begin {gather*} -\frac {2}{3 b \sqrt {a+b x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b*x^3)^(3/2),x]

[Out]

-2/(3*b*Sqrt[a + b*x^3])

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Maple [A]
time = 0.15, size = 15, normalized size = 0.83

method	result	size
gosper	\(-\frac {2}{3 b \sqrt {b \,x^{3}+a}}\)	\(15\)
derivativedivides	\(-\frac {2}{3 b \sqrt {b \,x^{3}+a}}\)	\(15\)
default	\(-\frac {2}{3 b \sqrt {b \,x^{3}+a}}\)	\(15\)
trager	\(-\frac {2}{3 b \sqrt {b \,x^{3}+a}}\)	\(15\)
elliptic	\(-\frac {2}{3 b \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3/b/(b*x^3+a)^(1/2)

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Maxima [A]
time = 0.30, size = 14, normalized size = 0.78 \begin {gather*} -\frac {2}{3 \, \sqrt {b x^{3} + a} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3/(sqrt(b*x^3 + a)*b)

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Fricas [A]
time = 0.36, size = 24, normalized size = 1.33 \begin {gather*} -\frac {2 \, \sqrt {b x^{3} + a}}{3 \, {\left (b^{2} x^{3} + a b\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*sqrt(b*x^3 + a)/(b^2*x^3 + a*b)

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Sympy [A]
time = 0.26, size = 26, normalized size = 1.44 \begin {gather*} \begin {cases} - \frac {2}{3 b \sqrt {a + b x^{3}}} & \text {for}\: b \neq 0 \\\frac {x^{3}}{3 a^{\frac {3}{2}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2/(3*b*sqrt(a + b*x**3)), Ne(b, 0)), (x**3/(3*a**(3/2)), True))

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Giac [A]
time = 1.02, size = 14, normalized size = 0.78 \begin {gather*} -\frac {2}{3 \, \sqrt {b x^{3} + a} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3/(sqrt(b*x^3 + a)*b)

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Mupad [B]
time = 1.10, size = 14, normalized size = 0.78 \begin {gather*} -\frac {2}{3\,b\,\sqrt {b\,x^3+a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a + b*x^3)^(3/2),x)

[Out]

-2/(3*b*(a + b*x^3)^(1/2))

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