3.2.70 \(\int \frac {x^2}{\sqrt {-b+a x^3}} \, dx\)

Optimal. Leaf size=20 \[ \frac {2 \sqrt {a x^3-b}}{3 a} \]

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Rubi [A]  time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {261} \begin {gather*} \frac {2 \sqrt {a x^3-b}}{3 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/Sqrt[-b + a*x^3],x]

[Out]

(2*Sqrt[-b + a*x^3])/(3*a)

Rule 261

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}{\sqrt {-b+a x^3}} \, dx &=\frac {2 \sqrt {-b+a x^3}}{3 a}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 20, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {a x^3-b}}{3 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/Sqrt[-b + a*x^3],x]

[Out]

(2*Sqrt[-b + a*x^3])/(3*a)

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IntegrateAlgebraic [A]  time = 0.02, size = 20, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {-b+a x^3}}{3 a} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^2/Sqrt[-b + a*x^3],x]

[Out]

(2*Sqrt[-b + a*x^3])/(3*a)

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fricas [A]  time = 0.46, size = 16, normalized size = 0.80 \begin {gather*} \frac {2 \, \sqrt {a x^{3} - b}}{3 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a*x^3-b)^(1/2),x, algorithm="fricas")

[Out]

2/3*sqrt(a*x^3 - b)/a

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giac [A]  time = 0.28, size = 16, normalized size = 0.80 \begin {gather*} \frac {2 \, \sqrt {a x^{3} - b}}{3 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a*x^3-b)^(1/2),x, algorithm="giac")

[Out]

2/3*sqrt(a*x^3 - b)/a

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maple [A]  time = 0.30, size = 17, normalized size = 0.85

method	result	size
gosper	\(\frac {2 \sqrt {a \,x^{3}-b}}{3 a}\)	\(17\)
derivativedivides	\(\frac {2 \sqrt {a \,x^{3}-b}}{3 a}\)	\(17\)
default	\(\frac {2 \sqrt {a \,x^{3}-b}}{3 a}\)	\(17\)
trager	\(\frac {2 \sqrt {a \,x^{3}-b}}{3 a}\)	\(17\)
risch	\(\frac {2 \sqrt {a \,x^{3}-b}}{3 a}\)	\(17\)
elliptic	\(\frac {2 \sqrt {a \,x^{3}-b}}{3 a}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a*x^3-b)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.42, size = 16, normalized size = 0.80 \begin {gather*} \frac {2 \, \sqrt {a x^{3} - b}}{3 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a*x^3-b)^(1/2),x, algorithm="maxima")

[Out]

2/3*sqrt(a*x^3 - b)/a

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mupad [B]  time = 0.30, size = 16, normalized size = 0.80 \begin {gather*} \frac {2\,\sqrt {a\,x^3-b}}{3\,a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a*x^3 - b)^(1/2),x)

[Out]

(2*(a*x^3 - b)^(1/2))/(3*a)

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sympy [A]  time = 0.36, size = 26, normalized size = 1.30 \begin {gather*} \begin {cases} \frac {2 \sqrt {a x^{3} - b}}{3 a} & \text {for}\: a \neq 0 \\\frac {x^{3}}{3 \sqrt {- b}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(a*x**3-b)**(1/2),x)

[Out]

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

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