\(\int \frac {x^2}{\sqrt {-b+a x^3}} \, dx\) [170]

Optimal result

Integrand size = 17, antiderivative size = 20 \[ \int \frac {x^2}{\sqrt {-b+a x^3}} \, dx=\frac {2 \sqrt {-b+a x^3}}{3 a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {267} \[ \int \frac {x^2}{\sqrt {-b+a x^3}} \, dx=\frac {2 \sqrt {a x^3-b}}{3 a} \]

[In]

Int[x^2/Sqrt[-b + a*x^3],x]

[Out]

(2*Sqrt[-b + a*x^3])/(3*a)

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{\sqrt {-b+a x^3}} \, dx=\frac {2 \sqrt {-b+a x^3}}{3 a} \]

[In]

Integrate[x^2/Sqrt[-b + a*x^3],x]

[Out]

(2*Sqrt[-b + a*x^3])/(3*a)

Maple [A] (verified)

Time = 3.61 (sec) , antiderivative size = 17, normalized size of antiderivative = 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	\(\frac {2 \sqrt {a \,x^{3}-b}}{3 a}\)	\(17\)
pseudoelliptic	\(\frac {2 \sqrt {a \,x^{3}-b}}{3 a}\)	\(17\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {x^2}{\sqrt {-b+a x^3}} \, dx=\frac {2 \, \sqrt {a x^{3} - b}}{3 \, a} \]

[In]

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

[Out]

2/3*sqrt(a*x^3 - b)/a

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.30 \[ \int \frac {x^2}{\sqrt {-b+a x^3}} \, dx=\begin {cases} \frac {2 \sqrt {a x^{3} - b}}{3 a} & \text {for}\: a \neq 0 \\\frac {x^{3}}{3 \sqrt {- b}} & \text {otherwise} \end {cases} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {x^2}{\sqrt {-b+a x^3}} \, dx=\frac {2 \, \sqrt {a x^{3} - b}}{3 \, a} \]

[In]

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

[Out]

2/3*sqrt(a*x^3 - b)/a

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {x^2}{\sqrt {-b+a x^3}} \, dx=\frac {2 \, \sqrt {a x^{3} - b}}{3 \, a} \]

[In]

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

[Out]

2/3*sqrt(a*x^3 - b)/a

Mupad [B] (verification not implemented)

Time = 5.21 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {x^2}{\sqrt {-b+a x^3}} \, dx=\frac {2\,\sqrt {a\,x^3-b}}{3\,a} \]

[In]

int(x^2/(a*x^3 - b)^(1/2),x)

[Out]

(2*(a*x^3 - b)^(1/2))/(3*a)