3.3.38 \(\int \frac {x^2}{(a+b x^3+c x^6)^{3/2}} \, dx\) [238]

Optimal. Leaf size=38 \[ -\frac {2 \left (b+2 c x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}} \]

[Out]

-2/3*(2*c*x^3+b)/(-4*a*c+b^2)/(c*x^6+b*x^3+a)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1366, 627} \begin {gather*} -\frac {2 \left (b+2 c x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*x^3 + c*x^6)^(3/2),x]

[Out]

(-2*(b + 2*c*x^3))/(3*(b^2 - 4*a*c)*Sqrt[a + b*x^3 + c*x^6])

Rule 627

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[-2*((b + 2*c*x)/((b^2 - 4*a*c)*Sqrt[a + b*x
+ c*x^2])), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 1366

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[(a + b*x +
c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[Simplify[m - n + 1], 0]

Rubi steps

\begin {align*} \int \frac {x^2}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {1}{\left (a+b x+c x^2\right )^{3/2}} \, dx,x,x^3\right )\\ &=-\frac {2 \left (b+2 c x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}}\\ \end {align*}

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Mathematica [A]
time = 0.25, size = 38, normalized size = 1.00 \begin {gather*} -\frac {2 \left (b+2 c x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b*x^3 + c*x^6)^(3/2),x]

[Out]

(-2*(b + 2*c*x^3))/(3*(b^2 - 4*a*c)*Sqrt[a + b*x^3 + c*x^6])

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Maple [A]
time = 0.03, size = 37, normalized size = 0.97

method	result	size
gosper	\(\frac {\frac {4 c \,x^{3}}{3}+\frac {2 b}{3}}{\sqrt {c \,x^{6}+b \,x^{3}+a}\, \left (4 a c -b^{2}\right )}\)	\(37\)
trager	\(\frac {\frac {4 c \,x^{3}}{3}+\frac {2 b}{3}}{\sqrt {c \,x^{6}+b \,x^{3}+a}\, \left (4 a c -b^{2}\right )}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(c*x^6+b*x^3+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

2/3/(c*x^6+b*x^3+a)^(1/2)*(2*c*x^3+b)/(4*a*c-b^2)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(c*x^6+b*x^3+a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more deta

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Fricas [A]
time = 0.37, size = 67, normalized size = 1.76 \begin {gather*} -\frac {2 \, \sqrt {c x^{6} + b x^{3} + a} {\left (2 \, c x^{3} + b\right )}}{3 \, {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{6} + {\left (b^{3} - 4 \, a b c\right )} x^{3} + a b^{2} - 4 \, a^{2} c\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(c*x^6+b*x^3+a)^(3/2),x, algorithm="fricas")

[Out]

-2/3*sqrt(c*x^6 + b*x^3 + a)*(2*c*x^3 + b)/((b^2*c - 4*a*c^2)*x^6 + (b^3 - 4*a*b*c)*x^3 + a*b^2 - 4*a^2*c)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\left (a + b x^{3} + c x^{6}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(c*x**6+b*x**3+a)**(3/2),x)

[Out]

Integral(x**2/(a + b*x**3 + c*x**6)**(3/2), x)

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Giac [A]
time = 2.96, size = 45, normalized size = 1.18 \begin {gather*} -\frac {2 \, {\left (\frac {2 \, c x^{3}}{b^{2} - 4 \, a c} + \frac {b}{b^{2} - 4 \, a c}\right )}}{3 \, \sqrt {c x^{6} + b x^{3} + a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(c*x^6+b*x^3+a)^(3/2),x, algorithm="giac")

[Out]

-2/3*(2*c*x^3/(b^2 - 4*a*c) + b/(b^2 - 4*a*c))/sqrt(c*x^6 + b*x^3 + a)

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Mupad [B]
time = 1.37, size = 37, normalized size = 0.97 \begin {gather*} \frac {4\,c\,x^3+2\,b}{\left (12\,a\,c-3\,b^2\right )\,\sqrt {c\,x^6+b\,x^3+a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a + b*x^3 + c*x^6)^(3/2),x)

[Out]

(2*b + 4*c*x^3)/((12*a*c - 3*b^2)*(a + b*x^3 + c*x^6)^(1/2))

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