3.1.100 \(\int \frac {x^2}{(a^2+2 a b x^3+b^2 x^6)^{3/2}} \, dx\)

Optimal. Leaf size=38 \[ -\frac {1}{6 b \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}} \]

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Rubi [A]  time = 0.03, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1352, 607} \begin {gather*} -\frac {1}{6 b \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(6*b*(a + b*x^3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])

Rule 607

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(2*(a + b*x + c*x^2)^(p + 1))/((2*p + 1)*(b + 2
*c*x)), x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && LtQ[p, -1]

Rule 1352

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

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Mathematica [A]  time = 0.01, size = 27, normalized size = 0.71 \begin {gather*} -\frac {a+b x^3}{6 b \left (\left (a+b x^3\right )^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [B]  time = 0.57, size = 137, normalized size = 3.61 \begin {gather*} \frac {\sqrt {b^2} \left (a-b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}+a^2 b+b^3 x^6}{3 b \sqrt {b^2} x^6 \left (2 a^2 b^2+4 a b^3 x^3+2 b^4 x^6\right )+3 b x^6 \left (-2 a b^3-2 b^4 x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.41, size = 26, normalized size = 0.68 \begin {gather*} -\frac {1}{6 \, {\left (b^{3} x^{6} + 2 \, a b^{2} x^{3} + a^{2} b\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6/(b^3*x^6 + 2*a*b^2*x^3 + a^2*b)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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maple [A]  time = 0.01, size = 24, normalized size = 0.63 \begin {gather*} -\frac {b \,x^{3}+a}{6 \left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {3}{2}} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(b^2*x^6+2*a*b*x^3+a^2)^(3/2),x)

[Out]

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

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maxima [A]  time = 0.89, size = 16, normalized size = 0.42 \begin {gather*} -\frac {1}{6 \, {\left (x^{3} + \frac {a}{b}\right )}^{2} b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6/((x^3 + a/b)^2*b^3)

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mupad [B]  time = 1.19, size = 34, normalized size = 0.89 \begin {gather*} -\frac {\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{6\,b\,{\left (b\,x^3+a\right )}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a^2 + b^2*x^6 + 2*a*b*x^3)^(3/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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