3.1.9 \(\int x^2 \sqrt {a^2+2 a b x^3+b^2 x^6} \, dx\)

Optimal. Leaf size=36 \[ \frac {\left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}{6 b} \]

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Rubi [A]  time = 0.03, antiderivative size = 36, 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, 609} \begin {gather*} \frac {\left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}{6 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6],x]

[Out]

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

Rule 609

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p + 1
)), x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && NeQ[p, -2^(-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 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \sqrt {a^2+2 a b x+b^2 x^2} \, dx,x,x^3\right )\\ &=\frac {\left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}{6 b}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 38, normalized size = 1.06 \begin {gather*} \frac {\sqrt {\left (a+b x^3\right )^2} \left (2 a x^3+b x^6\right )}{6 \left (a+b x^3\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6],x]

[Out]

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

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IntegrateAlgebraic [A]  time = 6.10, size = 38, normalized size = 1.06 \begin {gather*} \frac {\sqrt {\left (a+b x^3\right )^2} \left (2 a x^3+b x^6\right )}{6 \left (a+b x^3\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6],x]

[Out]

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

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fricas [A]  time = 1.67, size = 13, normalized size = 0.36 \begin {gather*} \frac {1}{6} \, b x^{6} + \frac {1}{3} \, a x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*b*x^6 + 1/3*a*x^3

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giac [A]  time = 0.36, size = 22, normalized size = 0.61 \begin {gather*} \frac {1}{6} \, {\left (b x^{6} + 2 \, a x^{3}\right )} \mathrm {sgn}\left (b x^{3} + a\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(b*x^6 + 2*a*x^3)*sgn(b*x^3 + a)

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maple [A]  time = 0.00, size = 35, normalized size = 0.97 \begin {gather*} \frac {\left (b \,x^{3}+2 a \right ) \sqrt {\left (b \,x^{3}+a \right )^{2}}\, x^{3}}{6 b \,x^{3}+6 a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.46, size = 52, normalized size = 1.44 \begin {gather*} \frac {1}{6} \, \sqrt {b^{2} x^{6} + 2 \, a b x^{3} + a^{2}} x^{3} + \frac {\sqrt {b^{2} x^{6} + 2 \, a b x^{3} + a^{2}} a}{6 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.23, size = 33, normalized size = 0.92 \begin {gather*} \left (\frac {a}{6\,b}+\frac {x^3}{6}\right )\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*((a + b*x^3)^2)^(1/2),x)

[Out]

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

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sympy [A]  time = 0.11, size = 12, normalized size = 0.33 \begin {gather*} \frac {a x^{3}}{3} + \frac {b x^{6}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x**3/3 + b*x**6/6

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