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

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

[Out]

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

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Rubi [A]  time = 0.0275715, antiderivative size = 36, normalized size of antiderivative = 1., 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} \[ \frac{\left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}{6 b} \]

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 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]

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)]

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*}

Mathematica [A]  time = 0.009384, size = 38, normalized size = 1.06 \[ \frac{\sqrt{\left (a+b x^3\right )^2} \left (2 a x^3+b x^6\right )}{6 \left (a+b x^3\right )} \]

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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Maple [A]  time = 0.003, size = 35, normalized size = 1. \begin{align*}{\frac{{x}^{3} \left ( b{x}^{3}+2\,a \right ) }{6\,b{x}^{3}+6\,a}\sqrt{ \left ( b{x}^{3}+a \right ) ^{2}}} \end{align*}

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

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]

Exception raised: ValueError

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Fricas [A]  time = 1.67638, size = 31, normalized size = 0.86 \begin{align*} \frac{1}{6} \, b x^{6} + \frac{1}{3} \, a x^{3} \end{align*}

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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Sympy [A]  time = 0.099698, size = 12, normalized size = 0.33 \begin{align*} \frac{a x^{3}}{3} + \frac{b x^{6}}{6} \end{align*}

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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Giac [A]  time = 1.11975, size = 30, normalized size = 0.83 \begin{align*} \frac{1}{6} \,{\left (b x^{6} + 2 \, a x^{3}\right )} \mathrm{sgn}\left (b x^{3} + a\right ) \end{align*}

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)