3.543 \(\int x \sqrt{a^2+2 a b x^2+b^2 x^4} \, dx\)

Optimal. Leaf size=36 \[ \frac{\left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 b} \]

[Out]

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

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Rubi [A]  time = 0.0255502, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {1107, 609} \[ \frac{\left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 b} \]

Antiderivative was successfully verified.

[In]

Int[x*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]

[Out]

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

Rule 1107

Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(a + b*x + c*x^2)^p, x],
x, x^2], x] /; FreeQ[{a, b, c, p}, x]

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 \sqrt{a^2+2 a b x^2+b^2 x^4} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \sqrt{a^2+2 a b x+b^2 x^2} \, dx,x,x^2\right )\\ &=\frac{\left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 b}\\ \end{align*}

Mathematica [A]  time = 0.0071594, size = 38, normalized size = 1.06 \[ \frac{\sqrt{\left (a+b x^2\right )^2} \left (2 a x^2+b x^4\right )}{4 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[x*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]

[Out]

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

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Maple [A]  time = 0.043, size = 35, normalized size = 1. \begin{align*}{\frac{{x}^{2} \left ( b{x}^{2}+2\,a \right ) }{4\,b{x}^{2}+4\,a}\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^2*(b*x^2+2*a)*((b*x^2+a)^2)^(1/2)/(b*x^2+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*((b*x^2+a)^2)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.45774, size = 31, normalized size = 0.86 \begin{align*} \frac{1}{4} \, b x^{4} + \frac{1}{2} \, a x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b*x^4 + 1/2*a*x^2

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Sympy [A]  time = 0.091266, size = 12, normalized size = 0.33 \begin{align*} \frac{a x^{2}}{2} + \frac{b x^{4}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x**2/2 + b*x**4/4

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Giac [A]  time = 1.14333, size = 30, normalized size = 0.83 \begin{align*} \frac{1}{4} \,{\left (b x^{4} + 2 \, a x^{2}\right )} \mathrm{sgn}\left (b x^{2} + a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(b*x^4 + 2*a*x^2)*sgn(b*x^2 + a)