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

Optimal. Leaf size=74 \[ \frac{b x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}+\frac{a x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \]

[Out]

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

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Rubi [A]  time = 0.0123292, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {1343} \[ \frac{b x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}+\frac{a x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1343

Int[((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_), x_Symbol] :> Dist[(a + b*x^n + c*x^(2*n))^p/(b + 2*c*x
^n)^(2*p), Int[(b + 2*c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0]

Rubi steps

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

Mathematica [A]  time = 0.0069461, size = 36, normalized size = 0.49 \[ \frac{\sqrt{\left (a+b x^3\right )^2} \left (4 a x+b x^4\right )}{4 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 33, normalized size = 0.5 \begin{align*}{\frac{x \left ( b{x}^{3}+4\,a \right ) }{4\,b{x}^{3}+4\,a}\sqrt{ \left ( b{x}^{3}+a \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02239, size = 14, normalized size = 0.19 \begin{align*} \frac{1}{4} \, b x^{4} + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b*x^4 + a*x

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Fricas [A]  time = 1.7091, size = 23, normalized size = 0.31 \begin{align*} \frac{1}{4} \, b x^{4} + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b*x^4 + a*x

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Sympy [A]  time = 0.096229, size = 8, normalized size = 0.11 \begin{align*} a x + \frac{b x^{4}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x + b*x**4/4

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Giac [A]  time = 1.10233, size = 27, normalized size = 0.36 \begin{align*} \frac{1}{4} \,{\left (b x^{4} + 4 \, a x\right )} \mathrm{sgn}\left (b x^{3} + a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(b*x^4 + 4*a*x)*sgn(b*x^3 + a)