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

Optimal. Leaf size=79 \[ \frac{b x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )}+\frac{a x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )} \]

[Out]

(a*x^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(2*(a + b*x^3)) + (b*x^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(5*(a + b*x^
3))

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Rubi [A]  time = 0.0180364, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {1355, 14} \[ \frac{b x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )}+\frac{a x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*x^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(2*(a + b*x^3)) + (b*x^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(5*(a + b*x^
3))

Rule 1355

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_), x_Symbol] :> Dist[(a + b*x^n + c*x^
(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^n)^(2*p), x], x] /; Fr
eeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int x \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 x \left (a b+b^2 x^3\right ) \, dx}{a b+b^2 x^3}\\ &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \int \left (a b x+b^2 x^4\right ) \, dx}{a b+b^2 x^3}\\ &=\frac{a x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac{b x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )}\\ \end{align*}

Mathematica [A]  time = 0.0089468, size = 39, normalized size = 0.49 \[ \frac{\sqrt{\left (a+b x^3\right )^2} \left (5 a x^2+2 b x^5\right )}{10 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[(a + b*x^3)^2]*(5*a*x^2 + 2*b*x^5))/(10*(a + b*x^3))

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Maple [A]  time = 0.008, size = 36, normalized size = 0.5 \begin{align*}{\frac{{x}^{2} \left ( 2\,b{x}^{3}+5\,a \right ) }{10\,b{x}^{3}+10\,a}\sqrt{ \left ( b{x}^{3}+a \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*x^2*(2*b*x^3+5*a)*((b*x^3+a)^2)^(1/2)/(b*x^3+a)

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Maxima [A]  time = 1.17017, size = 18, normalized size = 0.23 \begin{align*} \frac{1}{5} \, b x^{5} + \frac{1}{2} \, a x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b*x^5 + 1/2*a*x^2

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Fricas [A]  time = 1.62523, size = 31, normalized size = 0.39 \begin{align*} \frac{1}{5} \, b x^{5} + \frac{1}{2} \, a x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b*x^5 + 1/2*a*x^2

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Sympy [A]  time = 0.098539, size = 12, normalized size = 0.15 \begin{align*} \frac{a x^{2}}{2} + \frac{b x^{5}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x**2/2 + b*x**5/5

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Giac [A]  time = 1.10858, size = 39, normalized size = 0.49 \begin{align*} \frac{1}{5} \, b x^{5} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{1}{2} \, a x^{2} \mathrm{sgn}\left (b x^{3} + a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b*x^5*sgn(b*x^3 + a) + 1/2*a*x^2*sgn(b*x^3 + a)