3.6 \(\int x^5 \sqrt{a^2+2 a b x^3+b^2 x^6} \, dx\)

Optimal. Leaf size=79 \[ \frac{b x^9 \sqrt{a^2+2 a b x^3+b^2 x^6}}{9 \left (a+b x^3\right )}+\frac{a x^6 \sqrt{a^2+2 a b x^3+b^2 x^6}}{6 \left (a+b x^3\right )} \]

[Out]

(a*x^6*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(6*(a + b*x^3)) + (b*x^9*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(9*(a + b*x^
3))

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Rubi [A]  time = 0.0230703, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {1355, 14} \[ \frac{b x^9 \sqrt{a^2+2 a b x^3+b^2 x^6}}{9 \left (a+b x^3\right )}+\frac{a x^6 \sqrt{a^2+2 a b x^3+b^2 x^6}}{6 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

Int[x^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6],x]

[Out]

(a*x^6*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(6*(a + b*x^3)) + (b*x^9*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(9*(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^5 \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^5 \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^5+b^2 x^8\right ) \, dx}{a b+b^2 x^3}\\ &=\frac{a x^6 \sqrt{a^2+2 a b x^3+b^2 x^6}}{6 \left (a+b x^3\right )}+\frac{b x^9 \sqrt{a^2+2 a b x^3+b^2 x^6}}{9 \left (a+b x^3\right )}\\ \end{align*}

Mathematica [A]  time = 0.0136005, size = 39, normalized size = 0.49 \[ \frac{\sqrt{\left (a+b x^3\right )^2} \left (3 a x^6+2 b x^9\right )}{18 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6],x]

[Out]

(Sqrt[(a + b*x^3)^2]*(3*a*x^6 + 2*b*x^9))/(18*(a + b*x^3))

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Maple [A]  time = 0.004, size = 36, normalized size = 0.5 \begin{align*}{\frac{{x}^{6} \left ( 2\,b{x}^{3}+3\,a \right ) }{18\,b{x}^{3}+18\,a}\sqrt{ \left ( b{x}^{3}+a \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*x^6*(2*b*x^3+3*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^5*((b*x^3+a)^2)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.77177, size = 31, normalized size = 0.39 \begin{align*} \frac{1}{9} \, b x^{9} + \frac{1}{6} \, a x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*((b*x^3+a)^2)^(1/2),x, algorithm="fricas")

[Out]

1/9*b*x^9 + 1/6*a*x^6

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Sympy [A]  time = 0.10151, size = 12, normalized size = 0.15 \begin{align*} \frac{a x^{6}}{6} + \frac{b x^{9}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*((b*x**3+a)**2)**(1/2),x)

[Out]

a*x**6/6 + b*x**9/9

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Giac [A]  time = 1.12008, size = 31, normalized size = 0.39 \begin{align*} \frac{1}{18} \,{\left (2 \, b x^{9} + 3 \, a x^{6}\right )} \mathrm{sgn}\left (b x^{3} + a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*((b*x^3+a)^2)^(1/2),x, algorithm="giac")

[Out]

1/18*(2*b*x^9 + 3*a*x^6)*sgn(b*x^3 + a)