3.7 \(\int x^4 \sqrt{a^2+2 a b x^3+b^2 x^6} \, dx\)

Optimal. Leaf size=79 \[ \frac{b x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}+\frac{a x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )} \]

[Out]

(a*x^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(5*(a + b*x^3)) + (b*x^8*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(8*(a + b*x^
3))

________________________________________________________________________________________

Rubi [A]  time = 0.0233265, 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} \[ \frac{b x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}+\frac{a x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

Int[x^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6],x]

[Out]

(a*x^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(5*(a + b*x^3)) + (b*x^8*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(8*(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^4 \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^4 \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^4+b^2 x^7\right ) \, dx}{a b+b^2 x^3}\\ &=\frac{a x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )}+\frac{b x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}\\ \end{align*}

Mathematica [A]  time = 0.0077349, size = 39, normalized size = 0.49 \[ \frac{\sqrt{\left (a+b x^3\right )^2} \left (8 a x^5+5 b x^8\right )}{40 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6],x]

[Out]

(Sqrt[(a + b*x^3)^2]*(8*a*x^5 + 5*b*x^8))/(40*(a + b*x^3))

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 36, normalized size = 0.5 \begin{align*}{\frac{{x}^{5} \left ( 5\,b{x}^{3}+8\,a \right ) }{40\,b{x}^{3}+40\,a}\sqrt{ \left ( b{x}^{3}+a \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/40*x^5*(5*b*x^3+8*a)*((b*x^3+a)^2)^(1/2)/(b*x^3+a)

________________________________________________________________________________________

Maxima [A]  time = 1.12601, size = 18, normalized size = 0.23 \begin{align*} \frac{1}{8} \, b x^{8} + \frac{1}{5} \, a x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*((b*x^3+a)^2)^(1/2),x, algorithm="maxima")

[Out]

1/8*b*x^8 + 1/5*a*x^5

________________________________________________________________________________________

Fricas [A]  time = 1.70126, size = 31, normalized size = 0.39 \begin{align*} \frac{1}{8} \, b x^{8} + \frac{1}{5} \, a x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*((b*x^3+a)^2)^(1/2),x, algorithm="fricas")

[Out]

1/8*b*x^8 + 1/5*a*x^5

________________________________________________________________________________________

Sympy [A]  time = 0.101234, size = 12, normalized size = 0.15 \begin{align*} \frac{a x^{5}}{5} + \frac{b x^{8}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x**5/5 + b*x**8/8

________________________________________________________________________________________

Giac [A]  time = 1.13147, size = 39, normalized size = 0.49 \begin{align*} \frac{1}{8} \, b x^{8} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{1}{5} \, a x^{5} \mathrm{sgn}\left (b x^{3} + a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*((b*x^3+a)^2)^(1/2),x, algorithm="giac")

[Out]

1/8*b*x^8*sgn(b*x^3 + a) + 1/5*a*x^5*sgn(b*x^3 + a)