3.369 \(\int x^8 \sqrt{a+b x^3} \, dx\)

Optimal. Leaf size=59 \[ \frac{2 a^2 \left (a+b x^3\right )^{3/2}}{9 b^3}+\frac{2 \left (a+b x^3\right )^{7/2}}{21 b^3}-\frac{4 a \left (a+b x^3\right )^{5/2}}{15 b^3} \]

[Out]

(2*a^2*(a + b*x^3)^(3/2))/(9*b^3) - (4*a*(a + b*x^3)^(5/2))/(15*b^3) + (2*(a + b*x^3)^(7/2))/(21*b^3)

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Rubi [A]  time = 0.035611, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{2 a^2 \left (a+b x^3\right )^{3/2}}{9 b^3}+\frac{2 \left (a+b x^3\right )^{7/2}}{21 b^3}-\frac{4 a \left (a+b x^3\right )^{5/2}}{15 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^8*Sqrt[a + b*x^3],x]

[Out]

(2*a^2*(a + b*x^3)^(3/2))/(9*b^3) - (4*a*(a + b*x^3)^(5/2))/(15*b^3) + (2*(a + b*x^3)^(7/2))/(21*b^3)

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x^8 \sqrt{a+b x^3} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int x^2 \sqrt{a+b x} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{a^2 \sqrt{a+b x}}{b^2}-\frac{2 a (a+b x)^{3/2}}{b^2}+\frac{(a+b x)^{5/2}}{b^2}\right ) \, dx,x,x^3\right )\\ &=\frac{2 a^2 \left (a+b x^3\right )^{3/2}}{9 b^3}-\frac{4 a \left (a+b x^3\right )^{5/2}}{15 b^3}+\frac{2 \left (a+b x^3\right )^{7/2}}{21 b^3}\\ \end{align*}

Mathematica [A]  time = 0.0188507, size = 39, normalized size = 0.66 \[ \frac{2 \left (a+b x^3\right )^{3/2} \left (8 a^2-12 a b x^3+15 b^2 x^6\right )}{315 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^8*Sqrt[a + b*x^3],x]

[Out]

(2*(a + b*x^3)^(3/2)*(8*a^2 - 12*a*b*x^3 + 15*b^2*x^6))/(315*b^3)

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Maple [A]  time = 0.004, size = 36, normalized size = 0.6 \begin{align*}{\frac{30\,{b}^{2}{x}^{6}-24\,{x}^{3}ab+16\,{a}^{2}}{315\,{b}^{3}} \left ( b{x}^{3}+a \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(b*x^3+a)^(1/2),x)

[Out]

2/315*(b*x^3+a)^(3/2)*(15*b^2*x^6-12*a*b*x^3+8*a^2)/b^3

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Maxima [A]  time = 0.976512, size = 63, normalized size = 1.07 \begin{align*} \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{7}{2}}}{21 \, b^{3}} - \frac{4 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} a}{15 \, b^{3}} + \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a^{2}}{9 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(b*x^3+a)^(1/2),x, algorithm="maxima")

[Out]

2/21*(b*x^3 + a)^(7/2)/b^3 - 4/15*(b*x^3 + a)^(5/2)*a/b^3 + 2/9*(b*x^3 + a)^(3/2)*a^2/b^3

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Fricas [A]  time = 1.4712, size = 103, normalized size = 1.75 \begin{align*} \frac{2 \,{\left (15 \, b^{3} x^{9} + 3 \, a b^{2} x^{6} - 4 \, a^{2} b x^{3} + 8 \, a^{3}\right )} \sqrt{b x^{3} + a}}{315 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(b*x^3+a)^(1/2),x, algorithm="fricas")

[Out]

2/315*(15*b^3*x^9 + 3*a*b^2*x^6 - 4*a^2*b*x^3 + 8*a^3)*sqrt(b*x^3 + a)/b^3

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Sympy [A]  time = 1.71078, size = 90, normalized size = 1.53 \begin{align*} \begin{cases} \frac{16 a^{3} \sqrt{a + b x^{3}}}{315 b^{3}} - \frac{8 a^{2} x^{3} \sqrt{a + b x^{3}}}{315 b^{2}} + \frac{2 a x^{6} \sqrt{a + b x^{3}}}{105 b} + \frac{2 x^{9} \sqrt{a + b x^{3}}}{21} & \text{for}\: b \neq 0 \\\frac{\sqrt{a} x^{9}}{9} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8*(b*x**3+a)**(1/2),x)

[Out]

Piecewise((16*a**3*sqrt(a + b*x**3)/(315*b**3) - 8*a**2*x**3*sqrt(a + b*x**3)/(315*b**2) + 2*a*x**6*sqrt(a + b
*x**3)/(105*b) + 2*x**9*sqrt(a + b*x**3)/21, Ne(b, 0)), (sqrt(a)*x**9/9, True))

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Giac [A]  time = 1.09563, size = 58, normalized size = 0.98 \begin{align*} \frac{2 \,{\left (15 \,{\left (b x^{3} + a\right )}^{\frac{7}{2}} - 42 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a^{2}\right )}}{315 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(b*x^3+a)^(1/2),x, algorithm="giac")

[Out]

2/315*(15*(b*x^3 + a)^(7/2) - 42*(b*x^3 + a)^(5/2)*a + 35*(b*x^3 + a)^(3/2)*a^2)/b^3