3.404 \(\int \frac{x^{11}}{\sqrt{a+b x^3}} \, dx\)

Optimal. Leaf size=80 \[ \frac{2 a^2 \left (a+b x^3\right )^{3/2}}{3 b^4}-\frac{2 a^3 \sqrt{a+b x^3}}{3 b^4}+\frac{2 \left (a+b x^3\right )^{7/2}}{21 b^4}-\frac{2 a \left (a+b x^3\right )^{5/2}}{5 b^4} \]

[Out]

(-2*a^3*Sqrt[a + b*x^3])/(3*b^4) + (2*a^2*(a + b*x^3)^(3/2))/(3*b^4) - (2*a*(a + b*x^3)^(5/2))/(5*b^4) + (2*(a
+ b*x^3)^(7/2))/(21*b^4)

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Rubi [A]  time = 0.0442464, antiderivative size = 80, 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} \[ \frac{2 a^2 \left (a+b x^3\right )^{3/2}}{3 b^4}-\frac{2 a^3 \sqrt{a+b x^3}}{3 b^4}+\frac{2 \left (a+b x^3\right )^{7/2}}{21 b^4}-\frac{2 a \left (a+b x^3\right )^{5/2}}{5 b^4} \]

Antiderivative was successfully verified.

[In]

Int[x^11/Sqrt[a + b*x^3],x]

[Out]

(-2*a^3*Sqrt[a + b*x^3])/(3*b^4) + (2*a^2*(a + b*x^3)^(3/2))/(3*b^4) - (2*a*(a + b*x^3)^(5/2))/(5*b^4) + (2*(a
+ b*x^3)^(7/2))/(21*b^4)

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 \frac{x^{11}}{\sqrt{a+b x^3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^3}{\sqrt{a+b x}} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{a^3}{b^3 \sqrt{a+b x}}+\frac{3 a^2 \sqrt{a+b x}}{b^3}-\frac{3 a (a+b x)^{3/2}}{b^3}+\frac{(a+b x)^{5/2}}{b^3}\right ) \, dx,x,x^3\right )\\ &=-\frac{2 a^3 \sqrt{a+b x^3}}{3 b^4}+\frac{2 a^2 \left (a+b x^3\right )^{3/2}}{3 b^4}-\frac{2 a \left (a+b x^3\right )^{5/2}}{5 b^4}+\frac{2 \left (a+b x^3\right )^{7/2}}{21 b^4}\\ \end{align*}

Mathematica [A]  time = 0.0276513, size = 50, normalized size = 0.62 \[ \frac{2 \sqrt{a+b x^3} \left (8 a^2 b x^3-16 a^3-6 a b^2 x^6+5 b^3 x^9\right )}{105 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^11/Sqrt[a + b*x^3],x]

[Out]

(2*Sqrt[a + b*x^3]*(-16*a^3 + 8*a^2*b*x^3 - 6*a*b^2*x^6 + 5*b^3*x^9))/(105*b^4)

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Maple [A]  time = 0.006, size = 47, normalized size = 0.6 \begin{align*} -{\frac{-10\,{b}^{3}{x}^{9}+12\,a{b}^{2}{x}^{6}-16\,{a}^{2}b{x}^{3}+32\,{a}^{3}}{105\,{b}^{4}}\sqrt{b{x}^{3}+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11/(b*x^3+a)^(1/2),x)

[Out]

-2/105*(b*x^3+a)^(1/2)*(-5*b^3*x^9+6*a*b^2*x^6-8*a^2*b*x^3+16*a^3)/b^4

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Maxima [A]  time = 1.00705, size = 86, normalized size = 1.08 \begin{align*} \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{7}{2}}}{21 \, b^{4}} - \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} a}{5 \, b^{4}} + \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a^{2}}{3 \, b^{4}} - \frac{2 \, \sqrt{b x^{3} + a} a^{3}}{3 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(b*x^3+a)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.45942, size = 103, normalized size = 1.29 \begin{align*} \frac{2 \,{\left (5 \, b^{3} x^{9} - 6 \, a b^{2} x^{6} + 8 \, a^{2} b x^{3} - 16 \, a^{3}\right )} \sqrt{b x^{3} + a}}{105 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(b*x^3+a)^(1/2),x, algorithm="fricas")

[Out]

2/105*(5*b^3*x^9 - 6*a*b^2*x^6 + 8*a^2*b*x^3 - 16*a^3)*sqrt(b*x^3 + a)/b^4

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Sympy [A]  time = 3.4574, size = 94, normalized size = 1.18 \begin{align*} \begin{cases} - \frac{32 a^{3} \sqrt{a + b x^{3}}}{105 b^{4}} + \frac{16 a^{2} x^{3} \sqrt{a + b x^{3}}}{105 b^{3}} - \frac{4 a x^{6} \sqrt{a + b x^{3}}}{35 b^{2}} + \frac{2 x^{9} \sqrt{a + b x^{3}}}{21 b} & \text{for}\: b \neq 0 \\\frac{x^{12}}{12 \sqrt{a}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**11/(b*x**3+a)**(1/2),x)

[Out]

Piecewise((-32*a**3*sqrt(a + b*x**3)/(105*b**4) + 16*a**2*x**3*sqrt(a + b*x**3)/(105*b**3) - 4*a*x**6*sqrt(a +
b*x**3)/(35*b**2) + 2*x**9*sqrt(a + b*x**3)/(21*b), Ne(b, 0)), (x**12/(12*sqrt(a)), True))

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Giac [A]  time = 1.12384, size = 77, normalized size = 0.96 \begin{align*} \frac{2 \,{\left (5 \,{\left (b x^{3} + a\right )}^{\frac{7}{2}} - 21 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a^{2} - 35 \, \sqrt{b x^{3} + a} a^{3}\right )}}{105 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(b*x^3+a)^(1/2),x, algorithm="giac")

[Out]

2/105*(5*(b*x^3 + a)^(7/2) - 21*(b*x^3 + a)^(5/2)*a + 35*(b*x^3 + a)^(3/2)*a^2 - 35*sqrt(b*x^3 + a)*a^3)/b^4