3.831 \(\int \frac{x^{11}}{\sqrt{a-b x^4}} \, dx\)

Optimal. Leaf size=62 \[ -\frac{a^2 \sqrt{a-b x^4}}{2 b^3}-\frac{\left (a-b x^4\right )^{5/2}}{10 b^3}+\frac{a \left (a-b x^4\right )^{3/2}}{3 b^3} \]

[Out]

-(a^2*Sqrt[a - b*x^4])/(2*b^3) + (a*(a - b*x^4)^(3/2))/(3*b^3) - (a - b*x^4)^(5/2)/(10*b^3)

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Rubi [A]  time = 0.0351539, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {266, 43} \[ -\frac{a^2 \sqrt{a-b x^4}}{2 b^3}-\frac{\left (a-b x^4\right )^{5/2}}{10 b^3}+\frac{a \left (a-b x^4\right )^{3/2}}{3 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^11/Sqrt[a - b*x^4],x]

[Out]

-(a^2*Sqrt[a - b*x^4])/(2*b^3) + (a*(a - b*x^4)^(3/2))/(3*b^3) - (a - b*x^4)^(5/2)/(10*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 \frac{x^{11}}{\sqrt{a-b x^4}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a-b x}} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (\frac{a^2}{b^2 \sqrt{a-b x}}-\frac{2 a \sqrt{a-b x}}{b^2}+\frac{(a-b x)^{3/2}}{b^2}\right ) \, dx,x,x^4\right )\\ &=-\frac{a^2 \sqrt{a-b x^4}}{2 b^3}+\frac{a \left (a-b x^4\right )^{3/2}}{3 b^3}-\frac{\left (a-b x^4\right )^{5/2}}{10 b^3}\\ \end{align*}

Mathematica [A]  time = 0.0198333, size = 40, normalized size = 0.65 \[ -\frac{\sqrt{a-b x^4} \left (8 a^2+4 a b x^4+3 b^2 x^8\right )}{30 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^11/Sqrt[a - b*x^4],x]

[Out]

-(Sqrt[a - b*x^4]*(8*a^2 + 4*a*b*x^4 + 3*b^2*x^8))/(30*b^3)

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Maple [A]  time = 0.006, size = 37, normalized size = 0.6 \begin{align*} -{\frac{3\,{b}^{2}{x}^{8}+4\,ab{x}^{4}+8\,{a}^{2}}{30\,{b}^{3}}\sqrt{-b{x}^{4}+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11/(-b*x^4+a)^(1/2),x)

[Out]

-1/30*(-b*x^4+a)^(1/2)*(3*b^2*x^8+4*a*b*x^4+8*a^2)/b^3

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Maxima [A]  time = 0.950472, size = 68, normalized size = 1.1 \begin{align*} -\frac{{\left (-b x^{4} + a\right )}^{\frac{5}{2}}}{10 \, b^{3}} + \frac{{\left (-b x^{4} + a\right )}^{\frac{3}{2}} a}{3 \, b^{3}} - \frac{\sqrt{-b x^{4} + a} a^{2}}{2 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(-b*x^4+a)^(1/2),x, algorithm="maxima")

[Out]

-1/10*(-b*x^4 + a)^(5/2)/b^3 + 1/3*(-b*x^4 + a)^(3/2)*a/b^3 - 1/2*sqrt(-b*x^4 + a)*a^2/b^3

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Fricas [A]  time = 1.4096, size = 81, normalized size = 1.31 \begin{align*} -\frac{{\left (3 \, b^{2} x^{8} + 4 \, a b x^{4} + 8 \, a^{2}\right )} \sqrt{-b x^{4} + a}}{30 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(-b*x^4+a)^(1/2),x, algorithm="fricas")

[Out]

-1/30*(3*b^2*x^8 + 4*a*b*x^4 + 8*a^2)*sqrt(-b*x^4 + a)/b^3

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Sympy [A]  time = 3.69342, size = 70, normalized size = 1.13 \begin{align*} \begin{cases} - \frac{4 a^{2} \sqrt{a - b x^{4}}}{15 b^{3}} - \frac{2 a x^{4} \sqrt{a - b x^{4}}}{15 b^{2}} - \frac{x^{8} \sqrt{a - b x^{4}}}{10 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**4+a)**(1/2),x)

[Out]

Piecewise((-4*a**2*sqrt(a - b*x**4)/(15*b**3) - 2*a*x**4*sqrt(a - b*x**4)/(15*b**2) - x**8*sqrt(a - b*x**4)/(1
0*b), Ne(b, 0)), (x**12/(12*sqrt(a)), True))

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Giac [A]  time = 1.21427, size = 77, normalized size = 1.24 \begin{align*} -\frac{3 \,{\left (b x^{4} - a\right )}^{2} \sqrt{-b x^{4} + a} - 10 \,{\left (-b x^{4} + a\right )}^{\frac{3}{2}} a + 15 \, \sqrt{-b x^{4} + a} a^{2}}{30 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(-b*x^4+a)^(1/2),x, algorithm="giac")

[Out]

-1/30*(3*(b*x^4 - a)^2*sqrt(-b*x^4 + a) - 10*(-b*x^4 + a)^(3/2)*a + 15*sqrt(-b*x^4 + a)*a^2)/b^3