3.851 \(\int \frac{x^{11}}{(a+b x^4)^{3/2}} \, dx\)

Optimal. Leaf size=57 \[ -\frac{a^2}{2 b^3 \sqrt{a+b x^4}}-\frac{a \sqrt{a+b x^4}}{b^3}+\frac{\left (a+b x^4\right )^{3/2}}{6 b^3} \]

[Out]

-a^2/(2*b^3*Sqrt[a + b*x^4]) - (a*Sqrt[a + b*x^4])/b^3 + (a + b*x^4)^(3/2)/(6*b^3)

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Rubi [A]  time = 0.0326121, antiderivative size = 57, 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} \[ -\frac{a^2}{2 b^3 \sqrt{a+b x^4}}-\frac{a \sqrt{a+b x^4}}{b^3}+\frac{\left (a+b x^4\right )^{3/2}}{6 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^11/(a + b*x^4)^(3/2),x]

[Out]

-a^2/(2*b^3*Sqrt[a + b*x^4]) - (a*Sqrt[a + b*x^4])/b^3 + (a + b*x^4)^(3/2)/(6*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}}{\left (a+b x^4\right )^{3/2}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x^2}{(a+b x)^{3/2}} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (\frac{a^2}{b^2 (a+b x)^{3/2}}-\frac{2 a}{b^2 \sqrt{a+b x}}+\frac{\sqrt{a+b x}}{b^2}\right ) \, dx,x,x^4\right )\\ &=-\frac{a^2}{2 b^3 \sqrt{a+b x^4}}-\frac{a \sqrt{a+b x^4}}{b^3}+\frac{\left (a+b x^4\right )^{3/2}}{6 b^3}\\ \end{align*}

Mathematica [A]  time = 0.0176475, size = 38, normalized size = 0.67 \[ \frac{-8 a^2-4 a b x^4+b^2 x^8}{6 b^3 \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^11/(a + b*x^4)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.005, size = 36, normalized size = 0.6 \begin{align*} -{\frac{-{b}^{2}{x}^{8}+4\,ab{x}^{4}+8\,{a}^{2}}{6\,{b}^{3}}{\frac{1}{\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)^(3/2),x)

[Out]

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

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Maxima [A]  time = 0.972714, size = 63, normalized size = 1.11 \begin{align*} \frac{{\left (b x^{4} + a\right )}^{\frac{3}{2}}}{6 \, b^{3}} - \frac{\sqrt{b x^{4} + a} a}{b^{3}} - \frac{a^{2}}{2 \, \sqrt{b x^{4} + a} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(b*x^4+a)^(3/2),x, algorithm="maxima")

[Out]

1/6*(b*x^4 + a)^(3/2)/b^3 - sqrt(b*x^4 + a)*a/b^3 - 1/2*a^2/(sqrt(b*x^4 + a)*b^3)

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Fricas [A]  time = 1.47236, size = 93, normalized size = 1.63 \begin{align*} \frac{{\left (b^{2} x^{8} - 4 \, a b x^{4} - 8 \, a^{2}\right )} \sqrt{b x^{4} + a}}{6 \,{\left (b^{4} x^{4} + a b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(b*x^4+a)^(3/2),x, algorithm="fricas")

[Out]

1/6*(b^2*x^8 - 4*a*b*x^4 - 8*a^2)*sqrt(b*x^4 + a)/(b^4*x^4 + a*b^3)

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Sympy [A]  time = 4.02929, size = 68, normalized size = 1.19 \begin{align*} \begin{cases} - \frac{4 a^{2}}{3 b^{3} \sqrt{a + b x^{4}}} - \frac{2 a x^{4}}{3 b^{2} \sqrt{a + b x^{4}}} + \frac{x^{8}}{6 b \sqrt{a + b x^{4}}} & \text{for}\: b \neq 0 \\\frac{x^{12}}{12 a^{\frac{3}{2}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**11/(b*x**4+a)**(3/2),x)

[Out]

Piecewise((-4*a**2/(3*b**3*sqrt(a + b*x**4)) - 2*a*x**4/(3*b**2*sqrt(a + b*x**4)) + x**8/(6*b*sqrt(a + b*x**4)
), Ne(b, 0)), (x**12/(12*a**(3/2)), True))

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Giac [A]  time = 1.10191, size = 55, normalized size = 0.96 \begin{align*} \frac{{\left (b x^{4} + a\right )}^{\frac{3}{2}} - 6 \, \sqrt{b x^{4} + a} a - \frac{3 \, a^{2}}{\sqrt{b x^{4} + a}}}{6 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(b*x^4+a)^(3/2),x, algorithm="giac")

[Out]

1/6*((b*x^4 + a)^(3/2) - 6*sqrt(b*x^4 + a)*a - 3*a^2/sqrt(b*x^4 + a))/b^3