3.1112 \(\int \frac{x^{11}}{(a+b x^4)^{3/4}} \, dx\)

Optimal. Leaf size=56 \[ \frac{a^2 \sqrt [4]{a+b x^4}}{b^3}+\frac{\left (a+b x^4\right )^{9/4}}{9 b^3}-\frac{2 a \left (a+b x^4\right )^{5/4}}{5 b^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0322967, antiderivative size = 56, 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} \[ \frac{a^2 \sqrt [4]{a+b x^4}}{b^3}+\frac{\left (a+b x^4\right )^{9/4}}{9 b^3}-\frac{2 a \left (a+b x^4\right )^{5/4}}{5 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^11/(a + b*x^4)^(3/4),x]

[Out]

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

Mathematica [A]  time = 0.0168346, size = 39, normalized size = 0.7 \[ \frac{\sqrt [4]{a+b x^4} \left (32 a^2-8 a b x^4+5 b^2 x^8\right )}{45 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^11/(a + b*x^4)^(3/4),x]

[Out]

((a + b*x^4)^(1/4)*(32*a^2 - 8*a*b*x^4 + 5*b^2*x^8))/(45*b^3)

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 36, normalized size = 0.6 \begin{align*}{\frac{5\,{b}^{2}{x}^{8}-8\,ab{x}^{4}+32\,{a}^{2}}{45\,{b}^{3}}\sqrt [4]{b{x}^{4}+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11/(b*x^4+a)^(3/4),x)

[Out]

1/45*(b*x^4+a)^(1/4)*(5*b^2*x^8-8*a*b*x^4+32*a^2)/b^3

________________________________________________________________________________________

Maxima [A]  time = 0.990486, size = 62, normalized size = 1.11 \begin{align*} \frac{{\left (b x^{4} + a\right )}^{\frac{9}{4}}}{9 \, b^{3}} - \frac{2 \,{\left (b x^{4} + a\right )}^{\frac{5}{4}} a}{5 \, b^{3}} + \frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{2}}{b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.45409, size = 82, normalized size = 1.46 \begin{align*} \frac{{\left (5 \, b^{2} x^{8} - 8 \, a b x^{4} + 32 \, a^{2}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{45 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/45*(5*b^2*x^8 - 8*a*b*x^4 + 32*a^2)*(b*x^4 + a)^(1/4)/b^3

________________________________________________________________________________________

Sympy [A]  time = 2.84998, size = 68, normalized size = 1.21 \begin{align*} \begin{cases} \frac{32 a^{2} \sqrt [4]{a + b x^{4}}}{45 b^{3}} - \frac{8 a x^{4} \sqrt [4]{a + b x^{4}}}{45 b^{2}} + \frac{x^{8} \sqrt [4]{a + b x^{4}}}{9 b} & \text{for}\: b \neq 0 \\\frac{x^{12}}{12 a^{\frac{3}{4}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((32*a**2*(a + b*x**4)**(1/4)/(45*b**3) - 8*a*x**4*(a + b*x**4)**(1/4)/(45*b**2) + x**8*(a + b*x**4)*
*(1/4)/(9*b), Ne(b, 0)), (x**12/(12*a**(3/4)), True))

________________________________________________________________________________________

Giac [A]  time = 1.09452, size = 58, normalized size = 1.04 \begin{align*} \frac{5 \,{\left (b x^{4} + a\right )}^{\frac{9}{4}} - 18 \,{\left (b x^{4} + a\right )}^{\frac{5}{4}} a + 45 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{2}}{45 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/45*(5*(b*x^4 + a)^(9/4) - 18*(b*x^4 + a)^(5/4)*a + 45*(b*x^4 + a)^(1/4)*a^2)/b^3