3.1082 \(\int \frac{x^{11}}{\sqrt [4]{a+b x^4}} \, dx\)

Optimal. Leaf size=59 \[ \frac{a^2 \left (a+b x^4\right )^{3/4}}{3 b^3}+\frac{\left (a+b x^4\right )^{11/4}}{11 b^3}-\frac{2 a \left (a+b x^4\right )^{7/4}}{7 b^3} \]

[Out]

(a^2*(a + b*x^4)^(3/4))/(3*b^3) - (2*a*(a + b*x^4)^(7/4))/(7*b^3) + (a + b*x^4)^(11/4)/(11*b^3)

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Rubi [A]  time = 0.0327176, 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} \[ \frac{a^2 \left (a+b x^4\right )^{3/4}}{3 b^3}+\frac{\left (a+b x^4\right )^{11/4}}{11 b^3}-\frac{2 a \left (a+b x^4\right )^{7/4}}{7 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^11/(a + b*x^4)^(1/4),x]

[Out]

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

Mathematica [A]  time = 0.0186086, size = 39, normalized size = 0.66 \[ \frac{\left (a+b x^4\right )^{3/4} \left (32 a^2-24 a b x^4+21 b^2 x^8\right )}{231 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^11/(a + b*x^4)^(1/4),x]

[Out]

((a + b*x^4)^(3/4)*(32*a^2 - 24*a*b*x^4 + 21*b^2*x^8))/(231*b^3)

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Maple [A]  time = 0.004, size = 36, normalized size = 0.6 \begin{align*}{\frac{21\,{b}^{2}{x}^{8}-24\,ab{x}^{4}+32\,{a}^{2}}{231\,{b}^{3}} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{4}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/231*(b*x^4+a)^(3/4)*(21*b^2*x^8-24*a*b*x^4+32*a^2)/b^3

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Maxima [A]  time = 1.01119, size = 63, normalized size = 1.07 \begin{align*} \frac{{\left (b x^{4} + a\right )}^{\frac{11}{4}}}{11 \, b^{3}} - \frac{2 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}} a}{7 \, b^{3}} + \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}} a^{2}}{3 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*(b*x^4 + a)^(11/4)/b^3 - 2/7*(b*x^4 + a)^(7/4)*a/b^3 + 1/3*(b*x^4 + a)^(3/4)*a^2/b^3

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Fricas [A]  time = 1.49564, size = 86, normalized size = 1.46 \begin{align*} \frac{{\left (21 \, b^{2} x^{8} - 24 \, a b x^{4} + 32 \, a^{2}\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{231 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/231*(21*b^2*x^8 - 24*a*b*x^4 + 32*a^2)*(b*x^4 + a)^(3/4)/b^3

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Sympy [A]  time = 4.48652, size = 68, normalized size = 1.15 \begin{align*} \begin{cases} \frac{32 a^{2} \left (a + b x^{4}\right )^{\frac{3}{4}}}{231 b^{3}} - \frac{8 a x^{4} \left (a + b x^{4}\right )^{\frac{3}{4}}}{77 b^{2}} + \frac{x^{8} \left (a + b x^{4}\right )^{\frac{3}{4}}}{11 b} & \text{for}\: b \neq 0 \\\frac{x^{12}}{12 \sqrt [4]{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)**(1/4),x)

[Out]

Piecewise((32*a**2*(a + b*x**4)**(3/4)/(231*b**3) - 8*a*x**4*(a + b*x**4)**(3/4)/(77*b**2) + x**8*(a + b*x**4)
**(3/4)/(11*b), Ne(b, 0)), (x**12/(12*a**(1/4)), True))

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Giac [A]  time = 1.16014, size = 58, normalized size = 0.98 \begin{align*} \frac{21 \,{\left (b x^{4} + a\right )}^{\frac{11}{4}} - 66 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}} a + 77 \,{\left (b x^{4} + a\right )}^{\frac{3}{4}} a^{2}}{231 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/231*(21*(b*x^4 + a)^(11/4) - 66*(b*x^4 + a)^(7/4)*a + 77*(b*x^4 + a)^(3/4)*a^2)/b^3