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

Optimal. Leaf size=59 \[ \frac {a^2 \left (a+b x^4\right )^{5/4}}{5 b^3}+\frac {\left (a+b x^4\right )^{13/4}}{13 b^3}-\frac {2 a \left (a+b x^4\right )^{9/4}}{9 b^3} \]

[Out]

1/5*a^2*(b*x^4+a)^(5/4)/b^3-2/9*a*(b*x^4+a)^(9/4)/b^3+1/13*(b*x^4+a)^(13/4)/b^3

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 59, normalized size of antiderivative = 1.00, 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} \[ \frac {a^2 \left (a+b x^4\right )^{5/4}}{5 b^3}+\frac {\left (a+b x^4\right )^{13/4}}{13 b^3}-\frac {2 a \left (a+b x^4\right )^{9/4}}{9 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^11*(a + b*x^4)^(1/4),x]

[Out]

(a^2*(a + b*x^4)^(5/4))/(5*b^3) - (2*a*(a + b*x^4)^(9/4))/(9*b^3) + (a + b*x^4)^(13/4)/(13*b^3)

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])

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]]

Rubi steps

\begin {align*} \int x^{11} \sqrt [4]{a+b x^4} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int x^2 \sqrt [4]{a+b x} \, dx,x,x^4\right )\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \left (\frac {a^2 \sqrt [4]{a+b x}}{b^2}-\frac {2 a (a+b x)^{5/4}}{b^2}+\frac {(a+b x)^{9/4}}{b^2}\right ) \, dx,x,x^4\right )\\ &=\frac {a^2 \left (a+b x^4\right )^{5/4}}{5 b^3}-\frac {2 a \left (a+b x^4\right )^{9/4}}{9 b^3}+\frac {\left (a+b x^4\right )^{13/4}}{13 b^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 39, normalized size = 0.66 \[ \frac {\left (a+b x^4\right )^{5/4} \left (32 a^2-40 a b x^4+45 b^2 x^8\right )}{585 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^11*(a + b*x^4)^(1/4),x]

[Out]

((a + b*x^4)^(5/4)*(32*a^2 - 40*a*b*x^4 + 45*b^2*x^8))/(585*b^3)

________________________________________________________________________________________

fricas [A]  time = 0.47, size = 46, normalized size = 0.78 \[ \frac {{\left (45 \, b^{3} x^{12} + 5 \, a b^{2} x^{8} - 8 \, a^{2} b x^{4} + 32 \, a^{3}\right )} {\left (b x^{4} + a\right )}^{\frac {1}{4}}}{585 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/585*(45*b^3*x^12 + 5*a*b^2*x^8 - 8*a^2*b*x^4 + 32*a^3)*(b*x^4 + a)^(1/4)/b^3

________________________________________________________________________________________

giac [A]  time = 0.16, size = 43, normalized size = 0.73 \[ \frac {45 \, {\left (b x^{4} + a\right )}^{\frac {13}{4}} - 130 \, {\left (b x^{4} + a\right )}^{\frac {9}{4}} a + 117 \, {\left (b x^{4} + a\right )}^{\frac {5}{4}} a^{2}}{585 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/585*(45*(b*x^4 + a)^(13/4) - 130*(b*x^4 + a)^(9/4)*a + 117*(b*x^4 + a)^(5/4)*a^2)/b^3

________________________________________________________________________________________

maple [A]  time = 0.01, size = 36, normalized size = 0.61 \[ \frac {\left (b \,x^{4}+a \right )^{\frac {5}{4}} \left (45 b^{2} x^{8}-40 a b \,x^{4}+32 a^{2}\right )}{585 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/585*(b*x^4+a)^(5/4)*(45*b^2*x^8-40*a*b*x^4+32*a^2)/b^3

________________________________________________________________________________________

maxima [A]  time = 1.37, size = 47, normalized size = 0.80 \[ \frac {{\left (b x^{4} + a\right )}^{\frac {13}{4}}}{13 \, b^{3}} - \frac {2 \, {\left (b x^{4} + a\right )}^{\frac {9}{4}} a}{9 \, b^{3}} + \frac {{\left (b x^{4} + a\right )}^{\frac {5}{4}} a^{2}}{5 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*(b*x^4 + a)^(13/4)/b^3 - 2/9*(b*x^4 + a)^(9/4)*a/b^3 + 1/5*(b*x^4 + a)^(5/4)*a^2/b^3

________________________________________________________________________________________

mupad [B]  time = 1.12, size = 44, normalized size = 0.75 \[ {\left (b\,x^4+a\right )}^{1/4}\,\left (\frac {x^{12}}{13}+\frac {32\,a^3}{585\,b^3}+\frac {a\,x^8}{117\,b}-\frac {8\,a^2\,x^4}{585\,b^2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11*(a + b*x^4)^(1/4),x)

[Out]

(a + b*x^4)^(1/4)*(x^12/13 + (32*a^3)/(585*b^3) + (a*x^8)/(117*b) - (8*a^2*x^4)/(585*b^2))

________________________________________________________________________________________

sympy [A]  time = 5.62, size = 87, normalized size = 1.47 \[ \begin {cases} \frac {32 a^{3} \sqrt [4]{a + b x^{4}}}{585 b^{3}} - \frac {8 a^{2} x^{4} \sqrt [4]{a + b x^{4}}}{585 b^{2}} + \frac {a x^{8} \sqrt [4]{a + b x^{4}}}{117 b} + \frac {x^{12} \sqrt [4]{a + b x^{4}}}{13} & \text {for}\: b \neq 0 \\\frac {\sqrt [4]{a} x^{12}}{12} & \text {otherwise} \end {cases} \]

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**3*(a + b*x**4)**(1/4)/(585*b**3) - 8*a**2*x**4*(a + b*x**4)**(1/4)/(585*b**2) + a*x**8*(a + b
*x**4)**(1/4)/(117*b) + x**12*(a + b*x**4)**(1/4)/13, Ne(b, 0)), (a**(1/4)*x**12/12, True))

________________________________________________________________________________________