3.12.77 \(\int x^{11} \sqrt [4]{a-b x^4} \, dx\) [1177]

Optimal. Leaf size=62 \[ -\frac {a^2 \left (a-b x^4\right )^{5/4}}{5 b^3}+\frac {2 a \left (a-b x^4\right )^{9/4}}{9 b^3}-\frac {\left (a-b x^4\right )^{13/4}}{13 b^3} \]

[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 = 62, normalized size of antiderivative = 1.00, 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 = {272, 45} \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^11*(a - b*x^4)^(1/4),x]

[Out]

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

Rule 45

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 272

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} \text {Subst}\left (\int x^2 \sqrt [4]{a-b x} \, dx,x,x^4\right )\\ &=\frac {1}{4} \text {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 = 0.03, size = 51, normalized size = 0.82 \begin {gather*} \frac {\sqrt [4]{a-b x^4} \left (-32 a^3-8 a^2 b x^4-5 a b^2 x^8+45 b^3 x^{12}\right )}{585 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^11*(a - b*x^4)^(1/4),x]

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.17, size = 37, normalized size = 0.60

method	result	size
gosper	\(-\frac {\left (-b \,x^{4}+a \right )^{\frac {5}{4}} \left (45 b^{2} x^{8}+40 a b \,x^{4}+32 a^{2}\right )}{585 b^{3}}\)	\(37\)
trager	\(-\frac {\left (-45 b^{3} x^{12}+5 a \,b^{2} x^{8}+8 a^{2} b \,x^{4}+32 a^{3}\right ) \left (-b \,x^{4}+a \right )^{\frac {1}{4}}}{585 b^{3}}\)	\(48\)
risch	\(-\frac {\left (-b \,x^{4}+a \right )^{\frac {1}{4}} \left (\left (-b \,x^{4}+a \right )^{3}\right )^{\frac {1}{4}} \left (-45 b^{3} x^{12}+5 a \,b^{2} x^{8}+8 a^{2} b \,x^{4}+32 a^{3}\right )}{585 b^{3} \left (-\left (b \,x^{4}-a \right )^{3}\right )^{\frac {1}{4}}}\)	\(75\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11*(-b*x^4+a)^(1/4),x,method=_RETURNVERBOSE)

[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 = 0.30, size = 50, normalized size = 0.81 \begin {gather*} -\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}} \end {gather*}

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

________________________________________________________________________________________

Fricas [A]
time = 0.37, size = 47, normalized size = 0.76 \begin {gather*} \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}} \end {gather*}

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

________________________________________________________________________________________

Sympy [A]
time = 0.42, size = 87, normalized size = 1.40 \begin {gather*} \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} \end {gather*}

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))

________________________________________________________________________________________

Giac [A]
time = 2.42, size = 68, normalized size = 1.10 \begin {gather*} \frac {45 \, {\left (b x^{4} - a\right )}^{3} {\left (-b x^{4} + a\right )}^{\frac {1}{4}} + 130 \, {\left (b x^{4} - a\right )}^{2} {\left (-b x^{4} + a\right )}^{\frac {1}{4}} a - 117 \, {\left (-b x^{4} + a\right )}^{\frac {5}{4}} a^{2}}{585 \, b^{3}} \end {gather*}

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

________________________________________________________________________________________

Mupad [B]
time = 1.13, size = 46, normalized size = 0.74 \begin {gather*} -{\left (a-b\,x^4\right )}^{1/4}\,\left (\frac {32\,a^3}{585\,b^3}-\frac {x^{12}}{13}+\frac {a\,x^8}{117\,b}+\frac {8\,a^2\,x^4}{585\,b^2}\right ) \end {gather*}

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

________________________________________________________________________________________