3.1179 \(\int x^3 \sqrt [4]{a-b x^4} \, dx\)

Optimal. Leaf size=19 \[ -\frac {\left (a-b x^4\right )^{5/4}}{5 b} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {261} \[ -\frac {\left (a-b x^4\right )^{5/4}}{5 b} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(a - b*x^4)^(1/4),x]

[Out]

-(a - b*x^4)^(5/4)/(5*b)

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 19, normalized size = 1.00 \[ -\frac {\left (a-b x^4\right )^{5/4}}{5 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a - b*x^4)^(1/4),x]

[Out]

-1/5*(a - b*x^4)^(5/4)/b

________________________________________________________________________________________

fricas [A]  time = 1.04, size = 24, normalized size = 1.26 \[ \frac {{\left (b x^{4} - a\right )} {\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{5 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(-b*x^4+a)^(1/4),x, algorithm="fricas")

[Out]

1/5*(b*x^4 - a)*(-b*x^4 + a)^(1/4)/b

________________________________________________________________________________________

giac [A]  time = 0.15, size = 15, normalized size = 0.79 \[ -\frac {{\left (-b x^{4} + a\right )}^{\frac {5}{4}}}{5 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(-b*x^4+a)^(1/4),x, algorithm="giac")

[Out]

-1/5*(-b*x^4 + a)^(5/4)/b

________________________________________________________________________________________

maple [A]  time = 0.00, size = 16, normalized size = 0.84 \[ -\frac {\left (-b \,x^{4}+a \right )^{\frac {5}{4}}}{5 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(-b*x^4+a)^(1/4),x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.28, size = 15, normalized size = 0.79 \[ -\frac {{\left (-b x^{4} + a\right )}^{\frac {5}{4}}}{5 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(-b*x^4+a)^(1/4),x, algorithm="maxima")

[Out]

-1/5*(-b*x^4 + a)^(5/4)/b

________________________________________________________________________________________

mupad [B]  time = 1.11, size = 15, normalized size = 0.79 \[ -\frac {{\left (a-b\,x^4\right )}^{5/4}}{5\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a - b*x^4)^(1/4),x)

[Out]

-(a - b*x^4)^(5/4)/(5*b)

________________________________________________________________________________________

sympy [A]  time = 0.69, size = 39, normalized size = 2.05 \[ \begin {cases} - \frac {a \sqrt [4]{a - b x^{4}}}{5 b} + \frac {x^{4} \sqrt [4]{a - b x^{4}}}{5} & \text {for}\: b \neq 0 \\\frac {\sqrt [4]{a} x^{4}}{4} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(-b*x**4+a)**(1/4),x)

[Out]

Piecewise((-a*(a - b*x**4)**(1/4)/(5*b) + x**4*(a - b*x**4)**(1/4)/5, Ne(b, 0)), (a**(1/4)*x**4/4, True))

________________________________________________________________________________________