3.12.79 \(\int x^3 \sqrt [4]{a-b x^4} \, dx\) [1179]

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

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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 = {267} \begin {gather*} -\frac {\left (a-b x^4\right )^{5/4}}{5 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(a - b*x^4)^(1/4),x]

[Out]

-1/5*(a - b*x^4)^(5/4)/b

Rule 267

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*}

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Mathematica [A]
time = 0.02, size = 19, normalized size = 1.00 \begin {gather*} -\frac {\left (a-b x^4\right )^{5/4}}{5 b} \end {gather*}

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

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Maple [A]
time = 0.16, size = 16, normalized size = 0.84

method	result	size
gosper	\(-\frac {\left (-b \,x^{4}+a \right )^{\frac {5}{4}}}{5 b}\)	\(16\)
derivativedivides	\(-\frac {\left (-b \,x^{4}+a \right )^{\frac {5}{4}}}{5 b}\)	\(16\)
default	\(-\frac {\left (-b \,x^{4}+a \right )^{\frac {5}{4}}}{5 b}\)	\(16\)
trager	\(-\frac {\left (-b \,x^{4}+a \right )^{\frac {5}{4}}}{5 b}\)	\(16\)
risch	\(-\frac {\left (-b \,x^{4}+a \right )^{\frac {5}{4}} \left (\left (-b \,x^{4}+a \right )^{3}\right )^{\frac {1}{4}}}{5 b \left (-\left (b \,x^{4}-a \right )^{3}\right )^{\frac {1}{4}}}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 15, normalized size = 0.79 \begin {gather*} -\frac {{\left (-b x^{4} + a\right )}^{\frac {5}{4}}}{5 \, b} \end {gather*}

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

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Fricas [A]
time = 0.36, size = 24, normalized size = 1.26 \begin {gather*} \frac {{\left (b x^{4} - a\right )} {\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{5 \, b} \end {gather*}

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (14) = 28\).
time = 0.11, size = 39, normalized size = 2.05 \begin {gather*} \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} \end {gather*}

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))

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Giac [A]
time = 1.31, size = 15, normalized size = 0.79 \begin {gather*} -\frac {{\left (-b x^{4} + a\right )}^{\frac {5}{4}}}{5 \, b} \end {gather*}

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

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

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)

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