3.1177 \(\int x^3 \sqrt [4]{a-b x^4} \, dx\)

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

[Out]

-(a - b*x^4)^(5/4)/(5*b)

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Rubi [A]  time = 0.0108756, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ -\frac{\left (a-b x^4\right )^{5/4}}{5 b} \]

Antiderivative was successfully verified.

[In]  Int[x^3*(a - b*x^4)^(1/4),x]

[Out]

-(a - b*x^4)^(5/4)/(5*b)

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Rubi in Sympy [A]  time = 2.38956, size = 14, normalized size = 0.74 \[ - \frac{\left (a - b x^{4}\right )^{\frac{5}{4}}}{5 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(-b*x**4+a)**(1/4),x)

[Out]

-(a - b*x**4)**(5/4)/(5*b)

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Mathematica [A]  time = 0.0093243, size = 19, normalized size = 1. \[ -\frac{\left (a-b x^4\right )^{5/4}}{5 b} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*(a - b*x^4)^(1/4),x]

[Out]

-(a - b*x^4)^(5/4)/(5*b)

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Maple [A]  time = 0.005, size = 16, normalized size = 0.8 \[ -{\frac{1}{5\,b} \left ( -b{x}^{4}+a \right ) ^{{\frac{5}{4}}}} \]

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

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Maxima [A]  time = 1.41999, size = 20, normalized size = 1.05 \[ -\frac{{\left (-b x^{4} + a\right )}^{\frac{5}{4}}}{5 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5*(-b*x^4 + a)^(5/4)/b

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Fricas [A]  time = 0.234918, size = 32, normalized size = 1.68 \[ \frac{{\left (b x^{4} - a\right )}{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{5 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*(b*x^4 - a)*(-b*x^4 + a)^(1/4)/b

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Sympy [A]  time = 1.39672, 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))

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GIAC/XCAS [A]  time = 0.253951, size = 20, normalized size = 1.05 \[ -\frac{{\left (-b x^{4} + a\right )}^{\frac{5}{4}}}{5 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5*(-b*x^4 + a)^(5/4)/b