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

Optimal. Leaf size=18 \[ \frac{\left (a+b x^4\right )^{5/4}}{5 b} \]

[Out]

(a + b*x^4)^(5/4)/(5*b)

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Rubi [A]  time = 0.010355, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{\left (a+b x^4\right )^{5/4}}{5 b} \]

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 = 2.11942, size = 12, normalized size = 0.67 \[ \frac{\left (a + b x^{4}\right )^{\frac{5}{4}}}{5 b} \]

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 = 0.0085893, size = 18, 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 = 0.007, size = 15, 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 = 1.43418, size = 19, normalized size = 1.06 \[ \frac{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}{5 \, b} \]

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 = 0.269688, size = 19, normalized size = 1.06 \[ \frac{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}{5 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*(b*x^4 + a)^(5/4)/b

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Sympy [A]  time = 1.34232, size = 39, normalized size = 2.17 \[ \begin{cases} \frac{a \sqrt [4]{a + b x^{4}}}{5 b} + \frac{x^{4} \sqrt [4]{a + b x^{4}}}{5} & \text{for}\: b \neq 0 \\\frac{\sqrt [4]{a} x^{4}}{4} & \text{otherwise} \end{cases} \]

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 = 0.211964, size = 19, normalized size = 1.06 \[ \frac{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}{5 \, b} \]

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