3.811 \(\int \frac{x^3}{\sqrt{a+b x^4}} \, dx\)

Optimal. Leaf size=18 \[ \frac{\sqrt{a+b x^4}}{2 b} \]

[Out]

Sqrt[a + b*x^4]/(2*b)

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Rubi [A]  time = 0.0107421, 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 \[ \frac{\sqrt{a+b x^4}}{2 b} \]

Antiderivative was successfully verified.

[In]  Int[x^3/Sqrt[a + b*x^4],x]

[Out]

Sqrt[a + b*x^4]/(2*b)

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Rubi in Sympy [A]  time = 2.12308, size = 12, normalized size = 0.67 \[ \frac{\sqrt{a + b x^{4}}}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(b*x**4+a)**(1/2),x)

[Out]

sqrt(a + b*x**4)/(2*b)

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Mathematica [A]  time = 0.00628767, size = 18, normalized size = 1. \[ \frac{\sqrt{a+b x^4}}{2 b} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/Sqrt[a + b*x^4],x]

[Out]

Sqrt[a + b*x^4]/(2*b)

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Maple [A]  time = 0.008, size = 15, normalized size = 0.8 \[{\frac{1}{2\,b}\sqrt{b{x}^{4}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(b*x^4+a)^(1/2),x)

[Out]

1/2*(b*x^4+a)^(1/2)/b

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Maxima [A]  time = 1.43403, size = 19, normalized size = 1.06 \[ \frac{\sqrt{b x^{4} + a}}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(b*x^4 + a)/b

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Fricas [A]  time = 0.266061, size = 19, normalized size = 1.06 \[ \frac{\sqrt{b x^{4} + a}}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(b*x^4 + a)/b

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Sympy [A]  time = 1.66771, size = 22, normalized size = 1.22 \[ \begin{cases} \frac{\sqrt{a + b x^{4}}}{2 b} & \text{for}\: b \neq 0 \\\frac{x^{4}}{4 \sqrt{a}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(b*x**4+a)**(1/2),x)

[Out]

Piecewise((sqrt(a + b*x**4)/(2*b), Ne(b, 0)), (x**4/(4*sqrt(a)), True))

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GIAC/XCAS [A]  time = 0.213307, size = 19, normalized size = 1.06 \[ \frac{\sqrt{b x^{4} + a}}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(b*x^4 + a)/b