3.1082 \(\int \frac{x^3}{\sqrt [4]{a+b x^4}} \, dx\)

Optimal. Leaf size=18 \[ \frac{\left (a+b x^4\right )^{3/4}}{3 b} \]

[Out]

(a + b*x^4)^(3/4)/(3*b)

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Rubi [A]  time = 0.010962, 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 \[ \frac{\left (a+b x^4\right )^{3/4}}{3 b} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(a + b*x^4)^(1/4),x]

[Out]

(a + b*x^4)^(3/4)/(3*b)

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Rubi in Sympy [A]  time = 2.11894, size = 12, normalized size = 0.67 \[ \frac{\left (a + b x^{4}\right )^{\frac{3}{4}}}{3 b} \]

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)**(3/4)/(3*b)

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Mathematica [A]  time = 0.00691323, size = 18, normalized size = 1. \[ \frac{\left (a+b x^4\right )^{3/4}}{3 b} \]

Antiderivative was successfully verified.

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

[Out]

(a + b*x^4)^(3/4)/(3*b)

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Maple [A]  time = 0.005, size = 15, normalized size = 0.8 \[{\frac{1}{3\,b} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(b*x^4+a)^(1/4),x)

[Out]

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

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Maxima [A]  time = 1.44559, size = 19, normalized size = 1.06 \[ \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(b*x^4 + a)^(3/4)/b

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Fricas [A]  time = 0.300707, size = 19, normalized size = 1.06 \[ \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(b*x^4 + a)^(3/4)/b

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Sympy [A]  time = 1.8692, size = 22, normalized size = 1.22 \[ \begin{cases} \frac{\left (a + b x^{4}\right )^{\frac{3}{4}}}{3 b} & \text{for}\: b \neq 0 \\\frac{x^{4}}{4 \sqrt [4]{a}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise(((a + b*x**4)**(3/4)/(3*b), Ne(b, 0)), (x**4/(4*a**(1/4)), True))

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GIAC/XCAS [A]  time = 0.215182, size = 19, normalized size = 1.06 \[ \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(b*x^4 + a)^(3/4)/b