3.1049 \(\int x^3 \left (a+b x^4\right )^{5/4} \, dx\)

Optimal. Leaf size=18 \[ \frac{\left (a+b x^4\right )^{9/4}}{9 b} \]

[Out]

(a + b*x^4)^(9/4)/(9*b)

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Rubi [A]  time = 0.0108455, 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 \[ \frac{\left (a+b x^4\right )^{9/4}}{9 b} \]

Antiderivative was successfully verified.

[In]  Int[x^3*(a + b*x^4)^(5/4),x]

[Out]

(a + b*x^4)^(9/4)/(9*b)

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Rubi in Sympy [A]  time = 2.12816, size = 12, normalized size = 0.67 \[ \frac{\left (a + b x^{4}\right )^{\frac{9}{4}}}{9 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(b*x**4+a)**(5/4),x)

[Out]

(a + b*x**4)**(9/4)/(9*b)

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Mathematica [A]  time = 0.00843763, size = 18, normalized size = 1. \[ \frac{\left (a+b x^4\right )^{9/4}}{9 b} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*(a + b*x^4)^(5/4),x]

[Out]

(a + b*x^4)^(9/4)/(9*b)

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Maple [A]  time = 0.007, size = 15, normalized size = 0.8 \[{\frac{1}{9\,b} \left ( b{x}^{4}+a \right ) ^{{\frac{9}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(b*x^4+a)^(5/4),x)

[Out]

1/9*(b*x^4+a)^(9/4)/b

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Maxima [A]  time = 1.43556, size = 19, normalized size = 1.06 \[ \frac{{\left (b x^{4} + a\right )}^{\frac{9}{4}}}{9 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(5/4)*x^3,x, algorithm="maxima")

[Out]

1/9*(b*x^4 + a)^(9/4)/b

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Fricas [A]  time = 0.282678, size = 43, normalized size = 2.39 \[ \frac{{\left (b^{2} x^{8} + 2 \, a b x^{4} + a^{2}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{9 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(5/4)*x^3,x, algorithm="fricas")

[Out]

1/9*(b^2*x^8 + 2*a*b*x^4 + a^2)*(b*x^4 + a)^(1/4)/b

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Sympy [A]  time = 14.3177, size = 61, normalized size = 3.39 \[ \begin{cases} \frac{a^{2} \sqrt [4]{a + b x^{4}}}{9 b} + \frac{2 a x^{4} \sqrt [4]{a + b x^{4}}}{9} + \frac{b x^{8} \sqrt [4]{a + b x^{4}}}{9} & \text{for}\: b \neq 0 \\\frac{a^{\frac{5}{4}} x^{4}}{4} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(b*x**4+a)**(5/4),x)

[Out]

Piecewise((a**2*(a + b*x**4)**(1/4)/(9*b) + 2*a*x**4*(a + b*x**4)**(1/4)/9 + b*x
**8*(a + b*x**4)**(1/4)/9, Ne(b, 0)), (a**(5/4)*x**4/4, True))

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GIAC/XCAS [A]  time = 0.215666, size = 19, normalized size = 1.06 \[ \frac{{\left (b x^{4} + a\right )}^{\frac{9}{4}}}{9 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(5/4)*x^3,x, algorithm="giac")

[Out]

1/9*(b*x^4 + a)^(9/4)/b