3.764 \(\int x^3 \sqrt{a+c x^4} \, dx\)

Optimal. Leaf size=18 \[ \frac{\left (a+c x^4\right )^{3/2}}{6 c} \]

[Out]

(a + c*x^4)^(3/2)/(6*c)

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Rubi [A]  time = 0.0100257, 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 \[ \frac{\left (a+c x^4\right )^{3/2}}{6 c} \]

Antiderivative was successfully verified.

[In]  Int[x^3*Sqrt[a + c*x^4],x]

[Out]

(a + c*x^4)^(3/2)/(6*c)

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Rubi in Sympy [A]  time = 2.12814, size = 12, normalized size = 0.67 \[ \frac{\left (a + c x^{4}\right )^{\frac{3}{2}}}{6 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a + c*x**4)**(3/2)/(6*c)

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Mathematica [A]  time = 0.00862258, size = 18, normalized size = 1. \[ \frac{\left (a+c x^4\right )^{3/2}}{6 c} \]

Antiderivative was successfully verified.

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

[Out]

(a + c*x^4)^(3/2)/(6*c)

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Maple [A]  time = 0.007, size = 15, normalized size = 0.8 \[{\frac{1}{6\,c} \left ( c{x}^{4}+a \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(c*x^4+a)^(1/2),x)

[Out]

1/6*(c*x^4+a)^(3/2)/c

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Maxima [A]  time = 1.43666, size = 19, normalized size = 1.06 \[ \frac{{\left (c x^{4} + a\right )}^{\frac{3}{2}}}{6 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(c*x^4 + a)^(3/2)/c

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Fricas [A]  time = 0.244807, size = 19, normalized size = 1.06 \[ \frac{{\left (c x^{4} + a\right )}^{\frac{3}{2}}}{6 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(c*x^4 + a)^(3/2)/c

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Sympy [A]  time = 0.659839, size = 39, normalized size = 2.17 \[ \begin{cases} \frac{a \sqrt{a + c x^{4}}}{6 c} + \frac{x^{4} \sqrt{a + c x^{4}}}{6} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{4}}{4} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((a*sqrt(a + c*x**4)/(6*c) + x**4*sqrt(a + c*x**4)/6, Ne(c, 0)), (sqrt(
a)*x**4/4, True))

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GIAC/XCAS [A]  time = 0.214001, size = 19, normalized size = 1.06 \[ \frac{{\left (c x^{4} + a\right )}^{\frac{3}{2}}}{6 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(c*x^4 + a)^(3/2)/c