3.783 \(\int x^3 \left (a+c x^4\right )^{3/2} \, dx\)

Optimal. Leaf size=18 \[ \frac{\left (a+c x^4\right )^{5/2}}{10 c} \]

[Out]

(a + c*x^4)^(5/2)/(10*c)

_______________________________________________________________________________________

Rubi [A]  time = 0.0108471, 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 \[ \frac{\left (a+c x^4\right )^{5/2}}{10 c} \]

Antiderivative was successfully verified.

[In]  Int[x^3*(a + c*x^4)^(3/2),x]

[Out]

(a + c*x^4)^(5/2)/(10*c)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.16209, size = 12, normalized size = 0.67 \[ \frac{\left (a + c x^{4}\right )^{\frac{5}{2}}}{10 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a + c*x**4)**(5/2)/(10*c)

_______________________________________________________________________________________

Mathematica [A]  time = 0.00892337, size = 18, normalized size = 1. \[ \frac{\left (a+c x^4\right )^{5/2}}{10 c} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*(a + c*x^4)^(3/2),x]

[Out]

(a + c*x^4)^(5/2)/(10*c)

_______________________________________________________________________________________

Maple [A]  time = 0.005, size = 15, normalized size = 0.8 \[{\frac{1}{10\,c} \left ( c{x}^{4}+a \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(c*x^4+a)^(3/2),x)

[Out]

1/10*(c*x^4+a)^(5/2)/c

_______________________________________________________________________________________

Maxima [A]  time = 1.44181, size = 19, normalized size = 1.06 \[ \frac{{\left (c x^{4} + a\right )}^{\frac{5}{2}}}{10 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + a)^(3/2)*x^3,x, algorithm="maxima")

[Out]

1/10*(c*x^4 + a)^(5/2)/c

_______________________________________________________________________________________

Fricas [A]  time = 0.253964, size = 43, normalized size = 2.39 \[ \frac{{\left (c^{2} x^{8} + 2 \, a c x^{4} + a^{2}\right )} \sqrt{c x^{4} + a}}{10 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + a)^(3/2)*x^3,x, algorithm="fricas")

[Out]

1/10*(c^2*x^8 + 2*a*c*x^4 + a^2)*sqrt(c*x^4 + a)/c

_______________________________________________________________________________________

Sympy [A]  time = 2.78516, size = 60, normalized size = 3.33 \[ \begin{cases} \frac{a^{2} \sqrt{a + c x^{4}}}{10 c} + \frac{a x^{4} \sqrt{a + c x^{4}}}{5} + \frac{c x^{8} \sqrt{a + c x^{4}}}{10} & \text{for}\: c \neq 0 \\\frac{a^{\frac{3}{2}} x^{4}}{4} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((a**2*sqrt(a + c*x**4)/(10*c) + a*x**4*sqrt(a + c*x**4)/5 + c*x**8*sqr
t(a + c*x**4)/10, Ne(c, 0)), (a**(3/2)*x**4/4, True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.216249, size = 19, normalized size = 1.06 \[ \frac{{\left (c x^{4} + a\right )}^{\frac{5}{2}}}{10 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + a)^(3/2)*x^3,x, algorithm="giac")

[Out]

1/10*(c*x^4 + a)^(5/2)/c