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

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 = 0.0048617, 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, Rules used = {261} \[ \frac{\left (a+c x^4\right )^{5/2}}{10 c} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(a + c*x^4)^(3/2),x]

[Out]

(a + c*x^4)^(5/2)/(10*c)

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int x^3 \left (a+c x^4\right )^{3/2} \, dx &=\frac{\left (a+c x^4\right )^{5/2}}{10 c}\\ \end{align*}

Mathematica [A]  time = 0.0045495, 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 = 0.004, size = 15, normalized size = 0.8 \begin{align*}{\frac{1}{10\,c} \left ( c{x}^{4}+a \right ) ^{{\frac{5}{2}}}} \end{align*}

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 = 0.969078, size = 19, normalized size = 1.06 \begin{align*} \frac{{\left (c x^{4} + a\right )}^{\frac{5}{2}}}{10 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(c*x^4 + a)^(5/2)/c

________________________________________________________________________________________

Fricas [B]  time = 1.44254, size = 70, normalized size = 3.89 \begin{align*} \frac{{\left (c^{2} x^{8} + 2 \, a c x^{4} + a^{2}\right )} \sqrt{c x^{4} + a}}{10 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(c*x^4+a)^(3/2),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 = 1.28536, size = 60, normalized size = 3.33 \begin{align*} \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} \end{align*}

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*sqrt(a + c*x**4)/10, Ne(c, 0)), (
a**(3/2)*x**4/4, True))

________________________________________________________________________________________

Giac [A]  time = 1.10052, size = 19, normalized size = 1.06 \begin{align*} \frac{{\left (c x^{4} + a\right )}^{\frac{5}{2}}}{10 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(c*x^4 + a)^(5/2)/c