3.1004 \(\int \frac{x^3}{\sqrt{a+(2+2 b-2 (1+b)) x^2+c x^4}} \, dx\)

Optimal. Leaf size=18 \[ \frac{\sqrt{a+c x^4}}{2 c} \]

[Out]

Sqrt[a + c*x^4]/(2*c)

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Rubi [A]  time = 0.0044898, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {4, 261} \[ \frac{\sqrt{a+c x^4}}{2 c} \]

Antiderivative was successfully verified.

[In]

Int[x^3/Sqrt[a + (2 + 2*b - 2*(1 + b))*x^2 + c*x^4],x]

[Out]

Sqrt[a + c*x^4]/(2*c)

Rule 4

Int[(u_.)*((a_.) + (c_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[u*(a + c*x^(2*n))^p, x] /; Fre
eQ[{a, b, c, n, p}, x] && EqQ[j, 2*n] && EqQ[b, 0]

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 \frac{x^3}{\sqrt{a+(2+2 b-2 (1+b)) x^2+c x^4}} \, dx &=\int \frac{x^3}{\sqrt{a+c x^4}} \, dx\\ &=\frac{\sqrt{a+c x^4}}{2 c}\\ \end{align*}

Mathematica [A]  time = 0.003454, size = 18, normalized size = 1. \[ \frac{\sqrt{a+c x^4}}{2 c} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/Sqrt[a + (2 + 2*b - 2*(1 + b))*x^2 + c*x^4],x]

[Out]

Sqrt[a + c*x^4]/(2*c)

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Maple [A]  time = 0.044, size = 15, normalized size = 0.8 \begin{align*}{\frac{1}{2\,c}\sqrt{c{x}^{4}+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.967832, size = 19, normalized size = 1.06 \begin{align*} \frac{\sqrt{c x^{4} + a}}{2 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(c*x^4+a)^(1/2),x, algorithm="maxima")

[Out]

1/2*sqrt(c*x^4 + a)/c

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Fricas [A]  time = 1.44087, size = 31, normalized size = 1.72 \begin{align*} \frac{\sqrt{c x^{4} + a}}{2 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(c*x^4+a)^(1/2),x, algorithm="fricas")

[Out]

1/2*sqrt(c*x^4 + a)/c

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Sympy [A]  time = 0.457962, size = 22, normalized size = 1.22 \begin{align*} \begin{cases} \frac{\sqrt{a + c x^{4}}}{2 c} & \text{for}\: c \neq 0 \\\frac{x^{4}}{4 \sqrt{a}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.11893, size = 19, normalized size = 1.06 \begin{align*} \frac{\sqrt{c x^{4} + a}}{2 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(c*x^4+a)^(1/2),x, algorithm="giac")

[Out]

1/2*sqrt(c*x^4 + a)/c