3.1015 \(\int \frac{x}{\sqrt{a+b x^2+(2+2 c-2 (1+c)) x^4}} \, dx\)

Optimal. Leaf size=15 \[ \frac{\sqrt{a+b x^2}}{b} \]

[Out]

Sqrt[a + b*x^2]/b

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Rubi [A]  time = 0.0033684, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {5, 261} \[ \frac{\sqrt{a+b x^2}}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[a + b*x^2]/b

Rule 5

Int[(u_.)*((a_.) + (c_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[u*(a + b*x^n)^p, x] /; FreeQ[{
a, b, c, n, p}, x] && EqQ[j, 2*n] && EqQ[c, 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}{\sqrt{a+b x^2+(2+2 c-2 (1+c)) x^4}} \, dx &=\int \frac{x}{\sqrt{a+b x^2}} \, dx\\ &=\frac{\sqrt{a+b x^2}}{b}\\ \end{align*}

Mathematica [A]  time = 0.0020604, size = 15, normalized size = 1. \[ \frac{\sqrt{a+b x^2}}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[a + b*x^2]/b

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Maple [A]  time = 0.041, size = 14, normalized size = 0.9 \begin{align*}{\frac{1}{b}\sqrt{b{x}^{2}+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(b*x^2+a)^(1/2),x)

[Out]

(b*x^2+a)^(1/2)/b

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Maxima [A]  time = 0.9383, size = 18, normalized size = 1.2 \begin{align*} \frac{\sqrt{b x^{2} + a}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(b*x^2 + a)/b

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Fricas [A]  time = 1.5682, size = 26, normalized size = 1.73 \begin{align*} \frac{\sqrt{b x^{2} + a}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(b*x^2 + a)/b

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Sympy [A]  time = 0.368195, size = 20, normalized size = 1.33 \begin{align*} \begin{cases} \frac{\sqrt{a + b x^{2}}}{b} & \text{for}\: b \neq 0 \\\frac{x^{2}}{2 \sqrt{a}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x**2+a)**(1/2),x)

[Out]

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

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Giac [A]  time = 1.12855, size = 18, normalized size = 1.2 \begin{align*} \frac{\sqrt{b x^{2} + a}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(b*x^2 + a)/b