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

Optimal. Leaf size=23 \[ -\frac{2 \sqrt{a x+b x^4}}{3 a x^2} \]

[Out]

(-2*Sqrt[a*x + b*x^4])/(3*a*x^2)

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Rubi [A]  time = 0.0337861, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {2014} \[ -\frac{2 \sqrt{a x+b x^4}}{3 a x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*Sqrt[a*x + b*x^4]),x]

[Out]

(-2*Sqrt[a*x + b*x^4])/(3*a*x^2)

Rule 2014

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> -Simp[(c^(j - 1)*(c*x)^(m - j
+ 1)*(a*x^j + b*x^n)^(p + 1))/(a*(n - j)*(p + 1)), x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && N
eQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

Rubi steps

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

Mathematica [A]  time = 0.0106894, size = 23, normalized size = 1. \[ -\frac{2 \sqrt{x \left (a+b x^3\right )}}{3 a x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[x*(a + b*x^3)])/(3*a*x^2)

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Maple [A]  time = 0.003, size = 27, normalized size = 1.2 \begin{align*} -{\frac{2\,b{x}^{3}+2\,a}{3\,ax}{\frac{1}{\sqrt{b{x}^{4}+ax}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(b*x^3+a)/x/a/(b*x^4+a*x)^(1/2)

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Maxima [A]  time = 1.06738, size = 35, normalized size = 1.52 \begin{align*} -\frac{2 \,{\left (b x^{4} + a x\right )}}{3 \, \sqrt{b x^{3} + a} a x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(b*x^4 + a*x)/(sqrt(b*x^3 + a)*a*x^(5/2))

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Fricas [A]  time = 1.26757, size = 43, normalized size = 1.87 \begin{align*} -\frac{2 \, \sqrt{b x^{4} + a x}}{3 \, a x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*sqrt(b*x^4 + a*x)/(a*x^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{2} \sqrt{x \left (a + b x^{3}\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(x**2*sqrt(x*(a + b*x**3))), x)

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Giac [A]  time = 1.19666, size = 19, normalized size = 0.83 \begin{align*} -\frac{2 \, \sqrt{b + \frac{a}{x^{3}}}}{3 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*sqrt(b + a/x^3)/a