3.314 \(\int \frac{1}{\sqrt{a x^3+b x^4}} \, dx\)

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

[Out]

(-2*Sqrt[a*x^3 + b*x^4])/(a*x^2)

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Rubi [A]  time = 0.0050791, antiderivative size = 23, 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 = {2000} \[ -\frac{2 \sqrt{a x^3+b x^4}}{a x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a*x^3 + b*x^4],x]

[Out]

(-2*Sqrt[a*x^3 + b*x^4])/(a*x^2)

Rule 2000

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

Rubi steps

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

Mathematica [A]  time = 0.0071141, size = 21, normalized size = 0.91 \[ -\frac{2 \sqrt{x^3 (a+b x)}}{a x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a*x^3 + b*x^4],x]

[Out]

(-2*Sqrt[x^3*(a + b*x)])/(a*x^2)

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Maple [A]  time = 0.002, size = 25, normalized size = 1.1 \begin{align*} -2\,{\frac{x \left ( bx+a \right ) }{a\sqrt{b{x}^{4}+a{x}^{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b x^{4} + a x^{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(b*x^4 + a*x^3), x)

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Fricas [A]  time = 0.622897, 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/(b*x^4+a*x^3)^(1/2),x, algorithm="fricas")

[Out]

-2*sqrt(b*x^4 + a*x^3)/(a*x^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a x^{3} + b x^{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(a*x**3 + b*x**4), x)

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Giac [A]  time = 1.21362, size = 36, normalized size = 1.57 \begin{align*} \frac{2}{{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} \mathrm{sgn}\left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/((sqrt(b)*x - sqrt(b*x^2 + a*x))*sgn(x))