3.273 \(\int \frac{1}{\sqrt{x} \sqrt{a x^2+b x^3}} \, dx\)

Optimal. Leaf size=25 \[ -\frac{2 \sqrt{a x^2+b x^3}}{a x^{3/2}} \]

[Out]

(-2*Sqrt[a*x^2 + b*x^3])/(a*x^(3/2))

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Rubi [A]  time = 0.0377593, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {2014} \[ -\frac{2 \sqrt{a x^2+b x^3}}{a x^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[a*x^2 + b*x^3])/(a*x^(3/2))

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

Mathematica [A]  time = 0.0082031, size = 23, normalized size = 0.92 \[ -\frac{2 \sqrt{x^2 (a+b x)}}{a x^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[x^2*(a + b*x)])/(a*x^(3/2))

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Maple [A]  time = 0.003, size = 27, normalized size = 1.1 \begin{align*} -2\,{\frac{\sqrt{x} \left ( bx+a \right ) }{a\sqrt{b{x}^{3}+a{x}^{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b x^{3} + a x^{2}} \sqrt{x}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.713405, size = 49, normalized size = 1.96 \begin{align*} -\frac{2 \, \sqrt{b x^{3} + a x^{2}}}{a x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(b*x^3 + a*x^2)/(a*x^(3/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.26793, size = 41, normalized size = 1.64 \begin{align*} \frac{4 \, \sqrt{b}}{{\left (\sqrt{b} \sqrt{x} - \sqrt{b x + a}\right )}^{2} - a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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