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

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

[Out]

(2*Sqrt[a*x^2 + b*x^3])/(b*x)

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Rubi [A]  time = 0.0101149, 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 = {1588} \[ \frac{2 \sqrt{a x^2+b x^3}}{b x} \]

Antiderivative was successfully verified.

[In]

Int[x/Sqrt[a*x^2 + b*x^3],x]

[Out]

(2*Sqrt[a*x^2 + b*x^3])/(b*x)

Rule 1588

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*x^(p - q
+ 1)*Qq^(m + 1))/((p + m*q + 1)*Coeff[Qq, x, q]), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0073949, size = 21, normalized size = 0.91 \[ \frac{2 \sqrt{x^2 (a+b x)}}{b x} \]

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[a*x^2 + b*x^3],x]

[Out]

(2*Sqrt[x^2*(a + b*x)])/(b*x)

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Maple [A]  time = 0.001, size = 25, normalized size = 1.1 \begin{align*} 2\,{\frac{x \left ( bx+a \right ) }{b\sqrt{b{x}^{3}+a{x}^{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.16918, size = 16, normalized size = 0.7 \begin{align*} \frac{2 \, \sqrt{b x + a}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(b*x + a)/b

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Fricas [A]  time = 0.797656, size = 39, normalized size = 1.7 \begin{align*} \frac{2 \, \sqrt{b x^{3} + a x^{2}}}{b x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(b*x^3 + a*x^2)/(b*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x/sqrt(x**2*(a + b*x)), x)

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Giac [A]  time = 1.18953, size = 35, normalized size = 1.52 \begin{align*} \frac{2}{\sqrt{\frac{b}{x} + \frac{a}{x^{2}}} - \frac{\sqrt{a}}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(sqrt(b/x + a/x^2) - sqrt(a)/x)