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

Optimal. Leaf size=47 \[ \frac{2 \sqrt{a x+b x^{2/3}}}{a}-\frac{4 b \sqrt{a x+b x^{2/3}}}{a^2 \sqrt [3]{x}} \]

[Out]

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

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Rubi [A]  time = 0.0495725, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2002, 2014} \[ \frac{2 \sqrt{a x+b x^{2/3}}}{a}-\frac{4 b \sqrt{a x+b x^{2/3}}}{a^2 \sqrt [3]{x}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[b*x^(2/3) + a*x],x]

[Out]

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

Rule 2002

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

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{b x^{2/3}+a x}} \, dx &=\frac{2 \sqrt{b x^{2/3}+a x}}{a}-\frac{(2 b) \int \frac{1}{\sqrt [3]{x} \sqrt{b x^{2/3}+a x}} \, dx}{3 a}\\ &=\frac{2 \sqrt{b x^{2/3}+a x}}{a}-\frac{4 b \sqrt{b x^{2/3}+a x}}{a^2 \sqrt [3]{x}}\\ \end{align*}

Mathematica [A]  time = 0.0274036, size = 36, normalized size = 0.77 \[ \frac{2 \left (a \sqrt [3]{x}-2 b\right ) \sqrt{a x+b x^{2/3}}}{a^2 \sqrt [3]{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[b*x^(2/3) + a*x],x]

[Out]

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

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Maple [A]  time = 0.003, size = 36, normalized size = 0.8 \begin{align*} 2\,{\frac{\sqrt [3]{x} \left ( b+a\sqrt [3]{x} \right ) \left ( a\sqrt [3]{x}-2\,b \right ) }{\sqrt{b{x}^{2/3}+ax}{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(a*x + b*x^(2/3)), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(a*x + b*x**(2/3)), x)

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Giac [A]  time = 1.1424, size = 49, normalized size = 1.04 \begin{align*} \frac{4 \, b^{\frac{3}{2}}}{a^{2}} + \frac{2 \,{\left ({\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} - 3 \, \sqrt{a x^{\frac{1}{3}} + b} b\right )}}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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