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

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

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Rubi [A]  time = 0.05, antiderivative size = 47, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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}} \end {gather*}

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*}

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Mathematica [A]  time = 0.04, size = 36, normalized size = 0.77 \begin {gather*} \frac {2 \left (a \sqrt [3]{x}-2 b\right ) \sqrt {a x+b x^{2/3}}}{a^2 \sqrt [3]{x}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.04, size = 36, normalized size = 0.77 \begin {gather*} \frac {2 \left (a \sqrt [3]{x}-2 b\right ) \sqrt {a x+b x^{2/3}}}{a^2 \sqrt [3]{x}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[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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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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giac [A]  time = 0.49, size = 36, normalized size = 0.77 \begin {gather*} \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 {gather*}

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

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maple [A]  time = 0.04, size = 36, normalized size = 0.77 \begin {gather*} \frac {2 \left (a \,x^{\frac {1}{3}}+b \right ) \left (a \,x^{\frac {1}{3}}-2 b \right ) x^{\frac {1}{3}}}{\sqrt {a x +b \,x^{\frac {2}{3}}}\, a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a x + b x^{\frac {2}{3}}}}\,{d x} \end {gather*}

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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mupad [B]  time = 5.22, size = 40, normalized size = 0.85 \begin {gather*} \frac {3\,x\,\sqrt {\frac {a\,x^{1/3}}{b}+1}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},2;\ 3;\ -\frac {a\,x^{1/3}}{b}\right )}{2\,\sqrt {a\,x+b\,x^{2/3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x + b*x^(2/3))^(1/2),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a x + b x^{\frac {2}{3}}}}\, dx \end {gather*}

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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