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

Optimal. Leaf size=190 \[ \frac{3 b^2 \sqrt [3]{x} \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^3 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}-\frac{3 b x^{2/3} \left (a+\frac{b}{\sqrt [3]{x}}\right )}{2 a^2 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}+\frac{x \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}-\frac{3 b^3 \left (a+\frac{b}{\sqrt [3]{x}}\right ) \log \left (a \sqrt [3]{x}+b\right )}{a^4 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}} \]

[Out]

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

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Rubi [A]  time = 0.118063, antiderivative size = 190, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {1341, 1355, 263, 43} \[ \frac{3 b^2 \sqrt [3]{x} \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^3 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}-\frac{3 b x^{2/3} \left (a+\frac{b}{\sqrt [3]{x}}\right )}{2 a^2 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}+\frac{x \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}-\frac{3 b^3 \left (a+\frac{b}{\sqrt [3]{x}}\right ) \log \left (a \sqrt [3]{x}+b\right )}{a^4 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1341

Int[((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Dist[k, Subst[I
nt[x^(k - 1)*(a + b*x^(k*n) + c*x^(2*k*n))^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && EqQ[n2, 2*n] &
& FractionQ[n]

Rule 1355

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_), x_Symbol] :> Dist[(a + b*x^n + c*x^
(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^n)^(2*p), x], x] /; Fr
eeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

Rule 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0410827, size = 86, normalized size = 0.45 \[ \frac{\left (a \sqrt [3]{x}+b\right ) \left (-3 a^2 b x^{2/3}+2 a^3 x+6 a b^2 \sqrt [3]{x}-6 b^3 \log \left (a \sqrt [3]{x}+b\right )\right )}{2 a^4 \sqrt [3]{x} \sqrt{\frac{\left (a \sqrt [3]{x}+b\right )^2}{x^{2/3}}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b + a*x^(1/3))*(6*a*b^2*x^(1/3) - 3*a^2*b*x^(2/3) + 2*a^3*x - 6*b^3*Log[b + a*x^(1/3)]))/(2*a^4*Sqrt[(b + a*
x^(1/3))^2/x^(2/3)]*x^(1/3))

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Maple [A]  time = 0.006, size = 78, normalized size = 0.4 \begin{align*} -{\frac{1}{2\,{a}^{4}} \left ( b+a\sqrt [3]{x} \right ) \left ( 3\,{x}^{2/3}{a}^{2}b-6\,a{b}^{2}\sqrt [3]{x}+6\,{b}^{3}\ln \left ( b+a\sqrt [3]{x} \right ) -2\,{a}^{3}x \right ){\frac{1}{\sqrt{{ \left ({a}^{2}{x}^{{\frac{2}{3}}}+2\,ab\sqrt [3]{x}+{b}^{2} \right ){x}^{-{\frac{2}{3}}}}}}}{\frac{1}{\sqrt [3]{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/((a^2*x^(2/3)+2*a*b*x^(1/3)+b^2)/x^(2/3))^(1/2)/x^(1/3)*(b+a*x^(1/3))*(3*x^(2/3)*a^2*b-6*a*b^2*x^(1/3)+6*
b^3*ln(b+a*x^(1/3))-2*a^3*x)/a^4

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Maxima [A]  time = 1.01861, size = 59, normalized size = 0.31 \begin{align*} -\frac{3 \, b^{3} \log \left (a x^{\frac{1}{3}} + b\right )}{a^{4}} + \frac{2 \, a^{2} x - 3 \, a b x^{\frac{2}{3}} + 6 \, b^{2} x^{\frac{1}{3}}}{2 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*b^3*log(a*x^(1/3) + b)/a^4 + 1/2*(2*a^2*x - 3*a*b*x^(2/3) + 6*b^2*x^(1/3))/a^3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.20538, size = 104, normalized size = 0.55 \begin{align*} -\frac{3 \, b^{3} \log \left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{a^{4} \mathrm{sgn}\left (a x + b x^{\frac{2}{3}}\right ) \mathrm{sgn}\left (x\right )} + \frac{2 \, a^{2} x - 3 \, a b x^{\frac{2}{3}} + 6 \, b^{2} x^{\frac{1}{3}}}{2 \, a^{3} \mathrm{sgn}\left (a x + b x^{\frac{2}{3}}\right ) \mathrm{sgn}\left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*b^3*log(abs(a*x^(1/3) + b))/(a^4*sgn(a*x + b*x^(2/3))*sgn(x)) + 1/2*(2*a^2*x - 3*a*b*x^(2/3) + 6*b^2*x^(1/3
))/(a^3*sgn(a*x + b*x^(2/3))*sgn(x))