3.2019 \(\int \frac{1}{\sqrt{a+\frac{b}{x^3}} x} \, dx\)

Optimal. Leaf size=27 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^3}}}{\sqrt{a}}\right )}{3 \sqrt{a}} \]

[Out]

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

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Rubi [A]  time = 0.0176971, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {266, 63, 208} \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^3}}}{\sqrt{a}}\right )}{3 \sqrt{a}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b/x^3]*x),x]

[Out]

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

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

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

Mathematica [B]  time = 0.0155124, size = 59, normalized size = 2.19 \[ \frac{2 \sqrt{a x^3+b} \tanh ^{-1}\left (\frac{\sqrt{a} x^{3/2}}{\sqrt{a x^3+b}}\right )}{3 \sqrt{a} x^{3/2} \sqrt{a+\frac{b}{x^3}}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[a + b/x^3]*x),x]

[Out]

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

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Maple [C]  time = 0.008, size = 480, normalized size = 17.8 \begin{align*} -4\,{\frac{ \left ( a{x}^{3}+b \right ) \left ( -1+i\sqrt{3} \right ) \left ( -ax+\sqrt [3]{-b{a}^{2}} \right ) ^{2}}{{a}^{2}x\sqrt{x \left ( a{x}^{3}+b \right ) } \left ( i\sqrt{3}-3 \right ) }\sqrt{-{\frac{ \left ( i\sqrt{3}-3 \right ) xa}{ \left ( -1+i\sqrt{3} \right ) \left ( -ax+\sqrt [3]{-b{a}^{2}} \right ) }}}\sqrt{{\frac{i\sqrt{3}\sqrt [3]{-b{a}^{2}}+2\,ax+\sqrt [3]{-b{a}^{2}}}{ \left ( 1+i\sqrt{3} \right ) \left ( -ax+\sqrt [3]{-b{a}^{2}} \right ) }}}\sqrt{{\frac{i\sqrt{3}\sqrt [3]{-b{a}^{2}}-2\,ax-\sqrt [3]{-b{a}^{2}}}{ \left ( -1+i\sqrt{3} \right ) \left ( -ax+\sqrt [3]{-b{a}^{2}} \right ) }}} \left ({\it EllipticF} \left ( \sqrt{-{\frac{ \left ( i\sqrt{3}-3 \right ) xa}{ \left ( -1+i\sqrt{3} \right ) \left ( -ax+\sqrt [3]{-b{a}^{2}} \right ) }}},\sqrt{{\frac{ \left ( i\sqrt{3}+3 \right ) \left ( -1+i\sqrt{3} \right ) }{ \left ( i\sqrt{3}-3 \right ) \left ( 1+i\sqrt{3} \right ) }}} \right ) -{\it EllipticPi} \left ( \sqrt{-{\frac{ \left ( i\sqrt{3}-3 \right ) xa}{ \left ( -1+i\sqrt{3} \right ) \left ( -ax+\sqrt [3]{-b{a}^{2}} \right ) }}},{\frac{-1+i\sqrt{3}}{i\sqrt{3}-3}},\sqrt{{\frac{ \left ( i\sqrt{3}+3 \right ) \left ( -1+i\sqrt{3} \right ) }{ \left ( i\sqrt{3}-3 \right ) \left ( 1+i\sqrt{3} \right ) }}} \right ) \right ){\frac{1}{\sqrt{{\frac{a{x}^{3}+b}{{x}^{3}}}}}}{\frac{1}{\sqrt{{\frac{x \left ( -ax+\sqrt [3]{-b{a}^{2}} \right ) \left ( i\sqrt{3}\sqrt [3]{-b{a}^{2}}+2\,ax+\sqrt [3]{-b{a}^{2}} \right ) \left ( i\sqrt{3}\sqrt [3]{-b{a}^{2}}-2\,ax-\sqrt [3]{-b{a}^{2}} \right ) }{{a}^{2}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/((a*x^3+b)/x^3)^(1/2)/x*(a*x^3+b)*(-1+I*3^(1/2))*(-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^
(1/2)*(-a*x+(-b*a^2)^(1/3))^2*((I*3^(1/2)*(-b*a^2)^(1/3)+2*a*x+(-b*a^2)^(1/3))/(1+I*3^(1/2))/(-a*x+(-b*a^2)^(1
/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)-2*a*x-(-b*a^2)^(1/3))/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)/a^2*(
EllipticF((-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2),((I*3^(1/2)+3)*(-1+I*3^(1/2))/(1+I*3
^(1/2))/(I*3^(1/2)-3))^(1/2))-EllipticPi((-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2),(-1+I
*3^(1/2))/(I*3^(1/2)-3),((I*3^(1/2)+3)*(-1+I*3^(1/2))/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2)))/(x*(a*x^3+b))^(1/2)
/(I*3^(1/2)-3)/(1/a^2*x*(-a*x+(-b*a^2)^(1/3))*(I*3^(1/2)*(-b*a^2)^(1/3)+2*a*x+(-b*a^2)^(1/3))*(I*3^(1/2)*(-b*a
^2)^(1/3)-2*a*x-(-b*a^2)^(1/3)))^(1/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.35428, size = 242, normalized size = 8.96 \begin{align*} \left [\frac{\log \left (-8 \, a^{2} x^{6} - 8 \, a b x^{3} - b^{2} - 4 \,{\left (2 \, a x^{6} + b x^{3}\right )} \sqrt{a} \sqrt{\frac{a x^{3} + b}{x^{3}}}\right )}{6 \, \sqrt{a}}, -\frac{\sqrt{-a} \arctan \left (\frac{2 \, \sqrt{-a} x^{3} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{2 \, a x^{3} + b}\right )}{3 \, a}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6*log(-8*a^2*x^6 - 8*a*b*x^3 - b^2 - 4*(2*a*x^6 + b*x^3)*sqrt(a)*sqrt((a*x^3 + b)/x^3))/sqrt(a), -1/3*sqrt(
-a)*arctan(2*sqrt(-a)*x^3*sqrt((a*x^3 + b)/x^3)/(2*a*x^3 + b))/a]

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Sympy [A]  time = 1.26425, size = 24, normalized size = 0.89 \begin{align*} \frac{2 \operatorname{asinh}{\left (\frac{\sqrt{a} x^{\frac{3}{2}}}{\sqrt{b}} \right )}}{3 \sqrt{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*asinh(sqrt(a)*x**(3/2)/sqrt(b))/(3*sqrt(a))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(a + b/x^3)*x), x)