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

Optimal. Leaf size=107 \[ -\frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{2 a^{5/3}}+\frac{b \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^2}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{5/3}}+\frac{b \log (x)}{3 a^{5/3}}-\frac{\sqrt [3]{a+b x^2}}{2 a x^2} \]

[Out]

-(a + b*x^2)^(1/3)/(2*a*x^2) + (b*ArcTan[(a^(1/3) + 2*(a + b*x^2)^(1/3))/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(5/3))
+ (b*Log[x])/(3*a^(5/3)) - (b*Log[a^(1/3) - (a + b*x^2)^(1/3)])/(2*a^(5/3))

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Rubi [A]  time = 0.0658551, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {266, 51, 57, 617, 204, 31} \[ -\frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{2 a^{5/3}}+\frac{b \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^2}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{5/3}}+\frac{b \log (x)}{3 a^{5/3}}-\frac{\sqrt [3]{a+b x^2}}{2 a x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(a + b*x^2)^(2/3)),x]

[Out]

-(a + b*x^2)^(1/3)/(2*a*x^2) + (b*ArcTan[(a^(1/3) + 2*(a + b*x^2)^(1/3))/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(5/3))
+ (b*Log[x])/(3*a^(5/3)) - (b*Log[a^(1/3) - (a + b*x^2)^(1/3)])/(2*a^(5/3))

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 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 57

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, -Simp[L
og[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (-Dist[3/(2*b*q), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x
)^(1/3)], x] - Dist[3/(2*b*q^2), Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x]
&& PosQ[(b*c - a*d)/b]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{1}{x^3 \left (a+b x^2\right )^{2/3}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^2 (a+b x)^{2/3}} \, dx,x,x^2\right )\\ &=-\frac{\sqrt [3]{a+b x^2}}{2 a x^2}-\frac{b \operatorname{Subst}\left (\int \frac{1}{x (a+b x)^{2/3}} \, dx,x,x^2\right )}{3 a}\\ &=-\frac{\sqrt [3]{a+b x^2}}{2 a x^2}+\frac{b \log (x)}{3 a^{5/3}}+\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^2}\right )}{2 a^{5/3}}+\frac{b \operatorname{Subst}\left (\int \frac{1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x^2}\right )}{2 a^{4/3}}\\ &=-\frac{\sqrt [3]{a+b x^2}}{2 a x^2}+\frac{b \log (x)}{3 a^{5/3}}-\frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{2 a^{5/3}}-\frac{b \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{a+b x^2}}{\sqrt [3]{a}}\right )}{a^{5/3}}\\ &=-\frac{\sqrt [3]{a+b x^2}}{2 a x^2}+\frac{b \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{a+b x^2}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt{3} a^{5/3}}+\frac{b \log (x)}{3 a^{5/3}}-\frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{2 a^{5/3}}\\ \end{align*}

Mathematica [C]  time = 0.0072726, size = 37, normalized size = 0.35 \[ \frac{3 b \sqrt [3]{a+b x^2} \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{b x^2}{a}+1\right )}{2 a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*b*(a + b*x^2)^(1/3)*Hypergeometric2F1[1/3, 2, 4/3, 1 + (b*x^2)/a])/(2*a^2)

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Maple [F]  time = 0.026, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{2}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.8237, size = 466, normalized size = 4.36 \begin{align*} \frac{2 \, \sqrt{3} a b x^{2} \sqrt{-\left (-a^{2}\right )^{\frac{1}{3}}} \arctan \left (-\frac{{\left (\sqrt{3} \left (-a^{2}\right )^{\frac{1}{3}} a - 2 \, \sqrt{3}{\left (b x^{2} + a\right )}^{\frac{1}{3}} \left (-a^{2}\right )^{\frac{2}{3}}\right )} \sqrt{-\left (-a^{2}\right )^{\frac{1}{3}}}}{3 \, a^{2}}\right ) + \left (-a^{2}\right )^{\frac{2}{3}} b x^{2} \log \left ({\left (b x^{2} + a\right )}^{\frac{2}{3}} a - \left (-a^{2}\right )^{\frac{1}{3}} a +{\left (b x^{2} + a\right )}^{\frac{1}{3}} \left (-a^{2}\right )^{\frac{2}{3}}\right ) - 2 \, \left (-a^{2}\right )^{\frac{2}{3}} b x^{2} \log \left ({\left (b x^{2} + a\right )}^{\frac{1}{3}} a - \left (-a^{2}\right )^{\frac{2}{3}}\right ) - 3 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} a^{2}}{6 \, a^{3} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 1.33437, size = 41, normalized size = 0.38 \begin{align*} - \frac{\Gamma \left (\frac{5}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, \frac{5}{3} \\ \frac{8}{3} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{2}}} \right )}}{2 b^{\frac{2}{3}} x^{\frac{10}{3}} \Gamma \left (\frac{8}{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-gamma(5/3)*hyper((2/3, 5/3), (8/3,), a*exp_polar(I*pi)/(b*x**2))/(2*b**(2/3)*x**(10/3)*gamma(8/3))

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Giac [A]  time = 4.16456, size = 147, normalized size = 1.37 \begin{align*} \frac{1}{6} \, b{\left (\frac{2 \, \sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} + a^{\frac{1}{3}}\right )}}{3 \, a^{\frac{1}{3}}}\right )}{a^{\frac{5}{3}}} + \frac{\log \left ({\left (b x^{2} + a\right )}^{\frac{2}{3}} +{\left (b x^{2} + a\right )}^{\frac{1}{3}} a^{\frac{1}{3}} + a^{\frac{2}{3}}\right )}{a^{\frac{5}{3}}} - \frac{2 \, \log \left ({\left |{\left (b x^{2} + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right )}{a^{\frac{5}{3}}} - \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}}}{a b x^{2}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*b*(2*sqrt(3)*arctan(1/3*sqrt(3)*(2*(b*x^2 + a)^(1/3) + a^(1/3))/a^(1/3))/a^(5/3) + log((b*x^2 + a)^(2/3) +
(b*x^2 + a)^(1/3)*a^(1/3) + a^(2/3))/a^(5/3) - 2*log(abs((b*x^2 + a)^(1/3) - a^(1/3)))/a^(5/3) - 3*(b*x^2 + a
)^(1/3)/(a*b*x^2))