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

Optimal. Leaf size=101 \[ \frac{3}{4} a^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )+\frac{1}{2} \sqrt{3} a^{2/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^2}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )-\frac{1}{2} a^{2/3} \log (x)+\frac{3}{4} \left (a+b x^2\right )^{2/3} \]

[Out]

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

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Rubi [A]  time = 0.0664327, antiderivative size = 101, 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, 50, 55, 617, 204, 31} \[ \frac{3}{4} a^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )+\frac{1}{2} \sqrt{3} a^{2/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^2}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )-\frac{1}{2} a^{2/3} \log (x)+\frac{3}{4} \left (a+b x^2\right )^{2/3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(2/3)/x,x]

[Out]

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

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 50

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

Rule 55

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, -Simp[L
og[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/
3)], x] - Dist[3/(2*b*q), 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{\left (a+b x^2\right )^{2/3}}{x} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^{2/3}}{x} \, dx,x,x^2\right )\\ &=\frac{3}{4} \left (a+b x^2\right )^{2/3}+\frac{1}{2} a \operatorname{Subst}\left (\int \frac{1}{x \sqrt [3]{a+b x}} \, dx,x,x^2\right )\\ &=\frac{3}{4} \left (a+b x^2\right )^{2/3}-\frac{1}{2} a^{2/3} \log (x)-\frac{1}{4} \left (3 a^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^2}\right )+\frac{1}{4} (3 a) \operatorname{Subst}\left (\int \frac{1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x^2}\right )\\ &=\frac{3}{4} \left (a+b x^2\right )^{2/3}-\frac{1}{2} a^{2/3} \log (x)+\frac{3}{4} a^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )-\frac{1}{2} \left (3 a^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{a+b x^2}}{\sqrt [3]{a}}\right )\\ &=\frac{3}{4} \left (a+b x^2\right )^{2/3}+\frac{1}{2} \sqrt{3} a^{2/3} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{a+b x^2}}{\sqrt [3]{a}}}{\sqrt{3}}\right )-\frac{1}{2} a^{2/3} \log (x)+\frac{3}{4} a^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )\\ \end{align*}

Mathematica [A]  time = 0.0351165, size = 93, normalized size = 0.92 \[ \frac{1}{4} \left (3 \left (a^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )+\left (a+b x^2\right )^{2/3}\right )+2 \sqrt{3} a^{2/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{a+b x^2}}{\sqrt [3]{a}}+1}{\sqrt{3}}\right )-2 a^{2/3} \log (x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(2/3)/x,x]

[Out]

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

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Maple [F]  time = 0.023, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x} \left ( b{x}^{2}+a \right ) ^{{\frac{2}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.8373, size = 358, normalized size = 3.54 \begin{align*} \frac{1}{2} \, \sqrt{3}{\left (a^{2}\right )}^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3} a + 2 \, \sqrt{3}{\left (b x^{2} + a\right )}^{\frac{1}{3}}{\left (a^{2}\right )}^{\frac{1}{3}}}{3 \, a}\right ) - \frac{1}{4} \,{\left (a^{2}\right )}^{\frac{1}{3}} \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 ) + \frac{1}{2} \,{\left (a^{2}\right )}^{\frac{1}{3}} \log \left ({\left (b x^{2} + a\right )}^{\frac{1}{3}} a -{\left (a^{2}\right )}^{\frac{2}{3}}\right ) + \frac{3}{4} \,{\left (b x^{2} + a\right )}^{\frac{2}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 1.11012, size = 46, normalized size = 0.46 \begin{align*} - \frac{b^{\frac{2}{3}} x^{\frac{4}{3}} \Gamma \left (- \frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, - \frac{2}{3} \\ \frac{1}{3} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{2}}} \right )}}{2 \Gamma \left (\frac{1}{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b**(2/3)*x**(4/3)*gamma(-2/3)*hyper((-2/3, -2/3), (1/3,), a*exp_polar(I*pi)/(b*x**2))/(2*gamma(1/3))

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Giac [A]  time = 4.26245, size = 132, normalized size = 1.31 \begin{align*} \frac{1}{2} \, \sqrt{3} a^{\frac{2}{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 ) - \frac{1}{4} \, a^{\frac{2}{3}} \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 ) + \frac{1}{2} \, a^{\frac{2}{3}} \log \left ({\left |{\left (b x^{2} + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right ) + \frac{3}{4} \,{\left (b x^{2} + a\right )}^{\frac{2}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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