3.6.18 \(\int \frac {\sqrt [3]{a+b x^3}}{x^2} \, dx\) [518]

Optimal. Leaf size=88 \[ -\frac {\sqrt [3]{a+b x^3}}{x}-\frac {\sqrt [3]{b} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \sqrt [3]{b} \log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right ) \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 88, 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 = {283, 337} \begin {gather*} -\frac {\sqrt [3]{b} \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\sqrt [3]{a+b x^3}}{x}-\frac {1}{2} \sqrt [3]{b} \log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^(1/3)/x^2,x]

[Out]

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

Rule 283

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^p/(c*(m + 1
))), x] - Dist[b*n*(p/(c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 337

Int[(x_)/((a_) + (b_.)*(x_)^3)^(2/3), x_Symbol] :> With[{q = Rt[b, 3]}, Simp[-ArcTan[(1 + 2*q*(x/(a + b*x^3)^(
1/3)))/Sqrt[3]]/(Sqrt[3]*q^2), x] - Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*q^2), x]] /; FreeQ[{a, b}, x]

Rubi steps

\begin {align*} \int \frac {\sqrt [3]{a+b x^3}}{x^2} \, dx &=-\frac {\sqrt [3]{a+b x^3}}{x}+b \int \frac {x}{\left (a+b x^3\right )^{2/3}} \, dx\\ &=-\frac {\sqrt [3]{a+b x^3}}{x}+b \text {Subst}\left (\int \frac {x}{1-b x^3} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )\\ &=-\frac {\sqrt [3]{a+b x^3}}{x}+\frac {1}{3} b^{2/3} \text {Subst}\left (\int \frac {1}{1-\sqrt [3]{b} x} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )-\frac {1}{3} b^{2/3} \text {Subst}\left (\int \frac {1-\sqrt [3]{b} x}{1+\sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )\\ &=-\frac {\sqrt [3]{a+b x^3}}{x}-\frac {1}{3} \sqrt [3]{b} \log \left (1-\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )+\frac {1}{6} \sqrt [3]{b} \text {Subst}\left (\int \frac {\sqrt [3]{b}+2 b^{2/3} x}{1+\sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )-\frac {1}{2} b^{2/3} \text {Subst}\left (\int \frac {1}{1+\sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )\\ &=-\frac {\sqrt [3]{a+b x^3}}{x}-\frac {1}{3} \sqrt [3]{b} \log \left (1-\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )+\frac {1}{6} \sqrt [3]{b} \log \left (1+\frac {b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )+\sqrt [3]{b} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )\\ &=-\frac {\sqrt [3]{a+b x^3}}{x}-\frac {\sqrt [3]{b} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{3} \sqrt [3]{b} \log \left (1-\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )+\frac {1}{6} \sqrt [3]{b} \log \left (1+\frac {b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )\\ \end {align*}

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Mathematica [A]
time = 0.18, size = 144, normalized size = 1.64 \begin {gather*} -\frac {\sqrt [3]{a+b x^3}}{x}-\frac {\sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{a+b x^3}}\right )}{\sqrt {3}}-\frac {1}{3} \sqrt [3]{b} \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )+\frac {1}{6} \sqrt [3]{b} \log \left (b^{2/3} x^2+\sqrt [3]{b} x \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^(1/3)/x^2,x]

[Out]

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

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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.50, size = 114, normalized size = 1.30 \begin {gather*} \frac {1}{3} \, \sqrt {3} b^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (b^{\frac {1}{3}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, b^{\frac {1}{3}}}\right ) + \frac {1}{6} \, b^{\frac {1}{3}} \log \left (b^{\frac {2}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}}}{x} + \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right ) - \frac {1}{3} \, b^{\frac {1}{3}} \log \left (-b^{\frac {1}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) - \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 162 vs. \(2 (69) = 138\).
time = 73.34, size = 162, normalized size = 1.84 \begin {gather*} -\frac {2 \, \sqrt {3} \left (-b\right )^{\frac {1}{3}} x \arctan \left (\frac {2 \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b\right )^{\frac {2}{3}} x^{2} + 2 \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {2}{3}} \left (-b\right )^{\frac {1}{3}} x + \sqrt {3} a}{3 \, {\left (2 \, b x^{3} + a\right )}}\right ) + \left (-b\right )^{\frac {1}{3}} x \log \left (-\left (-b\right )^{\frac {1}{3}} b x^{2} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} b x + {\left (b x^{3} + a\right )}^{\frac {2}{3}} \left (-b\right )^{\frac {2}{3}}\right ) - 2 \, \left (-b\right )^{\frac {1}{3}} x \log \left (\left (-b\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right ) + 6 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{6 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 0.54, size = 41, normalized size = 0.47 \begin {gather*} \frac {\sqrt [3]{a} \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, - \frac {1}{3} \\ \frac {2}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x \Gamma \left (\frac {2}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^3 + a)^(1/3)/x^2, x)

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Mupad [B]
time = 1.21, size = 40, normalized size = 0.45 \begin {gather*} -\frac {{\left (b\,x^3+a\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{3},-\frac {1}{3};\ \frac {2}{3};\ -\frac {b\,x^3}{a}\right )}{x\,{\left (\frac {b\,x^3}{a}+1\right )}^{1/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^(1/3)/x^2,x)

[Out]

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

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