3.1969 \(\int \frac {x}{a+\frac {b}{x^3}} \, dx\)

Optimal. Leaf size=124 \[ -\frac {b^{2/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 a^{5/3}}+\frac {b^{2/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{5/3}}+\frac {b^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} a^{5/3}}+\frac {x^2}{2 a} \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {263, 321, 292, 31, 634, 617, 204, 628} \[ -\frac {b^{2/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 a^{5/3}}+\frac {b^{2/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{5/3}}+\frac {b^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} a^{5/3}}+\frac {x^2}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b/x^3),x]

[Out]

x^2/(2*a) + (b^(2/3)*ArcTan[(b^(1/3) - 2*a^(1/3)*x)/(Sqrt[3]*b^(1/3))])/(Sqrt[3]*a^(5/3)) + (b^(2/3)*Log[b^(1/
3) + a^(1/3)*x])/(3*a^(5/3)) - (b^(2/3)*Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2])/(6*a^(5/3))

Rule 31

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

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 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 292

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> -Dist[(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3
]^2*x^2), x], x] /; FreeQ[{a, b}, x]

Rule 321

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

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 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {x}{a+\frac {b}{x^3}} \, dx &=\int \frac {x^4}{b+a x^3} \, dx\\ &=\frac {x^2}{2 a}-\frac {b \int \frac {x}{b+a x^3} \, dx}{a}\\ &=\frac {x^2}{2 a}+\frac {b^{2/3} \int \frac {1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{3 a^{4/3}}-\frac {b^{2/3} \int \frac {\sqrt [3]{b}+\sqrt [3]{a} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{3 a^{4/3}}\\ &=\frac {x^2}{2 a}+\frac {b^{2/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 a^{5/3}}-\frac {b^{2/3} \int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 a^{2/3} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{6 a^{5/3}}-\frac {b \int \frac {1}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{2 a^{4/3}}\\ &=\frac {x^2}{2 a}+\frac {b^{2/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 a^{5/3}}-\frac {b^{2/3} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 a^{5/3}}-\frac {b^{2/3} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{a^{5/3}}\\ &=\frac {x^2}{2 a}+\frac {b^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} a^{5/3}}+\frac {b^{2/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 a^{5/3}}-\frac {b^{2/3} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 a^{5/3}}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 111, normalized size = 0.90 \[ \frac {-b^{2/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )+3 a^{2/3} x^2+2 b^{2/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )+2 \sqrt {3} b^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{6 a^{5/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b/x^3),x]

[Out]

(3*a^(2/3)*x^2 + 2*Sqrt[3]*b^(2/3)*ArcTan[(1 - (2*a^(1/3)*x)/b^(1/3))/Sqrt[3]] + 2*b^(2/3)*Log[b^(1/3) + a^(1/
3)*x] - b^(2/3)*Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2])/(6*a^(5/3))

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fricas [A]  time = 0.87, size = 123, normalized size = 0.99 \[ \frac {3 \, x^{2} - 2 \, \sqrt {3} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} - \sqrt {3} b}{3 \, b}\right ) - \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b x^{2} - a x \left (\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} + b \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right ) + 2 \, \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b x + a \left (\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}}\right )}{6 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.17, size = 114, normalized size = 0.92 \[ \frac {x^{2}}{2 \, a} + \frac {\left (-\frac {b}{a}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {b}{a}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a} + \frac {\sqrt {3} \left (-a^{2} b\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{3 \, a^{3}} - \frac {\left (-a^{2} b\right )^{\frac {2}{3}} \log \left (x^{2} + x \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 102, normalized size = 0.82 \[ \frac {x^{2}}{2 a}-\frac {\sqrt {3}\, b \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {b}{a}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {b}{a}\right )^{\frac {1}{3}} a^{2}}+\frac {b \ln \left (x +\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {b}{a}\right )^{\frac {1}{3}} a^{2}}-\frac {b \ln \left (x^{2}-\left (\frac {b}{a}\right )^{\frac {1}{3}} x +\left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {b}{a}\right )^{\frac {1}{3}} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.92, size = 109, normalized size = 0.88 \[ \frac {x^{2}}{2 \, a} - \frac {\sqrt {3} b \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{3 \, a^{2} \left (\frac {b}{a}\right )^{\frac {1}{3}}} - \frac {b \log \left (x^{2} - x \left (\frac {b}{a}\right )^{\frac {1}{3}} + \left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 \, a^{2} \left (\frac {b}{a}\right )^{\frac {1}{3}}} + \frac {b \log \left (x + \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 \, a^{2} \left (\frac {b}{a}\right )^{\frac {1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.19, size = 120, normalized size = 0.97 \[ \frac {x^2}{2\,a}+\frac {b^{2/3}\,\ln \left (\frac {b^{7/3}}{a^{4/3}}+\frac {b^2\,x}{a}\right )}{3\,a^{5/3}}-\frac {b^{2/3}\,\ln \left (\frac {b^2\,x}{a}+\frac {b^{7/3}\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{a^{4/3}}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,a^{5/3}}+\frac {b^{2/3}\,\ln \left (\frac {b^2\,x}{a}+\frac {9\,b^{7/3}\,{\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}^2}{a^{4/3}}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}{a^{5/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + b/x^3),x)

[Out]

x^2/(2*a) + (b^(2/3)*log(b^(7/3)/a^(4/3) + (b^2*x)/a))/(3*a^(5/3)) - (b^(2/3)*log((b^2*x)/a + (b^(7/3)*((3^(1/
2)*1i)/2 + 1/2)^2)/a^(4/3))*((3^(1/2)*1i)/2 + 1/2))/(3*a^(5/3)) + (b^(2/3)*log((b^2*x)/a + (9*b^(7/3)*((3^(1/2
)*1i)/6 - 1/6)^2)/a^(4/3))*((3^(1/2)*1i)/6 - 1/6))/a^(5/3)

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sympy [A]  time = 0.19, size = 32, normalized size = 0.26 \[ \operatorname {RootSum} {\left (27 t^{3} a^{5} - b^{2}, \left (t \mapsto t \log {\left (\frac {9 t^{2} a^{3}}{b} + x \right )} \right )\right )} + \frac {x^{2}}{2 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(27*_t**3*a**5 - b**2, Lambda(_t, _t*log(9*_t**2*a**3/b + x))) + x**2/(2*a)

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