\(\int \frac {x^3}{a+\frac {b}{x^3}} \, dx\) [1967]

Optimal result

Integrand size = 13, antiderivative size = 132 \[ \int \frac {x^3}{a+\frac {b}{x^3}} \, dx=-\frac {b x}{a^2}+\frac {x^4}{4 a}-\frac {b^{4/3} \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} a^{7/3}}+\frac {b^{4/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 a^{7/3}}-\frac {b^{4/3} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 a^{7/3}} \]

[Out]

-b*x/a^2+1/4*x^4/a+1/3*b^(4/3)*ln(b^(1/3)+a^(1/3)*x)/a^(7/3)-1/6*b^(4/3)*ln(b^(2/3)-a^(1/3)*b^(1/3)*x+a^(2/3)*
x^2)/a^(7/3)-1/3*b^(4/3)*arctan(1/3*(b^(1/3)-2*a^(1/3)*x)/b^(1/3)*3^(1/2))/a^(7/3)*3^(1/2)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {269, 308, 206, 31, 648, 631, 210, 642} \[ \int \frac {x^3}{a+\frac {b}{x^3}} \, dx=-\frac {b^{4/3} \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} a^{7/3}}-\frac {b^{4/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 a^{7/3}}+\frac {b^{4/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{7/3}}-\frac {b x}{a^2}+\frac {x^4}{4 a} \]

[In]

Int[x^3/(a + b/x^3),x]

[Out]

-((b*x)/a^2) + x^4/(4*a) - (b^(4/3)*ArcTan[(b^(1/3) - 2*a^(1/3)*x)/(Sqrt[3]*b^(1/3))])/(Sqrt[3]*a^(7/3)) + (b^
(4/3)*Log[b^(1/3) + a^(1/3)*x])/(3*a^(7/3)) - (b^(4/3)*Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2])/(6*a^(7
/3))

Rule 31

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

Rule 206

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*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 210

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

Rule 269

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 308

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 631

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 642

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 648

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*} \text {integral}& = \int \frac {x^6}{b+a x^3} \, dx \\ & = \int \left (-\frac {b}{a^2}+\frac {x^3}{a}+\frac {b^2}{a^2 \left (b+a x^3\right )}\right ) \, dx \\ & = -\frac {b x}{a^2}+\frac {x^4}{4 a}+\frac {b^2 \int \frac {1}{b+a x^3} \, dx}{a^2} \\ & = -\frac {b x}{a^2}+\frac {x^4}{4 a}+\frac {b^{4/3} \int \frac {1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{3 a^2}+\frac {b^{4/3} \int \frac {2 \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^2} \\ & = -\frac {b x}{a^2}+\frac {x^4}{4 a}+\frac {b^{4/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 a^{7/3}}-\frac {b^{4/3} \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^{7/3}}+\frac {b^{5/3} \int \frac {1}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{2 a^2} \\ & = -\frac {b x}{a^2}+\frac {x^4}{4 a}+\frac {b^{4/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 a^{7/3}}-\frac {b^{4/3} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 a^{7/3}}+\frac {b^{4/3} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{a^{7/3}} \\ & = -\frac {b x}{a^2}+\frac {x^4}{4 a}-\frac {b^{4/3} \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} a^{7/3}}+\frac {b^{4/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 a^{7/3}}-\frac {b^{4/3} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 a^{7/3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.91 \[ \int \frac {x^3}{a+\frac {b}{x^3}} \, dx=\frac {-12 \sqrt [3]{a} b x+3 a^{4/3} x^4-4 \sqrt {3} b^{4/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt {3}}\right )+4 b^{4/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )-2 b^{4/3} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{12 a^{7/3}} \]

[In]

Integrate[x^3/(a + b/x^3),x]

[Out]

(-12*a^(1/3)*b*x + 3*a^(4/3)*x^4 - 4*Sqrt[3]*b^(4/3)*ArcTan[(1 - (2*a^(1/3)*x)/b^(1/3))/Sqrt[3]] + 4*b^(4/3)*L
og[b^(1/3) + a^(1/3)*x] - 2*b^(4/3)*Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2])/(12*a^(7/3))

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.35

method	result	size
risch	\(\frac {x^{4}}{4 a}-\frac {b x}{a^{2}}+\frac {b^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{3}+b \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}\right )}{3 a^{3}}\)	\(46\)
default	\(\frac {\frac {1}{4} a \,x^{4}-b x}{a^{2}}+\frac {\left (\frac {\ln \left (x +\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {b}{a}\right )^{\frac {1}{3}} x +\left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 a \left (\frac {b}{a}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {b}{a}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {2}{3}}}\right ) b^{2}}{a^{2}}\)	\(114\)

[In]

int(x^3/(a+b/x^3),x,method=_RETURNVERBOSE)

[Out]

1/4*x^4/a-b/a^2*x+1/3/a^3*b^2*sum(1/_R^2*ln(x-_R),_R=RootOf(_Z^3*a+b))

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.82 \[ \int \frac {x^3}{a+\frac {b}{x^3}} \, dx=\frac {3 \, a x^{4} + 4 \, \sqrt {3} b \left (\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x \left (\frac {b}{a}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) - 2 \, b \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (x^{2} - x \left (\frac {b}{a}\right )^{\frac {1}{3}} + \left (\frac {b}{a}\right )^{\frac {2}{3}}\right ) + 4 \, b \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (x + \left (\frac {b}{a}\right )^{\frac {1}{3}}\right ) - 12 \, b x}{12 \, a^{2}} \]

[In]

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

[Out]

1/12*(3*a*x^4 + 4*sqrt(3)*b*(b/a)^(1/3)*arctan(1/3*(2*sqrt(3)*a*x*(b/a)^(2/3) - sqrt(3)*b)/b) - 2*b*(b/a)^(1/3
)*log(x^2 - x*(b/a)^(1/3) + (b/a)^(2/3)) + 4*b*(b/a)^(1/3)*log(x + (b/a)^(1/3)) - 12*b*x)/a^2

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.28 \[ \int \frac {x^3}{a+\frac {b}{x^3}} \, dx=\operatorname {RootSum} {\left (27 t^{3} a^{7} - b^{4}, \left ( t \mapsto t \log {\left (\frac {3 t a^{2}}{b} + x \right )} \right )\right )} + \frac {x^{4}}{4 a} - \frac {b x}{a^{2}} \]

[In]

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

[Out]

RootSum(27*_t**3*a**7 - b**4, Lambda(_t, _t*log(3*_t*a**2/b + x))) + x**4/(4*a) - b*x/a**2

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.92 \[ \int \frac {x^3}{a+\frac {b}{x^3}} \, dx=\frac {\sqrt {3} b^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{3 \, a^{3} \left (\frac {b}{a}\right )^{\frac {2}{3}}} - \frac {b^{2} \log \left (x^{2} - x \left (\frac {b}{a}\right )^{\frac {1}{3}} + \left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 \, a^{3} \left (\frac {b}{a}\right )^{\frac {2}{3}}} + \frac {b^{2} \log \left (x + \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 \, a^{3} \left (\frac {b}{a}\right )^{\frac {2}{3}}} + \frac {a x^{4} - 4 \, b x}{4 \, a^{2}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.98 \[ \int \frac {x^3}{a+\frac {b}{x^3}} \, dx=-\frac {b \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {b}{a}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a^{2}} + \frac {\sqrt {3} \left (-a^{2} b\right )^{\frac {1}{3}} b \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{3 \, a^{3}} + \frac {\left (-a^{2} b\right )^{\frac {1}{3}} b \log \left (x^{2} + x \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 \, a^{3}} + \frac {a^{3} x^{4} - 4 \, a^{2} b x}{4 \, a^{4}} \]

[In]

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

[Out]

-1/3*b*(-b/a)^(1/3)*log(abs(x - (-b/a)^(1/3)))/a^2 + 1/3*sqrt(3)*(-a^2*b)^(1/3)*b*arctan(1/3*sqrt(3)*(2*x + (-
b/a)^(1/3))/(-b/a)^(1/3))/a^3 + 1/6*(-a^2*b)^(1/3)*b*log(x^2 + x*(-b/a)^(1/3) + (-b/a)^(2/3))/a^3 + 1/4*(a^3*x
^4 - 4*a^2*b*x)/a^4

Mupad [B] (verification not implemented)

Time = 5.87 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.90 \[ \int \frac {x^3}{a+\frac {b}{x^3}} \, dx=\frac {x^4}{4\,a}+\frac {b^{4/3}\,\ln \left (3\,b^2\,x+\frac {3\,b^{7/3}}{a^{1/3}}\right )}{3\,a^{7/3}}-\frac {b\,x}{a^2}-\frac {b^{4/3}\,\ln \left (3\,b^2\,x-\frac {3\,b^{7/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{a^{1/3}}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,a^{7/3}}+\frac {b^{4/3}\,\ln \left (3\,b^2\,x+\frac {9\,b^{7/3}\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}{a^{1/3}}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}{a^{7/3}} \]

[In]

int(x^3/(a + b/x^3),x)

[Out]

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