\(\int \frac {x^4}{a-b x^3} \, dx\) [357]

Optimal result

Integrand size = 14, antiderivative size = 125 \[ \int \frac {x^4}{a-b x^3} \, dx=-\frac {x^2}{2 b}-\frac {a^{2/3} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{5/3}}-\frac {a^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 b^{5/3}}+\frac {a^{2/3} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{5/3}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {327, 298, 31, 648, 631, 210, 642} \[ \int \frac {x^4}{a-b x^3} \, dx=-\frac {a^{2/3} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{5/3}}+\frac {a^{2/3} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{5/3}}-\frac {a^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 b^{5/3}}-\frac {x^2}{2 b} \]

[In]

Int[x^4/(a - b*x^3),x]

[Out]

-1/2*x^2/b - (a^(2/3)*ArcTan[(a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*b^(5/3)) - (a^(2/3)*Log[a^(1
/3) - b^(1/3)*x])/(3*b^(5/3)) + (a^(2/3)*Log[a^(2/3) + a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*b^(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 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 298

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 327

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.89 \[ \int \frac {x^4}{a-b x^3} \, dx=-\frac {3 b^{2/3} x^2+2 \sqrt {3} a^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 a^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )-a^{2/3} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{5/3}} \]

[In]

Integrate[x^4/(a - b*x^3),x]

[Out]

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

Maple [C] (verified)

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

Time = 3.77 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.31

method	result	size
risch	\(-\frac {x^{2}}{2 b}-\frac {a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}}\right )}{3 b^{2}}\)	\(39\)
default	\(-\frac {x^{2}}{2 b}-\frac {\left (\frac {\ln \left (x -\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}-\frac {\ln \left (x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) a}{b}\)	\(107\)

[In]

int(x^4/(-b*x^3+a),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.04 \[ \int \frac {x^4}{a-b x^3} \, dx=-\frac {3 \, x^{2} + 2 \, \sqrt {3} \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} - \sqrt {3} a}{3 \, a}\right ) + \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \log \left (a x^{2} + b x \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}} - a \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}}\right ) - 2 \, \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \log \left (a x - b \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}}\right )}{6 \, b} \]

[In]

integrate(x^4/(-b*x^3+a),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.27 \[ \int \frac {x^4}{a-b x^3} \, dx=- \operatorname {RootSum} {\left (27 t^{3} b^{5} - a^{2}, \left ( t \mapsto t \log {\left (- \frac {9 t^{2} b^{3}}{a} + x \right )} \right )\right )} - \frac {x^{2}}{2 b} \]

[In]

integrate(x**4/(-b*x**3+a),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.86 \[ \int \frac {x^4}{a-b x^3} \, dx=-\frac {x^{2}}{2 \, b} - \frac {\sqrt {3} a \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {a \log \left (x^{2} + x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {a \log \left (x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]

[In]

integrate(x^4/(-b*x^3+a),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.85 \[ \int \frac {x^4}{a-b x^3} \, dx=-\frac {x^{2}}{2 \, b} - \frac {\left (\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, b} - \frac {\sqrt {3} \left (a b^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b^{3}} + \frac {\left (a b^{2}\right )^{\frac {2}{3}} \log \left (x^{2} + x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b^{3}} \]

[In]

integrate(x^4/(-b*x^3+a),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.61 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.07 \[ \int \frac {x^4}{a-b x^3} \, dx=\frac {a^{2/3}\,\ln \left (\frac {a^2\,x}{b}-\frac {a^{7/3}}{{\left (-b\right )}^{4/3}}\right )}{3\,{\left (-b\right )}^{5/3}}-\frac {x^2}{2\,b}-\frac {a^{2/3}\,\ln \left (\frac {a^2\,x}{b}-\frac {a^{7/3}\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{{\left (-b\right )}^{4/3}}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,{\left (-b\right )}^{5/3}}+\frac {a^{2/3}\,\ln \left (\frac {a^2\,x}{b}-\frac {9\,a^{7/3}\,{\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}^2}{{\left (-b\right )}^{4/3}}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}{{\left (-b\right )}^{5/3}} \]

[In]

int(x^4/(a - b*x^3),x)

[Out]

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