Integrand size = 14, antiderivative size = 120 \[ \int \frac {x^3}{a-b x^3} \, dx=-\frac {x}{b}+\frac {\sqrt [3]{a} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{4/3}}-\frac {\sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 b^{4/3}}+\frac {\sqrt [3]{a} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{4/3}} \]
-x/b-1/3*a^(1/3)*ln(a^(1/3)-b^(1/3)*x)/b^(4/3)+1/6*a^(1/3)*ln(a^(2/3)+a^(1 /3)*b^(1/3)*x+b^(2/3)*x^2)/b^(4/3)+1/3*a^(1/3)*arctan(1/3*(a^(1/3)+2*b^(1/ 3)*x)/a^(1/3)*3^(1/2))/b^(4/3)*3^(1/2)
Time = 0.01 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.90 \[ \int \frac {x^3}{a-b x^3} \, dx=\frac {-6 \sqrt [3]{b} x+2 \sqrt {3} \sqrt [3]{a} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )-2 \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )+\sqrt [3]{a} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{4/3}} \]
(-6*b^(1/3)*x + 2*Sqrt[3]*a^(1/3)*ArcTan[(1 + (2*b^(1/3)*x)/a^(1/3))/Sqrt[ 3]] - 2*a^(1/3)*Log[a^(1/3) - b^(1/3)*x] + a^(1/3)*Log[a^(2/3) + a^(1/3)*b ^(1/3)*x + b^(2/3)*x^2])/(6*b^(4/3))
Time = 0.28 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.02, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {843, 750, 16, 1142, 27, 1082, 217, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x^3}{a-b x^3} \, dx\) |
\(\Big \downarrow \) 843 |
\(\displaystyle \frac {a \int \frac {1}{a-b x^3}dx}{b}-\frac {x}{b}\) |
\(\Big \downarrow \) 750 |
\(\displaystyle \frac {a \left (\frac {\int \frac {\sqrt [3]{b} x+2 \sqrt [3]{a}}{b^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{a}-\sqrt [3]{b} x}dx}{3 a^{2/3}}\right )}{b}-\frac {x}{b}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {a \left (\frac {\int \frac {\sqrt [3]{b} x+2 \sqrt [3]{a}}{b^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}-\frac {\log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}-\frac {x}{b}\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle \frac {a \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {\int \frac {\sqrt [3]{b} \left (2 \sqrt [3]{b} x+\sqrt [3]{a}\right )}{b^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 a^{2/3}}-\frac {\log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}-\frac {x}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {a \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {1}{2} \int \frac {2 \sqrt [3]{b} x+\sqrt [3]{a}}{b^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}-\frac {\log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}-\frac {x}{b}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {a \left (\frac {\frac {1}{2} \int \frac {2 \sqrt [3]{b} x+\sqrt [3]{a}}{b^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {3 \int \frac {1}{-\left (\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}+1\right )^2-3}d\left (\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}+1\right )}{\sqrt [3]{b}}}{3 a^{2/3}}-\frac {\log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}-\frac {x}{b}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {a \left (\frac {\frac {1}{2} \int \frac {2 \sqrt [3]{b} x+\sqrt [3]{a}}{b^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}-\frac {\log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}-\frac {x}{b}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {a \left (\frac {\frac {\log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}+\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}-\frac {\log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}-\frac {x}{b}\) |
-(x/b) + (a*(-1/3*Log[a^(1/3) - b^(1/3)*x]/(a^(2/3)*b^(1/3)) + ((Sqrt[3]*A rcTan[(1 + (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3) + Log[a^(2/3) + a^(1/3 )*b^(1/3)*x + b^(2/3)*x^2]/(2*b^(1/3)))/(3*a^(2/3))))/b
3.4.58.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2) Int[1/ (Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[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]
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] - Simp[ 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]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[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])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[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]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.64 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.31
method | result | size |
risch | \(-\frac {x}{b}-\frac {a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}\right )}{3 b^{2}}\) | \(37\) |
default | \(-\frac {x}{b}-\frac {\left (\frac {\ln \left (x -\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right ) a}{b}\) | \(105\) |
Time = 0.27 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.86 \[ \int \frac {x^3}{a-b x^3} \, dx=-\frac {2 \, \sqrt {3} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x \left (-\frac {a}{b}\right )^{\frac {2}{3}} + \sqrt {3} a}{3 \, a}\right ) + \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x^{2} - x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right ) - 2 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right ) + 6 \, x}{6 \, b} \]
-1/6*(2*sqrt(3)*(-a/b)^(1/3)*arctan(1/3*(2*sqrt(3)*b*x*(-a/b)^(2/3) + sqrt (3)*a)/a) + (-a/b)^(1/3)*log(x^2 - x*(-a/b)^(1/3) + (-a/b)^(2/3)) - 2*(-a/ b)^(1/3)*log(x + (-a/b)^(1/3)) + 6*x)/b
Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.20 \[ \int \frac {x^3}{a-b x^3} \, dx=- \operatorname {RootSum} {\left (27 t^{3} b^{4} - a, \left ( t \mapsto t \log {\left (- 3 t b + x \right )} \right )\right )} - \frac {x}{b} \]
Time = 0.28 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.88 \[ \int \frac {x^3}{a-b x^3} \, dx=-\frac {x}{b} + \frac {\sqrt {3} a \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {a \log \left (x^{2} + x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {a \log \left (x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
-x/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)^(2/3)) + 1/6*a*log(x^2 + x*(a/b)^(1/3) + (a/b)^(2/3))/(b^2*(a/b) ^(2/3)) - 1/3*a*log(x - (a/b)^(1/3))/(b^2*(a/b)^(2/3))
Time = 0.27 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.87 \[ \int \frac {x^3}{a-b x^3} \, dx=-\frac {\left (\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, b} - \frac {x}{b} + \frac {\sqrt {3} \left (a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b^{2}} + \frac {\left (a b^{2}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b^{2}} \]
-1/3*(a/b)^(1/3)*log(abs(x - (a/b)^(1/3)))/b - x/b + 1/3*sqrt(3)*(a*b^2)^( 1/3)*arctan(1/3*sqrt(3)*(2*x + (a/b)^(1/3))/(a/b)^(1/3))/b^2 + 1/6*(a*b^2) ^(1/3)*log(x^2 + x*(a/b)^(1/3) + (a/b)^(2/3))/b^2
Time = 5.80 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.97 \[ \int \frac {x^3}{a-b x^3} \, dx=\frac {{\left (-a\right )}^{1/3}\,\ln \left ({\left (-a\right )}^{4/3}-a\,b^{1/3}\,x\right )}{3\,b^{4/3}}-\frac {x}{b}-\frac {{\left (-a\right )}^{1/3}\,\ln \left (3\,{\left (-a\right )}^{4/3}\,b^{2/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )+3\,a\,b\,x\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,b^{4/3}}+\frac {{\left (-a\right )}^{1/3}\,\ln \left (9\,{\left (-a\right )}^{4/3}\,b^{2/3}\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )-3\,a\,b\,x\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}{b^{4/3}} \]