Integrand size = 13, antiderivative size = 92 \[ \int \frac {(a+b x)^{2/3}}{x} \, dx=\frac {3}{2} (a+b x)^{2/3}+\sqrt {3} a^{2/3} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{a}}\right )-\frac {1}{2} a^{2/3} \log (x)+\frac {3}{2} a^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right ) \]
3/2*(b*x+a)^(2/3)-1/2*a^(2/3)*ln(x)+3/2*a^(2/3)*ln(a^(1/3)-(b*x+a)^(1/3))+ a^(2/3)*arctan(1/3*(a^(1/3)+2*(b*x+a)^(1/3))/a^(1/3)*3^(1/2))*3^(1/2)
Time = 0.06 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.24 \[ \int \frac {(a+b x)^{2/3}}{x} \, dx=\frac {3}{2} (a+b x)^{2/3}+\sqrt {3} a^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+a^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )-\frac {1}{2} a^{2/3} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right ) \]
(3*(a + b*x)^(2/3))/2 + Sqrt[3]*a^(2/3)*ArcTan[(1 + (2*(a + b*x)^(1/3))/a^ (1/3))/Sqrt[3]] + a^(2/3)*Log[a^(1/3) - (a + b*x)^(1/3)] - (a^(2/3)*Log[a^ (2/3) + a^(1/3)*(a + b*x)^(1/3) + (a + b*x)^(2/3)])/2
Time = 0.20 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {60, 67, 16, 1082, 217}
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.
\(\displaystyle \int \frac {(a+b x)^{2/3}}{x} \, dx\) |
\(\Big \downarrow \) 60 |
\(\displaystyle a \int \frac {1}{x \sqrt [3]{a+b x}}dx+\frac {3}{2} (a+b x)^{2/3}\) |
\(\Big \downarrow \) 67 |
\(\displaystyle a \left (\frac {3}{2} \int \frac {1}{a^{2/3}+\sqrt [3]{a+b x} \sqrt [3]{a}+(a+b x)^{2/3}}d\sqrt [3]{a+b x}-\frac {3 \int \frac {1}{\sqrt [3]{a}-\sqrt [3]{a+b x}}d\sqrt [3]{a+b x}}{2 \sqrt [3]{a}}-\frac {\log (x)}{2 \sqrt [3]{a}}\right )+\frac {3}{2} (a+b x)^{2/3}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle a \left (\frac {3}{2} \int \frac {1}{a^{2/3}+\sqrt [3]{a+b x} \sqrt [3]{a}+(a+b x)^{2/3}}d\sqrt [3]{a+b x}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 \sqrt [3]{a}}-\frac {\log (x)}{2 \sqrt [3]{a}}\right )+\frac {3}{2} (a+b x)^{2/3}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle a \left (-\frac {3 \int \frac {1}{-(a+b x)^{2/3}-3}d\left (\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}+1\right )}{\sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 \sqrt [3]{a}}-\frac {\log (x)}{2 \sqrt [3]{a}}\right )+\frac {3}{2} (a+b x)^{2/3}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle a \left (\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )}{\sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 \sqrt [3]{a}}-\frac {\log (x)}{2 \sqrt [3]{a}}\right )+\frac {3}{2} (a+b x)^{2/3}\) |
(3*(a + b*x)^(2/3))/2 + a*((Sqrt[3]*ArcTan[(1 + (2*(a + b*x)^(1/3))/a^(1/3 ))/Sqrt[3]])/a^(1/3) - Log[x]/(2*a^(1/3)) + (3*Log[a^(1/3) - (a + b*x)^(1/ 3)])/(2*a^(1/3)))
3.4.82.3.1 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_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[ {q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q), x ] + (Simp[3/(2*b) Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/3)], x] - Simp[3/(2*b*q) Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] / ; FreeQ[{a, b, c, d}, x] && PosQ[(b*c - a*d)/b]
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_) + (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]
Time = 0.08 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.93
method | result | size |
pseudoelliptic | \(\frac {3 \left (b x +a \right )^{\frac {2}{3}}}{2}+a^{\frac {2}{3}} \ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )-\frac {a^{\frac {2}{3}} \ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{2}+a^{\frac {2}{3}} \arctan \left (\frac {\left (a^{\frac {1}{3}}+2 \left (b x +a \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}\) | \(86\) |
derivativedivides | \(\frac {3 \left (b x +a \right )^{\frac {2}{3}}}{2}+3 \left (\frac {\ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{3 a^{\frac {1}{3}}}-\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{6 a^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{3 a^{\frac {1}{3}}}\right ) a\) | \(90\) |
default | \(\frac {3 \left (b x +a \right )^{\frac {2}{3}}}{2}+3 \left (\frac {\ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{3 a^{\frac {1}{3}}}-\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{6 a^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{3 a^{\frac {1}{3}}}\right ) a\) | \(90\) |
3/2*(b*x+a)^(2/3)+a^(2/3)*ln((b*x+a)^(1/3)-a^(1/3))-1/2*a^(2/3)*ln((b*x+a) ^(2/3)+a^(1/3)*(b*x+a)^(1/3)+a^(2/3))+a^(2/3)*arctan(1/3*(a^(1/3)+2*(b*x+a )^(1/3))/a^(1/3)*3^(1/2))*3^(1/2)
Time = 0.23 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.20 \[ \int \frac {(a+b x)^{2/3}}{x} \, dx=\sqrt {3} {\left (a^{2}\right )}^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} a + 2 \, \sqrt {3} {\left (a^{2}\right )}^{\frac {1}{3}} {\left (b x + a\right )}^{\frac {1}{3}}}{3 \, a}\right ) - \frac {1}{2} \, {\left (a^{2}\right )}^{\frac {1}{3}} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} a + {\left (a^{2}\right )}^{\frac {1}{3}} a + {\left (a^{2}\right )}^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {1}{3}}\right ) + {\left (a^{2}\right )}^{\frac {1}{3}} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} a - {\left (a^{2}\right )}^{\frac {2}{3}}\right ) + \frac {3}{2} \, {\left (b x + a\right )}^{\frac {2}{3}} \]
sqrt(3)*(a^2)^(1/3)*arctan(1/3*(sqrt(3)*a + 2*sqrt(3)*(a^2)^(1/3)*(b*x + a )^(1/3))/a) - 1/2*(a^2)^(1/3)*log((b*x + a)^(2/3)*a + (a^2)^(1/3)*a + (a^2 )^(2/3)*(b*x + a)^(1/3)) + (a^2)^(1/3)*log((b*x + a)^(1/3)*a - (a^2)^(2/3) ) + 3/2*(b*x + a)^(2/3)
Result contains complex when optimal does not.
Time = 1.74 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.98 \[ \int \frac {(a+b x)^{2/3}}{x} \, dx=\frac {5 a^{\frac {2}{3}} \log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{\frac {a}{b} + x}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac {5}{3}\right )}{3 \Gamma \left (\frac {8}{3}\right )} + \frac {5 a^{\frac {2}{3}} e^{\frac {2 i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{\frac {a}{b} + x} e^{\frac {2 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac {5}{3}\right )}{3 \Gamma \left (\frac {8}{3}\right )} + \frac {5 a^{\frac {2}{3}} e^{- \frac {2 i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{\frac {a}{b} + x} e^{\frac {4 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac {5}{3}\right )}{3 \Gamma \left (\frac {8}{3}\right )} + \frac {5 b^{\frac {2}{3}} \left (\frac {a}{b} + x\right )^{\frac {2}{3}} \Gamma \left (\frac {5}{3}\right )}{2 \Gamma \left (\frac {8}{3}\right )} \]
5*a**(2/3)*log(1 - b**(1/3)*(a/b + x)**(1/3)/a**(1/3))*gamma(5/3)/(3*gamma (8/3)) + 5*a**(2/3)*exp(2*I*pi/3)*log(1 - b**(1/3)*(a/b + x)**(1/3)*exp_po lar(2*I*pi/3)/a**(1/3))*gamma(5/3)/(3*gamma(8/3)) + 5*a**(2/3)*exp(-2*I*pi /3)*log(1 - b**(1/3)*(a/b + x)**(1/3)*exp_polar(4*I*pi/3)/a**(1/3))*gamma( 5/3)/(3*gamma(8/3)) + 5*b**(2/3)*(a/b + x)**(2/3)*gamma(5/3)/(2*gamma(8/3) )
Time = 0.30 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b x)^{2/3}}{x} \, dx=\sqrt {3} a^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right ) - \frac {1}{2} \, a^{\frac {2}{3}} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + a^{\frac {2}{3}} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) + \frac {3}{2} \, {\left (b x + a\right )}^{\frac {2}{3}} \]
sqrt(3)*a^(2/3)*arctan(1/3*sqrt(3)*(2*(b*x + a)^(1/3) + a^(1/3))/a^(1/3)) - 1/2*a^(2/3)*log((b*x + a)^(2/3) + (b*x + a)^(1/3)*a^(1/3) + a^(2/3)) + a ^(2/3)*log((b*x + a)^(1/3) - a^(1/3)) + 3/2*(b*x + a)^(2/3)
Time = 0.53 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b x)^{2/3}}{x} \, dx=\sqrt {3} a^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right ) - \frac {1}{2} \, a^{\frac {2}{3}} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + a^{\frac {2}{3}} \log \left ({\left | {\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right ) + \frac {3}{2} \, {\left (b x + a\right )}^{\frac {2}{3}} \]
sqrt(3)*a^(2/3)*arctan(1/3*sqrt(3)*(2*(b*x + a)^(1/3) + a^(1/3))/a^(1/3)) - 1/2*a^(2/3)*log((b*x + a)^(2/3) + (b*x + a)^(1/3)*a^(1/3) + a^(2/3)) + a ^(2/3)*log(abs((b*x + a)^(1/3) - a^(1/3))) + 3/2*(b*x + a)^(2/3)
Time = 0.12 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.27 \[ \int \frac {(a+b x)^{2/3}}{x} \, dx=\frac {3\,{\left (a+b\,x\right )}^{2/3}}{2}+a^{2/3}\,\ln \left (9\,a^2\,{\left (a+b\,x\right )}^{1/3}-9\,a^{7/3}\right )+\frac {a^{2/3}\,\ln \left (9\,a^2\,{\left (a+b\,x\right )}^{1/3}-\frac {9\,a^{7/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{4}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}-\frac {a^{2/3}\,\ln \left (9\,a^2\,{\left (a+b\,x\right )}^{1/3}-\frac {9\,a^{7/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{4}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2} \]
(3*(a + b*x)^(2/3))/2 + a^(2/3)*log(9*a^2*(a + b*x)^(1/3) - 9*a^(7/3)) + ( a^(2/3)*log(9*a^2*(a + b*x)^(1/3) - (9*a^(7/3)*(3^(1/2)*1i - 1)^2)/4)*(3^( 1/2)*1i - 1))/2 - (a^(2/3)*log(9*a^2*(a + b*x)^(1/3) - (9*a^(7/3)*(3^(1/2) *1i + 1)^2)/4)*(3^(1/2)*1i + 1))/2
Time = 0.00 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.73 \[ \int \frac {(a+b x)^{2/3}}{x} \, dx=\frac {-2 \sqrt {3}\, \mathit {atan} \left (\frac {2 \left (b x +a \right )^{\frac {1}{6}}+a^{\frac {1}{6}}}{a^{\frac {1}{6}} \sqrt {3}}\right ) a +2 \sqrt {3}\, \mathit {atan} \left (\frac {2 \left (b x +a \right )^{\frac {1}{6}}-a^{\frac {1}{6}}}{a^{\frac {1}{6}} \sqrt {3}}\right ) a +3 a^{\frac {1}{3}} \left (b x +a \right )^{\frac {2}{3}}+2 \,\mathrm {log}\left (\left (b x +a \right )^{\frac {1}{6}}+a^{\frac {1}{6}}\right ) a +2 \,\mathrm {log}\left (\left (b x +a \right )^{\frac {1}{6}}-a^{\frac {1}{6}}\right ) a -\mathrm {log}\left (-a^{\frac {1}{6}} \left (b x +a \right )^{\frac {1}{6}}+\left (b x +a \right )^{\frac {1}{3}}+a^{\frac {1}{3}}\right ) a -\mathrm {log}\left (a^{\frac {1}{6}} \left (b x +a \right )^{\frac {1}{6}}+\left (b x +a \right )^{\frac {1}{3}}+a^{\frac {1}{3}}\right ) a}{2 a^{\frac {1}{3}}} \]
( - 2*sqrt(3)*atan((2*(a + b*x)**(1/6) + a**(1/6))/(a**(1/6)*sqrt(3)))*a + 2*sqrt(3)*atan((2*(a + b*x)**(1/6) - a**(1/6))/(a**(1/6)*sqrt(3)))*a + 3* a**(1/3)*(a + b*x)**(2/3) + 2*log((a + b*x)**(1/6) + a**(1/6))*a + 2*log(( a + b*x)**(1/6) - a**(1/6))*a - log( - a**(1/6)*(a + b*x)**(1/6) + (a + b* x)**(1/3) + a**(1/3))*a - log(a**(1/6)*(a + b*x)**(1/6) + (a + b*x)**(1/3) + a**(1/3))*a)/(2*a**(1/3))