Integrand size = 13, antiderivative size = 91 \[ \int \frac {\sqrt [3]{a+b x}}{x} \, dx=3 \sqrt [3]{a+b x}-\sqrt {3} \sqrt [3]{a} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{a}}\right )-\frac {1}{2} \sqrt [3]{a} \log (x)+\frac {3}{2} \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right ) \]
3*(b*x+a)^(1/3)-1/2*a^(1/3)*ln(x)+3/2*a^(1/3)*ln(a^(1/3)-(b*x+a)^(1/3))-a^ (1/3)*arctan(1/3*(a^(1/3)+2*(b*x+a)^(1/3))/a^(1/3)*3^(1/2))*3^(1/2)
Time = 0.07 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.24 \[ \int \frac {\sqrt [3]{a+b x}}{x} \, dx=3 \sqrt [3]{a+b x}-\sqrt {3} \sqrt [3]{a} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+\sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )-\frac {1}{2} \sqrt [3]{a} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right ) \]
3*(a + b*x)^(1/3) - Sqrt[3]*a^(1/3)*ArcTan[(1 + (2*(a + b*x)^(1/3))/a^(1/3 ))/Sqrt[3]] + a^(1/3)*Log[a^(1/3) - (a + b*x)^(1/3)] - (a^(1/3)*Log[a^(2/3 ) + a^(1/3)*(a + b*x)^(1/3) + (a + b*x)^(2/3)])/2
Time = 0.20 (sec) , antiderivative size = 90, 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, 69, 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.
\(\displaystyle \int \frac {\sqrt [3]{a+b x}}{x} \, dx\) |
\(\Big \downarrow \) 60 |
\(\displaystyle a \int \frac {1}{x (a+b x)^{2/3}}dx+3 \sqrt [3]{a+b x}\) |
\(\Big \downarrow \) 69 |
\(\displaystyle a \left (-\frac {3 \int \frac {1}{\sqrt [3]{a}-\sqrt [3]{a+b x}}d\sqrt [3]{a+b x}}{2 a^{2/3}}-\frac {3 \int \frac {1}{a^{2/3}+\sqrt [3]{a+b x} \sqrt [3]{a}+(a+b x)^{2/3}}d\sqrt [3]{a+b x}}{2 \sqrt [3]{a}}-\frac {\log (x)}{2 a^{2/3}}\right )+3 \sqrt [3]{a+b x}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle a \left (-\frac {3 \int \frac {1}{a^{2/3}+\sqrt [3]{a+b x} \sqrt [3]{a}+(a+b x)^{2/3}}d\sqrt [3]{a+b x}}{2 \sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{2/3}}-\frac {\log (x)}{2 a^{2/3}}\right )+3 \sqrt [3]{a+b x}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle a \left (\frac {3 \int \frac {1}{-(a+b x)^{2/3}-3}d\left (\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}+1\right )}{a^{2/3}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{2/3}}-\frac {\log (x)}{2 a^{2/3}}\right )+3 \sqrt [3]{a+b x}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle a \left (-\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )}{a^{2/3}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{2/3}}-\frac {\log (x)}{2 a^{2/3}}\right )+3 \sqrt [3]{a+b x}\) |
3*(a + b*x)^(1/3) + a*(-((Sqrt[3]*ArcTan[(1 + (2*(a + b*x)^(1/3))/a^(1/3)) /Sqrt[3]])/a^(2/3)) - Log[x]/(2*a^(2/3)) + (3*Log[a^(1/3) - (a + b*x)^(1/3 )])/(2*a^(2/3)))
3.4.75.3.1 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_))^(2/3)), x_Symbol] :> With[ {q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (-Simp[3/(2*b*q) Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1 /3)], x] - Simp[3/(2*b*q^2) 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 = 0.45 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.96
method | result | size |
pseudoelliptic | \(3 \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {1}{3}} \ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )-\frac {a^{\frac {1}{3}} \ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{2}-a^{\frac {1}{3}} \arctan \left (\frac {\left (a^{\frac {1}{3}}+2 \left (b x +a \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}\) | \(87\) |
derivativedivides | \(3 \left (b x +a \right )^{\frac {1}{3}}+3 \left (\frac {\ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{3 a^{\frac {2}{3}}}-\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{6 a^{\frac {2}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{3 a^{\frac {2}{3}}}\right ) a\) | \(90\) |
default | \(3 \left (b x +a \right )^{\frac {1}{3}}+3 \left (\frac {\ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{3 a^{\frac {2}{3}}}-\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{6 a^{\frac {2}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{3 a^{\frac {2}{3}}}\right ) a\) | \(90\) |
3*(b*x+a)^(1/3)+a^(1/3)*ln((b*x+a)^(1/3)-a^(1/3))-1/2*a^(1/3)*ln((b*x+a)^( 2/3)+a^(1/3)*(b*x+a)^(1/3)+a^(2/3))-a^(1/3)*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 = 91, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{a+b x}}{x} \, dx=-\sqrt {3} a^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {2}{3}} + \sqrt {3} a}{3 \, a}\right ) - \frac {1}{2} \, a^{\frac {1}{3}} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + a^{\frac {1}{3}} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) + 3 \, {\left (b x + a\right )}^{\frac {1}{3}} \]
-sqrt(3)*a^(1/3)*arctan(1/3*(2*sqrt(3)*(b*x + a)^(1/3)*a^(2/3) + sqrt(3)*a )/a) - 1/2*a^(1/3)*log((b*x + a)^(2/3) + (b*x + a)^(1/3)*a^(1/3) + a^(2/3) ) + a^(1/3)*log((b*x + a)^(1/3) - a^(1/3)) + 3*(b*x + a)^(1/3)
Result contains complex when optimal does not.
Time = 1.94 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.98 \[ \int \frac {\sqrt [3]{a+b x}}{x} \, dx=\frac {4 \sqrt [3]{a} \log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{\frac {a}{b} + x}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac {4}{3}\right )}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {4 \sqrt [3]{a} e^{- \frac {2 i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{\frac {a}{b} + x} e^{\frac {2 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac {4}{3}\right )}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {4 \sqrt [3]{a} e^{\frac {2 i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{\frac {a}{b} + x} e^{\frac {4 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac {4}{3}\right )}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {4 \sqrt [3]{b} \sqrt [3]{\frac {a}{b} + x} \Gamma \left (\frac {4}{3}\right )}{\Gamma \left (\frac {7}{3}\right )} \]
4*a**(1/3)*log(1 - b**(1/3)*(a/b + x)**(1/3)/a**(1/3))*gamma(4/3)/(3*gamma (7/3)) + 4*a**(1/3)*exp(-2*I*pi/3)*log(1 - b**(1/3)*(a/b + x)**(1/3)*exp_p olar(2*I*pi/3)/a**(1/3))*gamma(4/3)/(3*gamma(7/3)) + 4*a**(1/3)*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( 4/3)/(3*gamma(7/3)) + 4*b**(1/3)*(a/b + x)**(1/3)*gamma(4/3)/gamma(7/3)
Time = 0.31 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt [3]{a+b x}}{x} \, dx=-\sqrt {3} a^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right ) - \frac {1}{2} \, a^{\frac {1}{3}} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + a^{\frac {1}{3}} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) + 3 \, {\left (b x + a\right )}^{\frac {1}{3}} \]
-sqrt(3)*a^(1/3)*arctan(1/3*sqrt(3)*(2*(b*x + a)^(1/3) + a^(1/3))/a^(1/3)) - 1/2*a^(1/3)*log((b*x + a)^(2/3) + (b*x + a)^(1/3)*a^(1/3) + a^(2/3)) + a^(1/3)*log((b*x + a)^(1/3) - a^(1/3)) + 3*(b*x + a)^(1/3)
Time = 0.52 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt [3]{a+b x}}{x} \, dx=-\sqrt {3} a^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right ) - \frac {1}{2} \, a^{\frac {1}{3}} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + a^{\frac {1}{3}} \log \left ({\left | {\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right ) + 3 \, {\left (b x + a\right )}^{\frac {1}{3}} \]
-sqrt(3)*a^(1/3)*arctan(1/3*sqrt(3)*(2*(b*x + a)^(1/3) + a^(1/3))/a^(1/3)) - 1/2*a^(1/3)*log((b*x + a)^(2/3) + (b*x + a)^(1/3)*a^(1/3) + a^(2/3)) + a^(1/3)*log(abs((b*x + a)^(1/3) - a^(1/3))) + 3*(b*x + a)^(1/3)
Time = 0.19 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.18 \[ \int \frac {\sqrt [3]{a+b x}}{x} \, dx=a^{1/3}\,\ln \left (9\,a\,{\left (a+b\,x\right )}^{1/3}-9\,a^{4/3}\right )+3\,{\left (a+b\,x\right )}^{1/3}+\frac {a^{1/3}\,\ln \left (9\,a\,{\left (a+b\,x\right )}^{1/3}-\frac {9\,a^{4/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}-\frac {a^{1/3}\,\ln \left (9\,a\,{\left (a+b\,x\right )}^{1/3}+\frac {9\,a^{4/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2} \]