Integrand size = 13, antiderivative size = 93 \[ \int \frac {1}{x (a+b x)^{4/3}} \, dx=\frac {3}{a \sqrt [3]{a+b x}}+\frac {\sqrt {3} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{a}}\right )}{a^{4/3}}-\frac {\log (x)}{2 a^{4/3}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{4/3}} \]
3/a/(b*x+a)^(1/3)-1/2*ln(x)/a^(4/3)+3/2*ln(a^(1/3)-(b*x+a)^(1/3))/a^(4/3)+ arctan(1/3*(a^(1/3)+2*(b*x+a)^(1/3))/a^(1/3)*3^(1/2))*3^(1/2)/a^(4/3)
Time = 0.09 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.19 \[ \int \frac {1}{x (a+b x)^{4/3}} \, dx=\frac {\frac {6 \sqrt [3]{a}}{\sqrt [3]{a+b x}}+2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )-\log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right )}{2 a^{4/3}} \]
((6*a^(1/3))/(a + b*x)^(1/3) + 2*Sqrt[3]*ArcTan[(1 + (2*(a + b*x)^(1/3))/a ^(1/3))/Sqrt[3]] + 2*Log[a^(1/3) - (a + b*x)^(1/3)] - Log[a^(2/3) + a^(1/3 )*(a + b*x)^(1/3) + (a + b*x)^(2/3)])/(2*a^(4/3))
Time = 0.20 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {61, 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.
| \(\displaystyle \int \frac {1}{x (a+b x)^{4/3}} \, dx\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {\int \frac {1}{x \sqrt [3]{a+b x}}dx}{a}+\frac {3}{a \sqrt [3]{a+b x}}\) |
| \(\Big \downarrow \) 67 |
| \(\displaystyle \frac {\frac {3}{2} \int \frac {1}{a^{2/3}+\sqrt [3]{a+b x} \sqrt [3]{a}+(a+b x)^{2/3}}d\sqrt [3]{a+b x}-\frac {3 \int \frac {1}{\sqrt [3]{a}-\sqrt [3]{a+b x}}d\sqrt [3]{a+b x}}{2 \sqrt [3]{a}}-\frac {\log (x)}{2 \sqrt [3]{a}}}{a}+\frac {3}{a \sqrt [3]{a+b x}}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {3}{2} \int \frac {1}{a^{2/3}+\sqrt [3]{a+b x} \sqrt [3]{a}+(a+b x)^{2/3}}d\sqrt [3]{a+b x}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 \sqrt [3]{a}}-\frac {\log (x)}{2 \sqrt [3]{a}}}{a}+\frac {3}{a \sqrt [3]{a+b x}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {-\frac {3 \int \frac {1}{-(a+b x)^{2/3}-3}d\left (\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}+1\right )}{\sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 \sqrt [3]{a}}-\frac {\log (x)}{2 \sqrt [3]{a}}}{a}+\frac {3}{a \sqrt [3]{a+b x}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )}{\sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 \sqrt [3]{a}}-\frac {\log (x)}{2 \sqrt [3]{a}}}{a}+\frac {3}{a \sqrt [3]{a+b x}}\) |
3/(a*(a + b*x)^(1/3)) + ((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)))/a
3.5.17.3.1 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 + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[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 = 0.10 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.99
| method | result | size |
| pseudoelliptic | \(\frac {\left (\arctan \left (\frac {\left (a^{\frac {1}{3}}+2 \left (b x +a \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}+\ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )-\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{2}\right ) \left (b x +a \right )^{\frac {1}{3}}+3 a^{\frac {1}{3}}}{a^{\frac {4}{3}} \left (b x +a \right )^{\frac {1}{3}}}\) | \(92\) |
| derivativedivides | \(\frac {\frac {\ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{a^{\frac {1}{3}}}-\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{2 a^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{a^{\frac {1}{3}}}}{a}+\frac {3}{a \left (b x +a \right )^{\frac {1}{3}}}\) | \(95\) |
| default | \(\frac {\frac {\ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{a^{\frac {1}{3}}}-\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{2 a^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{a^{\frac {1}{3}}}}{a}+\frac {3}{a \left (b x +a \right )^{\frac {1}{3}}}\) | \(95\) |
1/a^(4/3)/(b*x+a)^(1/3)*((arctan(1/3*(a^(1/3)+2*(b*x+a)^(1/3))/a^(1/3)*3^( 1/2))*3^(1/2)+ln((b*x+a)^(1/3)-a^(1/3))-1/2*ln((b*x+a)^(2/3)+a^(1/3)*(b*x+ a)^(1/3)+a^(2/3)))*(b*x+a)^(1/3)+3*a^(1/3))
Time = 0.23 (sec) , antiderivative size = 285, normalized size of antiderivative = 3.06 \[ \int \frac {1}{x (a+b x)^{4/3}} \, dx=\left [\frac {\sqrt {3} {\left (a b x + a^{2}\right )} \sqrt {-\frac {1}{a^{\frac {2}{3}}}} \log \left (\frac {2 \, b x + \sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {2}{3}} a^{\frac {2}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} a - a^{\frac {4}{3}}\right )} \sqrt {-\frac {1}{a^{\frac {2}{3}}}} - 3 \, {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {2}{3}} + 3 \, a}{x}\right ) - {\left (b x + a\right )} a^{\frac {2}{3}} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + 2 \, {\left (b x + a\right )} a^{\frac {2}{3}} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) + 6 \, {\left (b x + a\right )}^{\frac {2}{3}} a}{2 \, {\left (a^{2} b x + a^{3}\right )}}, -\frac {{\left (b x + a\right )} a^{\frac {2}{3}} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) - 2 \, {\left (b x + a\right )} a^{\frac {2}{3}} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) - \frac {2 \, \sqrt {3} {\left (a b x + a^{2}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {1}{3}}} - 6 \, {\left (b x + a\right )}^{\frac {2}{3}} a}{2 \, {\left (a^{2} b x + a^{3}\right )}}\right ] \]
[1/2*(sqrt(3)*(a*b*x + a^2)*sqrt(-1/a^(2/3))*log((2*b*x + sqrt(3)*(2*(b*x + a)^(2/3)*a^(2/3) - (b*x + a)^(1/3)*a - a^(4/3))*sqrt(-1/a^(2/3)) - 3*(b* x + a)^(1/3)*a^(2/3) + 3*a)/x) - (b*x + a)*a^(2/3)*log((b*x + a)^(2/3) + ( b*x + a)^(1/3)*a^(1/3) + a^(2/3)) + 2*(b*x + a)*a^(2/3)*log((b*x + a)^(1/3 ) - a^(1/3)) + 6*(b*x + a)^(2/3)*a)/(a^2*b*x + a^3), -1/2*((b*x + a)*a^(2/ 3)*log((b*x + a)^(2/3) + (b*x + a)^(1/3)*a^(1/3) + a^(2/3)) - 2*(b*x + a)* a^(2/3)*log((b*x + a)^(1/3) - a^(1/3)) - 2*sqrt(3)*(a*b*x + a^2)*arctan(1/ 3*sqrt(3)*(2*(b*x + a)^(1/3) + a^(1/3))/a^(1/3))/a^(1/3) - 6*(b*x + a)^(2/ 3)*a)/(a^2*b*x + a^3)]
Result contains complex when optimal does not.
Time = 2.04 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.98 \[ \int \frac {1}{x (a+b x)^{4/3}} \, dx=- \frac {\Gamma \left (- \frac {1}{3}\right )}{a \sqrt [3]{b} \sqrt [3]{\frac {a}{b} + x} \Gamma \left (\frac {2}{3}\right )} - \frac {\log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{\frac {a}{b} + x}}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac {1}{3}\right )}{3 a^{\frac {4}{3}} \Gamma \left (\frac {2}{3}\right )} - \frac {e^{\frac {2 i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{\frac {a}{b} + x} e^{\frac {2 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac {1}{3}\right )}{3 a^{\frac {4}{3}} \Gamma \left (\frac {2}{3}\right )} - \frac {e^{- \frac {2 i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{\frac {a}{b} + x} e^{\frac {4 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac {1}{3}\right )}{3 a^{\frac {4}{3}} \Gamma \left (\frac {2}{3}\right )} \]
-gamma(-1/3)/(a*b**(1/3)*(a/b + x)**(1/3)*gamma(2/3)) - log(1 - b**(1/3)*( a/b + x)**(1/3)/a**(1/3))*gamma(-1/3)/(3*a**(4/3)*gamma(2/3)) - exp(2*I*pi /3)*log(1 - b**(1/3)*(a/b + x)**(1/3)*exp_polar(2*I*pi/3)/a**(1/3))*gamma( -1/3)/(3*a**(4/3)*gamma(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(-1/3)/(3*a**(4/3)*gamma(2/3))
Time = 0.29 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.95 \[ \int \frac {1}{x (a+b x)^{4/3}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {4}{3}}} - \frac {\log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{2 \, a^{\frac {4}{3}}} + \frac {\log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{a^{\frac {4}{3}}} + \frac {3}{{\left (b x + a\right )}^{\frac {1}{3}} a} \]
sqrt(3)*arctan(1/3*sqrt(3)*(2*(b*x + a)^(1/3) + a^(1/3))/a^(1/3))/a^(4/3) - 1/2*log((b*x + a)^(2/3) + (b*x + a)^(1/3)*a^(1/3) + a^(2/3))/a^(4/3) + l og((b*x + a)^(1/3) - a^(1/3))/a^(4/3) + 3/((b*x + a)^(1/3)*a)
Time = 0.73 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x (a+b x)^{4/3}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {4}{3}}} - \frac {\log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{2 \, a^{\frac {4}{3}}} + \frac {\log \left ({\left | {\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{a^{\frac {4}{3}}} + \frac {3}{{\left (b x + a\right )}^{\frac {1}{3}} a} \]
sqrt(3)*arctan(1/3*sqrt(3)*(2*(b*x + a)^(1/3) + a^(1/3))/a^(1/3))/a^(4/3) - 1/2*log((b*x + a)^(2/3) + (b*x + a)^(1/3)*a^(1/3) + a^(2/3))/a^(4/3) + l og(abs((b*x + a)^(1/3) - a^(1/3)))/a^(4/3) + 3/((b*x + a)^(1/3)*a)
Time = 0.16 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.23 \[ \int \frac {1}{x (a+b x)^{4/3}} \, dx=\frac {\ln \left (9\,a\,{\left (a+b\,x\right )}^{1/3}-9\,a^{4/3}\right )}{a^{4/3}}+\frac {3}{a\,{\left (a+b\,x\right )}^{1/3}}+\frac {\ln \left (9\,a\,{\left (a+b\,x\right )}^{1/3}-\frac {9\,a^{4/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{4}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,a^{4/3}}-\frac {\ln \left (9\,a\,{\left (a+b\,x\right )}^{1/3}-\frac {9\,a^{4/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{4}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,a^{4/3}} \]
log(9*a*(a + b*x)^(1/3) - 9*a^(4/3))/a^(4/3) + 3/(a*(a + b*x)^(1/3)) + (lo g(9*a*(a + b*x)^(1/3) - (9*a^(4/3)*(3^(1/2)*1i - 1)^2)/4)*(3^(1/2)*1i - 1) )/(2*a^(4/3)) - (log(9*a*(a + b*x)^(1/3) - (9*a^(4/3)*(3^(1/2)*1i + 1)^2)/ 4)*(3^(1/2)*1i + 1))/(2*a^(4/3))
Time = 0.01 (sec) , antiderivative size = 195, normalized size of antiderivative = 2.10 \[ \int \frac {1}{x (a+b x)^{4/3}} \, dx=\frac {-2 \left (b x +a \right )^{\frac {1}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {2 \left (b x +a \right )^{\frac {1}{6}}+a^{\frac {1}{6}}}{a^{\frac {1}{6}} \sqrt {3}}\right )+2 \left (b x +a \right )^{\frac {1}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {2 \left (b x +a \right )^{\frac {1}{6}}-a^{\frac {1}{6}}}{a^{\frac {1}{6}} \sqrt {3}}\right )+2 \left (b x +a \right )^{\frac {1}{3}} \mathrm {log}\left (\left (b x +a \right )^{\frac {1}{6}}+a^{\frac {1}{6}}\right )+2 \left (b x +a \right )^{\frac {1}{3}} \mathrm {log}\left (\left (b x +a \right )^{\frac {1}{6}}-a^{\frac {1}{6}}\right )-\left (b x +a \right )^{\frac {1}{3}} \mathrm {log}\left (-a^{\frac {1}{6}} \left (b x +a \right )^{\frac {1}{6}}+\left (b x +a \right )^{\frac {1}{3}}+a^{\frac {1}{3}}\right )-\left (b x +a \right )^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {1}{6}} \left (b x +a \right )^{\frac {1}{6}}+\left (b x +a \right )^{\frac {1}{3}}+a^{\frac {1}{3}}\right )+6 a^{\frac {1}{3}}}{2 a^{\frac {4}{3}} \left (b x +a \right )^{\frac {1}{3}}} \]
( - 2*(a + b*x)**(1/3)*sqrt(3)*atan((2*(a + b*x)**(1/6) + a**(1/6))/(a**(1 /6)*sqrt(3))) + 2*(a + b*x)**(1/3)*sqrt(3)*atan((2*(a + b*x)**(1/6) - a**( 1/6))/(a**(1/6)*sqrt(3))) + 2*(a + b*x)**(1/3)*log((a + b*x)**(1/6) + a**( 1/6)) + 2*(a + b*x)**(1/3)*log((a + b*x)**(1/6) - a**(1/6)) - (a + b*x)**( 1/3)*log( - a**(1/6)*(a + b*x)**(1/6) + (a + b*x)**(1/3) + a**(1/3)) - (a + b*x)**(1/3)*log(a**(1/6)*(a + b*x)**(1/6) + (a + b*x)**(1/3) + a**(1/3)) + 6*a**(1/3))/(2*a**(1/3)*(a + b*x)**(1/3)*a)