Integrand size = 13, antiderivative size = 140 \[ \int \frac {\sqrt [3]{a+b x}}{x^3} \, dx=-\frac {b^2 \sqrt [3]{a+b x}}{3 a^2}+\left (-\frac {1}{2 a x^2}+\frac {b}{3 a^2 x}\right ) (a+b x)^{4/3}-\frac {b^2 \left (-\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{a+b x}}{2 \sqrt [3]{a}+\sqrt [3]{a+b x}}\right )+\frac {3}{2} \log \left (\frac {-\sqrt [3]{a}+\sqrt [3]{a+b x}}{\sqrt [3]{x}}\right )\right )}{9 a \sqrt [3]{a^2}} \]
(-1/2/a/x^2+1/3*b/a^2/x)*(b*x+a)^(4/3)-1/3*b^2/a^2*(b*x+a)^(1/3)-1/9*b^2/a /(a^2)^(1/3)*(3/2*ln(((b*x+a)^(1/3)-a^(1/3))/x^(1/3))-3^(1/2)*arctan(3^(1/ 2)*(b*x+a)^(1/3)/((b*x+a)^(1/3)+2*a^(1/3))))
Time = 0.34 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt [3]{a+b x}}{x^3} \, dx=\frac {-\frac {3 a^{2/3} \sqrt [3]{a+b x} (3 a+b x)}{x^2}+2 \sqrt {3} b^2 \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-2 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )+b^2 \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right )}{18 a^{5/3}} \]
((-3*a^(2/3)*(a + b*x)^(1/3)*(3*a + b*x))/x^2 + 2*Sqrt[3]*b^2*ArcTan[(1 + (2*(a + b*x)^(1/3))/a^(1/3))/Sqrt[3]] - 2*b^2*Log[a^(1/3) - (a + b*x)^(1/3 )] + b^2*Log[a^(2/3) + a^(1/3)*(a + b*x)^(1/3) + (a + b*x)^(2/3)])/(18*a^( 5/3))
Time = 0.22 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.89, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {51, 52, 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.
| \(\displaystyle \int \frac {\sqrt [3]{a+b x}}{x^3} \, dx\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {1}{6} b \int \frac {1}{x^2 (a+b x)^{2/3}}dx-\frac {\sqrt [3]{a+b x}}{2 x^2}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{6} b \left (-\frac {2 b \int \frac {1}{x (a+b x)^{2/3}}dx}{3 a}-\frac {\sqrt [3]{a+b x}}{a x}\right )-\frac {\sqrt [3]{a+b x}}{2 x^2}\) |
| \(\Big \downarrow \) 69 |
| \(\displaystyle \frac {1}{6} b \left (-\frac {2 b \left (-\frac {3 \int \frac {1}{\sqrt [3]{a}-\sqrt [3]{a+b x}}d\sqrt [3]{a+b x}}{2 a^{2/3}}-\frac {3 \int \frac {1}{a^{2/3}+\sqrt [3]{a+b x} \sqrt [3]{a}+(a+b x)^{2/3}}d\sqrt [3]{a+b x}}{2 \sqrt [3]{a}}-\frac {\log (x)}{2 a^{2/3}}\right )}{3 a}-\frac {\sqrt [3]{a+b x}}{a x}\right )-\frac {\sqrt [3]{a+b x}}{2 x^2}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {1}{6} b \left (-\frac {2 b \left (-\frac {3 \int \frac {1}{a^{2/3}+\sqrt [3]{a+b x} \sqrt [3]{a}+(a+b x)^{2/3}}d\sqrt [3]{a+b x}}{2 \sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{2/3}}-\frac {\log (x)}{2 a^{2/3}}\right )}{3 a}-\frac {\sqrt [3]{a+b x}}{a x}\right )-\frac {\sqrt [3]{a+b x}}{2 x^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{6} b \left (-\frac {2 b \left (\frac {3 \int \frac {1}{-(a+b x)^{2/3}-3}d\left (\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}+1\right )}{a^{2/3}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{2/3}}-\frac {\log (x)}{2 a^{2/3}}\right )}{3 a}-\frac {\sqrt [3]{a+b x}}{a x}\right )-\frac {\sqrt [3]{a+b x}}{2 x^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{6} b \left (-\frac {2 b \left (-\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )}{a^{2/3}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{2/3}}-\frac {\log (x)}{2 a^{2/3}}\right )}{3 a}-\frac {\sqrt [3]{a+b x}}{a x}\right )-\frac {\sqrt [3]{a+b x}}{2 x^2}\) |
-1/2*(a + b*x)^(1/3)/x^2 + (b*(-((a + b*x)^(1/3)/(a*x)) - (2*b*(-((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*a)))/6
3.2.54.3.1 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 + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
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] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
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 = 0.14 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.84
| method | result | size |
| derivativedivides | \(3 b^{2} \left (-\frac {\frac {\left (b x +a \right )^{\frac {4}{3}}}{18 a}+\frac {\left (b x +a \right )^{\frac {1}{3}}}{9}}{b^{2} x^{2}}-\frac {\frac {\ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{3 a^{\frac {2}{3}}}-\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+\left (b x +a \right )^{\frac {1}{3}} a^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{6 a^{\frac {2}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{3 a^{\frac {2}{3}}}}{9 a}\right )\) | \(118\) |
| default | \(3 b^{2} \left (-\frac {\frac {\left (b x +a \right )^{\frac {4}{3}}}{18 a}+\frac {\left (b x +a \right )^{\frac {1}{3}}}{9}}{b^{2} x^{2}}-\frac {\frac {\ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{3 a^{\frac {2}{3}}}-\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+\left (b x +a \right )^{\frac {1}{3}} a^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{6 a^{\frac {2}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{3 a^{\frac {2}{3}}}}{9 a}\right )\) | \(118\) |
| pseudoelliptic | \(\frac {2 b^{2} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (2 \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {1}{3}}\right )}{3 a^{\frac {1}{3}}}\right ) x^{2}-2 b^{2} \ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right ) x^{2}+b^{2} \ln \left (\left (b x +a \right )^{\frac {2}{3}}+\left (b x +a \right )^{\frac {1}{3}} a^{\frac {1}{3}}+a^{\frac {2}{3}}\right ) x^{2}-3 b x \left (b x +a \right )^{\frac {1}{3}} a^{\frac {2}{3}}-9 \left (b x +a \right )^{\frac {1}{3}} a^{\frac {5}{3}}}{18 a^{\frac {5}{3}} x^{2}}\) | \(121\) |
3*b^2*(-(1/18/a*(b*x+a)^(4/3)+1/9*(b*x+a)^(1/3))/b^2/x^2-1/9/a*(1/3/a^(2/3 )*ln((b*x+a)^(1/3)-a^(1/3))-1/6/a^(2/3)*ln((b*x+a)^(2/3)+(b*x+a)^(1/3)*a^( 1/3)+a^(2/3))-1/3/a^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/a^(1/3)*(b*x+a)^(1 /3)+1))))
Time = 0.25 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.34 \[ \int \frac {\sqrt [3]{a+b x}}{x^3} \, dx=\frac {2 \, \sqrt {3} a b^{2} x^{2} \sqrt {-\left (-a^{2}\right )^{\frac {1}{3}}} \arctan \left (-\frac {{\left (\sqrt {3} \left (-a^{2}\right )^{\frac {1}{3}} a - 2 \, \sqrt {3} \left (-a^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {1}{3}}\right )} \sqrt {-\left (-a^{2}\right )^{\frac {1}{3}}}}{3 \, a^{2}}\right ) + \left (-a^{2}\right )^{\frac {2}{3}} b^{2} x^{2} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} a - \left (-a^{2}\right )^{\frac {1}{3}} a + \left (-a^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {1}{3}}\right ) - 2 \, \left (-a^{2}\right )^{\frac {2}{3}} b^{2} x^{2} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} a - \left (-a^{2}\right )^{\frac {2}{3}}\right ) - 3 \, {\left (a^{2} b x + 3 \, a^{3}\right )} {\left (b x + a\right )}^{\frac {1}{3}}}{18 \, a^{3} x^{2}} \]
1/18*(2*sqrt(3)*a*b^2*x^2*sqrt(-(-a^2)^(1/3))*arctan(-1/3*(sqrt(3)*(-a^2)^ (1/3)*a - 2*sqrt(3)*(-a^2)^(2/3)*(b*x + a)^(1/3))*sqrt(-(-a^2)^(1/3))/a^2) + (-a^2)^(2/3)*b^2*x^2*log((b*x + a)^(2/3)*a - (-a^2)^(1/3)*a + (-a^2)^(2 /3)*(b*x + a)^(1/3)) - 2*(-a^2)^(2/3)*b^2*x^2*log((b*x + a)^(1/3)*a - (-a^ 2)^(2/3)) - 3*(a^2*b*x + 3*a^3)*(b*x + a)^(1/3))/(a^3*x^2)
Result contains complex when optimal does not.
Time = 1.91 (sec) , antiderivative size = 2266, normalized size of antiderivative = 16.19 \[ \int \frac {\sqrt [3]{a+b x}}{x^3} \, dx=\text {Too large to display} \]
-4*a**(16/3)*b**2*exp(2*I*pi/3)*log(1 - b**(1/3)*(a/b + x)**(1/3)/a**(1/3) )*gamma(4/3)/(27*a**7*exp(2*I*pi/3)*gamma(7/3) - 81*a**6*b*(a/b + x)*exp(2 *I*pi/3)*gamma(7/3) + 81*a**5*b**2*(a/b + x)**2*exp(2*I*pi/3)*gamma(7/3) - 27*a**4*b**3*(a/b + x)**3*exp(2*I*pi/3)*gamma(7/3)) - 4*a**(16/3)*b**2*lo g(1 - b**(1/3)*(a/b + x)**(1/3)*exp_polar(2*I*pi/3)/a**(1/3))*gamma(4/3)/( 27*a**7*exp(2*I*pi/3)*gamma(7/3) - 81*a**6*b*(a/b + x)*exp(2*I*pi/3)*gamma (7/3) + 81*a**5*b**2*(a/b + x)**2*exp(2*I*pi/3)*gamma(7/3) - 27*a**4*b**3* (a/b + x)**3*exp(2*I*pi/3)*gamma(7/3)) - 4*a**(16/3)*b**2*exp(-2*I*pi/3)*l og(1 - b**(1/3)*(a/b + x)**(1/3)*exp_polar(4*I*pi/3)/a**(1/3))*gamma(4/3)/ (27*a**7*exp(2*I*pi/3)*gamma(7/3) - 81*a**6*b*(a/b + x)*exp(2*I*pi/3)*gamm a(7/3) + 81*a**5*b**2*(a/b + x)**2*exp(2*I*pi/3)*gamma(7/3) - 27*a**4*b**3 *(a/b + x)**3*exp(2*I*pi/3)*gamma(7/3)) + 12*a**(13/3)*b**3*(a/b + x)*exp( 2*I*pi/3)*log(1 - b**(1/3)*(a/b + x)**(1/3)/a**(1/3))*gamma(4/3)/(27*a**7* exp(2*I*pi/3)*gamma(7/3) - 81*a**6*b*(a/b + x)*exp(2*I*pi/3)*gamma(7/3) + 81*a**5*b**2*(a/b + x)**2*exp(2*I*pi/3)*gamma(7/3) - 27*a**4*b**3*(a/b + x )**3*exp(2*I*pi/3)*gamma(7/3)) + 12*a**(13/3)*b**3*(a/b + x)*log(1 - b**(1 /3)*(a/b + x)**(1/3)*exp_polar(2*I*pi/3)/a**(1/3))*gamma(4/3)/(27*a**7*exp (2*I*pi/3)*gamma(7/3) - 81*a**6*b*(a/b + x)*exp(2*I*pi/3)*gamma(7/3) + 81* a**5*b**2*(a/b + x)**2*exp(2*I*pi/3)*gamma(7/3) - 27*a**4*b**3*(a/b + x)** 3*exp(2*I*pi/3)*gamma(7/3)) + 12*a**(13/3)*b**3*(a/b + x)*exp(-2*I*pi/3...
Time = 0.30 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.99 \[ \int \frac {\sqrt [3]{a+b x}}{x^3} \, dx=\frac {\sqrt {3} b^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{9 \, a^{\frac {5}{3}}} + \frac {b^{2} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{18 \, a^{\frac {5}{3}}} - \frac {b^{2} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{9 \, a^{\frac {5}{3}}} - \frac {{\left (b x + a\right )}^{\frac {4}{3}} b^{2} + 2 \, {\left (b x + a\right )}^{\frac {1}{3}} a b^{2}}{6 \, {\left ({\left (b x + a\right )}^{2} a - 2 \, {\left (b x + a\right )} a^{2} + a^{3}\right )}} \]
1/9*sqrt(3)*b^2*arctan(1/3*sqrt(3)*(2*(b*x + a)^(1/3) + a^(1/3))/a^(1/3))/ a^(5/3) + 1/18*b^2*log((b*x + a)^(2/3) + (b*x + a)^(1/3)*a^(1/3) + a^(2/3) )/a^(5/3) - 1/9*b^2*log((b*x + a)^(1/3) - a^(1/3))/a^(5/3) - 1/6*((b*x + a )^(4/3)*b^2 + 2*(b*x + a)^(1/3)*a*b^2)/((b*x + a)^2*a - 2*(b*x + a)*a^2 + a^3)
Time = 0.49 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt [3]{a+b x}}{x^3} \, dx=\frac {\frac {2 \, \sqrt {3} b^{3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {5}{3}}} + \frac {b^{3} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{a^{\frac {5}{3}}} - \frac {2 \, b^{3} \log \left ({\left | {\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{a^{\frac {5}{3}}} - \frac {3 \, {\left ({\left (b x + a\right )}^{\frac {4}{3}} b^{3} + 2 \, {\left (b x + a\right )}^{\frac {1}{3}} a b^{3}\right )}}{a b^{2} x^{2}}}{18 \, b} \]
1/18*(2*sqrt(3)*b^3*arctan(1/3*sqrt(3)*(2*(b*x + a)^(1/3) + a^(1/3))/a^(1/ 3))/a^(5/3) + b^3*log((b*x + a)^(2/3) + (b*x + a)^(1/3)*a^(1/3) + a^(2/3)) /a^(5/3) - 2*b^3*log(abs((b*x + a)^(1/3) - a^(1/3)))/a^(5/3) - 3*((b*x + a )^(4/3)*b^3 + 2*(b*x + a)^(1/3)*a*b^3)/(a*b^2*x^2))/b
Time = 20.99 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.40 \[ \int \frac {\sqrt [3]{a+b x}}{x^3} \, dx=\frac {b^2\,\ln \left (\frac {b^2}{{\left (-a\right )}^{2/3}}-\frac {b^2\,{\left (a+b\,x\right )}^{1/3}}{a}\right )}{9\,{\left (-a\right )}^{5/3}}-\frac {\ln \left (\frac {b^2+\sqrt {3}\,b^2\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{2/3}}+\frac {b^2\,{\left (a+b\,x\right )}^{1/3}}{a}\right )\,\left (b^2+\sqrt {3}\,b^2\,1{}\mathrm {i}\right )}{18\,{\left (-a\right )}^{5/3}}-\frac {\frac {b^2\,{\left (a+b\,x\right )}^{1/3}}{3}+\frac {b^2\,{\left (a+b\,x\right )}^{4/3}}{6\,a}}{{\left (a+b\,x\right )}^2-2\,a\,\left (a+b\,x\right )+a^2}+\frac {b^2\,\ln \left (\frac {b^2\,{\left (a+b\,x\right )}^{1/3}}{a}-\frac {b^2\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{{\left (-a\right )}^{2/3}}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{9\,{\left (-a\right )}^{5/3}} \]
(b^2*log(b^2/(-a)^(2/3) - (b^2*(a + b*x)^(1/3))/a))/(9*(-a)^(5/3)) - (log( (3^(1/2)*b^2*1i + b^2)/(2*(-a)^(2/3)) + (b^2*(a + b*x)^(1/3))/a)*(3^(1/2)* b^2*1i + b^2))/(18*(-a)^(5/3)) - ((b^2*(a + b*x)^(1/3))/3 + (b^2*(a + b*x) ^(4/3))/(6*a))/((a + b*x)^2 - 2*a*(a + b*x) + a^2) + (b^2*log((b^2*(a + b* x)^(1/3))/a - (b^2*((3^(1/2)*1i)/2 - 1/2))/(-a)^(2/3))*((3^(1/2)*1i)/2 - 1 /2))/(9*(-a)^(5/3))