Integrand size = 13, antiderivative size = 97 \[ \int \frac {\sqrt [3]{a+b x}}{x^2} \, dx=-\frac {\sqrt [3]{a+b x}}{x}-\frac {b \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3}}-\frac {b \log (x)}{6 a^{2/3}}+\frac {b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{2/3}} \]
-(b*x+a)^(1/3)/x-1/6*b*ln(x)/a^(2/3)+1/2*b*ln(a^(1/3)-(b*x+a)^(1/3))/a^(2/ 3)-1/3*b*arctan(1/3*(a^(1/3)+2*(b*x+a)^(1/3))/a^(1/3)*3^(1/2))/a^(2/3)*3^( 1/2)
Time = 0.14 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.23 \[ \int \frac {\sqrt [3]{a+b x}}{x^2} \, dx=-\frac {6 a^{2/3} \sqrt [3]{a+b x}+2 \sqrt {3} b x \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-2 b x \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )+b x \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right )}{6 a^{2/3} x} \]
-1/6*(6*a^(2/3)*(a + b*x)^(1/3) + 2*Sqrt[3]*b*x*ArcTan[(1 + (2*(a + b*x)^( 1/3))/a^(1/3))/Sqrt[3]] - 2*b*x*Log[a^(1/3) - (a + b*x)^(1/3)] + b*x*Log[a ^(2/3) + a^(1/3)*(a + b*x)^(1/3) + (a + b*x)^(2/3)])/(a^(2/3)*x)
Time = 0.20 (sec) , antiderivative size = 96, 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 = {51, 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.
\(\displaystyle \int \frac {\sqrt [3]{a+b x}}{x^2} \, dx\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {1}{3} b \int \frac {1}{x (a+b x)^{2/3}}dx-\frac {\sqrt [3]{a+b x}}{x}\) |
\(\Big \downarrow \) 69 |
\(\displaystyle \frac {1}{3} b \left (-\frac {3 \int \frac {1}{\sqrt [3]{a}-\sqrt [3]{a+b x}}d\sqrt [3]{a+b x}}{2 a^{2/3}}-\frac {3 \int \frac {1}{a^{2/3}+\sqrt [3]{a+b x} \sqrt [3]{a}+(a+b x)^{2/3}}d\sqrt [3]{a+b x}}{2 \sqrt [3]{a}}-\frac {\log (x)}{2 a^{2/3}}\right )-\frac {\sqrt [3]{a+b x}}{x}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {1}{3} b \left (-\frac {3 \int \frac {1}{a^{2/3}+\sqrt [3]{a+b x} \sqrt [3]{a}+(a+b x)^{2/3}}d\sqrt [3]{a+b x}}{2 \sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{2/3}}-\frac {\log (x)}{2 a^{2/3}}\right )-\frac {\sqrt [3]{a+b x}}{x}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {1}{3} b \left (\frac {3 \int \frac {1}{-(a+b x)^{2/3}-3}d\left (\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}+1\right )}{a^{2/3}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{2/3}}-\frac {\log (x)}{2 a^{2/3}}\right )-\frac {\sqrt [3]{a+b x}}{x}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {1}{3} b \left (-\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )}{a^{2/3}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{2/3}}-\frac {\log (x)}{2 a^{2/3}}\right )-\frac {\sqrt [3]{a+b x}}{x}\) |
-((a + b*x)^(1/3)/x) + (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
3.4.76.3.1 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[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 = 0.13 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.98
method | result | size |
derivativedivides | \(3 b \left (-\frac {\left (b x +a \right )^{\frac {1}{3}}}{3 b x}+\frac {\ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{9 a^{\frac {2}{3}}}-\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{18 a^{\frac {2}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{9 a^{\frac {2}{3}}}\right )\) | \(95\) |
default | \(3 b \left (-\frac {\left (b x +a \right )^{\frac {1}{3}}}{3 b x}+\frac {\ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{9 a^{\frac {2}{3}}}-\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{18 a^{\frac {2}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{9 a^{\frac {2}{3}}}\right )\) | \(95\) |
pseudoelliptic | \(\frac {-\arctan \left (\frac {\left (a^{\frac {1}{3}}+2 \left (b x +a \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}\, b x +\ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right ) b x -\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right ) b x}{2}-3 \left (b x +a \right )^{\frac {1}{3}} a^{\frac {2}{3}}}{3 a^{\frac {2}{3}} x}\) | \(95\) |
3*b*(-1/3*(b*x+a)^(1/3)/b/x+1/9/a^(2/3)*ln((b*x+a)^(1/3)-a^(1/3))-1/18/a^( 2/3)*ln((b*x+a)^(2/3)+a^(1/3)*(b*x+a)^(1/3)+a^(2/3))-1/9/a^(2/3)*3^(1/2)*a rctan(1/3*3^(1/2)*(2/a^(1/3)*(b*x+a)^(1/3)+1)))
Time = 0.23 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.43 \[ \int \frac {\sqrt [3]{a+b x}}{x^2} \, dx=-\frac {2 \, \sqrt {3} {\left (a^{2}\right )}^{\frac {1}{6}} a b x \arctan \left (\frac {{\left (a^{2}\right )}^{\frac {1}{6}} {\left (\sqrt {3} {\left (a^{2}\right )}^{\frac {1}{3}} a + 2 \, \sqrt {3} {\left (a^{2}\right )}^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {1}{3}}\right )}}{3 \, a^{2}}\right ) + {\left (a^{2}\right )}^{\frac {2}{3}} b x \log \left ({\left (b x + a\right )}^{\frac {2}{3}} a + {\left (a^{2}\right )}^{\frac {1}{3}} a + {\left (a^{2}\right )}^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {1}{3}}\right ) - 2 \, {\left (a^{2}\right )}^{\frac {2}{3}} b x \log \left ({\left (b x + a\right )}^{\frac {1}{3}} a - {\left (a^{2}\right )}^{\frac {2}{3}}\right ) + 6 \, {\left (b x + a\right )}^{\frac {1}{3}} a^{2}}{6 \, a^{2} x} \]
-1/6*(2*sqrt(3)*(a^2)^(1/6)*a*b*x*arctan(1/3*(a^2)^(1/6)*(sqrt(3)*(a^2)^(1 /3)*a + 2*sqrt(3)*(a^2)^(2/3)*(b*x + a)^(1/3))/a^2) + (a^2)^(2/3)*b*x*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*x*log((b*x + a)^(1/3)*a - (a^2)^(2/3)) + 6*(b*x + a)^(1/3)*a^2)/( a^2*x)
Result contains complex when optimal does not.
Time = 2.04 (sec) , antiderivative size = 643, normalized size of antiderivative = 6.63 \[ \int \frac {\sqrt [3]{a+b x}}{x^2} \, dx =\text {Too large to display} \]
4*a**(7/3)*b*exp(2*I*pi/3)*log(1 - b**(1/3)*(a/b + x)**(1/3)/a**(1/3))*gam ma(4/3)/(9*a**3*exp(2*I*pi/3)*gamma(7/3) - 9*a**2*b*(a/b + x)*exp(2*I*pi/3 )*gamma(7/3)) + 4*a**(7/3)*b*log(1 - b**(1/3)*(a/b + x)**(1/3)*exp_polar(2 *I*pi/3)/a**(1/3))*gamma(4/3)/(9*a**3*exp(2*I*pi/3)*gamma(7/3) - 9*a**2*b* (a/b + x)*exp(2*I*pi/3)*gamma(7/3)) + 4*a**(7/3)*b*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)/(9*a**3 *exp(2*I*pi/3)*gamma(7/3) - 9*a**2*b*(a/b + x)*exp(2*I*pi/3)*gamma(7/3)) - 4*a**(4/3)*b**2*(a/b + x)*exp(2*I*pi/3)*log(1 - b**(1/3)*(a/b + x)**(1/3) /a**(1/3))*gamma(4/3)/(9*a**3*exp(2*I*pi/3)*gamma(7/3) - 9*a**2*b*(a/b + x )*exp(2*I*pi/3)*gamma(7/3)) - 4*a**(4/3)*b**2*(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)/(9*a**3*exp(2*I*p i/3)*gamma(7/3) - 9*a**2*b*(a/b + x)*exp(2*I*pi/3)*gamma(7/3)) - 4*a**(4/3 )*b**2*(a/b + x)*exp(-2*I*pi/3)*log(1 - b**(1/3)*(a/b + x)**(1/3)*exp_pola r(4*I*pi/3)/a**(1/3))*gamma(4/3)/(9*a**3*exp(2*I*pi/3)*gamma(7/3) - 9*a**2 *b*(a/b + x)*exp(2*I*pi/3)*gamma(7/3)) + 12*a**2*b**(4/3)*(a/b + x)**(1/3) *exp(2*I*pi/3)*gamma(4/3)/(9*a**3*exp(2*I*pi/3)*gamma(7/3) - 9*a**2*b*(a/b + x)*exp(2*I*pi/3)*gamma(7/3))
Time = 0.29 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt [3]{a+b x}}{x^2} \, dx=-\frac {\sqrt {3} b \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{3 \, a^{\frac {2}{3}}} - \frac {b \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{6 \, a^{\frac {2}{3}}} + \frac {b \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{3 \, a^{\frac {2}{3}}} - \frac {{\left (b x + a\right )}^{\frac {1}{3}}}{x} \]
-1/3*sqrt(3)*b*arctan(1/3*sqrt(3)*(2*(b*x + a)^(1/3) + a^(1/3))/a^(1/3))/a ^(2/3) - 1/6*b*log((b*x + a)^(2/3) + (b*x + a)^(1/3)*a^(1/3) + a^(2/3))/a^ (2/3) + 1/3*b*log((b*x + a)^(1/3) - a^(1/3))/a^(2/3) - (b*x + a)^(1/3)/x
Time = 0.53 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt [3]{a+b x}}{x^2} \, dx=-\frac {\frac {2 \, \sqrt {3} b^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {2}{3}}} + \frac {b^{2} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{a^{\frac {2}{3}}} - \frac {2 \, b^{2} \log \left ({\left | {\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{a^{\frac {2}{3}}} + \frac {6 \, {\left (b x + a\right )}^{\frac {1}{3}} b}{x}}{6 \, b} \]
-1/6*(2*sqrt(3)*b^2*arctan(1/3*sqrt(3)*(2*(b*x + a)^(1/3) + a^(1/3))/a^(1/ 3))/a^(2/3) + b^2*log((b*x + a)^(2/3) + (b*x + a)^(1/3)*a^(1/3) + a^(2/3)) /a^(2/3) - 2*b^2*log(abs((b*x + a)^(1/3) - a^(1/3)))/a^(2/3) + 6*(b*x + a) ^(1/3)*b/x)/b
Time = 0.08 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.21 \[ \int \frac {\sqrt [3]{a+b x}}{x^2} \, dx=\frac {b\,\ln \left (3\,b\,{\left (a+b\,x\right )}^{1/3}-3\,a^{1/3}\,b\right )}{3\,a^{2/3}}-\frac {{\left (a+b\,x\right )}^{1/3}}{x}-\frac {\ln \left (\frac {3\,a^{1/3}\,\left (b-\sqrt {3}\,b\,1{}\mathrm {i}\right )}{2}+3\,b\,{\left (a+b\,x\right )}^{1/3}\right )\,\left (b-\sqrt {3}\,b\,1{}\mathrm {i}\right )}{6\,a^{2/3}}-\frac {\ln \left (\frac {3\,a^{1/3}\,\left (b+\sqrt {3}\,b\,1{}\mathrm {i}\right )}{2}+3\,b\,{\left (a+b\,x\right )}^{1/3}\right )\,\left (b+\sqrt {3}\,b\,1{}\mathrm {i}\right )}{6\,a^{2/3}} \]
(b*log(3*b*(a + b*x)^(1/3) - 3*a^(1/3)*b))/(3*a^(2/3)) - (a + b*x)^(1/3)/x - (log((3*a^(1/3)*(b - 3^(1/2)*b*1i))/2 + 3*b*(a + b*x)^(1/3))*(b - 3^(1/ 2)*b*1i))/(6*a^(2/3)) - (log((3*a^(1/3)*(b + 3^(1/2)*b*1i))/2 + 3*b*(a + b *x)^(1/3))*(b + 3^(1/2)*b*1i))/(6*a^(2/3))
Time = 0.01 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.73 \[ \int \frac {\sqrt [3]{a+b x}}{x^2} \, dx=\frac {2 \sqrt {3}\, \mathit {atan} \left (\frac {2 \left (b x +a \right )^{\frac {1}{6}}+a^{\frac {1}{6}}}{a^{\frac {1}{6}} \sqrt {3}}\right ) b x -2 \sqrt {3}\, \mathit {atan} \left (\frac {2 \left (b x +a \right )^{\frac {1}{6}}-a^{\frac {1}{6}}}{a^{\frac {1}{6}} \sqrt {3}}\right ) b x -6 a^{\frac {2}{3}} \left (b x +a \right )^{\frac {1}{3}}+2 \,\mathrm {log}\left (\left (b x +a \right )^{\frac {1}{6}}+a^{\frac {1}{6}}\right ) b x +2 \,\mathrm {log}\left (\left (b x +a \right )^{\frac {1}{6}}-a^{\frac {1}{6}}\right ) b x -\mathrm {log}\left (-a^{\frac {1}{6}} \left (b x +a \right )^{\frac {1}{6}}+\left (b x +a \right )^{\frac {1}{3}}+a^{\frac {1}{3}}\right ) b x -\mathrm {log}\left (a^{\frac {1}{6}} \left (b x +a \right )^{\frac {1}{6}}+\left (b x +a \right )^{\frac {1}{3}}+a^{\frac {1}{3}}\right ) b x}{6 a^{\frac {2}{3}} x} \]
(2*sqrt(3)*atan((2*(a + b*x)**(1/6) + a**(1/6))/(a**(1/6)*sqrt(3)))*b*x - 2*sqrt(3)*atan((2*(a + b*x)**(1/6) - a**(1/6))/(a**(1/6)*sqrt(3)))*b*x - 6 *a**(2/3)*(a + b*x)**(1/3) + 2*log((a + b*x)**(1/6) + a**(1/6))*b*x + 2*lo g((a + b*x)**(1/6) - a**(1/6))*b*x - log( - a**(1/6)*(a + b*x)**(1/6) + (a + b*x)**(1/3) + a**(1/3))*b*x - log(a**(1/6)*(a + b*x)**(1/6) + (a + b*x) **(1/3) + a**(1/3))*b*x)/(6*a**(2/3)*x)