Integrand size = 13, antiderivative size = 100 \[ \int \frac {1}{x^2 \sqrt [3]{a+b x}} \, dx=-\frac {(a+b x)^{2/3}}{a x}-\frac {b \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{4/3}}+\frac {b \log (x)}{6 a^{4/3}}-\frac {b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{4/3}} \]
-(b*x+a)^(2/3)/a/x+1/6*b*ln(x)/a^(4/3)-1/2*b*ln(a^(1/3)-(b*x+a)^(1/3))/a^( 4/3)-1/3*b*arctan(1/3*(a^(1/3)+2*(b*x+a)^(1/3))/a^(1/3)*3^(1/2))/a^(4/3)*3 ^(1/2)
Time = 0.14 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.20 \[ \int \frac {1}{x^2 \sqrt [3]{a+b x}} \, dx=-\frac {6 \sqrt [3]{a} (a+b x)^{2/3}+2 \sqrt {3} b x \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 b x \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )-b x \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right )}{6 a^{4/3} x} \]
-1/6*(6*a^(1/3)*(a + b*x)^(2/3) + 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^(4/3)*x)
Time = 0.20 (sec) , antiderivative size = 101, 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 = {52, 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.
| \(\displaystyle \int \frac {1}{x^2 \sqrt [3]{a+b x}} \, dx\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {b \int \frac {1}{x \sqrt [3]{a+b x}}dx}{3 a}-\frac {(a+b x)^{2/3}}{a x}\) |
| \(\Big \downarrow \) 67 |
| \(\displaystyle -\frac {b \left (\frac {3}{2} \int \frac {1}{a^{2/3}+\sqrt [3]{a+b x} \sqrt [3]{a}+(a+b x)^{2/3}}d\sqrt [3]{a+b x}-\frac {3 \int \frac {1}{\sqrt [3]{a}-\sqrt [3]{a+b x}}d\sqrt [3]{a+b x}}{2 \sqrt [3]{a}}-\frac {\log (x)}{2 \sqrt [3]{a}}\right )}{3 a}-\frac {(a+b x)^{2/3}}{a x}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle -\frac {b \left (\frac {3}{2} \int \frac {1}{a^{2/3}+\sqrt [3]{a+b x} \sqrt [3]{a}+(a+b x)^{2/3}}d\sqrt [3]{a+b x}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 \sqrt [3]{a}}-\frac {\log (x)}{2 \sqrt [3]{a}}\right )}{3 a}-\frac {(a+b x)^{2/3}}{a x}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {b \left (-\frac {3 \int \frac {1}{-(a+b x)^{2/3}-3}d\left (\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}+1\right )}{\sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 \sqrt [3]{a}}-\frac {\log (x)}{2 \sqrt [3]{a}}\right )}{3 a}-\frac {(a+b x)^{2/3}}{a x}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {b \left (\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )}{\sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 \sqrt [3]{a}}-\frac {\log (x)}{2 \sqrt [3]{a}}\right )}{3 a}-\frac {(a+b x)^{2/3}}{a x}\) |
-((a + b*x)^(2/3)/(a*x)) - (b*((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*a)
3.4.97.3.1 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] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
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 = 0.09 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.95
| method | result | size |
| risch | \(-\frac {\left (b x +a \right )^{\frac {2}{3}}}{a x}-\frac {b \ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{3 a^{\frac {4}{3}}}+\frac {b \ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{6 a^{\frac {4}{3}}}-\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{3 a^{\frac {4}{3}}}\) | \(95\) |
| pseudoelliptic | \(\frac {-2 \arctan \left (\frac {\left (a^{\frac {1}{3}}+2 \left (b x +a \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}\, b x -6 \left (b x +a \right )^{\frac {2}{3}} a^{\frac {1}{3}}-2 \ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right ) b x +\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right ) b x}{6 a^{\frac {4}{3}} x}\) | \(95\) |
| derivativedivides | \(3 b \left (-\frac {\left (b x +a \right )^{\frac {2}{3}}}{3 a b x}+\frac {-\frac {\ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{3 a^{\frac {1}{3}}}+\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{6 a^{\frac {1}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{3 a^{\frac {1}{3}}}}{3 a}\right )\) | \(104\) |
| default | \(3 b \left (-\frac {\left (b x +a \right )^{\frac {2}{3}}}{3 a b x}+\frac {-\frac {\ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{3 a^{\frac {1}{3}}}+\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{6 a^{\frac {1}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{3 a^{\frac {1}{3}}}}{3 a}\right )\) | \(104\) |
-(b*x+a)^(2/3)/a/x-1/3*b/a^(4/3)*ln((b*x+a)^(1/3)-a^(1/3))+1/6*b/a^(4/3)*l n((b*x+a)^(2/3)+a^(1/3)*(b*x+a)^(1/3)+a^(2/3))-1/3*b/a^(4/3)*3^(1/2)*arcta n(1/3*3^(1/2)*(2/a^(1/3)*(b*x+a)^(1/3)+1))
Time = 0.24 (sec) , antiderivative size = 306, normalized size of antiderivative = 3.06 \[ \int \frac {1}{x^2 \sqrt [3]{a+b x}} \, dx=\left [\frac {3 \, \sqrt {\frac {1}{3}} a b x \sqrt {\frac {\left (-a\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b x - 3 \, \sqrt {\frac {1}{3}} {\left (2 \, {\left (b x + a\right )}^{\frac {2}{3}} \left (-a\right )^{\frac {2}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} a + \left (-a\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (-a\right )^{\frac {1}{3}}}{a}} - 3 \, {\left (b x + a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {2}{3}} + 3 \, a}{x}\right ) + \left (-a\right )^{\frac {2}{3}} b x \log \left ({\left (b x + a\right )}^{\frac {2}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} + \left (-a\right )^{\frac {2}{3}}\right ) - 2 \, \left (-a\right )^{\frac {2}{3}} b x \log \left ({\left (b x + a\right )}^{\frac {1}{3}} + \left (-a\right )^{\frac {1}{3}}\right ) - 6 \, {\left (b x + a\right )}^{\frac {2}{3}} a}{6 \, a^{2} x}, -\frac {6 \, \sqrt {\frac {1}{3}} a b x \sqrt {-\frac {\left (-a\right )^{\frac {1}{3}}}{a}} \arctan \left (\sqrt {\frac {1}{3}} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} - \left (-a\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-a\right )^{\frac {1}{3}}}{a}}\right ) - \left (-a\right )^{\frac {2}{3}} b x \log \left ({\left (b x + a\right )}^{\frac {2}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} + \left (-a\right )^{\frac {2}{3}}\right ) + 2 \, \left (-a\right )^{\frac {2}{3}} b x \log \left ({\left (b x + a\right )}^{\frac {1}{3}} + \left (-a\right )^{\frac {1}{3}}\right ) + 6 \, {\left (b x + a\right )}^{\frac {2}{3}} a}{6 \, a^{2} x}\right ] \]
[1/6*(3*sqrt(1/3)*a*b*x*sqrt((-a)^(1/3)/a)*log((2*b*x - 3*sqrt(1/3)*(2*(b* x + a)^(2/3)*(-a)^(2/3) - (b*x + a)^(1/3)*a + (-a)^(1/3)*a)*sqrt((-a)^(1/3 )/a) - 3*(b*x + a)^(1/3)*(-a)^(2/3) + 3*a)/x) + (-a)^(2/3)*b*x*log((b*x + a)^(2/3) - (b*x + a)^(1/3)*(-a)^(1/3) + (-a)^(2/3)) - 2*(-a)^(2/3)*b*x*log ((b*x + a)^(1/3) + (-a)^(1/3)) - 6*(b*x + a)^(2/3)*a)/(a^2*x), -1/6*(6*sqr t(1/3)*a*b*x*sqrt(-(-a)^(1/3)/a)*arctan(sqrt(1/3)*(2*(b*x + a)^(1/3) - (-a )^(1/3))*sqrt(-(-a)^(1/3)/a)) - (-a)^(2/3)*b*x*log((b*x + a)^(2/3) - (b*x + a)^(1/3)*(-a)^(1/3) + (-a)^(2/3)) + 2*(-a)^(2/3)*b*x*log((b*x + a)^(1/3) + (-a)^(1/3)) + 6*(b*x + a)^(2/3)*a)/(a^2*x)]
Result contains complex when optimal does not.
Time = 1.90 (sec) , antiderivative size = 831, normalized size of antiderivative = 8.31 \[ \int \frac {1}{x^2 \sqrt [3]{a+b x}} \, dx=\text {Too large to display} \]
-2*a**(5/3)*b**(7/3)*(a/b + x)**(4/3)*exp(2*I*pi/3)*log(1 - b**(1/3)*(a/b + x)**(1/3)/a**(1/3))*gamma(2/3)/(9*a**3*b**(4/3)*(a/b + x)**(4/3)*exp(2*I *pi/3)*gamma(5/3) - 9*a**2*b**(7/3)*(a/b + x)**(7/3)*exp(2*I*pi/3)*gamma(5 /3)) - 2*a**(5/3)*b**(7/3)*(a/b + x)**(4/3)*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(2/3)/(9*a**3*b**(4/ 3)*(a/b + x)**(4/3)*exp(2*I*pi/3)*gamma(5/3) - 9*a**2*b**(7/3)*(a/b + x)** (7/3)*exp(2*I*pi/3)*gamma(5/3)) - 2*a**(5/3)*b**(7/3)*(a/b + x)**(4/3)*log (1 - b**(1/3)*(a/b + x)**(1/3)*exp_polar(4*I*pi/3)/a**(1/3))*gamma(2/3)/(9 *a**3*b**(4/3)*(a/b + x)**(4/3)*exp(2*I*pi/3)*gamma(5/3) - 9*a**2*b**(7/3) *(a/b + x)**(7/3)*exp(2*I*pi/3)*gamma(5/3)) + 2*a**(2/3)*b**(10/3)*(a/b + x)**(7/3)*exp(2*I*pi/3)*log(1 - b**(1/3)*(a/b + x)**(1/3)/a**(1/3))*gamma( 2/3)/(9*a**3*b**(4/3)*(a/b + x)**(4/3)*exp(2*I*pi/3)*gamma(5/3) - 9*a**2*b **(7/3)*(a/b + x)**(7/3)*exp(2*I*pi/3)*gamma(5/3)) + 2*a**(2/3)*b**(10/3)* (a/b + x)**(7/3)*exp(-2*I*pi/3)*log(1 - b**(1/3)*(a/b + x)**(1/3)*exp_pola r(2*I*pi/3)/a**(1/3))*gamma(2/3)/(9*a**3*b**(4/3)*(a/b + x)**(4/3)*exp(2*I *pi/3)*gamma(5/3) - 9*a**2*b**(7/3)*(a/b + x)**(7/3)*exp(2*I*pi/3)*gamma(5 /3)) + 2*a**(2/3)*b**(10/3)*(a/b + x)**(7/3)*log(1 - b**(1/3)*(a/b + x)**( 1/3)*exp_polar(4*I*pi/3)/a**(1/3))*gamma(2/3)/(9*a**3*b**(4/3)*(a/b + x)** (4/3)*exp(2*I*pi/3)*gamma(5/3) - 9*a**2*b**(7/3)*(a/b + x)**(7/3)*exp(2*I* pi/3)*gamma(5/3)) + 6*a*b**3*(a/b + x)**2*exp(2*I*pi/3)*gamma(2/3)/(9*a...
Time = 0.31 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.06 \[ \int \frac {1}{x^2 \sqrt [3]{a+b x}} \, dx=-\frac {\sqrt {3} b \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{3 \, a^{\frac {4}{3}}} - \frac {{\left (b x + a\right )}^{\frac {2}{3}} b}{{\left (b x + a\right )} a - a^{2}} + \frac {b \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{6 \, a^{\frac {4}{3}}} - \frac {b \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{3 \, a^{\frac {4}{3}}} \]
-1/3*sqrt(3)*b*arctan(1/3*sqrt(3)*(2*(b*x + a)^(1/3) + a^(1/3))/a^(1/3))/a ^(4/3) - (b*x + a)^(2/3)*b/((b*x + a)*a - a^2) + 1/6*b*log((b*x + a)^(2/3) + (b*x + a)^(1/3)*a^(1/3) + a^(2/3))/a^(4/3) - 1/3*b*log((b*x + a)^(1/3) - a^(1/3))/a^(4/3)
Time = 0.56 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^2 \sqrt [3]{a+b x}} \, dx=-\frac {\frac {2 \, \sqrt {3} b^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {4}{3}}} - \frac {b^{2} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{a^{\frac {4}{3}}} + \frac {2 \, b^{2} \log \left ({\left | {\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{a^{\frac {4}{3}}} + \frac {6 \, {\left (b x + a\right )}^{\frac {2}{3}} b}{a x}}{6 \, b} \]
-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^(4/3) - b^2*log((b*x + a)^(2/3) + (b*x + a)^(1/3)*a^(1/3) + a^(2/3)) /a^(4/3) + 2*b^2*log(abs((b*x + a)^(1/3) - a^(1/3)))/a^(4/3) + 6*(b*x + a) ^(2/3)*b/(a*x))/b
Time = 0.17 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.30 \[ \int \frac {1}{x^2 \sqrt [3]{a+b x}} \, dx=-\frac {{\left (a+b\,x\right )}^{2/3}}{a\,x}+\frac {\ln \left (\frac {{\left (b-\sqrt {3}\,b\,1{}\mathrm {i}\right )}^2}{4\,a^{5/3}}-\frac {b^2\,{\left (a+b\,x\right )}^{1/3}}{a^2}\right )\,\left (b-\sqrt {3}\,b\,1{}\mathrm {i}\right )}{6\,a^{4/3}}+\frac {\ln \left (\frac {{\left (b+\sqrt {3}\,b\,1{}\mathrm {i}\right )}^2}{4\,a^{5/3}}-\frac {b^2\,{\left (a+b\,x\right )}^{1/3}}{a^2}\right )\,\left (b+\sqrt {3}\,b\,1{}\mathrm {i}\right )}{6\,a^{4/3}}-\frac {b\,\ln \left ({\left (a+b\,x\right )}^{1/3}-a^{1/3}\right )}{3\,a^{4/3}} \]