Integrand size = 33, antiderivative size = 118 \[ \int \left (-\frac {C x^2}{a+b x^3}+\frac {A+C x^2}{a+b x^3}\right ) \, dx=-\frac {A \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} \sqrt [3]{b}}+\frac {A \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}-\frac {A \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} \sqrt [3]{b}} \]
1/3*A*ln(a^(1/3)+b^(1/3)*x)/a^(2/3)/b^(1/3)-1/6*A*ln(a^(2/3)-a^(1/3)*b^(1/ 3)*x+b^(2/3)*x^2)/a^(2/3)/b^(1/3)-1/3*A*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a ^(1/3)*3^(1/2))/a^(2/3)/b^(1/3)*3^(1/2)
Time = 0.01 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.76 \[ \int \left (-\frac {C x^2}{a+b x^3}+\frac {A+C x^2}{a+b x^3}\right ) \, dx=-\frac {A \left (2 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )\right )}{6 a^{2/3} \sqrt [3]{b}} \]
-1/6*(A*(2*Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] - 2*Log[a^( 1/3) + b^(1/3)*x] + Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]))/(a^(2 /3)*b^(1/3))
Time = 0.28 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {2009}
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.
\(\displaystyle \int \left (\frac {A+C x^2}{a+b x^3}-\frac {C x^2}{a+b x^3}\right ) \, dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {A \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} \sqrt [3]{b}}-\frac {A \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} \sqrt [3]{b}}+\frac {A \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\) |
-((A*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(2/3)*b ^(1/3))) + (A*Log[a^(1/3) + b^(1/3)*x])/(3*a^(2/3)*b^(1/3)) - (A*Log[a^(2/ 3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(2/3)*b^(1/3))
3.1.23.3.1 Defintions of rubi rules used
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.62 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.42
method | result | size |
risch | \(-\frac {C \ln \left (b \,x^{3}+a \right )}{3 b}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (C \,\textit {\_R}^{2}+A \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{3 b}\) | \(49\) |
default | \(\frac {A \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {A \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {A \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\) | \(94\) |
Time = 0.29 (sec) , antiderivative size = 305, normalized size of antiderivative = 2.58 \[ \int \left (-\frac {C x^2}{a+b x^3}+\frac {A+C x^2}{a+b x^3}\right ) \, dx=\left [\frac {3 \, \sqrt {\frac {1}{3}} A a b \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, a b x^{3} - 3 \, \left (a^{2} b\right )^{\frac {1}{3}} a x - a^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b x^{2} + \left (a^{2} b\right )^{\frac {2}{3}} x - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{b x^{3} + a}\right ) - \left (a^{2} b\right )^{\frac {2}{3}} A \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac {2}{3}} x + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) + 2 \, \left (a^{2} b\right )^{\frac {2}{3}} A \log \left (a b x + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{6 \, a^{2} b}, \frac {6 \, \sqrt {\frac {1}{3}} A a b \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (a^{2} b\right )^{\frac {2}{3}} x - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{a^{2}}\right ) - \left (a^{2} b\right )^{\frac {2}{3}} A \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac {2}{3}} x + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) + 2 \, \left (a^{2} b\right )^{\frac {2}{3}} A \log \left (a b x + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{6 \, a^{2} b}\right ] \]
[1/6*(3*sqrt(1/3)*A*a*b*sqrt(-(a^2*b)^(1/3)/b)*log((2*a*b*x^3 - 3*(a^2*b)^ (1/3)*a*x - a^2 + 3*sqrt(1/3)*(2*a*b*x^2 + (a^2*b)^(2/3)*x - (a^2*b)^(1/3) *a)*sqrt(-(a^2*b)^(1/3)/b))/(b*x^3 + a)) - (a^2*b)^(2/3)*A*log(a*b*x^2 - ( a^2*b)^(2/3)*x + (a^2*b)^(1/3)*a) + 2*(a^2*b)^(2/3)*A*log(a*b*x + (a^2*b)^ (2/3)))/(a^2*b), 1/6*(6*sqrt(1/3)*A*a*b*sqrt((a^2*b)^(1/3)/b)*arctan(sqrt( 1/3)*(2*(a^2*b)^(2/3)*x - (a^2*b)^(1/3)*a)*sqrt((a^2*b)^(1/3)/b)/a^2) - (a ^2*b)^(2/3)*A*log(a*b*x^2 - (a^2*b)^(2/3)*x + (a^2*b)^(1/3)*a) + 2*(a^2*b) ^(2/3)*A*log(a*b*x + (a^2*b)^(2/3)))/(a^2*b)]
Time = 0.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.19 \[ \int \left (-\frac {C x^2}{a+b x^3}+\frac {A+C x^2}{a+b x^3}\right ) \, dx=A \operatorname {RootSum} {\left (27 t^{3} a^{2} b - 1, \left ( t \mapsto t \log {\left (3 t a + x \right )} \right )\right )} \]
Time = 0.27 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.35 \[ \int \left (-\frac {C x^2}{a+b x^3}+\frac {A+C x^2}{a+b x^3}\right ) \, dx=-\frac {C \log \left (b x^{3} + a\right )}{3 \, b} - \frac {\sqrt {3} {\left (2 \, C a - {\left (3 \, A \left (\frac {a}{b}\right )^{\frac {1}{3}} + \frac {2 \, C a}{b}\right )} b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a b} + \frac {{\left (2 \, C \left (\frac {a}{b}\right )^{\frac {2}{3}} - A\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (C \left (\frac {a}{b}\right )^{\frac {2}{3}} + A\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
-1/3*C*log(b*x^3 + a)/b - 1/9*sqrt(3)*(2*C*a - (3*A*(a/b)^(1/3) + 2*C*a/b) *b)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a*b) + 1/6*(2*C*( a/b)^(2/3) - A)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(b*(a/b)^(2/3)) + 1 /3*(C*(a/b)^(2/3) + A)*log(x + (a/b)^(1/3))/(b*(a/b)^(2/3))
Time = 0.27 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.97 \[ \int \left (-\frac {C x^2}{a+b x^3}+\frac {A+C x^2}{a+b x^3}\right ) \, dx=-\frac {A \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a} + \frac {\sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} A \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a b} + \frac {\left (-a b^{2}\right )^{\frac {1}{3}} A \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a b} \]
-1/3*A*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/a + 1/3*sqrt(3)*(-a*b^2)^(1 /3)*A*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a*b) + 1/6*(- a*b^2)^(1/3)*A*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a*b)
Time = 9.41 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.81 \[ \int \left (-\frac {C x^2}{a+b x^3}+\frac {A+C x^2}{a+b x^3}\right ) \, dx=\frac {A\,\ln \left (b^{1/3}\,x+a^{1/3}\right )}{3\,a^{2/3}\,b^{1/3}}-\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x-\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (A-\sqrt {3}\,A\,1{}\mathrm {i}\right )}{6\,a^{2/3}\,b^{1/3}}-\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}-\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (A+\sqrt {3}\,A\,1{}\mathrm {i}\right )}{6\,a^{2/3}\,b^{1/3}} \]