Integrand size = 36, antiderivative size = 161 \[ \int \left (-\frac {C x^2}{a+b x^3}+\frac {A+B x+C x^2}{a+b x^3}\right ) \, dx=-\frac {\left (A \sqrt [3]{b}+\sqrt [3]{a} B\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} b^{2/3}}+\frac {\left (A \sqrt [3]{b}-\sqrt [3]{a} B\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{2/3}}-\frac {\left (A-\frac {\sqrt [3]{a} B}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} \sqrt [3]{b}} \] Output:
-1/3*(A*b^(1/3)+a^(1/3)*B)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)*3^(1/2)/a^(1/3 ))*3^(1/2)/a^(2/3)/b^(2/3)+1/3*(A*b^(1/3)-a^(1/3)*B)*ln(a^(1/3)+b^(1/3)*x) /a^(2/3)/b^(2/3)-1/6*(A-a^(1/3)*B/b^(1/3))*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^ (2/3)*x^2)/a^(2/3)/b^(1/3)
Time = 0.03 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.77 \[ \int \left (-\frac {C x^2}{a+b x^3}+\frac {A+B x+C x^2}{a+b x^3}\right ) \, dx=\frac {-2 \sqrt {3} \left (A \sqrt [3]{b}+\sqrt [3]{a} B\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+\left (A \sqrt [3]{b}-\sqrt [3]{a} B\right ) \left (2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )\right )}{6 a^{2/3} b^{2/3}} \] Input:
Integrate[-((C*x^2)/(a + b*x^3)) + (A + B*x + C*x^2)/(a + b*x^3),x]
Output:
(-2*Sqrt[3]*(A*b^(1/3) + a^(1/3)*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqr t[3]] + (A*b^(1/3) - a^(1/3)*B)*(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]))/(6*a^(2/3)*b^(2/3))
Time = 0.57 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, 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.
\(\displaystyle \int \left (\frac {A+B x+C x^2}{a+b x^3}-\frac {C x^2}{a+b x^3}\right ) \, dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\left (\sqrt [3]{a} B+A \sqrt [3]{b}\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} b^{2/3}}-\frac {\left (A-\frac {\sqrt [3]{a} B}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} \sqrt [3]{b}}+\frac {\left (A \sqrt [3]{b}-\sqrt [3]{a} B\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{2/3}}\) |
Input:
Int[-((C*x^2)/(a + b*x^3)) + (A + B*x + C*x^2)/(a + b*x^3),x]
Output:
-(((A*b^(1/3) + a^(1/3)*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3) )])/(Sqrt[3]*a^(2/3)*b^(2/3))) + ((A*b^(1/3) - a^(1/3)*B)*Log[a^(1/3) + b^ (1/3)*x])/(3*a^(2/3)*b^(2/3)) - ((A - (a^(1/3)*B)/b^(1/3))*Log[a^(2/3) - a ^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(2/3)*b^(1/3))
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.10 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.32
method | result | size |
risch | \(-\frac {C \ln \left (b \,x^{3}+a \right )}{3 b}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b +a \right )}{\sum }\frac {\left (C \,\textit {\_R}^{2}+B \textit {\_R} +A \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{3 b}\) | \(52\) |
default | \(\frac {A \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {A \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {A \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {B \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {B \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {B \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\) | \(186\) |
Input:
int(-C*x^2/(b*x^3+a)+(C*x^2+B*x+A)/(b*x^3+a),x,method=_RETURNVERBOSE)
Output:
-1/3*C/b*ln(b*x^3+a)+1/3/b*sum((C*_R^2+B*_R+A)/_R^2*ln(x-_R),_R=RootOf(_Z^ 3*b+a))
Result contains complex when optimal does not.
Time = 0.87 (sec) , antiderivative size = 1961, normalized size of antiderivative = 12.18 \[ \int \left (-\frac {C x^2}{a+b x^3}+\frac {A+B x+C x^2}{a+b x^3}\right ) \, dx=\text {Too large to display} \] Input:
integrate(-C*x^2/(b*x^3+a)+(C*x^2+B*x+A)/(b*x^3+a),x, algorithm="fricas")
Output:
-1/6*((1/2)^(1/3)*(I*sqrt(3) + 1)*((B^3*a + A^3*b)/(a^2*b^2) - (B^3*a - A^ 3*b)/(a^2*b^2))^(1/3) - 2*(1/2)^(2/3)*A*B*(-I*sqrt(3) + 1)/(a*b*((B^3*a + A^3*b)/(a^2*b^2) - (B^3*a - A^3*b)/(a^2*b^2))^(1/3)))*log(1/4*((1/2)^(1/3) *(I*sqrt(3) + 1)*((B^3*a + A^3*b)/(a^2*b^2) - (B^3*a - A^3*b)/(a^2*b^2))^( 1/3) - 2*(1/2)^(2/3)*A*B*(-I*sqrt(3) + 1)/(a*b*((B^3*a + A^3*b)/(a^2*b^2) - (B^3*a - A^3*b)/(a^2*b^2))^(1/3)))^2*B*a^2*b - 1/2*((1/2)^(1/3)*(I*sqrt( 3) + 1)*((B^3*a + A^3*b)/(a^2*b^2) - (B^3*a - A^3*b)/(a^2*b^2))^(1/3) - 2* (1/2)^(2/3)*A*B*(-I*sqrt(3) + 1)/(a*b*((B^3*a + A^3*b)/(a^2*b^2) - (B^3*a - A^3*b)/(a^2*b^2))^(1/3)))*A^2*a*b + 2*A*B^2*a + (B^3*a + A^3*b)*x) + 1/1 2*((1/2)^(1/3)*(I*sqrt(3) + 1)*((B^3*a + A^3*b)/(a^2*b^2) - (B^3*a - A^3*b )/(a^2*b^2))^(1/3) - 2*(1/2)^(2/3)*A*B*(-I*sqrt(3) + 1)/(a*b*((B^3*a + A^3 *b)/(a^2*b^2) - (B^3*a - A^3*b)/(a^2*b^2))^(1/3)) + 3*sqrt(1/3)*sqrt(-(((1 /2)^(1/3)*(I*sqrt(3) + 1)*((B^3*a + A^3*b)/(a^2*b^2) - (B^3*a - A^3*b)/(a^ 2*b^2))^(1/3) - 2*(1/2)^(2/3)*A*B*(-I*sqrt(3) + 1)/(a*b*((B^3*a + A^3*b)/( a^2*b^2) - (B^3*a - A^3*b)/(a^2*b^2))^(1/3)))^2*a*b + 16*A*B)/(a*b)))*log( -1/4*((1/2)^(1/3)*(I*sqrt(3) + 1)*((B^3*a + A^3*b)/(a^2*b^2) - (B^3*a - A^ 3*b)/(a^2*b^2))^(1/3) - 2*(1/2)^(2/3)*A*B*(-I*sqrt(3) + 1)/(a*b*((B^3*a + A^3*b)/(a^2*b^2) - (B^3*a - A^3*b)/(a^2*b^2))^(1/3)))^2*B*a^2*b + 1/2*((1/ 2)^(1/3)*(I*sqrt(3) + 1)*((B^3*a + A^3*b)/(a^2*b^2) - (B^3*a - A^3*b)/(a^2 *b^2))^(1/3) - 2*(1/2)^(2/3)*A*B*(-I*sqrt(3) + 1)/(a*b*((B^3*a + A^3*b)...
Time = 0.53 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.47 \[ \int \left (-\frac {C x^2}{a+b x^3}+\frac {A+B x+C x^2}{a+b x^3}\right ) \, dx=\operatorname {RootSum} {\left (27 t^{3} a^{2} b^{2} + 9 t A B a b - A^{3} b + B^{3} a, \left ( t \mapsto t \log {\left (x + \frac {9 t^{2} B a^{2} b + 3 t A^{2} a b + 2 A B^{2} a}{A^{3} b + B^{3} a} \right )} \right )\right )} \] Input:
integrate(-C*x**2/(b*x**3+a)+(C*x**2+B*x+A)/(b*x**3+a),x)
Output:
RootSum(27*_t**3*a**2*b**2 + 9*_t*A*B*a*b - A**3*b + B**3*a, Lambda(_t, _t *log(x + (9*_t**2*B*a**2*b + 3*_t*A**2*a*b + 2*A*B**2*a)/(A**3*b + B**3*a) )))
Time = 0.11 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.17 \[ \int \left (-\frac {C x^2}{a+b x^3}+\frac {A+B x+C x^2}{a+b x^3}\right ) \, dx=-\frac {C \log \left (b x^{3} + a\right )}{3 \, b} - \frac {\sqrt {3} {\left (2 \, C a - {\left (3 \, B \left (\frac {a}{b}\right )^{\frac {2}{3}} + 3 \, A \left (\frac {a}{b}\right )^{\frac {1}{3}} + \frac {2 \, C a}{b}\right )} b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a b} + \frac {{\left (2 \, C \left (\frac {a}{b}\right )^{\frac {2}{3}} + B \left (\frac {a}{b}\right )^{\frac {1}{3}} - A\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (C \left (\frac {a}{b}\right )^{\frac {2}{3}} - B \left (\frac {a}{b}\right )^{\frac {1}{3}} + A\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \] Input:
integrate(-C*x^2/(b*x^3+a)+(C*x^2+B*x+A)/(b*x^3+a),x, algorithm="maxima")
Output:
-1/3*C*log(b*x^3 + a)/b - 1/9*sqrt(3)*(2*C*a - (3*B*(a/b)^(2/3) + 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) + B*(a/b)^(1/3) - 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) - B*(a/b)^(1/3) + A)*l og(x + (a/b)^(1/3))/(b*(a/b)^(2/3))
Time = 0.13 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.91 \[ \int \left (-\frac {C x^2}{a+b x^3}+\frac {A+B x+C x^2}{a+b x^3}\right ) \, dx=-\frac {\sqrt {3} {\left (A b - \left (-a b^{2}\right )^{\frac {1}{3}} B\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, \left (-a b^{2}\right )^{\frac {2}{3}}} - \frac {{\left (A b + \left (-a b^{2}\right )^{\frac {1}{3}} B\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, \left (-a b^{2}\right )^{\frac {2}{3}}} - \frac {{\left (B b \left (-\frac {a}{b}\right )^{\frac {1}{3}} + A b\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a b} \] Input:
integrate(-C*x^2/(b*x^3+a)+(C*x^2+B*x+A)/(b*x^3+a),x, algorithm="giac")
Output:
-1/3*sqrt(3)*(A*b - (-a*b^2)^(1/3)*B)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/ 3))/(-a/b)^(1/3))/(-a*b^2)^(2/3) - 1/6*(A*b + (-a*b^2)^(1/3)*B)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(-a*b^2)^(2/3) - 1/3*(B*b*(-a/b)^(1/3) + A* b)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b)
Time = 5.89 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.79 \[ \int \left (-\frac {C x^2}{a+b x^3}+\frac {A+B x+C x^2}{a+b x^3}\right ) \, dx=\sum _{k=1}^3\ln \left (b\,\left (B^2\,x+A\,B+{\mathrm {root}\left (27\,a^2\,b^2\,z^3+9\,A\,B\,a\,b\,z+B^3\,a-A^3\,b,z,k\right )}^2\,a\,b\,9+A\,\mathrm {root}\left (27\,a^2\,b^2\,z^3+9\,A\,B\,a\,b\,z+B^3\,a-A^3\,b,z,k\right )\,b\,x\,3\right )\right )\,\mathrm {root}\left (27\,a^2\,b^2\,z^3+9\,A\,B\,a\,b\,z+B^3\,a-A^3\,b,z,k\right ) \] Input:
int((A + B*x + C*x^2)/(a + b*x^3) - (C*x^2)/(a + b*x^3),x)
Output:
symsum(log(b*(B^2*x + A*B + 9*root(27*a^2*b^2*z^3 + 9*A*B*a*b*z + B^3*a - A^3*b, z, k)^2*a*b + 3*A*root(27*a^2*b^2*z^3 + 9*A*B*a*b*z + B^3*a - A^3*b , z, k)*b*x))*root(27*a^2*b^2*z^3 + 9*A*B*a*b*z + B^3*a - A^3*b, z, k), k, 1, 3)
Time = 0.15 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.90 \[ \int \left (-\frac {C x^2}{a+b x^3}+\frac {A+B x+C x^2}{a+b x^3}\right ) \, dx=\frac {-2 b^{\frac {1}{3}} a^{\frac {2}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right )-2 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) b -b^{\frac {1}{3}} a^{\frac {2}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right )+2 b^{\frac {1}{3}} a^{\frac {2}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )+\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) b -2 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) b}{6 b^{\frac {2}{3}} a^{\frac {1}{3}}} \] Input:
int(-C*x^2/(b*x^3+a)+(C*x^2+B*x+A)/(b*x^3+a),x)
Output:
( - 2*b**(1/3)*a**(2/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*s qrt(3))) - 2*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*b - b**(1/3)*a**(2/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2) + 2*b**(1/3)*a**(2/3)*log(a**(1/3) + b**(1/3)*x) + log(a**(2/3) - b**(1/3)*a **(1/3)*x + b**(2/3)*x**2)*b - 2*log(a**(1/3) + b**(1/3)*x)*b)/(6*b**(2/3) *a**(1/3))