Integrand size = 21, antiderivative size = 183 \[ \int \frac {x \left (c+d x+e x^2\right )}{a+b x^3} \, dx=\frac {e x}{b}-\frac {\left (b^{2/3} c-a^{2/3} e\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{a} b^{4/3}}-\frac {\left (b^{2/3} c+a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{4/3}}+\frac {\left (b^{2/3} c+a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{a} b^{4/3}}+\frac {d \log \left (a+b x^3\right )}{3 b} \]
e*x/b-1/3*(b^(2/3)*c+a^(2/3)*e)*ln(a^(1/3)+b^(1/3)*x)/a^(1/3)/b^(4/3)+1/6* (b^(2/3)*c+a^(2/3)*e)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(1/3)/b^ (4/3)+1/3*d*ln(b*x^3+a)/b-1/3*(b^(2/3)*c-a^(2/3)*e)*arctan(1/3*(a^(1/3)-2* b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(1/3)/b^(4/3)*3^(1/2)
Time = 0.06 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.09 \[ \int \frac {x \left (c+d x+e x^2\right )}{a+b x^3} \, dx=\frac {e x}{b}+\frac {\left (a^{2/3} b c-a^{4/3} \sqrt [3]{b} e\right ) \arctan \left (\frac {-\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a b^{5/3}}+\frac {\left (-a^{2/3} b c-a^{4/3} \sqrt [3]{b} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a b^{5/3}}-\frac {\left (-a^{2/3} b c-a^{4/3} \sqrt [3]{b} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a b^{5/3}}+\frac {d \log \left (a+b x^3\right )}{3 b} \]
(e*x)/b + ((a^(2/3)*b*c - a^(4/3)*b^(1/3)*e)*ArcTan[(-a^(1/3) + 2*b^(1/3)* x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a*b^(5/3)) + ((-(a^(2/3)*b*c) - a^(4/3)*b^ (1/3)*e)*Log[a^(1/3) + b^(1/3)*x])/(3*a*b^(5/3)) - ((-(a^(2/3)*b*c) - a^(4 /3)*b^(1/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a*b^(5/3 )) + (d*Log[a + b*x^3])/(3*b)
Time = 0.41 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2426, 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.
\(\displaystyle \int \frac {x \left (c+d x+e x^2\right )}{a+b x^3} \, dx\) |
\(\Big \downarrow \) 2426 |
\(\displaystyle \int \left (\frac {e}{b}-\frac {a e-b c x-b d x^2}{b \left (a+b x^3\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (b^{2/3} c-a^{2/3} e\right )}{\sqrt {3} \sqrt [3]{a} b^{4/3}}+\frac {\left (a^{2/3} e+b^{2/3} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{a} b^{4/3}}-\frac {\left (a^{2/3} e+b^{2/3} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{4/3}}+\frac {d \log \left (a+b x^3\right )}{3 b}+\frac {e x}{b}\) |
(e*x)/b - ((b^(2/3)*c - a^(2/3)*e)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3] *a^(1/3))])/(Sqrt[3]*a^(1/3)*b^(4/3)) - ((b^(2/3)*c + a^(2/3)*e)*Log[a^(1/ 3) + b^(1/3)*x])/(3*a^(1/3)*b^(4/3)) + ((b^(2/3)*c + a^(2/3)*e)*Log[a^(2/3 ) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(1/3)*b^(4/3)) + (d*Log[a + b*x ^3])/(3*b)
3.4.39.3.1 Defintions of rubi rules used
Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[Pq/(a + b*x^n), x], x] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IntegerQ[n]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.54 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.27
method | result | size |
risch | \(\frac {e x}{b}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (\textit {\_R}^{2} b d +\textit {\_R} b c -a e \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{3 b^{2}}\) | \(49\) |
default | \(\frac {e x}{b}+\frac {-a e \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+b c \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )+\frac {d \ln \left (b \,x^{3}+a \right )}{3}}{b}\) | \(211\) |
Result contains complex when optimal does not.
Time = 1.22 (sec) , antiderivative size = 4628, normalized size of antiderivative = 25.29 \[ \int \frac {x \left (c+d x+e x^2\right )}{a+b x^3} \, dx=\text {Too large to display} \]
-1/12*(2*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(d^2/b^2 - (d^2 - c*e)/b^2)/(2*d^ 3/b^3 - 3*(d^2 - c*e)*d/b^3 - (b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a *b^4) - (b^2*c^3 - a^2*e^3)/(a*b^4))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*( 2*d^3/b^3 - 3*(d^2 - c*e)*d/b^3 - (b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b )/(a*b^4) - (b^2*c^3 - a^2*e^3)/(a*b^4))^(1/3) - 2*d/b)*b*log(-1/4*(2*(1/2 )^(2/3)*(-I*sqrt(3) + 1)*(d^2/b^2 - (d^2 - c*e)/b^2)/(2*d^3/b^3 - 3*(d^2 - c*e)*d/b^3 - (b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a*b^4) - (b^2*c^3 - a^2*e^3)/(a*b^4))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(2*d^3/b^3 - 3*(d ^2 - c*e)*d/b^3 - (b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a*b^4) - (b^2 *c^3 - a^2*e^3)/(a*b^4))^(1/3) - 2*d/b)^2*a*b^3*c - a*b*c*d^2 + 2*a*b*c^2* e + a^2*d*e^2 - 1/2*(2*a*b^2*c*d - a^2*b*e^2)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(d^2/b^2 - (d^2 - c*e)/b^2)/(2*d^3/b^3 - 3*(d^2 - c*e)*d/b^3 - (b^2*c^ 3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a*b^4) - (b^2*c^3 - a^2*e^3)/(a*b^4))^ (1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(2*d^3/b^3 - 3*(d^2 - c*e)*d/b^3 - (b^ 2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a*b^4) - (b^2*c^3 - a^2*e^3)/(a*b^ 4))^(1/3) - 2*d/b) - (b^2*c^3 - a^2*e^3)*x) - 12*e*x - ((2*(1/2)^(2/3)*(-I *sqrt(3) + 1)*(d^2/b^2 - (d^2 - c*e)/b^2)/(2*d^3/b^3 - 3*(d^2 - c*e)*d/b^3 - (b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a*b^4) - (b^2*c^3 - a^2*e^3) /(a*b^4))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(2*d^3/b^3 - 3*(d^2 - c*e)*d /b^3 - (b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a*b^4) - (b^2*c^3 - a...
Time = 0.75 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.87 \[ \int \frac {x \left (c+d x+e x^2\right )}{a+b x^3} \, dx=\operatorname {RootSum} {\left (27 t^{3} a b^{4} - 27 t^{2} a b^{3} d + t \left (- 9 a b^{2} c e + 9 a b^{2} d^{2}\right ) + a^{2} e^{3} + 3 a b c d e - a b d^{3} + b^{2} c^{3}, \left ( t \mapsto t \log {\left (x + \frac {- 9 t^{2} a b^{3} c - 3 t a^{2} b e^{2} + 6 t a b^{2} c d + a^{2} d e^{2} + 2 a b c^{2} e - a b c d^{2}}{a^{2} e^{3} - b^{2} c^{3}} \right )} \right )\right )} + \frac {e x}{b} \]
RootSum(27*_t**3*a*b**4 - 27*_t**2*a*b**3*d + _t*(-9*a*b**2*c*e + 9*a*b**2 *d**2) + a**2*e**3 + 3*a*b*c*d*e - a*b*d**3 + b**2*c**3, Lambda(_t, _t*log (x + (-9*_t**2*a*b**3*c - 3*_t*a**2*b*e**2 + 6*_t*a*b**2*c*d + a**2*d*e**2 + 2*a*b*c**2*e - a*b*c*d**2)/(a**2*e**3 - b**2*c**3)))) + e*x/b
Time = 0.28 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.95 \[ \int \frac {x \left (c+d x+e x^2\right )}{a+b x^3} \, dx=\frac {e x}{b} + \frac {\sqrt {3} {\left (b c \left (\frac {a}{b}\right )^{\frac {2}{3}} - a e \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a b} + \frac {{\left (2 \, b d \left (\frac {a}{b}\right )^{\frac {2}{3}} + b c \left (\frac {a}{b}\right )^{\frac {1}{3}} + a e\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (b d \left (\frac {a}{b}\right )^{\frac {2}{3}} - b c \left (\frac {a}{b}\right )^{\frac {1}{3}} - a e\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
e*x/b + 1/3*sqrt(3)*(b*c*(a/b)^(2/3) - a*e*(a/b)^(1/3))*arctan(1/3*sqrt(3) *(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a*b) + 1/6*(2*b*d*(a/b)^(2/3) + b*c*(a/ b)^(1/3) + a*e)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(b^2*(a/b)^(2/3)) + 1/3*(b*d*(a/b)^(2/3) - b*c*(a/b)^(1/3) - a*e)*log(x + (a/b)^(1/3))/(b^2*( a/b)^(2/3))
Time = 0.28 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.95 \[ \int \frac {x \left (c+d x+e x^2\right )}{a+b x^3} \, dx=\frac {\sqrt {3} {\left (a e + \left (-a b^{2}\right )^{\frac {1}{3}} c\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, \left (-a b^{2}\right )^{\frac {2}{3}}} + \frac {{\left (a e - \left (-a b^{2}\right )^{\frac {1}{3}} c\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, \left (-a b^{2}\right )^{\frac {2}{3}}} + \frac {e x}{b} + \frac {d \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b} - \frac {{\left (b^{3} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a b^{2} e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a b^{3}} \]
1/3*sqrt(3)*(a*e + (-a*b^2)^(1/3)*c)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3 ))/(-a/b)^(1/3))/(-a*b^2)^(2/3) + 1/6*(a*e - (-a*b^2)^(1/3)*c)*log(x^2 + x *(-a/b)^(1/3) + (-a/b)^(2/3))/(-a*b^2)^(2/3) + e*x/b + 1/3*d*log(abs(b*x^3 + a))/b - 1/3*(b^3*c*(-a/b)^(1/3) - a*b^2*e)*(-a/b)^(1/3)*log(abs(x - (-a /b)^(1/3)))/(a*b^3)
Time = 9.11 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.45 \[ \int \frac {x \left (c+d x+e x^2\right )}{a+b x^3} \, dx=\left (\sum _{k=1}^3\ln \left (x\,\left (b\,c^2+a\,d\,e\right )-\mathrm {root}\left (27\,a\,b^4\,z^3-27\,a\,b^3\,d\,z^2-9\,a\,b^2\,c\,e\,z+9\,a\,b^2\,d^2\,z+3\,a\,b\,c\,d\,e-a\,b\,d^3+a^2\,e^3+b^2\,c^3,z,k\right )\,\left (6\,a\,b\,d-\mathrm {root}\left (27\,a\,b^4\,z^3-27\,a\,b^3\,d\,z^2-9\,a\,b^2\,c\,e\,z+9\,a\,b^2\,d^2\,z+3\,a\,b\,c\,d\,e-a\,b\,d^3+a^2\,e^3+b^2\,c^3,z,k\right )\,a\,b^2\,9+3\,a\,b\,e\,x\right )+a\,d^2-a\,c\,e\right )\,\mathrm {root}\left (27\,a\,b^4\,z^3-27\,a\,b^3\,d\,z^2-9\,a\,b^2\,c\,e\,z+9\,a\,b^2\,d^2\,z+3\,a\,b\,c\,d\,e-a\,b\,d^3+a^2\,e^3+b^2\,c^3,z,k\right )\right )+\frac {e\,x}{b} \]
symsum(log(x*(b*c^2 + a*d*e) - root(27*a*b^4*z^3 - 27*a*b^3*d*z^2 - 9*a*b^ 2*c*e*z + 9*a*b^2*d^2*z + 3*a*b*c*d*e - a*b*d^3 + a^2*e^3 + b^2*c^3, z, k) *(6*a*b*d - 9*root(27*a*b^4*z^3 - 27*a*b^3*d*z^2 - 9*a*b^2*c*e*z + 9*a*b^2 *d^2*z + 3*a*b*c*d*e - a*b*d^3 + a^2*e^3 + b^2*c^3, z, k)*a*b^2 + 3*a*b*e* x) + a*d^2 - a*c*e)*root(27*a*b^4*z^3 - 27*a*b^3*d*z^2 - 9*a*b^2*c*e*z + 9 *a*b^2*d^2*z + 3*a*b*c*d*e - a*b*d^3 + a^2*e^3 + b^2*c^3, z, k), k, 1, 3) + (e*x)/b