Integrand size = 21, antiderivative size = 183 \[ \int \frac {x \left (c+d x+e x^2\right )}{a+b x^3} \, dx=\frac {e x}{b}-\frac {\left (b^{2/3} c-a^{2/3} e\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{a} b^{4/3}}-\frac {\left (b^{2/3} c+a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{4/3}}+\frac {\left (b^{2/3} c+a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{a} b^{4/3}}+\frac {d \log \left (a+b x^3\right )}{3 b} \] Output:
e*x/b-1/3*(b^(2/3)*c-a^(2/3)*e)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)*3^(1/2)/a ^(1/3))*3^(1/2)/a^(1/3)/b^(4/3)-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
Time = 0.04 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.09 \[ \int \frac {x \left (c+d x+e x^2\right )}{a+b x^3} \, dx=\frac {e x}{b}+\frac {\left (a^{2/3} b c-a^{4/3} \sqrt [3]{b} e\right ) \arctan \left (\frac {-\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a b^{5/3}}+\frac {\left (-a^{2/3} b c-a^{4/3} \sqrt [3]{b} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a b^{5/3}}-\frac {\left (-a^{2/3} b c-a^{4/3} \sqrt [3]{b} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a b^{5/3}}+\frac {d \log \left (a+b x^3\right )}{3 b} \] Input:
Integrate[(x*(c + d*x + e*x^2))/(a + b*x^3),x]
Output:
(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 = 0.70 (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 |
\(\displaystyle -\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (b^{2/3} c-a^{2/3} e\right )}{\sqrt {3} \sqrt [3]{a} b^{4/3}}+\frac {\left (a^{2/3} e+b^{2/3} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{a} b^{4/3}}-\frac {\left (a^{2/3} e+b^{2/3} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{4/3}}+\frac {d \log \left (a+b x^3\right )}{3 b}+\frac {e x}{b}\) |
Input:
Int[(x*(c + d*x + e*x^2))/(a + b*x^3),x]
Output:
(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)
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 = 0.10 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.27
method | result | size |
risch | \(\frac {e x}{b}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b +a \right )}{\sum }\frac {\left (\textit {\_R}^{2} b d +\textit {\_R} b c -a e \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{3 b^{2}}\) | \(49\) |
default | \(\frac {e x}{b}+\frac {-a e \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+c b \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )+\frac {d \ln \left (b \,x^{3}+a \right )}{3}}{b}\) | \(211\) |
Input:
int(x*(e*x^2+d*x+c)/(b*x^3+a),x,method=_RETURNVERBOSE)
Output:
e*x/b+1/3/b^2*sum((_R^2*b*d+_R*b*c-a*e)/_R^2*ln(x-_R),_R=RootOf(_Z^3*b+a))
Result contains complex when optimal does not.
Time = 0.78 (sec) , antiderivative size = 4628, normalized size of antiderivative = 25.29 \[ \int \frac {x \left (c+d x+e x^2\right )}{a+b x^3} \, dx=\text {Too large to display} \] Input:
integrate(x*(e*x^2+d*x+c)/(b*x^3+a),x, algorithm="fricas")
Output:
Too large to include
Time = 0.73 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.87 \[ \int \frac {x \left (c+d x+e x^2\right )}{a+b x^3} \, dx=\operatorname {RootSum} {\left (27 t^{3} a b^{4} - 27 t^{2} a b^{3} d + t \left (- 9 a b^{2} c e + 9 a b^{2} d^{2}\right ) + a^{2} e^{3} + 3 a b c d e - a b d^{3} + b^{2} c^{3}, \left ( t \mapsto t \log {\left (x + \frac {- 9 t^{2} a b^{3} c - 3 t a^{2} b e^{2} + 6 t a b^{2} c d + a^{2} d e^{2} + 2 a b c^{2} e - a b c d^{2}}{a^{2} e^{3} - b^{2} c^{3}} \right )} \right )\right )} + \frac {e x}{b} \] Input:
integrate(x*(e*x**2+d*x+c)/(b*x**3+a),x)
Output:
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 = 0.11 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.95 \[ \int \frac {x \left (c+d x+e x^2\right )}{a+b x^3} \, dx=\frac {e x}{b} + \frac {\sqrt {3} {\left (b c \left (\frac {a}{b}\right )^{\frac {2}{3}} - a e \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a b} + \frac {{\left (2 \, b d \left (\frac {a}{b}\right )^{\frac {2}{3}} + b c \left (\frac {a}{b}\right )^{\frac {1}{3}} + a e\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (b d \left (\frac {a}{b}\right )^{\frac {2}{3}} - b c \left (\frac {a}{b}\right )^{\frac {1}{3}} - a e\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \] Input:
integrate(x*(e*x^2+d*x+c)/(b*x^3+a),x, algorithm="maxima")
Output:
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 = 0.13 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.95 \[ \int \frac {x \left (c+d x+e x^2\right )}{a+b x^3} \, dx=\frac {\sqrt {3} {\left (a e + \left (-a b^{2}\right )^{\frac {1}{3}} c\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, \left (-a b^{2}\right )^{\frac {2}{3}}} + \frac {{\left (a e - \left (-a b^{2}\right )^{\frac {1}{3}} c\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, \left (-a b^{2}\right )^{\frac {2}{3}}} + \frac {e x}{b} + \frac {d \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b} - \frac {{\left (b^{3} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a b^{2} e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a b^{3}} \] Input:
integrate(x*(e*x^2+d*x+c)/(b*x^3+a),x, algorithm="giac")
Output:
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 = 6.21 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.45 \[ \int \frac {x \left (c+d x+e x^2\right )}{a+b x^3} \, dx=\left (\sum _{k=1}^3\ln \left (x\,\left (b\,c^2+a\,d\,e\right )-\mathrm {root}\left (27\,a\,b^4\,z^3-27\,a\,b^3\,d\,z^2-9\,a\,b^2\,c\,e\,z+9\,a\,b^2\,d^2\,z+3\,a\,b\,c\,d\,e-a\,b\,d^3+a^2\,e^3+b^2\,c^3,z,k\right )\,\left (6\,a\,b\,d-\mathrm {root}\left (27\,a\,b^4\,z^3-27\,a\,b^3\,d\,z^2-9\,a\,b^2\,c\,e\,z+9\,a\,b^2\,d^2\,z+3\,a\,b\,c\,d\,e-a\,b\,d^3+a^2\,e^3+b^2\,c^3,z,k\right )\,a\,b^2\,9+3\,a\,b\,e\,x\right )+a\,d^2-a\,c\,e\right )\,\mathrm {root}\left (27\,a\,b^4\,z^3-27\,a\,b^3\,d\,z^2-9\,a\,b^2\,c\,e\,z+9\,a\,b^2\,d^2\,z+3\,a\,b\,c\,d\,e-a\,b\,d^3+a^2\,e^3+b^2\,c^3,z,k\right )\right )+\frac {e\,x}{b} \] Input:
int((x*(c + d*x + e*x^2))/(a + b*x^3),x)
Output:
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
Time = 0.15 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.14 \[ \int \frac {x \left (c+d x+e x^2\right )}{a+b x^3} \, dx=\frac {2 b^{\frac {1}{3}} a^{\frac {2}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) e -2 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) b c +b^{\frac {1}{3}} a^{\frac {2}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) e -2 b^{\frac {1}{3}} a^{\frac {2}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) e +2 b^{\frac {2}{3}} a^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) d +2 b^{\frac {2}{3}} a^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) d +6 b^{\frac {2}{3}} a^{\frac {1}{3}} e x +\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) b c -2 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) b c}{6 b^{\frac {5}{3}} a^{\frac {1}{3}}} \] Input:
int(x*(e*x^2+d*x+c)/(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)*sqrt (3)))*e - 2*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*b*c + b**(1/3)*a**(2/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*e - 2*b**(1/3)*a**(2/3)*log(a**(1/3) + b**(1/3)*x)*e + 2*b**(2/3)*a**(1/3)* log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*d + 2*b**(2/3)*a**(1/3 )*log(a**(1/3) + b**(1/3)*x)*d + 6*b**(2/3)*a**(1/3)*e*x + log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*b*c - 2*log(a**(1/3) + b**(1/3)*x)*b* c)/(6*b**(2/3)*a**(1/3)*b)