Integrand size = 22, antiderivative size = 416 \[ \int \frac {\left (a+b x+c x^2\right )^3}{d+e x^3} \, dx=-\frac {\left (c^3 d-b^3 e-6 a b c e\right ) x}{e^2}+\frac {3 c \left (b^2+a c\right ) x^2}{2 e}+\frac {b c^2 x^3}{e}+\frac {c^3 x^4}{4 e}-\frac {\left (c^3 d^2-3 b^2 c d^{4/3} e^{2/3}-3 a c^2 d^{4/3} e^{2/3}-b^3 d e-6 a b c d e+3 a^2 b \sqrt [3]{d} e^{5/3}+a^3 e^2\right ) \arctan \left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{\sqrt {3} d^{2/3} e^{7/3}}+\frac {\left (c^3 d^2-6 a b c d e-e \left (b^3 d-a^3 e\right )+3 \sqrt [3]{d} e^{2/3} \left (b^2 c d+a c^2 d-a^2 b e\right )\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} e^{7/3}}-\frac {\left (c^3 d^2-6 a b c d e-e \left (b^3 d-a^3 e\right )+3 \sqrt [3]{d} e^{2/3} \left (b^2 c d+a c^2 d-a^2 b e\right )\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{2/3} e^{7/3}}-\frac {\left (b c^2 d-a b^2 e-a^2 c e\right ) \log \left (d+e x^3\right )}{e^2} \]
-(-6*a*b*c*e-b^3*e+c^3*d)*x/e^2+3/2*c*(a*c+b^2)*x^2/e+b*c^2*x^3/e+1/4*c^3* x^4/e+1/3*(c^3*d^2-6*a*b*c*d*e-e*(-a^3*e+b^3*d)+3*d^(1/3)*e^(2/3)*(-a^2*b* e+a*c^2*d+b^2*c*d))*ln(d^(1/3)+e^(1/3)*x)/d^(2/3)/e^(7/3)-1/6*(c^3*d^2-6*a *b*c*d*e-e*(-a^3*e+b^3*d)+3*d^(1/3)*e^(2/3)*(-a^2*b*e+a*c^2*d+b^2*c*d))*ln (d^(2/3)-d^(1/3)*e^(1/3)*x+e^(2/3)*x^2)/d^(2/3)/e^(7/3)-(-a^2*c*e-a*b^2*e+ b*c^2*d)*ln(e*x^3+d)/e^2-1/3*(c^3*d^2-3*b^2*c*d^(4/3)*e^(2/3)-3*a*c^2*d^(4 /3)*e^(2/3)-b^3*d*e-6*a*b*c*d*e+3*a^2*b*d^(1/3)*e^(5/3)+a^3*e^2)*arctan(1/ 3*(d^(1/3)-2*e^(1/3)*x)/d^(1/3)*3^(1/2))/d^(2/3)/e^(7/3)*3^(1/2)
Time = 0.42 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a+b x+c x^2\right )^3}{d+e x^3} \, dx=\frac {12 \sqrt [3]{e} \left (-c^3 d+b^3 e+6 a b c e\right ) x+18 c \left (b^2+a c\right ) e^{4/3} x^2+12 b c^2 e^{4/3} x^3+3 c^3 e^{4/3} x^4-\frac {4 \sqrt {3} \left (c^3 d^2-3 a c^2 d^{4/3} e^{2/3}+e \left (-b^3 d+3 a^2 b \sqrt [3]{d} e^{2/3}+a^3 e\right )-3 c \left (b^2 d^{4/3} e^{2/3}+2 a b d e\right )\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}}{\sqrt {3}}\right )}{d^{2/3}}+\frac {4 \left (c^3 d^2+3 b^2 c d^{4/3} e^{2/3}+3 a c^2 d^{4/3} e^{2/3}-b^3 d e-6 a b c d e-3 a^2 b \sqrt [3]{d} e^{5/3}+a^3 e^2\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{d^{2/3}}-\frac {2 \left (c^3 d^2+3 b^2 c d^{4/3} e^{2/3}+3 a c^2 d^{4/3} e^{2/3}-b^3 d e-6 a b c d e-3 a^2 b \sqrt [3]{d} e^{5/3}+a^3 e^2\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{d^{2/3}}+12 \sqrt [3]{e} \left (-b c^2 d+a b^2 e+a^2 c e\right ) \log \left (d+e x^3\right )}{12 e^{7/3}} \]
(12*e^(1/3)*(-(c^3*d) + b^3*e + 6*a*b*c*e)*x + 18*c*(b^2 + a*c)*e^(4/3)*x^ 2 + 12*b*c^2*e^(4/3)*x^3 + 3*c^3*e^(4/3)*x^4 - (4*Sqrt[3]*(c^3*d^2 - 3*a*c ^2*d^(4/3)*e^(2/3) + e*(-(b^3*d) + 3*a^2*b*d^(1/3)*e^(2/3) + a^3*e) - 3*c* (b^2*d^(4/3)*e^(2/3) + 2*a*b*d*e))*ArcTan[(1 - (2*e^(1/3)*x)/d^(1/3))/Sqrt [3]])/d^(2/3) + (4*(c^3*d^2 + 3*b^2*c*d^(4/3)*e^(2/3) + 3*a*c^2*d^(4/3)*e^ (2/3) - b^3*d*e - 6*a*b*c*d*e - 3*a^2*b*d^(1/3)*e^(5/3) + a^3*e^2)*Log[d^( 1/3) + e^(1/3)*x])/d^(2/3) - (2*(c^3*d^2 + 3*b^2*c*d^(4/3)*e^(2/3) + 3*a*c ^2*d^(4/3)*e^(2/3) - b^3*d*e - 6*a*b*c*d*e - 3*a^2*b*d^(1/3)*e^(5/3) + a^3 *e^2)*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/d^(2/3) + 12*e^(1/3) *(-(b*c^2*d) + a*b^2*e + a^2*c*e)*Log[d + e*x^3])/(12*e^(7/3))
Time = 0.84 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, 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.
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right )^3}{d+e x^3} \, dx\) |
| \(\Big \downarrow \) 2426 |
| \(\displaystyle \int \left (\frac {-e \left (b^3 d-a^3 e\right )-3 e x^2 \left (a^2 (-c) e-a b^2 e+b c^2 d\right )-3 e x \left (a^2 (-b) e+a c^2 d+b^2 c d\right )-6 a b c d e+c^3 d^2}{e^2 \left (d+e x^3\right )}-\frac {-6 a b c e+b^3 (-e)+c^3 d}{e^2}+\frac {3 c x \left (a c+b^2\right )}{e}+\frac {3 b c^2 x^2}{e}+\frac {c^3 x^3}{e}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\log \left (d+e x^3\right ) \left (a^2 (-c) e-a b^2 e+b c^2 d\right )}{e^2}-\frac {\arctan \left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right ) \left (a^3 e^2+3 a^2 b \sqrt [3]{d} e^{5/3}-6 a b c d e-3 a c^2 d^{4/3} e^{2/3}-b^3 d e-3 b^2 c d^{4/3} e^{2/3}+c^3 d^2\right )}{\sqrt {3} d^{2/3} e^{7/3}}-\frac {\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (-e \left (b^3 d-a^3 e\right )+3 \sqrt [3]{d} e^{2/3} \left (a^2 (-b) e+a c^2 d+b^2 c d\right )-6 a b c d e+c^3 d^2\right )}{6 d^{2/3} e^{7/3}}+\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (-e \left (b^3 d-a^3 e\right )+3 \sqrt [3]{d} e^{2/3} \left (a^2 (-b) e+a c^2 d+b^2 c d\right )-6 a b c d e+c^3 d^2\right )}{3 d^{2/3} e^{7/3}}-\frac {x \left (-6 a b c e+b^3 (-e)+c^3 d\right )}{e^2}+\frac {3 c x^2 \left (a c+b^2\right )}{2 e}+\frac {b c^2 x^3}{e}+\frac {c^3 x^4}{4 e}\) |
-(((c^3*d - b^3*e - 6*a*b*c*e)*x)/e^2) + (3*c*(b^2 + a*c)*x^2)/(2*e) + (b* c^2*x^3)/e + (c^3*x^4)/(4*e) - ((c^3*d^2 - 3*b^2*c*d^(4/3)*e^(2/3) - 3*a*c ^2*d^(4/3)*e^(2/3) - b^3*d*e - 6*a*b*c*d*e + 3*a^2*b*d^(1/3)*e^(5/3) + a^3 *e^2)*ArcTan[(d^(1/3) - 2*e^(1/3)*x)/(Sqrt[3]*d^(1/3))])/(Sqrt[3]*d^(2/3)* e^(7/3)) + ((c^3*d^2 - 6*a*b*c*d*e - e*(b^3*d - a^3*e) + 3*d^(1/3)*e^(2/3) *(b^2*c*d + a*c^2*d - a^2*b*e))*Log[d^(1/3) + e^(1/3)*x])/(3*d^(2/3)*e^(7/ 3)) - ((c^3*d^2 - 6*a*b*c*d*e - e*(b^3*d - a^3*e) + 3*d^(1/3)*e^(2/3)*(b^2 *c*d + a*c^2*d - a^2*b*e))*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2]) /(6*d^(2/3)*e^(7/3)) - ((b*c^2*d - a*b^2*e - a^2*c*e)*Log[d + e*x^3])/e^2
3.1.74.3.1 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 = 1.69 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.44
| method | result | size |
| risch | \(\frac {c^{3} x^{4}}{4 e}+\frac {b \,c^{2} x^{3}}{e}+\frac {3 a \,c^{2} x^{2}}{2 e}+\frac {3 b^{2} c \,x^{2}}{2 e}+\frac {6 a b c x}{e}+\frac {b^{3} x}{e}-\frac {c^{3} d x}{e^{2}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} e +d \right )}{\sum }\frac {\left (3 e \left (a^{2} c e +a \,b^{2} e -b \,c^{2} d \right ) \textit {\_R}^{2}+3 e \left (a^{2} b e -a \,c^{2} d -b^{2} c d \right ) \textit {\_R} +a^{3} e^{2}-6 a b c d e -b^{3} d e +c^{3} d^{2}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{3 e^{3}}\) | \(181\) |
| default | \(\frac {\frac {1}{4} c^{3} x^{4} e +b \,c^{2} x^{3} e +\frac {3}{2} a \,c^{2} e \,x^{2}+\frac {3}{2} b^{2} c e \,x^{2}+6 a b c e x +b^{3} e x -c^{3} d x}{e^{2}}+\frac {\left (a^{3} e^{2}-6 a b c d e -b^{3} d e +c^{3} d^{2}\right ) \left (\frac {\ln \left (x +\left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}{3 e \left (\frac {d}{e}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {d}{e}\right )^{\frac {1}{3}} x +\left (\frac {d}{e}\right )^{\frac {2}{3}}\right )}{6 e \left (\frac {d}{e}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {d}{e}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 e \left (\frac {d}{e}\right )^{\frac {2}{3}}}\right )+\left (3 a^{2} b \,e^{2}-3 a \,c^{2} d e -3 b^{2} d c e \right ) \left (-\frac {\ln \left (x +\left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}{3 e \left (\frac {d}{e}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {d}{e}\right )^{\frac {1}{3}} x +\left (\frac {d}{e}\right )^{\frac {2}{3}}\right )}{6 e \left (\frac {d}{e}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {d}{e}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 e \left (\frac {d}{e}\right )^{\frac {1}{3}}}\right )+\frac {\left (3 a^{2} c \,e^{2}+3 a \,b^{2} e^{2}-3 b \,c^{2} d e \right ) \ln \left (e \,x^{3}+d \right )}{3 e}}{e^{2}}\) | \(347\) |
1/4*c^3*x^4/e+b*c^2*x^3/e+3/2/e*a*c^2*x^2+3/2/e*b^2*c*x^2+6/e*a*b*c*x+1/e* b^3*x-1/e^2*c^3*d*x+1/3/e^3*sum((3*e*(a^2*c*e+a*b^2*e-b*c^2*d)*_R^2+3*e*(a ^2*b*e-a*c^2*d-b^2*c*d)*_R+a^3*e^2-6*a*b*c*d*e-b^3*d*e+c^3*d^2)/_R^2*ln(x- _R),_R=RootOf(_Z^3*e+d))
Result contains complex when optimal does not.
Time = 11.41 (sec) , antiderivative size = 29479, normalized size of antiderivative = 70.86 \[ \int \frac {\left (a+b x+c x^2\right )^3}{d+e x^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^3}{d+e x^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^3}{d+e x^3} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Time = 0.27 (sec) , antiderivative size = 462, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a+b x+c x^2\right )^3}{d+e x^3} \, dx=-\frac {\sqrt {3} {\left (c^{3} d^{2} - b^{3} d e - 6 \, a b c d e + a^{3} e^{2} + 3 \, \left (-d e^{2}\right )^{\frac {1}{3}} b^{2} c d + 3 \, \left (-d e^{2}\right )^{\frac {1}{3}} a c^{2} d - 3 \, \left (-d e^{2}\right )^{\frac {1}{3}} a^{2} b e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {d}{e}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {d}{e}\right )^{\frac {1}{3}}}\right )}{3 \, \left (-d e^{2}\right )^{\frac {2}{3}} e} - \frac {{\left (c^{3} d^{2} - b^{3} d e - 6 \, a b c d e + a^{3} e^{2} - 3 \, \left (-d e^{2}\right )^{\frac {1}{3}} b^{2} c d - 3 \, \left (-d e^{2}\right )^{\frac {1}{3}} a c^{2} d + 3 \, \left (-d e^{2}\right )^{\frac {1}{3}} a^{2} b e\right )} \log \left (x^{2} + x \left (-\frac {d}{e}\right )^{\frac {1}{3}} + \left (-\frac {d}{e}\right )^{\frac {2}{3}}\right )}{6 \, \left (-d e^{2}\right )^{\frac {2}{3}} e} - \frac {{\left (b c^{2} d - a b^{2} e - a^{2} c e\right )} \log \left ({\left | e x^{3} + d \right |}\right )}{e^{2}} + \frac {c^{3} e^{3} x^{4} + 4 \, b c^{2} e^{3} x^{3} + 6 \, b^{2} c e^{3} x^{2} + 6 \, a c^{2} e^{3} x^{2} - 4 \, c^{3} d e^{2} x + 4 \, b^{3} e^{3} x + 24 \, a b c e^{3} x}{4 \, e^{4}} + \frac {{\left (3 \, b^{2} c d e^{8} \left (-\frac {d}{e}\right )^{\frac {1}{3}} + 3 \, a c^{2} d e^{8} \left (-\frac {d}{e}\right )^{\frac {1}{3}} - 3 \, a^{2} b e^{9} \left (-\frac {d}{e}\right )^{\frac {1}{3}} - c^{3} d^{2} e^{7} + b^{3} d e^{8} + 6 \, a b c d e^{8} - a^{3} e^{9}\right )} \left (-\frac {d}{e}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {d}{e}\right )^{\frac {1}{3}} \right |}\right )}{3 \, d e^{9}} \]
-1/3*sqrt(3)*(c^3*d^2 - b^3*d*e - 6*a*b*c*d*e + a^3*e^2 + 3*(-d*e^2)^(1/3) *b^2*c*d + 3*(-d*e^2)^(1/3)*a*c^2*d - 3*(-d*e^2)^(1/3)*a^2*b*e)*arctan(1/3 *sqrt(3)*(2*x + (-d/e)^(1/3))/(-d/e)^(1/3))/((-d*e^2)^(2/3)*e) - 1/6*(c^3* d^2 - b^3*d*e - 6*a*b*c*d*e + a^3*e^2 - 3*(-d*e^2)^(1/3)*b^2*c*d - 3*(-d*e ^2)^(1/3)*a*c^2*d + 3*(-d*e^2)^(1/3)*a^2*b*e)*log(x^2 + x*(-d/e)^(1/3) + ( -d/e)^(2/3))/((-d*e^2)^(2/3)*e) - (b*c^2*d - a*b^2*e - a^2*c*e)*log(abs(e* x^3 + d))/e^2 + 1/4*(c^3*e^3*x^4 + 4*b*c^2*e^3*x^3 + 6*b^2*c*e^3*x^2 + 6*a *c^2*e^3*x^2 - 4*c^3*d*e^2*x + 4*b^3*e^3*x + 24*a*b*c*e^3*x)/e^4 + 1/3*(3* b^2*c*d*e^8*(-d/e)^(1/3) + 3*a*c^2*d*e^8*(-d/e)^(1/3) - 3*a^2*b*e^9*(-d/e) ^(1/3) - c^3*d^2*e^7 + b^3*d*e^8 + 6*a*b*c*d*e^8 - a^3*e^9)*(-d/e)^(1/3)*l og(abs(x - (-d/e)^(1/3)))/(d*e^9)
Time = 9.18 (sec) , antiderivative size = 1700, normalized size of antiderivative = 4.09 \[ \int \frac {\left (a+b x+c x^2\right )^3}{d+e x^3} \, dx=\text {Too large to display} \]
x*((b^3 + 6*a*b*c)/e - (c^3*d)/e^2) + symsum(log(root(27*d^2*e^7*z^3 + 81* b*c^2*d^3*e^5*z^2 - 81*a^2*c*d^2*e^6*z^2 - 81*a*b^2*d^2*e^6*z^2 - 27*a^3*b ^2*c*d^2*e^5*z + 27*a^2*b*c^3*d^3*e^4*z + 27*a*b^3*c^2*d^3*e^4*z + 54*b^2* c^4*d^4*e^3*z + 54*a^4*c^2*d^2*e^5*z + 54*a^2*b^4*d^2*e^5*z + 27*b^5*c*d^3 *e^4*z - 27*a*c^5*d^4*e^3*z + 27*a^5*b*d*e^6*z + 18*a^4*b^4*c*d^2*e^4 - 18 *a^4*b*c^4*d^3*e^3 + 18*a*b^4*c^4*d^4*e^2 - 9*a*b^7*c*d^3*e^3 - 27*a^5*b^2 *c^2*d^2*e^4 + 27*a^2*b^5*c^2*d^3*e^3 - 27*a^2*b^2*c^5*d^4*e^2 - 21*a^3*b^ 3*c^3*d^3*e^3 - 9*a^7*b*c*d*e^5 - 9*a*b*c^7*d^5*e - 3*b^6*c^3*d^4*e^2 - 3* a^6*c^3*d^2*e^4 - 3*a^3*c^6*d^4*e^2 - 3*a^3*b^6*d^2*e^4 + 3*b^3*c^6*d^5*e + 3*a^6*b^3*d*e^5 + b^9*d^3*e^3 - c^9*d^6 - a^9*e^6, z, k)*((3*x*(a^3*e^4 - b^3*d*e^3 + c^3*d^2*e^2 - 6*a*b*c*d*e^3))/e^2 - (3*(6*a*b^2*d*e^3 - 6*b* c^2*d^2*e^2 + 6*a^2*c*d*e^3))/e^2 + 9*root(27*d^2*e^7*z^3 + 81*b*c^2*d^3*e ^5*z^2 - 81*a^2*c*d^2*e^6*z^2 - 81*a*b^2*d^2*e^6*z^2 - 27*a^3*b^2*c*d^2*e^ 5*z + 27*a^2*b*c^3*d^3*e^4*z + 27*a*b^3*c^2*d^3*e^4*z + 54*b^2*c^4*d^4*e^3 *z + 54*a^4*c^2*d^2*e^5*z + 54*a^2*b^4*d^2*e^5*z + 27*b^5*c*d^3*e^4*z - 27 *a*c^5*d^4*e^3*z + 27*a^5*b*d*e^6*z + 18*a^4*b^4*c*d^2*e^4 - 18*a^4*b*c^4* d^3*e^3 + 18*a*b^4*c^4*d^4*e^2 - 9*a*b^7*c*d^3*e^3 - 27*a^5*b^2*c^2*d^2*e^ 4 + 27*a^2*b^5*c^2*d^3*e^3 - 27*a^2*b^2*c^5*d^4*e^2 - 21*a^3*b^3*c^3*d^3*e ^3 - 9*a^7*b*c*d*e^5 - 9*a*b*c^7*d^5*e - 3*b^6*c^3*d^4*e^2 - 3*a^6*c^3*d^2 *e^4 - 3*a^3*c^6*d^4*e^2 - 3*a^3*b^6*d^2*e^4 + 3*b^3*c^6*d^5*e + 3*a^6*...