Integrand size = 22, antiderivative size = 270 \[ \int \frac {\left (a+b x+c x^2\right )^2}{d+e x^3} \, dx=\frac {2 b c x}{e}+\frac {c^2 x^2}{2 e}+\frac {\left (c^2 d^{4/3}+2 b c d \sqrt [3]{e}-a \left (2 b \sqrt [3]{d}+a \sqrt [3]{e}\right ) e\right ) \arctan \left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{\sqrt {3} d^{2/3} e^{5/3}}-\frac {\left (\sqrt [3]{e} \left (2 b c d-a^2 e\right )-\sqrt [3]{d} \left (c^2 d-2 a b e\right )\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} e^{5/3}}+\frac {\left (2 b c d-a^2 e-\frac {\sqrt [3]{d} \left (c^2 d-2 a b e\right )}{\sqrt [3]{e}}\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{2/3} e^{4/3}}+\frac {\left (b^2+2 a c\right ) \log \left (d+e x^3\right )}{3 e} \] Output:
2*b*c*x/e+1/2*c^2*x^2/e+1/3*(c^2*d^(4/3)+2*b*c*d*e^(1/3)-a*(2*b*d^(1/3)+a* e^(1/3))*e)*arctan(1/3*(d^(1/3)-2*e^(1/3)*x)*3^(1/2)/d^(1/3))*3^(1/2)/d^(2 /3)/e^(5/3)-1/3*(e^(1/3)*(-a^2*e+2*b*c*d)-d^(1/3)*(-2*a*b*e+c^2*d))*ln(d^( 1/3)+e^(1/3)*x)/d^(2/3)/e^(5/3)+1/6*(2*b*c*d-a^2*e-d^(1/3)*(-2*a*b*e+c^2*d )/e^(1/3))*ln(d^(2/3)-d^(1/3)*e^(1/3)*x+e^(2/3)*x^2)/d^(2/3)/e^(4/3)+1/3*( 2*a*c+b^2)*ln(e*x^3+d)/e
Time = 0.20 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x+c x^2\right )^2}{d+e x^3} \, dx=\frac {12 b c e^{2/3} x+3 c^2 e^{2/3} x^2+\frac {2 \sqrt {3} \left (c d^{2/3}-a e^{2/3}\right ) \left (c d^{2/3}+2 b \sqrt [3]{d} \sqrt [3]{e}+a e^{2/3}\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}}{\sqrt {3}}\right )}{d^{2/3}}+\frac {2 \left (c^2 d^{4/3}-2 b c d \sqrt [3]{e}+a \left (-2 b \sqrt [3]{d}+a \sqrt [3]{e}\right ) e\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{d^{2/3}}-\frac {\left (c^2 d^{4/3}-2 b c d \sqrt [3]{e}+a \left (-2 b \sqrt [3]{d}+a \sqrt [3]{e}\right ) e\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{d^{2/3}}+2 \left (b^2+2 a c\right ) e^{2/3} \log \left (d+e x^3\right )}{6 e^{5/3}} \] Input:
Integrate[(a + b*x + c*x^2)^2/(d + e*x^3),x]
Output:
(12*b*c*e^(2/3)*x + 3*c^2*e^(2/3)*x^2 + (2*Sqrt[3]*(c*d^(2/3) - a*e^(2/3)) *(c*d^(2/3) + 2*b*d^(1/3)*e^(1/3) + a*e^(2/3))*ArcTan[(1 - (2*e^(1/3)*x)/d ^(1/3))/Sqrt[3]])/d^(2/3) + (2*(c^2*d^(4/3) - 2*b*c*d*e^(1/3) + a*(-2*b*d^ (1/3) + a*e^(1/3))*e)*Log[d^(1/3) + e^(1/3)*x])/d^(2/3) - ((c^2*d^(4/3) - 2*b*c*d*e^(1/3) + a*(-2*b*d^(1/3) + a*e^(1/3))*e)*Log[d^(2/3) - d^(1/3)*e^ (1/3)*x + e^(2/3)*x^2])/d^(2/3) + 2*(b^2 + 2*a*c)*e^(2/3)*Log[d + e*x^3])/ (6*e^(5/3))
Time = 1.03 (sec) , antiderivative size = 270, 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.
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right )^2}{d+e x^3} \, dx\) |
| \(\Big \downarrow \) 2426 |
| \(\displaystyle \int \left (-\frac {a^2 (-e)-e x^2 \left (2 a c+b^2\right )+x \left (c^2 d-2 a b e\right )+2 b c d}{e \left (d+e x^3\right )}+\frac {2 b c}{e}+\frac {c^2 x}{e}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (a^2 (-e)-\frac {\sqrt [3]{d} \left (c^2 d-2 a b e\right )}{\sqrt [3]{e}}+2 b c d\right )}{6 d^{2/3} e^{4/3}}-\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (\sqrt [3]{e} \left (2 b c d-a^2 e\right )-\sqrt [3]{d} \left (c^2 d-2 a b e\right )\right )}{3 d^{2/3} e^{5/3}}+\frac {\arctan \left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right ) \left (-a e \left (a \sqrt [3]{e}+2 b \sqrt [3]{d}\right )+2 b c d \sqrt [3]{e}+c^2 d^{4/3}\right )}{\sqrt {3} d^{2/3} e^{5/3}}+\frac {\left (2 a c+b^2\right ) \log \left (d+e x^3\right )}{3 e}+\frac {2 b c x}{e}+\frac {c^2 x^2}{2 e}\) |
Input:
Int[(a + b*x + c*x^2)^2/(d + e*x^3),x]
Output:
(2*b*c*x)/e + (c^2*x^2)/(2*e) + ((c^2*d^(4/3) + 2*b*c*d*e^(1/3) - a*(2*b*d ^(1/3) + a*e^(1/3))*e)*ArcTan[(d^(1/3) - 2*e^(1/3)*x)/(Sqrt[3]*d^(1/3))])/ (Sqrt[3]*d^(2/3)*e^(5/3)) - ((e^(1/3)*(2*b*c*d - a^2*e) - d^(1/3)*(c^2*d - 2*a*b*e))*Log[d^(1/3) + e^(1/3)*x])/(3*d^(2/3)*e^(5/3)) + ((2*b*c*d - a^2 *e - (d^(1/3)*(c^2*d - 2*a*b*e))/e^(1/3))*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/(6*d^(2/3)*e^(4/3)) + ((b^2 + 2*a*c)*Log[d + e*x^3])/(3*e)
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 = 0.21 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.31
| method | result | size |
| risch | \(\frac {c^{2} x^{2}}{2 e}+\frac {2 b c x}{e}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{3}+d \right )}{\sum }\frac {\left (e \left (2 a c +b^{2}\right ) \textit {\_R}^{2}+\left (2 a b e -c^{2} d \right ) \textit {\_R} +a^{2} e -2 d b c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{3 e^{2}}\) | \(85\) |
| default | \(\frac {c \left (\frac {1}{2} c \,x^{2}+2 b x \right )}{e}+\frac {\left (a^{2} e -2 d b c \right ) \left (\frac {\ln \left (x +\left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}{3 e \left (\frac {d}{e}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {d}{e}\right )^{\frac {1}{3}} x +\left (\frac {d}{e}\right )^{\frac {2}{3}}\right )}{6 e \left (\frac {d}{e}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {d}{e}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 e \left (\frac {d}{e}\right )^{\frac {2}{3}}}\right )+\left (2 a b e -c^{2} d \right ) \left (-\frac {\ln \left (x +\left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}{3 e \left (\frac {d}{e}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {d}{e}\right )^{\frac {1}{3}} x +\left (\frac {d}{e}\right )^{\frac {2}{3}}\right )}{6 e \left (\frac {d}{e}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {d}{e}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 e \left (\frac {d}{e}\right )^{\frac {1}{3}}}\right )+\frac {\left (2 a c e +b^{2} e \right ) \ln \left (x^{3} e +d \right )}{3 e}}{e}\) | \(252\) |
Input:
int((c*x^2+b*x+a)^2/(e*x^3+d),x,method=_RETURNVERBOSE)
Output:
1/2*c^2*x^2/e+2*b*c*x/e+1/3/e^2*sum((e*(2*a*c+b^2)*_R^2+(2*a*b*e-c^2*d)*_R +a^2*e-2*d*b*c)/_R^2*ln(x-_R),_R=RootOf(_Z^3*e+d))
Result contains complex when optimal does not.
Time = 1.42 (sec) , antiderivative size = 12827, normalized size of antiderivative = 47.51 \[ \int \frac {\left (a+b x+c x^2\right )^2}{d+e x^3} \, dx=\text {Too large to display} \] Input:
integrate((c*x^2+b*x+a)^2/(e*x^3+d),x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^2}{d+e x^3} \, dx=\text {Timed out} \] Input:
integrate((c*x**2+b*x+a)**2/(e*x**3+d),x)
Output:
Timed out
Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^2}{d+e x^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((c*x^2+b*x+a)^2/(e*x^3+d),x, algorithm="maxima")
Output:
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 = 0.13 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.04 \[ \int \frac {\left (a+b x+c x^2\right )^2}{d+e x^3} \, dx=\frac {{\left (b^{2} + 2 \, a c\right )} \log \left ({\left | e x^{3} + d \right |}\right )}{3 \, e} + \frac {\sqrt {3} {\left (2 \, b c d e - a^{2} e^{2} - \left (-d e^{2}\right )^{\frac {1}{3}} c^{2} d + 2 \, \left (-d e^{2}\right )^{\frac {1}{3}} a b e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {d}{e}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {d}{e}\right )^{\frac {1}{3}}}\right )}{3 \, \left (-d e^{2}\right )^{\frac {2}{3}} e} + \frac {{\left (2 \, b c d e - a^{2} e^{2} + \left (-d e^{2}\right )^{\frac {1}{3}} c^{2} d - 2 \, \left (-d e^{2}\right )^{\frac {1}{3}} a b e\right )} \log \left (x^{2} + x \left (-\frac {d}{e}\right )^{\frac {1}{3}} + \left (-\frac {d}{e}\right )^{\frac {2}{3}}\right )}{6 \, \left (-d e^{2}\right )^{\frac {2}{3}} e} + \frac {c^{2} e x^{2} + 4 \, b c e x}{2 \, e^{2}} + \frac {{\left (c^{2} d e^{4} \left (-\frac {d}{e}\right )^{\frac {1}{3}} - 2 \, a b e^{5} \left (-\frac {d}{e}\right )^{\frac {1}{3}} + 2 \, b c d e^{4} - a^{2} e^{5}\right )} \left (-\frac {d}{e}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {d}{e}\right )^{\frac {1}{3}} \right |}\right )}{3 \, d e^{5}} \] Input:
integrate((c*x^2+b*x+a)^2/(e*x^3+d),x, algorithm="giac")
Output:
1/3*(b^2 + 2*a*c)*log(abs(e*x^3 + d))/e + 1/3*sqrt(3)*(2*b*c*d*e - a^2*e^2 - (-d*e^2)^(1/3)*c^2*d + 2*(-d*e^2)^(1/3)*a*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*(2*b*c*d*e - a^2*e^ 2 + (-d*e^2)^(1/3)*c^2*d - 2*(-d*e^2)^(1/3)*a*b*e)*log(x^2 + x*(-d/e)^(1/3 ) + (-d/e)^(2/3))/((-d*e^2)^(2/3)*e) + 1/2*(c^2*e*x^2 + 4*b*c*e*x)/e^2 + 1 /3*(c^2*d*e^4*(-d/e)^(1/3) - 2*a*b*e^5*(-d/e)^(1/3) + 2*b*c*d*e^4 - a^2*e^ 5)*(-d/e)^(1/3)*log(abs(x - (-d/e)^(1/3)))/(d*e^5)
Time = 6.06 (sec) , antiderivative size = 769, normalized size of antiderivative = 2.85 \[ \int \frac {\left (a+b x+c x^2\right )^2}{d+e x^3} \, dx=\left (\sum _{k=1}^3\ln \left (\frac {2\,a^3\,b\,e^2+3\,a^2\,c^2\,d\,e+b^4\,d\,e+2\,b\,c^3\,d^2}{e}+\frac {x\,\left (-2\,a^3\,c\,e^2+3\,a^2\,b^2\,e^2+2\,b^3\,c\,d\,e+c^4\,d^2\right )}{e}-\mathrm {root}\left (27\,d^2\,e^5\,z^3-54\,a\,c\,d^2\,e^4\,z^2-27\,b^2\,d^2\,e^4\,z^2+27\,a^2\,c^2\,d^2\,e^3\,z+18\,b\,c^3\,d^3\,e^2\,z+18\,a^3\,b\,d\,e^4\,z+9\,b^4\,d^2\,e^3\,z+6\,a\,b^4\,c\,d^2\,e^2-9\,a^2\,b^2\,c^2\,d^2\,e^2-6\,a^4\,b\,c\,d\,e^3-6\,a\,b\,c^4\,d^3\,e-2\,a^3\,c^3\,d^2\,e^2+2\,b^3\,c^3\,d^3\,e+2\,a^3\,b^3\,d\,e^3-b^6\,d^2\,e^2-c^6\,d^4-a^6\,e^4,z,k\right )\,e\,\left (2\,b^2\,d-\mathrm {root}\left (27\,d^2\,e^5\,z^3-54\,a\,c\,d^2\,e^4\,z^2-27\,b^2\,d^2\,e^4\,z^2+27\,a^2\,c^2\,d^2\,e^3\,z+18\,b\,c^3\,d^3\,e^2\,z+18\,a^3\,b\,d\,e^4\,z+9\,b^4\,d^2\,e^3\,z+6\,a\,b^4\,c\,d^2\,e^2-9\,a^2\,b^2\,c^2\,d^2\,e^2-6\,a^4\,b\,c\,d\,e^3-6\,a\,b\,c^4\,d^3\,e-2\,a^3\,c^3\,d^2\,e^2+2\,b^3\,c^3\,d^3\,e+2\,a^3\,b^3\,d\,e^3-b^6\,d^2\,e^2-c^6\,d^4-a^6\,e^4,z,k\right )\,d\,e\,3+4\,a\,c\,d-a^2\,e\,x+2\,b\,c\,d\,x\right )\,3\right )\,\mathrm {root}\left (27\,d^2\,e^5\,z^3-54\,a\,c\,d^2\,e^4\,z^2-27\,b^2\,d^2\,e^4\,z^2+27\,a^2\,c^2\,d^2\,e^3\,z+18\,b\,c^3\,d^3\,e^2\,z+18\,a^3\,b\,d\,e^4\,z+9\,b^4\,d^2\,e^3\,z+6\,a\,b^4\,c\,d^2\,e^2-9\,a^2\,b^2\,c^2\,d^2\,e^2-6\,a^4\,b\,c\,d\,e^3-6\,a\,b\,c^4\,d^3\,e-2\,a^3\,c^3\,d^2\,e^2+2\,b^3\,c^3\,d^3\,e+2\,a^3\,b^3\,d\,e^3-b^6\,d^2\,e^2-c^6\,d^4-a^6\,e^4,z,k\right )\right )+\frac {c^2\,x^2}{2\,e}+\frac {2\,b\,c\,x}{e} \] Input:
int((a + b*x + c*x^2)^2/(d + e*x^3),x)
Output:
symsum(log((2*a^3*b*e^2 + 2*b*c^3*d^2 + b^4*d*e + 3*a^2*c^2*d*e)/e + (x*(c ^4*d^2 - 2*a^3*c*e^2 + 3*a^2*b^2*e^2 + 2*b^3*c*d*e))/e - 3*root(27*d^2*e^5 *z^3 - 54*a*c*d^2*e^4*z^2 - 27*b^2*d^2*e^4*z^2 + 27*a^2*c^2*d^2*e^3*z + 18 *b*c^3*d^3*e^2*z + 18*a^3*b*d*e^4*z + 9*b^4*d^2*e^3*z + 6*a*b^4*c*d^2*e^2 - 9*a^2*b^2*c^2*d^2*e^2 - 6*a^4*b*c*d*e^3 - 6*a*b*c^4*d^3*e - 2*a^3*c^3*d^ 2*e^2 + 2*b^3*c^3*d^3*e + 2*a^3*b^3*d*e^3 - b^6*d^2*e^2 - c^6*d^4 - a^6*e^ 4, z, k)*e*(2*b^2*d - 3*root(27*d^2*e^5*z^3 - 54*a*c*d^2*e^4*z^2 - 27*b^2* d^2*e^4*z^2 + 27*a^2*c^2*d^2*e^3*z + 18*b*c^3*d^3*e^2*z + 18*a^3*b*d*e^4*z + 9*b^4*d^2*e^3*z + 6*a*b^4*c*d^2*e^2 - 9*a^2*b^2*c^2*d^2*e^2 - 6*a^4*b*c *d*e^3 - 6*a*b*c^4*d^3*e - 2*a^3*c^3*d^2*e^2 + 2*b^3*c^3*d^3*e + 2*a^3*b^3 *d*e^3 - b^6*d^2*e^2 - c^6*d^4 - a^6*e^4, z, k)*d*e + 4*a*c*d - a^2*e*x + 2*b*c*d*x))*root(27*d^2*e^5*z^3 - 54*a*c*d^2*e^4*z^2 - 27*b^2*d^2*e^4*z^2 + 27*a^2*c^2*d^2*e^3*z + 18*b*c^3*d^3*e^2*z + 18*a^3*b*d*e^4*z + 9*b^4*d^2 *e^3*z + 6*a*b^4*c*d^2*e^2 - 9*a^2*b^2*c^2*d^2*e^2 - 6*a^4*b*c*d*e^3 - 6*a *b*c^4*d^3*e - 2*a^3*c^3*d^2*e^2 + 2*b^3*c^3*d^3*e + 2*a^3*b^3*d*e^3 - b^6 *d^2*e^2 - c^6*d^4 - a^6*e^4, z, k), k, 1, 3) + (c^2*x^2)/(2*e) + (2*b*c*x )/e
Time = 0.15 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.67 \[ \int \frac {\left (a+b x+c x^2\right )^2}{d+e x^3} \, dx=\frac {-2 e^{\frac {4}{3}} d^{\frac {2}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {d^{\frac {1}{3}}-2 e^{\frac {1}{3}} x}{d^{\frac {1}{3}} \sqrt {3}}\right ) a^{2}+4 e^{\frac {1}{3}} d^{\frac {5}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {d^{\frac {1}{3}}-2 e^{\frac {1}{3}} x}{d^{\frac {1}{3}} \sqrt {3}}\right ) b c -4 \sqrt {3}\, \mathit {atan} \left (\frac {d^{\frac {1}{3}}-2 e^{\frac {1}{3}} x}{d^{\frac {1}{3}} \sqrt {3}}\right ) a b d e +2 \sqrt {3}\, \mathit {atan} \left (\frac {d^{\frac {1}{3}}-2 e^{\frac {1}{3}} x}{d^{\frac {1}{3}} \sqrt {3}}\right ) c^{2} d^{2}-e^{\frac {4}{3}} d^{\frac {2}{3}} \mathrm {log}\left (d^{\frac {2}{3}}-e^{\frac {1}{3}} d^{\frac {1}{3}} x +e^{\frac {2}{3}} x^{2}\right ) a^{2}+2 e^{\frac {1}{3}} d^{\frac {5}{3}} \mathrm {log}\left (d^{\frac {2}{3}}-e^{\frac {1}{3}} d^{\frac {1}{3}} x +e^{\frac {2}{3}} x^{2}\right ) b c +2 e^{\frac {4}{3}} d^{\frac {2}{3}} \mathrm {log}\left (d^{\frac {1}{3}}+e^{\frac {1}{3}} x \right ) a^{2}-4 e^{\frac {1}{3}} d^{\frac {5}{3}} \mathrm {log}\left (d^{\frac {1}{3}}+e^{\frac {1}{3}} x \right ) b c +4 e^{\frac {2}{3}} d^{\frac {4}{3}} \mathrm {log}\left (d^{\frac {2}{3}}-e^{\frac {1}{3}} d^{\frac {1}{3}} x +e^{\frac {2}{3}} x^{2}\right ) a c +2 e^{\frac {2}{3}} d^{\frac {4}{3}} \mathrm {log}\left (d^{\frac {2}{3}}-e^{\frac {1}{3}} d^{\frac {1}{3}} x +e^{\frac {2}{3}} x^{2}\right ) b^{2}+4 e^{\frac {2}{3}} d^{\frac {4}{3}} \mathrm {log}\left (d^{\frac {1}{3}}+e^{\frac {1}{3}} x \right ) a c +2 e^{\frac {2}{3}} d^{\frac {4}{3}} \mathrm {log}\left (d^{\frac {1}{3}}+e^{\frac {1}{3}} x \right ) b^{2}+12 e^{\frac {2}{3}} d^{\frac {4}{3}} b c x +3 e^{\frac {2}{3}} d^{\frac {4}{3}} c^{2} x^{2}+2 \,\mathrm {log}\left (d^{\frac {2}{3}}-e^{\frac {1}{3}} d^{\frac {1}{3}} x +e^{\frac {2}{3}} x^{2}\right ) a b d e -\mathrm {log}\left (d^{\frac {2}{3}}-e^{\frac {1}{3}} d^{\frac {1}{3}} x +e^{\frac {2}{3}} x^{2}\right ) c^{2} d^{2}-4 \,\mathrm {log}\left (d^{\frac {1}{3}}+e^{\frac {1}{3}} x \right ) a b d e +2 \,\mathrm {log}\left (d^{\frac {1}{3}}+e^{\frac {1}{3}} x \right ) c^{2} d^{2}}{6 e^{\frac {5}{3}} d^{\frac {4}{3}}} \] Input:
int((c*x^2+b*x+a)^2/(e*x^3+d),x)
Output:
( - 2*e**(1/3)*d**(2/3)*sqrt(3)*atan((d**(1/3) - 2*e**(1/3)*x)/(d**(1/3)*s qrt(3)))*a**2*e + 4*e**(1/3)*d**(2/3)*sqrt(3)*atan((d**(1/3) - 2*e**(1/3)* x)/(d**(1/3)*sqrt(3)))*b*c*d - 4*sqrt(3)*atan((d**(1/3) - 2*e**(1/3)*x)/(d **(1/3)*sqrt(3)))*a*b*d*e + 2*sqrt(3)*atan((d**(1/3) - 2*e**(1/3)*x)/(d**( 1/3)*sqrt(3)))*c**2*d**2 - e**(1/3)*d**(2/3)*log(d**(2/3) - e**(1/3)*d**(1 /3)*x + e**(2/3)*x**2)*a**2*e + 2*e**(1/3)*d**(2/3)*log(d**(2/3) - e**(1/3 )*d**(1/3)*x + e**(2/3)*x**2)*b*c*d + 2*e**(1/3)*d**(2/3)*log(d**(1/3) + e **(1/3)*x)*a**2*e - 4*e**(1/3)*d**(2/3)*log(d**(1/3) + e**(1/3)*x)*b*c*d + 4*e**(2/3)*d**(1/3)*log(d**(2/3) - e**(1/3)*d**(1/3)*x + e**(2/3)*x**2)*a *c*d + 2*e**(2/3)*d**(1/3)*log(d**(2/3) - e**(1/3)*d**(1/3)*x + e**(2/3)*x **2)*b**2*d + 4*e**(2/3)*d**(1/3)*log(d**(1/3) + e**(1/3)*x)*a*c*d + 2*e** (2/3)*d**(1/3)*log(d**(1/3) + e**(1/3)*x)*b**2*d + 12*e**(2/3)*d**(1/3)*b* c*d*x + 3*e**(2/3)*d**(1/3)*c**2*d*x**2 + 2*log(d**(2/3) - e**(1/3)*d**(1/ 3)*x + e**(2/3)*x**2)*a*b*d*e - log(d**(2/3) - e**(1/3)*d**(1/3)*x + e**(2 /3)*x**2)*c**2*d**2 - 4*log(d**(1/3) + e**(1/3)*x)*a*b*d*e + 2*log(d**(1/3 ) + e**(1/3)*x)*c**2*d**2)/(6*e**(2/3)*d**(1/3)*d*e)