\(\int \frac {x^3 (d+e x)}{a+b x+c x^2} \, dx\) [40]

Optimal result

Integrand size = 21, antiderivative size = 169 \[ \int \frac {x^3 (d+e x)}{a+b x+c x^2} \, dx=-\frac {\left (b c d-b^2 e+a c e\right ) x}{c^3}+\frac {(c d-b e) x^2}{2 c^2}+\frac {e x^3}{3 c}+\frac {\left (b^3 c d-3 a b c^2 d-b^4 e+4 a b^2 c e-2 a^2 c^2 e\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^4 \sqrt {b^2-4 a c}}+\frac {\left (b^2 c d-a c^2 d-b^3 e+2 a b c e\right ) \log \left (a+b x+c x^2\right )}{2 c^4} \] Output:

-(a*c*e-b^2*e+b*c*d)*x/c^3+1/2*(-b*e+c*d)*x^2/c^2+1/3*e*x^3/c+(-2*a^2*c^2* 
e+4*a*b^2*c*e-3*a*b*c^2*d-b^4*e+b^3*c*d)*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1 
/2))/c^4/(-4*a*c+b^2)^(1/2)+1/2*(2*a*b*c*e-a*c^2*d-b^3*e+b^2*c*d)*ln(c*x^2 
+b*x+a)/c^4
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.98 \[ \int \frac {x^3 (d+e x)}{a+b x+c x^2} \, dx=\frac {-6 c \left (b c d-b^2 e+a c e\right ) x+3 c^2 (c d-b e) x^2+2 c^3 e x^3+\frac {6 \left (-b^3 c d+3 a b c^2 d+b^4 e-4 a b^2 c e+2 a^2 c^2 e\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}-3 \left (-b^2 c d+a c^2 d+b^3 e-2 a b c e\right ) \log (a+x (b+c x))}{6 c^4} \] Input:

Integrate[(x^3*(d + e*x))/(a + b*x + c*x^2),x]
 

Output:

(-6*c*(b*c*d - b^2*e + a*c*e)*x + 3*c^2*(c*d - b*e)*x^2 + 2*c^3*e*x^3 + (6 
*(-(b^3*c*d) + 3*a*b*c^2*d + b^4*e - 4*a*b^2*c*e + 2*a^2*c^2*e)*ArcTan[(b 
+ 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] - 3*(-(b^2*c*d) + a*c^2*d 
 + b^3*e - 2*a*b*c*e)*Log[a + x*(b + c*x)])/(6*c^4)
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 169, 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 = {1200, 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.

\(\displaystyle \int \frac {x^3 (d+e x)}{a+b x+c x^2} \, dx\)

\(\Big \downarrow \) 1200

\(\displaystyle \int \left (-\frac {a c e+b^2 (-e)+b c d}{c^3}+\frac {a \left (a c e+b^2 (-e)+b c d\right )+x \left (2 a b c e-a c^2 d+b^3 (-e)+b^2 c d\right )}{c^3 \left (a+b x+c x^2\right )}+\frac {x (c d-b e)}{c^2}+\frac {e x^2}{c}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (-2 a^2 c^2 e+4 a b^2 c e-3 a b c^2 d+b^4 (-e)+b^3 c d\right )}{c^4 \sqrt {b^2-4 a c}}-\frac {x \left (a c e+b^2 (-e)+b c d\right )}{c^3}+\frac {\left (2 a b c e-a c^2 d+b^3 (-e)+b^2 c d\right ) \log \left (a+b x+c x^2\right )}{2 c^4}+\frac {x^2 (c d-b e)}{2 c^2}+\frac {e x^3}{3 c}\)

Input:

Int[(x^3*(d + e*x))/(a + b*x + c*x^2),x]
 

Output:

-(((b*c*d - b^2*e + a*c*e)*x)/c^3) + ((c*d - b*e)*x^2)/(2*c^2) + (e*x^3)/( 
3*c) + ((b^3*c*d - 3*a*b*c^2*d - b^4*e + 4*a*b^2*c*e - 2*a^2*c^2*e)*ArcTan 
h[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^4*Sqrt[b^2 - 4*a*c]) + ((b^2*c*d - a* 
c^2*d - b^3*e + 2*a*b*c*e)*Log[a + b*x + c*x^2])/(2*c^4)
 

Defintions of rubi rules used

rule 1200

Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* 
(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g* 
x)^n/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && In 
tegersQ[n]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 1.36 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.08

method	result	size
default	\(-\frac {-\frac {1}{3} e \,x^{3} c^{2}+\frac {1}{2} b c e \,x^{2}-\frac {1}{2} c^{2} d \,x^{2}+a c e x -b^{2} e x +b c d x}{c^{3}}+\frac {\frac {\left (2 a b c e -a \,c^{2} d -e \,b^{3}+c d \,b^{2}\right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (a^{2} c e -e a \,b^{2}+a b c d -\frac {\left (2 a b c e -a \,c^{2} d -e \,b^{3}+c d \,b^{2}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{c^{3}}\)	\(183\)
risch	\(\text {Expression too large to display}\)	\(2940\)

Input:

int(x^3*(e*x+d)/(c*x^2+b*x+a),x,method=_RETURNVERBOSE)
 

Output:

-1/c^3*(-1/3*e*x^3*c^2+1/2*b*c*e*x^2-1/2*c^2*d*x^2+a*c*e*x-b^2*e*x+b*c*d*x 
)+1/c^3*(1/2*(2*a*b*c*e-a*c^2*d-b^3*e+b^2*c*d)/c*ln(c*x^2+b*x+a)+2*(a^2*c* 
e-e*a*b^2+a*b*c*d-1/2*(2*a*b*c*e-a*c^2*d-b^3*e+b^2*c*d)*b/c)/(4*a*c-b^2)^( 
1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 563, normalized size of antiderivative = 3.33 \[ \int \frac {x^3 (d+e x)}{a+b x+c x^2} \, dx=\left [\frac {2 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} e x^{3} + 3 \, {\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d - {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} e\right )} x^{2} - 3 \, \sqrt {b^{2} - 4 \, a c} {\left ({\left (b^{3} c - 3 \, a b c^{2}\right )} d - {\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} e\right )} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c - \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) - 6 \, {\left ({\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d - {\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} e\right )} x + 3 \, {\left ({\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} d - {\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} e\right )} \log \left (c x^{2} + b x + a\right )}{6 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )}}, \frac {2 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} e x^{3} + 3 \, {\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d - {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} e\right )} x^{2} + 6 \, \sqrt {-b^{2} + 4 \, a c} {\left ({\left (b^{3} c - 3 \, a b c^{2}\right )} d - {\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} e\right )} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) - 6 \, {\left ({\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d - {\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} e\right )} x + 3 \, {\left ({\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} d - {\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} e\right )} \log \left (c x^{2} + b x + a\right )}{6 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )}}\right ] \] Input:

integrate(x^3*(e*x+d)/(c*x^2+b*x+a),x, algorithm="fricas")
 

Output:

[1/6*(2*(b^2*c^3 - 4*a*c^4)*e*x^3 + 3*((b^2*c^3 - 4*a*c^4)*d - (b^3*c^2 - 
4*a*b*c^3)*e)*x^2 - 3*sqrt(b^2 - 4*a*c)*((b^3*c - 3*a*b*c^2)*d - (b^4 - 4* 
a*b^2*c + 2*a^2*c^2)*e)*log((2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c - sqrt(b^2 
- 4*a*c)*(2*c*x + b))/(c*x^2 + b*x + a)) - 6*((b^3*c^2 - 4*a*b*c^3)*d - (b 
^4*c - 5*a*b^2*c^2 + 4*a^2*c^3)*e)*x + 3*((b^4*c - 5*a*b^2*c^2 + 4*a^2*c^3 
)*d - (b^5 - 6*a*b^3*c + 8*a^2*b*c^2)*e)*log(c*x^2 + b*x + a))/(b^2*c^4 - 
4*a*c^5), 1/6*(2*(b^2*c^3 - 4*a*c^4)*e*x^3 + 3*((b^2*c^3 - 4*a*c^4)*d - (b 
^3*c^2 - 4*a*b*c^3)*e)*x^2 + 6*sqrt(-b^2 + 4*a*c)*((b^3*c - 3*a*b*c^2)*d - 
 (b^4 - 4*a*b^2*c + 2*a^2*c^2)*e)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/( 
b^2 - 4*a*c)) - 6*((b^3*c^2 - 4*a*b*c^3)*d - (b^4*c - 5*a*b^2*c^2 + 4*a^2* 
c^3)*e)*x + 3*((b^4*c - 5*a*b^2*c^2 + 4*a^2*c^3)*d - (b^5 - 6*a*b^3*c + 8* 
a^2*b*c^2)*e)*log(c*x^2 + b*x + a))/(b^2*c^4 - 4*a*c^5)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 840 vs. \(2 (168) = 336\).

Time = 1.71 (sec) , antiderivative size = 840, normalized size of antiderivative = 4.97 \[ \int \frac {x^3 (d+e x)}{a+b x+c x^2} \, dx=x^{2} \left (- \frac {b e}{2 c^{2}} + \frac {d}{2 c}\right ) + x \left (- \frac {a e}{c^{2}} + \frac {b^{2} e}{c^{3}} - \frac {b d}{c^{2}}\right ) + \left (- \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a^{2} c^{2} e - 4 a b^{2} c e + 3 a b c^{2} d + b^{4} e - b^{3} c d\right )}{2 c^{4} \cdot \left (4 a c - b^{2}\right )} + \frac {2 a b c e - a c^{2} d - b^{3} e + b^{2} c d}{2 c^{4}}\right ) \log {\left (x + \frac {- 3 a^{2} b c e + 2 a^{2} c^{2} d + a b^{3} e - a b^{2} c d + 4 a c^{4} \left (- \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a^{2} c^{2} e - 4 a b^{2} c e + 3 a b c^{2} d + b^{4} e - b^{3} c d\right )}{2 c^{4} \cdot \left (4 a c - b^{2}\right )} + \frac {2 a b c e - a c^{2} d - b^{3} e + b^{2} c d}{2 c^{4}}\right ) - b^{2} c^{3} \left (- \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a^{2} c^{2} e - 4 a b^{2} c e + 3 a b c^{2} d + b^{4} e - b^{3} c d\right )}{2 c^{4} \cdot \left (4 a c - b^{2}\right )} + \frac {2 a b c e - a c^{2} d - b^{3} e + b^{2} c d}{2 c^{4}}\right )}{2 a^{2} c^{2} e - 4 a b^{2} c e + 3 a b c^{2} d + b^{4} e - b^{3} c d} \right )} + \left (\frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a^{2} c^{2} e - 4 a b^{2} c e + 3 a b c^{2} d + b^{4} e - b^{3} c d\right )}{2 c^{4} \cdot \left (4 a c - b^{2}\right )} + \frac {2 a b c e - a c^{2} d - b^{3} e + b^{2} c d}{2 c^{4}}\right ) \log {\left (x + \frac {- 3 a^{2} b c e + 2 a^{2} c^{2} d + a b^{3} e - a b^{2} c d + 4 a c^{4} \left (\frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a^{2} c^{2} e - 4 a b^{2} c e + 3 a b c^{2} d + b^{4} e - b^{3} c d\right )}{2 c^{4} \cdot \left (4 a c - b^{2}\right )} + \frac {2 a b c e - a c^{2} d - b^{3} e + b^{2} c d}{2 c^{4}}\right ) - b^{2} c^{3} \left (\frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a^{2} c^{2} e - 4 a b^{2} c e + 3 a b c^{2} d + b^{4} e - b^{3} c d\right )}{2 c^{4} \cdot \left (4 a c - b^{2}\right )} + \frac {2 a b c e - a c^{2} d - b^{3} e + b^{2} c d}{2 c^{4}}\right )}{2 a^{2} c^{2} e - 4 a b^{2} c e + 3 a b c^{2} d + b^{4} e - b^{3} c d} \right )} + \frac {e x^{3}}{3 c} \] Input:

integrate(x**3*(e*x+d)/(c*x**2+b*x+a),x)
                                                                                    
                                                                                    
 

Output:

x**2*(-b*e/(2*c**2) + d/(2*c)) + x*(-a*e/c**2 + b**2*e/c**3 - b*d/c**2) + 
(-sqrt(-4*a*c + b**2)*(2*a**2*c**2*e - 4*a*b**2*c*e + 3*a*b*c**2*d + b**4* 
e - b**3*c*d)/(2*c**4*(4*a*c - b**2)) + (2*a*b*c*e - a*c**2*d - b**3*e + b 
**2*c*d)/(2*c**4))*log(x + (-3*a**2*b*c*e + 2*a**2*c**2*d + a*b**3*e - a*b 
**2*c*d + 4*a*c**4*(-sqrt(-4*a*c + b**2)*(2*a**2*c**2*e - 4*a*b**2*c*e + 3 
*a*b*c**2*d + b**4*e - b**3*c*d)/(2*c**4*(4*a*c - b**2)) + (2*a*b*c*e - a* 
c**2*d - b**3*e + b**2*c*d)/(2*c**4)) - b**2*c**3*(-sqrt(-4*a*c + b**2)*(2 
*a**2*c**2*e - 4*a*b**2*c*e + 3*a*b*c**2*d + b**4*e - b**3*c*d)/(2*c**4*(4 
*a*c - b**2)) + (2*a*b*c*e - a*c**2*d - b**3*e + b**2*c*d)/(2*c**4)))/(2*a 
**2*c**2*e - 4*a*b**2*c*e + 3*a*b*c**2*d + b**4*e - b**3*c*d)) + (sqrt(-4* 
a*c + b**2)*(2*a**2*c**2*e - 4*a*b**2*c*e + 3*a*b*c**2*d + b**4*e - b**3*c 
*d)/(2*c**4*(4*a*c - b**2)) + (2*a*b*c*e - a*c**2*d - b**3*e + b**2*c*d)/( 
2*c**4))*log(x + (-3*a**2*b*c*e + 2*a**2*c**2*d + a*b**3*e - a*b**2*c*d + 
4*a*c**4*(sqrt(-4*a*c + b**2)*(2*a**2*c**2*e - 4*a*b**2*c*e + 3*a*b*c**2*d 
 + b**4*e - b**3*c*d)/(2*c**4*(4*a*c - b**2)) + (2*a*b*c*e - a*c**2*d - b* 
*3*e + b**2*c*d)/(2*c**4)) - b**2*c**3*(sqrt(-4*a*c + b**2)*(2*a**2*c**2*e 
 - 4*a*b**2*c*e + 3*a*b*c**2*d + b**4*e - b**3*c*d)/(2*c**4*(4*a*c - b**2) 
) + (2*a*b*c*e - a*c**2*d - b**3*e + b**2*c*d)/(2*c**4)))/(2*a**2*c**2*e - 
 4*a*b**2*c*e + 3*a*b*c**2*d + b**4*e - b**3*c*d)) + e*x**3/(3*c)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^3 (d+e x)}{a+b x+c x^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(x^3*(e*x+d)/(c*x^2+b*x+a),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(4*a*c-b^2>0)', see `assume?` for 
 more deta
 

Giac [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.00 \[ \int \frac {x^3 (d+e x)}{a+b x+c x^2} \, dx=\frac {2 \, c^{2} e x^{3} + 3 \, c^{2} d x^{2} - 3 \, b c e x^{2} - 6 \, b c d x + 6 \, b^{2} e x - 6 \, a c e x}{6 \, c^{3}} + \frac {{\left (b^{2} c d - a c^{2} d - b^{3} e + 2 \, a b c e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{4}} - \frac {{\left (b^{3} c d - 3 \, a b c^{2} d - b^{4} e + 4 \, a b^{2} c e - 2 \, a^{2} c^{2} e\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c} c^{4}} \] Input:

integrate(x^3*(e*x+d)/(c*x^2+b*x+a),x, algorithm="giac")
 

Output:

1/6*(2*c^2*e*x^3 + 3*c^2*d*x^2 - 3*b*c*e*x^2 - 6*b*c*d*x + 6*b^2*e*x - 6*a 
*c*e*x)/c^3 + 1/2*(b^2*c*d - a*c^2*d - b^3*e + 2*a*b*c*e)*log(c*x^2 + b*x 
+ a)/c^4 - (b^3*c*d - 3*a*b*c^2*d - b^4*e + 4*a*b^2*c*e - 2*a^2*c^2*e)*arc 
tan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*c^4)
 

Mupad [B] (verification not implemented)

Time = 10.48 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.31 \[ \int \frac {x^3 (d+e x)}{a+b x+c x^2} \, dx=x^2\,\left (\frac {d}{2\,c}-\frac {b\,e}{2\,c^2}\right )-x\,\left (\frac {b\,\left (\frac {d}{c}-\frac {b\,e}{c^2}\right )}{c}+\frac {a\,e}{c^2}\right )+\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (8\,e\,a^2\,b\,c^2-4\,d\,a^2\,c^3-6\,e\,a\,b^3\,c+5\,d\,a\,b^2\,c^2+e\,b^5-d\,b^4\,c\right )}{2\,\left (4\,a\,c^5-b^2\,c^4\right )}+\frac {e\,x^3}{3\,c}+\frac {\mathrm {atan}\left (\frac {b}{\sqrt {4\,a\,c-b^2}}+\frac {2\,c\,x}{\sqrt {4\,a\,c-b^2}}\right )\,\left (2\,e\,a^2\,c^2-4\,e\,a\,b^2\,c+3\,d\,a\,b\,c^2+e\,b^4-d\,b^3\,c\right )}{c^4\,\sqrt {4\,a\,c-b^2}} \] Input:

int((x^3*(d + e*x))/(a + b*x + c*x^2),x)
 

Output:

x^2*(d/(2*c) - (b*e)/(2*c^2)) - x*((b*(d/c - (b*e)/c^2))/c + (a*e)/c^2) + 
(log(a + b*x + c*x^2)*(b^5*e - 4*a^2*c^3*d - b^4*c*d - 6*a*b^3*c*e + 5*a*b 
^2*c^2*d + 8*a^2*b*c^2*e))/(2*(4*a*c^5 - b^2*c^4)) + (e*x^3)/(3*c) + (atan 
(b/(4*a*c - b^2)^(1/2) + (2*c*x)/(4*a*c - b^2)^(1/2))*(b^4*e + 2*a^2*c^2*e 
 - b^3*c*d + 3*a*b*c^2*d - 4*a*b^2*c*e))/(c^4*(4*a*c - b^2)^(1/2))
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 447, normalized size of antiderivative = 2.64 \[ \int \frac {x^3 (d+e x)}{a+b x+c x^2} \, dx=\frac {12 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a^{2} c^{2} e -24 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a \,b^{2} c e +18 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a b \,c^{2} d +6 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b^{4} e -6 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b^{3} c d +24 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) a^{2} b \,c^{2} e -12 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) a^{2} c^{3} d -18 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) a \,b^{3} c e +15 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) a \,b^{2} c^{2} d +3 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) b^{5} e -3 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) b^{4} c d -24 a^{2} c^{3} e x +30 a \,b^{2} c^{2} e x -24 a b \,c^{3} d x -12 a b \,c^{3} e \,x^{2}+12 a \,c^{4} d \,x^{2}+8 a \,c^{4} e \,x^{3}-6 b^{4} c e x +6 b^{3} c^{2} d x +3 b^{3} c^{2} e \,x^{2}-3 b^{2} c^{3} d \,x^{2}-2 b^{2} c^{3} e \,x^{3}}{6 c^{4} \left (4 a c -b^{2}\right )} \] Input:

int(x^3*(e*x+d)/(c*x^2+b*x+a),x)
 

Output:

(12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*c**2*e - 
24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**2*c*e + 18 
*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b*c**2*d + 6*sq 
rt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**4*e - 6*sqrt(4*a* 
c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**3*c*d + 24*log(a + b*x + 
 c*x**2)*a**2*b*c**2*e - 12*log(a + b*x + c*x**2)*a**2*c**3*d - 18*log(a + 
 b*x + c*x**2)*a*b**3*c*e + 15*log(a + b*x + c*x**2)*a*b**2*c**2*d + 3*log 
(a + b*x + c*x**2)*b**5*e - 3*log(a + b*x + c*x**2)*b**4*c*d - 24*a**2*c** 
3*e*x + 30*a*b**2*c**2*e*x - 24*a*b*c**3*d*x - 12*a*b*c**3*e*x**2 + 12*a*c 
**4*d*x**2 + 8*a*c**4*e*x**3 - 6*b**4*c*e*x + 6*b**3*c**2*d*x + 3*b**3*c** 
2*e*x**2 - 3*b**2*c**3*d*x**2 - 2*b**2*c**3*e*x**3)/(6*c**4*(4*a*c - b**2) 
)