\(\int \frac {x^3 (d+e x)}{(a+b x+c x^2)^2} \, dx\) [49]

Optimal result

Integrand size = 21, antiderivative size = 202 \[ \int \frac {x^3 (d+e x)}{\left (a+b x+c x^2\right )^2} \, dx=\frac {e x}{c^2}+\frac {a \left (b^2 c d-2 a c^2 d-b^3 e+3 a b c e\right )+\left (b^3 c d-3 a b c^2 d-b^4 e+4 a b^2 c e-2 a^2 c^2 e\right ) x}{c^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\left (b^3 c d-6 a b c^2 d-2 b^4 e+12 a b^2 c e-12 a^2 c^2 e\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{3/2}}+\frac {(c d-2 b e) \log \left (a+b x+c x^2\right )}{2 c^3} \] Output:

e*x/c^2+(a*(3*a*b*c*e-2*a*c^2*d-b^3*e+b^2*c*d)+(-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)*x)/c^3/(-4*a*c+b^2)/(c*x^2+b*x+a)+(-12*a^2*c^2*e 
+12*a*b^2*c*e-6*a*b*c^2*d-2*b^4*e+b^3*c*d)*arctanh((2*c*x+b)/(-4*a*c+b^2)^ 
(1/2))/c^3/(-4*a*c+b^2)^(3/2)+1/2*(-2*b*e+c*d)*ln(c*x^2+b*x+a)/c^3
 

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.94 \[ \int \frac {x^3 (d+e x)}{\left (a+b x+c x^2\right )^2} \, dx=\frac {2 c e x+\frac {2 \left (b^3 (c d-b e) x+a^2 c (3 b e-2 c (d+e x))+a b \left (-b^2 e-3 c^2 d x+b c (d+4 e x)\right )\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}-\frac {2 \left (-b^3 c d+6 a b c^2 d+2 b^4 e-12 a b^2 c e+12 a^2 c^2 e\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2}}+(c d-2 b e) \log (a+x (b+c x))}{2 c^3} \] Input:

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

Output:

(2*c*e*x + (2*(b^3*(c*d - b*e)*x + a^2*c*(3*b*e - 2*c*(d + e*x)) + a*b*(-( 
b^2*e) - 3*c^2*d*x + b*c*(d + 4*e*x))))/((b^2 - 4*a*c)*(a + x*(b + c*x))) 
- (2*(-(b^3*c*d) + 6*a*b*c^2*d + 2*b^4*e - 12*a*b^2*c*e + 12*a^2*c^2*e)*Ar 
cTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + (c*d - 2*b*e) 
*Log[a + x*(b + c*x)])/(2*c^3)
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {1233, 25, 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.

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

\(\Big \downarrow \) 1233

\(\displaystyle \frac {\int -\frac {x \left (2 a (2 c d-b e)+\left (-2 e b^2+c d b+6 a c e\right ) x\right )}{c x^2+b x+a}dx}{c \left (b^2-4 a c\right )}+\frac {x^2 \left (x \left (2 a c e+b^2 (-e)+b c d\right )+a (2 c d-b e)\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {x^2 \left (x \left (2 a c e+b^2 (-e)+b c d\right )+a (2 c d-b e)\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\int \frac {x \left (2 a (2 c d-b e)+\left (-2 e b^2+c d b+6 a c e\right ) x\right )}{c x^2+b x+a}dx}{c \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 1200

\(\displaystyle \frac {x^2 \left (x \left (2 a c e+b^2 (-e)+b c d\right )+a (2 c d-b e)\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\int \left (-\frac {2 e b^2}{c}+d b+6 a e-\frac {a \left (-2 e b^2+c d b+6 a c e\right )+\left (b^2-4 a c\right ) (c d-2 b e) x}{c \left (c x^2+b x+a\right )}\right )dx}{c \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {x^2 \left (x \left (2 a c e+b^2 (-e)+b c d\right )+a (2 c d-b e)\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {-\frac {\text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (-12 a^2 c^2 e+12 a b^2 c e-6 a b c^2 d-2 b^4 e+b^3 c d\right )}{c^2 \sqrt {b^2-4 a c}}-\frac {\left (b^2-4 a c\right ) (c d-2 b e) \log \left (a+b x+c x^2\right )}{2 c^2}+x \left (6 a e-\frac {2 b^2 e}{c}+b d\right )}{c \left (b^2-4 a c\right )}\)

Input:

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

Output:

(x^2*(a*(2*c*d - b*e) + (b*c*d - b^2*e + 2*a*c*e)*x))/(c*(b^2 - 4*a*c)*(a 
+ b*x + c*x^2)) - ((b*d + 6*a*e - (2*b^2*e)/c)*x - ((b^3*c*d - 6*a*b*c^2*d 
 - 2*b^4*e + 12*a*b^2*c*e - 12*a^2*c^2*e)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4 
*a*c]])/(c^2*Sqrt[b^2 - 4*a*c]) - ((b^2 - 4*a*c)*(c*d - 2*b*e)*Log[a + b*x 
 + c*x^2])/(2*c^2))/(c*(b^2 - 4*a*c))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 1233

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m - 1))*(a + b*x + c*x^2) 
^(p + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g - c 
*(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Simp[1/(c*( 
p + 1)*(b^2 - 4*a*c))   Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Sim 
p[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a*e*(e*f 
*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*( 
m + p + 1) + 2*c^2*d*f*(m + 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2* 
p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1] && 
GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, b, c, d, e, f, g]) | 
|  !ILtQ[m + 2*p + 3, 0])
 

rule 2009

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

Maple [A] (verified)

Time = 1.30 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.33

method	result	size
default	\(\frac {e x}{c^{2}}-\frac {\frac {-\frac {\left (2 a^{2} c^{2} e -4 a \,b^{2} c e +3 a b \,c^{2} d +b^{4} e -b^{3} c d \right ) x}{\left (4 a c -b^{2}\right ) c}+\frac {a \left (3 a b c e -2 a \,c^{2} d -e \,b^{3}+c d \,b^{2}\right )}{\left (4 a c -b^{2}\right ) c}}{c \,x^{2}+b x +a}+\frac {\frac {\left (8 a b c e -4 a \,c^{2} d -2 e \,b^{3}+c d \,b^{2}\right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (6 a^{2} c e -2 e a \,b^{2}+a b c d -\frac {\left (8 a b c e -4 a \,c^{2} d -2 e \,b^{3}+c d \,b^{2}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{4 a c -b^{2}}}{c^{2}}\)	\(269\)
risch	\(\text {Expression too large to display}\)	\(3047\)

Input:

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

Output:

e*x/c^2-1/c^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)/(4*a* 
c-b^2)/c*x+a*(3*a*b*c*e-2*a*c^2*d-b^3*e+b^2*c*d)/(4*a*c-b^2)/c)/(c*x^2+b*x 
+a)+1/(4*a*c-b^2)*(1/2*(8*a*b*c*e-4*a*c^2*d-2*b^3*e+b^2*c*d)/c*ln(c*x^2+b* 
x+a)+2*(6*a^2*c*e-2*e*a*b^2+a*b*c*d-1/2*(8*a*b*c*e-4*a*c^2*d-2*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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 632 vs. \(2 (196) = 392\).

Time = 0.10 (sec) , antiderivative size = 1283, normalized size of antiderivative = 6.35 \[ \int \frac {x^3 (d+e x)}{\left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/2*(2*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*e*x^3 + 2*(b^5*c - 8*a*b^3*c^ 
2 + 16*a^2*b*c^3)*e*x^2 + (((b^3*c^2 - 6*a*b*c^3)*d - 2*(b^4*c - 6*a*b^2*c 
^2 + 6*a^2*c^3)*e)*x^2 + (a*b^3*c - 6*a^2*b*c^2)*d - 2*(a*b^4 - 6*a^2*b^2* 
c + 6*a^3*c^2)*e + ((b^4*c - 6*a*b^2*c^2)*d - 2*(b^5 - 6*a*b^3*c + 6*a^2*b 
*c^2)*e)*x)*sqrt(b^2 - 4*a*c)*log((2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c + sqr 
t(b^2 - 4*a*c)*(2*c*x + b))/(c*x^2 + b*x + a)) + 2*(a*b^4*c - 6*a^2*b^2*c^ 
2 + 8*a^3*c^3)*d - 2*(a*b^5 - 7*a^2*b^3*c + 12*a^3*b*c^2)*e + 2*((b^5*c - 
7*a*b^3*c^2 + 12*a^2*b*c^3)*d - (b^6 - 9*a*b^4*c + 26*a^2*b^2*c^2 - 24*a^3 
*c^3)*e)*x + (((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d - 2*(b^5*c - 8*a*b^3 
*c^2 + 16*a^2*b*c^3)*e)*x^2 + (a*b^4*c - 8*a^2*b^2*c^2 + 16*a^3*c^3)*d - 2 
*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*e + ((b^5*c - 8*a*b^3*c^2 + 16*a^2*b 
*c^3)*d - 2*(b^6 - 8*a*b^4*c + 16*a^2*b^2*c^2)*e)*x)*log(c*x^2 + b*x + a)) 
/(a*b^4*c^3 - 8*a^2*b^2*c^4 + 16*a^3*c^5 + (b^4*c^4 - 8*a*b^2*c^5 + 16*a^2 
*c^6)*x^2 + (b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*x), 1/2*(2*(b^4*c^2 - 8 
*a*b^2*c^3 + 16*a^2*c^4)*e*x^3 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*e* 
x^2 + 2*(((b^3*c^2 - 6*a*b*c^3)*d - 2*(b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*e) 
*x^2 + (a*b^3*c - 6*a^2*b*c^2)*d - 2*(a*b^4 - 6*a^2*b^2*c + 6*a^3*c^2)*e + 
 ((b^4*c - 6*a*b^2*c^2)*d - 2*(b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*e)*x)*sqrt(- 
b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + 2*(a* 
b^4*c - 6*a^2*b^2*c^2 + 8*a^3*c^3)*d - 2*(a*b^5 - 7*a^2*b^3*c + 12*a^3*...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1248 vs. \(2 (206) = 412\).

Time = 2.89 (sec) , antiderivative size = 1248, normalized size of antiderivative = 6.18 \[ \int \frac {x^3 (d+e x)}{\left (a+b x+c x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(-sqrt(-(4*a*c - b**2)**3)*(12*a**2*c**2*e - 12*a*b**2*c*e + 6*a*b*c**2*d 
+ 2*b**4*e - b**3*c*d)/(2*c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b* 
*4*c - b**6)) - (2*b*e - c*d)/(2*c**3))*log(x + (-10*a**2*b*c*e - 16*a**2* 
c**4*(-sqrt(-(4*a*c - b**2)**3)*(12*a**2*c**2*e - 12*a*b**2*c*e + 6*a*b*c* 
*2*d + 2*b**4*e - b**3*c*d)/(2*c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12 
*a*b**4*c - b**6)) - (2*b*e - c*d)/(2*c**3)) + 8*a**2*c**2*d + 2*a*b**3*e 
+ 8*a*b**2*c**3*(-sqrt(-(4*a*c - b**2)**3)*(12*a**2*c**2*e - 12*a*b**2*c*e 
 + 6*a*b*c**2*d + 2*b**4*e - b**3*c*d)/(2*c**3*(64*a**3*c**3 - 48*a**2*b** 
2*c**2 + 12*a*b**4*c - b**6)) - (2*b*e - c*d)/(2*c**3)) - a*b**2*c*d - b** 
4*c**2*(-sqrt(-(4*a*c - b**2)**3)*(12*a**2*c**2*e - 12*a*b**2*c*e + 6*a*b* 
c**2*d + 2*b**4*e - b**3*c*d)/(2*c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 
12*a*b**4*c - b**6)) - (2*b*e - c*d)/(2*c**3)))/(12*a**2*c**2*e - 12*a*b** 
2*c*e + 6*a*b*c**2*d + 2*b**4*e - b**3*c*d)) + (sqrt(-(4*a*c - b**2)**3)*( 
12*a**2*c**2*e - 12*a*b**2*c*e + 6*a*b*c**2*d + 2*b**4*e - b**3*c*d)/(2*c* 
*3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (2*b*e - c*d 
)/(2*c**3))*log(x + (-10*a**2*b*c*e - 16*a**2*c**4*(sqrt(-(4*a*c - b**2)** 
3)*(12*a**2*c**2*e - 12*a*b**2*c*e + 6*a*b*c**2*d + 2*b**4*e - b**3*c*d)/( 
2*c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (2*b*e - 
 c*d)/(2*c**3)) + 8*a**2*c**2*d + 2*a*b**3*e + 8*a*b**2*c**3*(sqrt(-(4*a*c 
 - b**2)**3)*(12*a**2*c**2*e - 12*a*b**2*c*e + 6*a*b*c**2*d + 2*b**4*e ...
 

Maxima [F(-2)]

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

integrate(x^3*(e*x+d)/(c*x^2+b*x+a)^2,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 = 0.20 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.11 \[ \int \frac {x^3 (d+e x)}{\left (a+b x+c x^2\right )^2} \, dx=-\frac {{\left (b^{3} c d - 6 \, a b c^{2} d - 2 \, b^{4} e + 12 \, a b^{2} c e - 12 \, a^{2} c^{2} e\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {e x}{c^{2}} + \frac {{\left (c d - 2 \, b e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{3}} + \frac {\frac {{\left (b^{3} c d - 3 \, a b c^{2} d - b^{4} e + 4 \, a b^{2} c e - 2 \, a^{2} c^{2} e\right )} x}{c} + \frac {a b^{2} c d - 2 \, a^{2} c^{2} d - a b^{3} e + 3 \, a^{2} b c e}{c}}{{\left (c x^{2} + b x + a\right )} {\left (b^{2} - 4 \, a c\right )} c^{2}} \] Input:

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

Output:

-(b^3*c*d - 6*a*b*c^2*d - 2*b^4*e + 12*a*b^2*c*e - 12*a^2*c^2*e)*arctan((2 
*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^2*c^3 - 4*a*c^4)*sqrt(-b^2 + 4*a*c)) + e 
*x/c^2 + 1/2*(c*d - 2*b*e)*log(c*x^2 + b*x + a)/c^3 + ((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)*x/c + (a*b^2*c*d - 2*a^2*c^2*d - 
a*b^3*e + 3*a^2*b*c*e)/c)/((c*x^2 + b*x + a)*(b^2 - 4*a*c)*c^2)
 

Mupad [B] (verification not implemented)

Time = 11.50 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.78 \[ \int \frac {x^3 (d+e x)}{\left (a+b x+c x^2\right )^2} \, dx=\frac {\frac {a\,\left (e\,b^3-d\,b^2\,c-3\,a\,e\,b\,c+2\,a\,d\,c^2\right )}{c\,\left (4\,a\,c-b^2\right )}+\frac {x\,\left (2\,e\,a^2\,c^2-4\,e\,a\,b^2\,c+3\,d\,a\,b\,c^2+e\,b^4-d\,b^3\,c\right )}{c\,\left (4\,a\,c-b^2\right )}}{c^3\,x^2+b\,c^2\,x+a\,c^2}+\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (-128\,e\,a^3\,b\,c^3+64\,d\,a^3\,c^4+96\,e\,a^2\,b^3\,c^2-48\,d\,a^2\,b^2\,c^3-24\,e\,a\,b^5\,c+12\,d\,a\,b^4\,c^2+2\,e\,b^7-d\,b^6\,c\right )}{2\,\left (64\,a^3\,c^6-48\,a^2\,b^2\,c^5+12\,a\,b^4\,c^4-b^6\,c^3\right )}+\frac {e\,x}{c^2}-\frac {\mathrm {atan}\left (\frac {2\,c\,x}{\sqrt {4\,a\,c-b^2}}-\frac {b^3\,c^2-4\,a\,b\,c^3}{c^2\,{\left (4\,a\,c-b^2\right )}^{3/2}}\right )\,\left (12\,e\,a^2\,c^2-12\,e\,a\,b^2\,c+6\,d\,a\,b\,c^2+2\,e\,b^4-d\,b^3\,c\right )}{c^3\,{\left (4\,a\,c-b^2\right )}^{3/2}} \] Input:

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

Output:

((a*(b^3*e + 2*a*c^2*d - b^2*c*d - 3*a*b*c*e))/(c*(4*a*c - b^2)) + (x*(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*a*c - b^2)) 
)/(a*c^2 + c^3*x^2 + b*c^2*x) + (log(a + b*x + c*x^2)*(2*b^7*e + 64*a^3*c^ 
4*d - b^6*c*d - 48*a^2*b^2*c^3*d + 96*a^2*b^3*c^2*e - 24*a*b^5*c*e + 12*a* 
b^4*c^2*d - 128*a^3*b*c^3*e))/(2*(64*a^3*c^6 - b^6*c^3 + 12*a*b^4*c^4 - 48 
*a^2*b^2*c^5)) + (e*x)/c^2 - (atan((2*c*x)/(4*a*c - b^2)^(1/2) - (b^3*c^2 
- 4*a*b*c^3)/(c^2*(4*a*c - b^2)^(3/2)))*(2*b^4*e + 12*a^2*c^2*e - b^3*c*d 
+ 6*a*b*c^2*d - 12*a*b^2*c*e))/(c^3*(4*a*c - b^2)^(3/2))
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 1294, normalized size of antiderivative = 6.41 \[ \int \frac {x^3 (d+e x)}{\left (a+b x+c x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b*c**2 
*e + 24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**3* 
c*e - 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**2 
*c**2*d - 24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2* 
b**2*c**2*e*x - 24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2)) 
*a**2*b*c**3*e*x**2 - 4*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b 
**2))*a*b**5*e + 2*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2)) 
*a*b**4*c*d + 24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a 
*b**4*c*e*x - 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a 
*b**3*c**2*d*x + 24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2) 
)*a*b**3*c**2*e*x**2 - 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - 
 b**2))*a*b**2*c**3*d*x**2 - 4*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4* 
a*c - b**2))*b**6*e*x + 2*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - 
 b**2))*b**5*c*d*x - 4*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b* 
*2))*b**5*c*e*x**2 + 2*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b* 
*2))*b**4*c**2*d*x**2 - 32*log(a + b*x + c*x**2)*a**3*b**2*c**2*e + 16*log 
(a + b*x + c*x**2)*a**3*b*c**3*d + 16*log(a + b*x + c*x**2)*a**2*b**4*c*e 
- 8*log(a + b*x + c*x**2)*a**2*b**3*c**2*d - 32*log(a + b*x + c*x**2)*a**2 
*b**3*c**2*e*x + 16*log(a + b*x + c*x**2)*a**2*b**2*c**3*d*x - 32*log(a + 
b*x + c*x**2)*a**2*b**2*c**3*e*x**2 + 16*log(a + b*x + c*x**2)*a**2*b*c...