\(\int \frac {1}{(d+e x) (a+b x+c x^2)^3} \, dx\) [491]

Optimal result

Integrand size = 20, antiderivative size = 433 \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^3} \, dx=-\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}+\frac {b^3 c d e^2+2 b^4 e^3+16 a^2 c^2 e^3-3 b^2 c e \left (3 c d^2+5 a e^2\right )+2 b c^2 d \left (3 c d^2+7 a e^2\right )+2 c (2 c d-b e) \left (3 c^2 d^2-b^2 e^2-c e (3 b d-7 a e)\right ) x}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}-\frac {\left (12 c^5 d^5-b^5 e^5+10 a b^3 c e^5-30 a^2 b c^2 e^5-10 c^4 d^3 e (3 b d-4 a e)+20 c^3 d e^2 \left (b^2 d^2-3 a b d e+3 a^2 e^2\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2} \left (c d^2-b d e+a e^2\right )^3}+\frac {e^5 \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}-\frac {e^5 \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^3} \] Output:

-1/2*(b*c*d-b^2*e+2*a*c*e+c*(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^ 
2)/(c*x^2+b*x+a)^2+1/2*(b^3*c*d*e^2+2*b^4*e^3+16*a^2*c^2*e^3-3*b^2*c*e*(5* 
a*e^2+3*c*d^2)+2*b*c^2*d*(7*a*e^2+3*c*d^2)+2*c*(-b*e+2*c*d)*(3*c^2*d^2-b^2 
*e^2-c*e*(-7*a*e+3*b*d))*x)/(-4*a*c+b^2)^2/(a*e^2-b*d*e+c*d^2)^2/(c*x^2+b* 
x+a)-(12*c^5*d^5-b^5*e^5+10*a*b^3*c*e^5-30*a^2*b*c^2*e^5-10*c^4*d^3*e*(-4* 
a*e+3*b*d)+20*c^3*d*e^2*(3*a^2*e^2-3*a*b*d*e+b^2*d^2))*arctanh((2*c*x+b)/( 
-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(5/2)/(a*e^2-b*d*e+c*d^2)^3+e^5*ln(e*x+d)/ 
(a*e^2-b*d*e+c*d^2)^3-1/2*e^5*ln(c*x^2+b*x+a)/(a*e^2-b*d*e+c*d^2)^3
 

Mathematica [A] (verified)

Time = 0.77 (sec) , antiderivative size = 429, normalized size of antiderivative = 0.99 \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^3} \, dx=\frac {1}{2} \left (\frac {-b^2 e+2 c (a e+c d x)+b c (d-e x)}{\left (b^2-4 a c\right ) \left (-c d^2+e (b d-a e)\right ) (a+x (b+c x))^2}+\frac {2 b^4 e^3+b^3 c e^2 (d+2 e x)+4 c^2 \left (4 a^2 e^3+3 c^2 d^3 x+7 a c d e^2 x\right )+2 b c^2 \left (3 c d^2 (d-3 e x)+7 a e^2 (d-e x)\right )+b^2 c e \left (-15 a e^2+c d (-9 d+2 e x)\right )}{\left (b^2-4 a c\right )^2 \left (c d^2+e (-b d+a e)\right )^2 (a+x (b+c x))}+\frac {2 \left (-12 c^5 d^5+b^5 e^5-10 a b^3 c e^5+30 a^2 b c^2 e^5+10 c^4 d^3 e (3 b d-4 a e)-20 c^3 d e^2 \left (b^2 d^2-3 a b d e+3 a^2 e^2\right )\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{5/2} \left (-c d^2+e (b d-a e)\right )^3}+\frac {2 e^5 \log (d+e x)}{\left (c d^2+e (-b d+a e)\right )^3}-\frac {e^5 \log (a+x (b+c x))}{\left (c d^2+e (-b d+a e)\right )^3}\right ) \] Input:

Integrate[1/((d + e*x)*(a + b*x + c*x^2)^3),x]
 

Output:

((-(b^2*e) + 2*c*(a*e + c*d*x) + b*c*(d - e*x))/((b^2 - 4*a*c)*(-(c*d^2) + 
 e*(b*d - a*e))*(a + x*(b + c*x))^2) + (2*b^4*e^3 + b^3*c*e^2*(d + 2*e*x) 
+ 4*c^2*(4*a^2*e^3 + 3*c^2*d^3*x + 7*a*c*d*e^2*x) + 2*b*c^2*(3*c*d^2*(d - 
3*e*x) + 7*a*e^2*(d - e*x)) + b^2*c*e*(-15*a*e^2 + c*d*(-9*d + 2*e*x)))/(( 
b^2 - 4*a*c)^2*(c*d^2 + e*(-(b*d) + a*e))^2*(a + x*(b + c*x))) + (2*(-12*c 
^5*d^5 + b^5*e^5 - 10*a*b^3*c*e^5 + 30*a^2*b*c^2*e^5 + 10*c^4*d^3*e*(3*b*d 
 - 4*a*e) - 20*c^3*d*e^2*(b^2*d^2 - 3*a*b*d*e + 3*a^2*e^2))*ArcTan[(b + 2* 
c*x)/Sqrt[-b^2 + 4*a*c]])/((-b^2 + 4*a*c)^(5/2)*(-(c*d^2) + e*(b*d - a*e)) 
^3) + (2*e^5*Log[d + e*x])/(c*d^2 + e*(-(b*d) + a*e))^3 - (e^5*Log[a + x*( 
b + c*x)])/(c*d^2 + e*(-(b*d) + a*e))^3)/2
 

Rubi [A] (verified)

Time = 1.06 (sec) , antiderivative size = 510, normalized size of antiderivative = 1.18, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1165, 1235, 27, 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.

\(\displaystyle \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^3} \, dx\)

\(\Big \downarrow \) 1165

\(\displaystyle -\frac {\int \frac {6 c^2 d^2-2 b^2 e^2-c e (3 b d-8 a e)+3 c e (2 c d-b e) x}{(d+e x) \left (c x^2+b x+a\right )^2}dx}{2 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1235

\(\displaystyle -\frac {\frac {-2 c x (2 c d-b e) \left (-c e (3 b d-7 a e)-b^2 e^2+3 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (-c e (3 b d-8 a e)-2 b^2 e^2+6 c^2 d^2\right )+3 a c e (2 c d-b e)^2}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}-\frac {\int \frac {2 \left (6 c^4 d^4-c^3 e (9 b d-14 a e) d^2+b^4 e^4+b^2 c e^3 (b d-8 a e)+c^2 e^2 \left (b^2 d^2-7 a b e d+16 a^2 e^2\right )+c e (2 c d-b e) \left (3 c^2 d^2-b^2 e^2-c e (3 b d-7 a e)\right ) x\right )}{(d+e x) \left (c x^2+b x+a\right )}dx}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}}{2 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {-2 c x (2 c d-b e) \left (-c e (3 b d-7 a e)-b^2 e^2+3 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (-c e (3 b d-8 a e)-2 b^2 e^2+6 c^2 d^2\right )+3 a c e (2 c d-b e)^2}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \int \frac {6 c^4 d^4-c^3 e (9 b d-14 a e) d^2+b^4 e^4+b^2 c e^3 (b d-8 a e)+c^2 e^2 \left (b^2 d^2-7 a b e d+16 a^2 e^2\right )+c e (2 c d-b e) \left (3 c^2 d^2-b^2 e^2-c e (3 b d-7 a e)\right ) x}{(d+e x) \left (c x^2+b x+a\right )}dx}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}}{2 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1200

\(\displaystyle -\frac {\frac {-2 c x (2 c d-b e) \left (-c e (3 b d-7 a e)-b^2 e^2+3 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (-c e (3 b d-8 a e)-2 b^2 e^2+6 c^2 d^2\right )+3 a c e (2 c d-b e)^2}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \int \left (\frac {\left (b^2-4 a c\right )^2 e^6}{\left (c d^2-b e d+a e^2\right ) (d+e x)}+\frac {6 c^5 d^5-5 c^4 e (3 b d-4 a e) d^3+10 c^3 e^2 \left (b^2 d^2-3 a b e d+3 a^2 e^2\right ) d-b^5 e^5-23 a^2 b c^2 e^5+9 a b^3 c e^5-c \left (b^2-4 a c\right )^2 e^5 x}{\left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )}\right )dx}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}}{2 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\frac {-2 c x (2 c d-b e) \left (-c e (3 b d-7 a e)-b^2 e^2+3 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (-c e (3 b d-8 a e)-2 b^2 e^2+6 c^2 d^2\right )+3 a c e (2 c d-b e)^2}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (-\frac {\text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (20 c^3 d e^2 \left (3 a^2 e^2-3 a b d e+b^2 d^2\right )-30 a^2 b c^2 e^5+10 a b^3 c e^5-10 c^4 d^3 e (3 b d-4 a e)-b^5 e^5+12 c^5 d^5\right )}{\sqrt {b^2-4 a c} \left (a e^2-b d e+c d^2\right )}-\frac {e^5 \left (b^2-4 a c\right )^2 \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )}+\frac {e^5 \left (b^2-4 a c\right )^2 \log (d+e x)}{a e^2-b d e+c d^2}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}}{2 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}\)

Input:

Int[1/((d + e*x)*(a + b*x + c*x^2)^3),x]
 

Output:

-1/2*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)/((b^2 - 4*a*c)*(c*d^2 - 
 b*d*e + a*e^2)*(a + b*x + c*x^2)^2) - ((3*a*c*e*(2*c*d - b*e)^2 - (b*c*d 
- b^2*e + 2*a*c*e)*(6*c^2*d^2 - 2*b^2*e^2 - c*e*(3*b*d - 8*a*e)) - 2*c*(2* 
c*d - b*e)*(3*c^2*d^2 - b^2*e^2 - c*e*(3*b*d - 7*a*e))*x)/((b^2 - 4*a*c)*( 
c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)) - (2*(-(((12*c^5*d^5 - b^5*e^5 + 
 10*a*b^3*c*e^5 - 30*a^2*b*c^2*e^5 - 10*c^4*d^3*e*(3*b*d - 4*a*e) + 20*c^3 
*d*e^2*(b^2*d^2 - 3*a*b*d*e + 3*a^2*e^2))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4 
*a*c]])/(Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2))) + ((b^2 - 4*a*c)^2*e^ 
5*Log[d + e*x])/(c*d^2 - b*d*e + a*e^2) - ((b^2 - 4*a*c)^2*e^5*Log[a + b*x 
 + c*x^2])/(2*(c*d^2 - b*d*e + a*e^2))))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a 
*e^2)))/(2*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 1165

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m + 1)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e) 
*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^ 
2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))   Int[(d 
+ e*x)^m*Simp[b*c*d*e*(2*p - m + 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p 
+ 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x, x]*(a + 
 b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[p, -1] 
 && IntQuadraticQ[a, b, c, d, e, m, p, 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 1235

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

rule 2009

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1101\) vs. \(2(423)=846\).

Time = 1.26 (sec) , antiderivative size = 1102, normalized size of antiderivative = 2.55

method	result	size
default	\(\text {Expression too large to display}\)	\(1102\)
risch	\(\text {Expression too large to display}\)	\(5780\)

Input:

int(1/(e*x+d)/(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)
 

Output:

e^5*ln(e*x+d)/(a*e^2-b*d*e+c*d^2)^3-1/(a*e^2-b*d*e+c*d^2)^3*((c^2*(7*a^2*b 
*c*e^5-14*a^2*c^2*d*e^4-a*b^3*e^5-8*a*b^2*c*d*e^4+30*a*b*c^2*d^2*e^3-20*a* 
c^3*d^3*e^2+b^4*d*e^4-10*b^2*c^2*d^3*e^2+15*b*c^3*d^4*e-6*c^4*d^5)/(16*a^2 
*c^2-8*a*b^2*c+b^4)*x^3-1/2*c*(16*a^3*c^2*e^5-29*a^2*b^2*c*e^5+26*a^2*b*c^ 
2*d*e^4+16*a^2*c^3*d^2*e^3+4*a*b^4*e^5+32*a*b^3*c*d*e^4-98*a*b^2*c^2*d^2*e 
^3+60*a*b*c^3*d^3*e^2-4*b^5*d*e^4+b^4*c*d^2*e^3+30*b^3*c^2*d^3*e^2-45*b^2* 
c^3*d^4*e+18*b*c^4*d^5)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2+(a^3*b*c^2*e^5-18*a 
^3*c^3*d*e^4+6*a^2*b^3*c*e^5-10*a^2*b^2*c^2*d*e^4+34*a^2*b*c^3*d^2*e^3-28* 
a^2*c^4*d^3*e^2-a*b^5*e^5-6*a*b^4*c*d*e^4+18*a*b^3*c^2*d^2*e^3-26*a*b^2*c^ 
3*d^3*e^2+25*a*b*c^4*d^4*e-10*a*c^5*d^5+b^6*d*e^4-b^5*c*d^2*e^3-3*b^4*c^2* 
d^3*e^2+5*b^3*c^3*d^4*e-2*b^2*c^4*d^5)/(16*a^2*c^2-8*a*b^2*c+b^4)*x-1/2*(2 
4*a^4*c^2*e^5-21*a^3*b^2*c*e^5-14*a^3*b*c^2*d*e^4+32*a^3*c^3*d^2*e^3+3*a^2 
*b^4*e^5+27*a^2*b^3*c*d*e^4-50*a^2*b^2*c^2*d^2*e^3+12*a^2*b*c^3*d^3*e^2+8* 
a^2*c^4*d^4*e-4*a*b^5*d*e^4-a*b^4*c*d^2*e^3+24*a*b^3*c^2*d^3*e^2-29*a*b^2* 
c^3*d^4*e+10*a*b*c^4*d^5+b^6*d^2*e^3-3*b^5*c*d^3*e^2+3*b^4*c^2*d^4*e-b^3*c 
^3*d^5)/(16*a^2*c^2-8*a*b^2*c+b^4))/(c*x^2+b*x+a)^2+1/(16*a^2*c^2-8*a*b^2* 
c+b^4)*(1/2*(16*a^2*c^3*e^5-8*a*b^2*c^2*e^5+b^4*c*e^5)/c*ln(c*x^2+b*x+a)+2 
*(23*a^2*b*c^2*e^5-30*a^2*c^3*d*e^4-9*a*b^3*c*e^5+30*a*b*c^3*d^2*e^3-20*a* 
c^4*d^3*e^2+b^5*e^5-10*b^2*c^3*d^3*e^2+15*b*c^4*d^4*e-6*d^5*c^5-1/2*(16*a^ 
2*c^3*e^5-8*a*b^2*c^2*e^5+b^4*c*e^5)*b/c)/(4*a*c-b^2)^(1/2)*arctan((2*c...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3778 vs. \(2 (423) = 846\).

Time = 165.60 (sec) , antiderivative size = 7575, normalized size of antiderivative = 17.49 \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(1/(e*x+d)/(c*x^2+b*x+a)^3,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate(1/(e*x+d)/(c*x**2+b*x+a)**3,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(1/(e*x+d)/(c*x^2+b*x+a)^3,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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1461 vs. \(2 (423) = 846\).

Time = 0.35 (sec) , antiderivative size = 1461, normalized size of antiderivative = 3.37 \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(1/(e*x+d)/(c*x^2+b*x+a)^3,x, algorithm="giac")
 

Output:

e^6*log(abs(e*x + d))/(c^3*d^6*e - 3*b*c^2*d^5*e^2 + 3*b^2*c*d^4*e^3 + 3*a 
*c^2*d^4*e^3 - b^3*d^3*e^4 - 6*a*b*c*d^3*e^4 + 3*a*b^2*d^2*e^5 + 3*a^2*c*d 
^2*e^5 - 3*a^2*b*d*e^6 + a^3*e^7) - 1/2*e^5*log(c*x^2 + b*x + a)/(c^3*d^6 
- 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 + 3*a*c^2*d^4*e^2 - b^3*d^3*e^3 - 6*a*b* 
c*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a^2*c*d^2*e^4 - 3*a^2*b*d*e^5 + a^3*e^6) + 
 (12*c^5*d^5 - 30*b*c^4*d^4*e + 20*b^2*c^3*d^3*e^2 + 40*a*c^4*d^3*e^2 - 60 
*a*b*c^3*d^2*e^3 + 60*a^2*c^3*d*e^4 - b^5*e^5 + 10*a*b^3*c*e^5 - 30*a^2*b* 
c^2*e^5)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^4*c^3*d^6 - 8*a*b^2*c^ 
4*d^6 + 16*a^2*c^5*d^6 - 3*b^5*c^2*d^5*e + 24*a*b^3*c^3*d^5*e - 48*a^2*b*c 
^4*d^5*e + 3*b^6*c*d^4*e^2 - 21*a*b^4*c^2*d^4*e^2 + 24*a^2*b^2*c^3*d^4*e^2 
 + 48*a^3*c^4*d^4*e^2 - b^7*d^3*e^3 + 2*a*b^5*c*d^3*e^3 + 32*a^2*b^3*c^2*d 
^3*e^3 - 96*a^3*b*c^3*d^3*e^3 + 3*a*b^6*d^2*e^4 - 21*a^2*b^4*c*d^2*e^4 + 2 
4*a^3*b^2*c^2*d^2*e^4 + 48*a^4*c^3*d^2*e^4 - 3*a^2*b^5*d*e^5 + 24*a^3*b^3* 
c*d*e^5 - 48*a^4*b*c^2*d*e^5 + a^3*b^4*e^6 - 8*a^4*b^2*c*e^6 + 16*a^5*c^2* 
e^6)*sqrt(-b^2 + 4*a*c)) - 1/2*(b^3*c^3*d^5 - 10*a*b*c^4*d^5 - 3*b^4*c^2*d 
^4*e + 29*a*b^2*c^3*d^4*e - 8*a^2*c^4*d^4*e + 3*b^5*c*d^3*e^2 - 24*a*b^3*c 
^2*d^3*e^2 - 12*a^2*b*c^3*d^3*e^2 - b^6*d^2*e^3 + a*b^4*c*d^2*e^3 + 50*a^2 
*b^2*c^2*d^2*e^3 - 32*a^3*c^3*d^2*e^3 + 4*a*b^5*d*e^4 - 27*a^2*b^3*c*d*e^4 
 + 14*a^3*b*c^2*d*e^4 - 3*a^2*b^4*e^5 + 21*a^3*b^2*c*e^5 - 24*a^4*c^2*e^5 
- 2*(6*c^6*d^5 - 15*b*c^5*d^4*e + 10*b^2*c^4*d^3*e^2 + 20*a*c^5*d^3*e^2...
 

Mupad [B] (verification not implemented)

Time = 11.09 (sec) , antiderivative size = 3292, normalized size of antiderivative = 7.60 \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

int(1/((d + e*x)*(a + b*x + c*x^2)^3),x)
 

Output:

((3*a*b^4*e^3 - b^5*d*e^2 + 24*a^3*c^2*e^3 - b^3*c^2*d^3 - 21*a^2*b^2*c*e^ 
3 + 8*a^2*c^3*d^2*e + 10*a*b*c^3*d^3 + 2*b^4*c*d^2*e + 6*a*b^3*c*d*e^2 - 1 
9*a*b^2*c^2*d^2*e + 10*a^2*b*c^2*d*e^2)/(2*(a^2*b^4*e^4 + 16*a^2*c^4*d^4 + 
 16*a^4*c^2*e^4 + b^4*c^2*d^4 + b^6*d^2*e^2 - 8*a*b^2*c^3*d^4 - 8*a^3*b^2* 
c*e^4 + 32*a^3*c^3*d^2*e^2 - 2*a*b^5*d*e^3 - 2*b^5*c*d^3*e + 16*a*b^3*c^2* 
d^3*e - 6*a*b^4*c*d^2*e^2 - 32*a^2*b*c^3*d^3*e + 16*a^2*b^3*c*d*e^3 - 32*a 
^3*b*c^2*d*e^3)) + (x^3*(6*c^5*d^3 + b^3*c^2*e^3 + b^2*c^3*d*e^2 - 7*a*b*c 
^3*e^3 + 14*a*c^4*d*e^2 - 9*b*c^4*d^2*e))/(a^2*b^4*e^4 + 16*a^2*c^4*d^4 + 
16*a^4*c^2*e^4 + b^4*c^2*d^4 + b^6*d^2*e^2 - 8*a*b^2*c^3*d^4 - 8*a^3*b^2*c 
*e^4 + 32*a^3*c^3*d^2*e^2 - 2*a*b^5*d*e^3 - 2*b^5*c*d^3*e + 16*a*b^3*c^2*d 
^3*e - 6*a*b^4*c*d^2*e^2 - 32*a^2*b*c^3*d^3*e + 16*a^2*b^3*c*d*e^3 - 32*a^ 
3*b*c^2*d*e^3) + (x*(b^5*e^3 + 10*a*c^4*d^3 + 2*b^2*c^3*d^3 - a^2*b*c^2*e^ 
3 + 18*a^2*c^3*d*e^2 - 3*b^3*c^2*d^2*e - 6*a*b^3*c*e^3 - 15*a*b*c^3*d^2*e 
+ 9*a*b^2*c^2*d*e^2))/(a^2*b^4*e^4 + 16*a^2*c^4*d^4 + 16*a^4*c^2*e^4 + b^4 
*c^2*d^4 + b^6*d^2*e^2 - 8*a*b^2*c^3*d^4 - 8*a^3*b^2*c*e^4 + 32*a^3*c^3*d^ 
2*e^2 - 2*a*b^5*d*e^3 - 2*b^5*c*d^3*e + 16*a*b^3*c^2*d^3*e - 6*a*b^4*c*d^2 
*e^2 - 32*a^2*b*c^3*d^3*e + 16*a^2*b^3*c*d*e^3 - 32*a^3*b*c^2*d*e^3) + (x^ 
2*(18*b*c^4*d^3 + 4*b^4*c*e^3 + 16*a^2*c^3*e^3 - 29*a*b^2*c^2*e^3 - 27*b^2 
*c^3*d^2*e + 3*b^3*c^2*d*e^2 + 42*a*b*c^3*d*e^2))/(2*(a^2*b^4*e^4 + 16*a^2 
*c^4*d^4 + 16*a^4*c^2*e^4 + b^4*c^2*d^4 + b^6*d^2*e^2 - 8*a*b^2*c^3*d^4...
 

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 7351, normalized size of antiderivative = 16.98 \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int(1/(e*x+d)/(c*x^2+b*x+a)^3,x)
 

Output:

( - 60*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**4*b**2*c 
**2*e**5 + 120*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a** 
4*b*c**3*d*e**4 + 20*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2 
))*a**3*b**4*c*e**5 - 120*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - 
 b**2))*a**3*b**3*c**2*e**5*x - 120*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sq 
rt(4*a*c - b**2))*a**3*b**2*c**3*d**2*e**3 + 240*sqrt(4*a*c - b**2)*atan(( 
b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b**2*c**3*d*e**4*x - 120*sqrt(4*a*c - 
b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b**2*c**3*e**5*x**2 + 80*s 
qrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b*c**4*d**3*e* 
*2 + 240*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b*c* 
*4*d*e**4*x**2 - 2*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2)) 
*a**2*b**6*e**5 + 40*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2 
))*a**2*b**5*c*e**5*x - 20*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c 
- b**2))*a**2*b**4*c**2*e**5*x**2 + 40*sqrt(4*a*c - b**2)*atan((b + 2*c*x) 
/sqrt(4*a*c - b**2))*a**2*b**3*c**3*d**3*e**2 - 240*sqrt(4*a*c - b**2)*ata 
n((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**3*c**3*d**2*e**3*x + 120*sqrt(4* 
a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**3*c**3*d*e**4*x** 
2 - 120*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**3* 
c**3*e**5*x**3 - 60*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2) 
)*a**2*b**2*c**4*d**4*e + 160*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(...