\(\int \frac {(b+2 c x) (d+e x)^3}{(a+b x+c x^2)^3} \, dx\) [562]

Optimal result

Integrand size = 26, antiderivative size = 126 \[ \int \frac {(b+2 c x) (d+e x)^3}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {(d+e x)^3}{2 \left (a+b x+c x^2\right )^2}-\frac {3 e (d+e x) (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {6 e \left (c d^2-b d e+a e^2\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.71 \[ \int \frac {(b+2 c x) (d+e x)^3}{\left (a+b x+c x^2\right )^3} \, dx=\frac {1}{2} \left (\frac {-b e^3 (a+b x)-c^2 d^2 (d+3 e x)+c e^2 (3 a d+3 b d x+a e x)}{c^2 (a+x (b+c x))^2}+\frac {e \left (-b^3 e^2+b^2 c e (3 d+4 e x)+b c \left (7 a e^2+3 c d (d-2 e x)\right )+2 c^2 \left (3 c d^2 x-a e (12 d+5 e x)\right )\right )}{c^2 \left (-b^2+4 a c\right ) (a+x (b+c x))}+\frac {12 e \left (c d^2+e (-b d+a e)\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2}}\right ) \] Input:

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

Output:

((-(b*e^3*(a + b*x)) - c^2*d^2*(d + 3*e*x) + c*e^2*(3*a*d + 3*b*d*x + a*e* 
x))/(c^2*(a + x*(b + c*x))^2) + (e*(-(b^3*e^2) + b^2*c*e*(3*d + 4*e*x) + b 
*c*(7*a*e^2 + 3*c*d*(d - 2*e*x)) + 2*c^2*(3*c*d^2*x - a*e*(12*d + 5*e*x))) 
)/(c^2*(-b^2 + 4*a*c)*(a + x*(b + c*x))) + (12*e*(c*d^2 + e*(-(b*d) + a*e) 
)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2))/2
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1222, 1153, 1083, 219}

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.

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

\(\Big \downarrow \) 1222

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

\(\Big \downarrow \) 1153

\(\displaystyle \frac {3}{2} e \left (-\frac {2 \left (a e^2-b d e+c d^2\right ) \int \frac {1}{c x^2+b x+a}dx}{b^2-4 a c}-\frac {(d+e x) (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )-\frac {(d+e x)^3}{2 \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {3}{2} e \left (\frac {4 \left (a e^2-b d e+c d^2\right ) \int \frac {1}{b^2-(b+2 c x)^2-4 a c}d(b+2 c x)}{b^2-4 a c}-\frac {(d+e x) (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )-\frac {(d+e x)^3}{2 \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 219

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

Input:

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

Output:

-1/2*(d + e*x)^3/(a + b*x + c*x^2)^2 + (3*e*(-(((d + e*x)*(b*d - 2*a*e + ( 
2*c*d - b*e)*x))/((b^2 - 4*a*c)*(a + b*x + c*x^2))) + (4*(c*d^2 - b*d*e + 
a*e^2)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(3/2)))/2
 

Defintions of rubi rules used

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

rule 1153

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m - 1)*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x 
+ c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] - Simp[2*(2*p + 3)*((c*d^2 - 
b*d*e + a*e^2)/((p + 1)*(b^2 - 4*a*c)))   Int[(d + e*x)^(m - 2)*(a + b*x + 
c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + 2*p + 2, 0] 
&& LtQ[p, -1]
 

rule 1222

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(299\) vs. \(2(118)=236\).

Time = 1.28 (sec) , antiderivative size = 300, normalized size of antiderivative = 2.38

method	result	size
default	\(\frac {-\frac {e \left (5 a c \,e^{2}-2 b^{2} e^{2}+3 b c d e -3 c^{2} d^{2}\right ) x^{3}}{4 a c -b^{2}}-\frac {3 e \left (a b c \,e^{2}+8 a \,c^{2} d e -e^{2} b^{3}+b^{2} c d e -3 b \,c^{2} d^{2}\right ) x^{2}}{2 \left (4 a c -b^{2}\right ) c}-\frac {3 e \left (a^{2} c \,e^{2}-a \,b^{2} e^{2}+3 a b c d e +a \,c^{2} d^{2}-b^{2} c \,d^{2}\right ) x}{c \left (4 a c -b^{2}\right )}+\frac {3 a^{2} b \,e^{3}-12 a^{2} c d \,e^{2}+3 a b c \,d^{2} e -4 d^{3} a \,c^{2}+b^{2} c \,d^{3}}{2 \left (4 a c -b^{2}\right ) c}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {6 e \left (a \,e^{2}-b d e +c \,d^{2}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}}}\)	\(300\)
risch	\(\frac {-\frac {e \left (5 a c \,e^{2}-2 b^{2} e^{2}+3 b c d e -3 c^{2} d^{2}\right ) x^{3}}{4 a c -b^{2}}-\frac {3 e \left (a b c \,e^{2}+8 a \,c^{2} d e -e^{2} b^{3}+b^{2} c d e -3 b \,c^{2} d^{2}\right ) x^{2}}{2 \left (4 a c -b^{2}\right ) c}-\frac {3 e \left (a^{2} c \,e^{2}-a \,b^{2} e^{2}+3 a b c d e +a \,c^{2} d^{2}-b^{2} c \,d^{2}\right ) x}{c \left (4 a c -b^{2}\right )}+\frac {3 a^{2} b \,e^{3}-12 a^{2} c d \,e^{2}+3 a b c \,d^{2} e -4 d^{3} a \,c^{2}+b^{2} c \,d^{3}}{2 \left (4 a c -b^{2}\right ) c}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {3 e^{3} \ln \left (\left (-8 a \,c^{2}+2 b^{2} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {3}{2}}-4 a b c +b^{3}\right ) a}{\left (-4 a c +b^{2}\right )^{\frac {3}{2}}}-\frac {3 e^{2} \ln \left (\left (-8 a \,c^{2}+2 b^{2} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {3}{2}}-4 a b c +b^{3}\right ) b d}{\left (-4 a c +b^{2}\right )^{\frac {3}{2}}}+\frac {3 e \ln \left (\left (-8 a \,c^{2}+2 b^{2} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {3}{2}}-4 a b c +b^{3}\right ) c \,d^{2}}{\left (-4 a c +b^{2}\right )^{\frac {3}{2}}}-\frac {3 e^{3} \ln \left (\left (8 a \,c^{2}-2 b^{2} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {3}{2}}+4 a b c -b^{3}\right ) a}{\left (-4 a c +b^{2}\right )^{\frac {3}{2}}}+\frac {3 e^{2} \ln \left (\left (8 a \,c^{2}-2 b^{2} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {3}{2}}+4 a b c -b^{3}\right ) b d}{\left (-4 a c +b^{2}\right )^{\frac {3}{2}}}-\frac {3 e \ln \left (\left (8 a \,c^{2}-2 b^{2} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {3}{2}}+4 a b c -b^{3}\right ) c \,d^{2}}{\left (-4 a c +b^{2}\right )^{\frac {3}{2}}}\)	\(565\)

Input:

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

Output:

(-e*(5*a*c*e^2-2*b^2*e^2+3*b*c*d*e-3*c^2*d^2)/(4*a*c-b^2)*x^3-3/2*e*(a*b*c 
*e^2+8*a*c^2*d*e-b^3*e^2+b^2*c*d*e-3*b*c^2*d^2)/(4*a*c-b^2)/c*x^2-3/c*e*(a 
^2*c*e^2-a*b^2*e^2+3*a*b*c*d*e+a*c^2*d^2-b^2*c*d^2)/(4*a*c-b^2)*x+1/2*(3*a 
^2*b*e^3-12*a^2*c*d*e^2+3*a*b*c*d^2*e-4*a*c^2*d^3+b^2*c*d^3)/(4*a*c-b^2)/c 
)/(c*x^2+b*x+a)^2+6*e*(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)^(3/2)*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. 717 vs. \(2 (118) = 236\).

Time = 0.10 (sec) , antiderivative size = 1455, normalized size of antiderivative = 11.55 \[ \int \frac {(b+2 c x) (d+e x)^3}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/2*((b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^3 + 3*(a*b^3*c - 4*a^2*b*c^2)* 
d^2*e - 12*(a^2*b^2*c - 4*a^3*c^2)*d*e^2 + 3*(a^2*b^3 - 4*a^3*b*c)*e^3 + 2 
*(3*(b^2*c^3 - 4*a*c^4)*d^2*e - 3*(b^3*c^2 - 4*a*b*c^3)*d*e^2 + (2*b^4*c - 
 13*a*b^2*c^2 + 20*a^2*c^3)*e^3)*x^3 + 3*(3*(b^3*c^2 - 4*a*b*c^3)*d^2*e - 
(b^4*c + 4*a*b^2*c^2 - 32*a^2*c^3)*d*e^2 + (b^5 - 5*a*b^3*c + 4*a^2*b*c^2) 
*e^3)*x^2 + 6*(a^2*c^2*d^2*e - a^2*b*c*d*e^2 + a^3*c*e^3 + (c^4*d^2*e - b* 
c^3*d*e^2 + a*c^3*e^3)*x^4 + 2*(b*c^3*d^2*e - b^2*c^2*d*e^2 + a*b*c^2*e^3) 
*x^3 + ((b^2*c^2 + 2*a*c^3)*d^2*e - (b^3*c + 2*a*b*c^2)*d*e^2 + (a*b^2*c + 
 2*a^2*c^2)*e^3)*x^2 + 2*(a*b*c^2*d^2*e - a*b^2*c*d*e^2 + a^2*b*c*e^3)*x)* 
sqrt(b^2 - 4*a*c)*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^4*c - 5*a*b^2*c^2 + 4*a^2*c^3)* 
d^2*e - 3*(a*b^3*c - 4*a^2*b*c^2)*d*e^2 + (a*b^4 - 5*a^2*b^2*c + 4*a^3*c^2 
)*e^3)*x)/(a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3 + (b^4*c^3 - 8*a*b^2*c^4 
 + 16*a^2*c^5)*x^4 + 2*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*x^3 + (b^6*c 
 - 6*a*b^4*c^2 + 32*a^3*c^4)*x^2 + 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c 
^3)*x), -1/2*((b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^3 + 3*(a*b^3*c - 4*a^2* 
b*c^2)*d^2*e - 12*(a^2*b^2*c - 4*a^3*c^2)*d*e^2 + 3*(a^2*b^3 - 4*a^3*b*c)* 
e^3 + 2*(3*(b^2*c^3 - 4*a*c^4)*d^2*e - 3*(b^3*c^2 - 4*a*b*c^3)*d*e^2 + (2* 
b^4*c - 13*a*b^2*c^2 + 20*a^2*c^3)*e^3)*x^3 + 3*(3*(b^3*c^2 - 4*a*b*c^3)*d 
^2*e - (b^4*c + 4*a*b^2*c^2 - 32*a^2*c^3)*d*e^2 + (b^5 - 5*a*b^3*c + 4*...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 762 vs. \(2 (117) = 234\).

Time = 17.11 (sec) , antiderivative size = 762, normalized size of antiderivative = 6.05 \[ \int \frac {(b+2 c x) (d+e x)^3}{\left (a+b x+c x^2\right )^3} \, dx=- 3 e \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) \log {\left (x + \frac {- 48 a^{2} c^{2} e \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) + 24 a b^{2} c e \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) + 3 a b e^{3} - 3 b^{4} e \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) - 3 b^{2} d e^{2} + 3 b c d^{2} e}{6 a c e^{3} - 6 b c d e^{2} + 6 c^{2} d^{2} e} \right )} + 3 e \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) \log {\left (x + \frac {48 a^{2} c^{2} e \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) - 24 a b^{2} c e \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) + 3 a b e^{3} + 3 b^{4} e \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) - 3 b^{2} d e^{2} + 3 b c d^{2} e}{6 a c e^{3} - 6 b c d e^{2} + 6 c^{2} d^{2} e} \right )} + \frac {3 a^{2} b e^{3} - 12 a^{2} c d e^{2} + 3 a b c d^{2} e - 4 a c^{2} d^{3} + b^{2} c d^{3} + x^{3} \left (- 10 a c^{2} e^{3} + 4 b^{2} c e^{3} - 6 b c^{2} d e^{2} + 6 c^{3} d^{2} e\right ) + x^{2} \left (- 3 a b c e^{3} - 24 a c^{2} d e^{2} + 3 b^{3} e^{3} - 3 b^{2} c d e^{2} + 9 b c^{2} d^{2} e\right ) + x \left (- 6 a^{2} c e^{3} + 6 a b^{2} e^{3} - 18 a b c d e^{2} - 6 a c^{2} d^{2} e + 6 b^{2} c d^{2} e\right )}{8 a^{3} c^{2} - 2 a^{2} b^{2} c + x^{4} \cdot \left (8 a c^{4} - 2 b^{2} c^{3}\right ) + x^{3} \cdot \left (16 a b c^{3} - 4 b^{3} c^{2}\right ) + x^{2} \cdot \left (16 a^{2} c^{3} + 4 a b^{2} c^{2} - 2 b^{4} c\right ) + x \left (16 a^{2} b c^{2} - 4 a b^{3} c\right )} \] Input:

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

Output:

-3*e*sqrt(-1/(4*a*c - b**2)**3)*(a*e**2 - b*d*e + c*d**2)*log(x + (-48*a** 
2*c**2*e*sqrt(-1/(4*a*c - b**2)**3)*(a*e**2 - b*d*e + c*d**2) + 24*a*b**2* 
c*e*sqrt(-1/(4*a*c - b**2)**3)*(a*e**2 - b*d*e + c*d**2) + 3*a*b*e**3 - 3* 
b**4*e*sqrt(-1/(4*a*c - b**2)**3)*(a*e**2 - b*d*e + c*d**2) - 3*b**2*d*e** 
2 + 3*b*c*d**2*e)/(6*a*c*e**3 - 6*b*c*d*e**2 + 6*c**2*d**2*e)) + 3*e*sqrt( 
-1/(4*a*c - b**2)**3)*(a*e**2 - b*d*e + c*d**2)*log(x + (48*a**2*c**2*e*sq 
rt(-1/(4*a*c - b**2)**3)*(a*e**2 - b*d*e + c*d**2) - 24*a*b**2*c*e*sqrt(-1 
/(4*a*c - b**2)**3)*(a*e**2 - b*d*e + c*d**2) + 3*a*b*e**3 + 3*b**4*e*sqrt 
(-1/(4*a*c - b**2)**3)*(a*e**2 - b*d*e + c*d**2) - 3*b**2*d*e**2 + 3*b*c*d 
**2*e)/(6*a*c*e**3 - 6*b*c*d*e**2 + 6*c**2*d**2*e)) + (3*a**2*b*e**3 - 12* 
a**2*c*d*e**2 + 3*a*b*c*d**2*e - 4*a*c**2*d**3 + b**2*c*d**3 + x**3*(-10*a 
*c**2*e**3 + 4*b**2*c*e**3 - 6*b*c**2*d*e**2 + 6*c**3*d**2*e) + x**2*(-3*a 
*b*c*e**3 - 24*a*c**2*d*e**2 + 3*b**3*e**3 - 3*b**2*c*d*e**2 + 9*b*c**2*d* 
*2*e) + x*(-6*a**2*c*e**3 + 6*a*b**2*e**3 - 18*a*b*c*d*e**2 - 6*a*c**2*d** 
2*e + 6*b**2*c*d**2*e))/(8*a**3*c**2 - 2*a**2*b**2*c + x**4*(8*a*c**4 - 2* 
b**2*c**3) + x**3*(16*a*b*c**3 - 4*b**3*c**2) + x**2*(16*a**2*c**3 + 4*a*b 
**2*c**2 - 2*b**4*c) + x*(16*a**2*b*c**2 - 4*a*b**3*c))
 

Maxima [F(-2)]

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

integrate((2*c*x+b)*(e*x+d)^3/(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. 300 vs. \(2 (118) = 236\).

Time = 0.23 (sec) , antiderivative size = 300, normalized size of antiderivative = 2.38 \[ \int \frac {(b+2 c x) (d+e x)^3}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {6 \, {\left (c d^{2} e - b d e^{2} + a e^{3}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} - 4 \, a c\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {6 \, c^{3} d^{2} e x^{3} - 6 \, b c^{2} d e^{2} x^{3} + 4 \, b^{2} c e^{3} x^{3} - 10 \, a c^{2} e^{3} x^{3} + 9 \, b c^{2} d^{2} e x^{2} - 3 \, b^{2} c d e^{2} x^{2} - 24 \, a c^{2} d e^{2} x^{2} + 3 \, b^{3} e^{3} x^{2} - 3 \, a b c e^{3} x^{2} + 6 \, b^{2} c d^{2} e x - 6 \, a c^{2} d^{2} e x - 18 \, a b c d e^{2} x + 6 \, a b^{2} e^{3} x - 6 \, a^{2} c e^{3} x + b^{2} c d^{3} - 4 \, a c^{2} d^{3} + 3 \, a b c d^{2} e - 12 \, a^{2} c d e^{2} + 3 \, a^{2} b e^{3}}{2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} {\left (c x^{2} + b x + a\right )}^{2}} \] Input:

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

Output:

-6*(c*d^2*e - b*d*e^2 + a*e^3)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^ 
2 - 4*a*c)*sqrt(-b^2 + 4*a*c)) - 1/2*(6*c^3*d^2*e*x^3 - 6*b*c^2*d*e^2*x^3 
+ 4*b^2*c*e^3*x^3 - 10*a*c^2*e^3*x^3 + 9*b*c^2*d^2*e*x^2 - 3*b^2*c*d*e^2*x 
^2 - 24*a*c^2*d*e^2*x^2 + 3*b^3*e^3*x^2 - 3*a*b*c*e^3*x^2 + 6*b^2*c*d^2*e* 
x - 6*a*c^2*d^2*e*x - 18*a*b*c*d*e^2*x + 6*a*b^2*e^3*x - 6*a^2*c*e^3*x + b 
^2*c*d^3 - 4*a*c^2*d^3 + 3*a*b*c*d^2*e - 12*a^2*c*d*e^2 + 3*a^2*b*e^3)/((b 
^2*c - 4*a*c^2)*(c*x^2 + b*x + a)^2)
 

Mupad [B] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 412, normalized size of antiderivative = 3.27 \[ \int \frac {(b+2 c x) (d+e x)^3}{\left (a+b x+c x^2\right )^3} \, dx=\frac {\frac {3\,a^2\,b\,e^3-12\,a^2\,c\,d\,e^2+3\,a\,b\,c\,d^2\,e-4\,a\,c^2\,d^3+b^2\,c\,d^3}{2\,c\,\left (4\,a\,c-b^2\right )}+\frac {e\,x^3\,\left (2\,b^2\,e^2-3\,b\,c\,d\,e+3\,c^2\,d^2-5\,a\,c\,e^2\right )}{4\,a\,c-b^2}-\frac {3\,e\,x\,\left (a^2\,c\,e^2-a\,b^2\,e^2+3\,a\,b\,c\,d\,e+a\,c^2\,d^2-b^2\,c\,d^2\right )}{c\,\left (4\,a\,c-b^2\right )}-\frac {3\,e\,x^2\,\left (-b^3\,e^2+b^2\,c\,d\,e-3\,b\,c^2\,d^2+a\,b\,c\,e^2+8\,a\,c^2\,d\,e\right )}{2\,c\,\left (4\,a\,c-b^2\right )}}{x^2\,\left (b^2+2\,a\,c\right )+a^2+c^2\,x^4+2\,a\,b\,x+2\,b\,c\,x^3}-\frac {6\,e\,\mathrm {atan}\left (\frac {\left (\frac {3\,e\,\left (b^3-4\,a\,b\,c\right )\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{{\left (4\,a\,c-b^2\right )}^{5/2}}-\frac {6\,c\,e\,x\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{{\left (4\,a\,c-b^2\right )}^{3/2}}\right )\,\left (4\,a\,c-b^2\right )}{3\,c\,d^2\,e-3\,b\,d\,e^2+3\,a\,e^3}\right )\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{{\left (4\,a\,c-b^2\right )}^{3/2}} \] Input:

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

Output:

((3*a^2*b*e^3 - 4*a*c^2*d^3 + b^2*c*d^3 - 12*a^2*c*d*e^2 + 3*a*b*c*d^2*e)/ 
(2*c*(4*a*c - b^2)) + (e*x^3*(2*b^2*e^2 + 3*c^2*d^2 - 5*a*c*e^2 - 3*b*c*d* 
e))/(4*a*c - b^2) - (3*e*x*(a*c^2*d^2 - a*b^2*e^2 + a^2*c*e^2 - b^2*c*d^2 
+ 3*a*b*c*d*e))/(c*(4*a*c - b^2)) - (3*e*x^2*(a*b*c*e^2 - 3*b*c^2*d^2 - b^ 
3*e^2 + 8*a*c^2*d*e + b^2*c*d*e))/(2*c*(4*a*c - b^2)))/(x^2*(2*a*c + b^2) 
+ a^2 + c^2*x^4 + 2*a*b*x + 2*b*c*x^3) - (6*e*atan((((3*e*(b^3 - 4*a*b*c)* 
(a*e^2 + c*d^2 - b*d*e))/(4*a*c - b^2)^(5/2) - (6*c*e*x*(a*e^2 + c*d^2 - b 
*d*e))/(4*a*c - b^2)^(3/2))*(4*a*c - b^2))/(3*a*e^3 - 3*b*d*e^2 + 3*c*d^2* 
e))*(a*e^2 + c*d^2 - b*d*e))/(4*a*c - b^2)^(3/2)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 1447, normalized size of antiderivative = 11.48 \[ \int \frac {(b+2 c x) (d+e x)^3}{\left (a+b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b*c*e**3 
- 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**2*c*d 
*e**2 + 24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b* 
*2*c*e**3*x + 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a 
**2*b*c**2*d**2*e + 24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b* 
*2))*a**2*b*c**2*e**3*x**2 - 24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4 
*a*c - b**2))*a*b**3*c*d*e**2*x + 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/s 
qrt(4*a*c - b**2))*a*b**3*c*e**3*x**2 + 24*sqrt(4*a*c - b**2)*atan((b + 2* 
c*x)/sqrt(4*a*c - b**2))*a*b**2*c**2*d**2*e*x - 24*sqrt(4*a*c - b**2)*atan 
((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**2*c**2*d*e**2*x**2 + 24*sqrt(4*a*c - 
 b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**2*c**2*e**3*x**3 + 24*sqr 
t(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b*c**3*d**2*e*x**2 
+ 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b*c**3*e**3 
*x**4 - 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**4*c* 
d*e**2*x**2 + 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b 
**3*c**2*d**2*e*x**2 - 24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - 
 b**2))*b**3*c**2*d*e**2*x**3 + 24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqr 
t(4*a*c - b**2))*b**2*c**3*d**2*e*x**3 - 12*sqrt(4*a*c - b**2)*atan((b + 2 
*c*x)/sqrt(4*a*c - b**2))*b**2*c**3*d*e**2*x**4 + 12*sqrt(4*a*c - b**2)*at 
an((b + 2*c*x)/sqrt(4*a*c - b**2))*b*c**4*d**2*e*x**4 + 20*a**4*c**2*e*...