\(\int \frac {f+g x}{(d+e x)^3 (a+b x+c x^2)} \, dx\) [755]

Optimal result

Integrand size = 25, antiderivative size = 390 \[ \int \frac {f+g x}{(d+e x)^3 \left (a+b x+c x^2\right )} \, dx=-\frac {e f-d g}{2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}+\frac {e^2 (b f-a g)-c d (2 e f-d g)}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)}-\frac {\left (2 c^3 d^3 f-b^2 e^3 (b f-a g)+c e^2 \left (3 b^2 d f+3 a b e f-3 a b d g-2 a^2 e g\right )-c^2 d (6 a e (e f-d g)+b d (3 e f+d g))\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )^3}+\frac {\left (b e^3 (b f-a g)+c^2 d^2 (3 e f-d g)-c e^2 (3 b d f+a e f-3 a d g)\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}-\frac {\left (b e^3 (b f-a g)+c^2 d^2 (3 e f-d g)-c e^2 (3 b d f+a e f-3 a d g)\right ) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 386, normalized size of antiderivative = 0.99 \[ \int \frac {f+g x}{(d+e x)^3 \left (a+b x+c x^2\right )} \, dx=\frac {-e f+d g}{2 \left (c d^2+e (-b d+a e)\right ) (d+e x)^2}+\frac {e^2 (b f-a g)+c d (-2 e f+d g)}{\left (c d^2+e (-b d+a e)\right )^2 (d+e x)}+\frac {\left (-2 c^3 d^3 f+b^2 e^3 (b f-a g)+c e^2 \left (-3 b^2 d f-3 a b e f+3 a b d g+2 a^2 e g\right )+c^2 d (6 a e (e f-d g)+b d (3 e f+d g))\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c} \left (-c d^2+e (b d-a e)\right )^3}-\frac {\left (b e^3 (-b f+a g)+c^2 d^2 (-3 e f+d g)+c e^2 (3 b d f+a e f-3 a d g)\right ) \log (d+e x)}{\left (c d^2+e (-b d+a e)\right )^3}+\frac {\left (b e^3 (-b f+a g)+c^2 d^2 (-3 e f+d g)+c e^2 (3 b d f+a e f-3 a d g)\right ) \log (a+x (b+c x))}{2 \left (c d^2+e (-b d+a e)\right )^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.68 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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.

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

\(\Big \downarrow \) 1200

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-1/2*(e*f - d*g)/((c*d^2 - b*d*e + a*e^2)*(d + e*x)^2) + (e^2*(b*f - a*g) 
- c*d*(2*e*f - d*g))/((c*d^2 - b*d*e + a*e^2)^2*(d + e*x)) - ((2*c^3*d^3*f 
 - b^2*e^3*(b*f - a*g) + c*e^2*(3*b^2*d*f + 3*a*b*e*f - 3*a*b*d*g - 2*a^2* 
e*g) - c^2*d*(6*a*e*(e*f - d*g) + b*d*(3*e*f + d*g)))*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)^3) + ((b*e^ 
3*(b*f - a*g) + c^2*d^2*(3*e*f - d*g) - c*e^2*(3*b*d*f + a*e*f - 3*a*d*g)) 
*Log[d + e*x])/(c*d^2 - b*d*e + a*e^2)^3 - ((b*e^3*(b*f - a*g) + c^2*d^2*( 
3*e*f - d*g) - c*e^2*(3*b*d*f + a*e*f - 3*a*d*g))*Log[a + b*x + c*x^2])/(2 
*(c*d^2 - b*d*e + a*e^2)^3)
 

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 = 2.27 (sec) , antiderivative size = 489, normalized size of antiderivative = 1.25

method	result	size
default	\(-\frac {a \,e^{2} g -b \,e^{2} f -c \,d^{2} g +2 c d e f}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (e x +d \right )}-\frac {\left (a b \,e^{3} g -3 a c d \,e^{2} g +a c \,e^{3} f -b^{2} e^{3} f +3 b c d \,e^{2} f +c^{2} d^{3} g -3 c^{2} d^{2} e f \right ) \ln \left (e x +d \right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{3}}+\frac {d g -e f}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (e x +d \right )^{2}}+\frac {\frac {\left (a b c \,e^{3} g -3 a \,c^{2} d \,e^{2} g +a \,c^{2} e^{3} f -b^{2} c \,e^{3} f +3 b \,c^{2} d \,e^{2} f +c^{3} d^{3} g -3 d^{2} f \,c^{3} e \right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (-a^{2} c \,e^{3} g +a \,b^{2} e^{3} g -3 a b c d \,e^{2} g +2 a b c \,e^{3} f +3 a \,c^{2} d^{2} e g -3 a \,c^{2} d \,e^{2} f -b^{3} e^{3} f +3 b^{2} c d \,e^{2} f -3 b \,c^{2} d^{2} e f +c^{3} d^{3} f -\frac {\left (a b c \,e^{3} g -3 a \,c^{2} d \,e^{2} g +a \,c^{2} e^{3} f -b^{2} c \,e^{3} f +3 b \,c^{2} d \,e^{2} f +c^{3} d^{3} g -3 d^{2} f \,c^{3} e \right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{3}}\)	\(489\)
risch	\(\text {Expression too large to display}\)	\(215586\)

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((g*x+f)/(e*x+d)^3/(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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 846 vs. \(2 (383) = 766\).

Time = 0.19 (sec) , antiderivative size = 846, normalized size of antiderivative = 2.17 \[ \int \frac {f+g x}{(d+e x)^3 \left (a+b x+c x^2\right )} \, dx=-\frac {{\left (3 \, c^{2} d^{2} e f - 3 \, b c d e^{2} f + b^{2} e^{3} f - a c e^{3} f - c^{2} d^{3} g + 3 \, a c d e^{2} g - a b e^{3} g\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (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}\right )}} + \frac {{\left (3 \, c^{2} d^{2} e^{2} f - 3 \, b c d e^{3} f + b^{2} e^{4} f - a c e^{4} f - c^{2} d^{3} e g + 3 \, a c d e^{3} g - a b e^{4} g\right )} \log \left ({\left | e x + d \right |}\right )}{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}} + \frac {{\left (2 \, c^{3} d^{3} f - 3 \, b c^{2} d^{2} e f + 3 \, b^{2} c d e^{2} f - 6 \, a c^{2} d e^{2} f - b^{3} e^{3} f + 3 \, a b c e^{3} f - b c^{2} d^{3} g + 6 \, a c^{2} d^{2} e g - 3 \, a b c d e^{2} g + a b^{2} e^{3} g - 2 \, a^{2} c e^{3} g\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (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}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {5 \, c^{2} d^{4} e f - 8 \, b c d^{3} e^{2} f + 3 \, b^{2} d^{2} e^{3} f + 6 \, a c d^{2} e^{3} f - 4 \, a b d e^{4} f + a^{2} e^{5} f - 3 \, c^{2} d^{5} g + 4 \, b c d^{4} e g - b^{2} d^{3} e^{2} g - 2 \, a c d^{3} e^{2} g + a^{2} d e^{4} g + 2 \, {\left (2 \, c^{2} d^{3} e^{2} f - 3 \, b c d^{2} e^{3} f + b^{2} d e^{4} f + 2 \, a c d e^{4} f - a b e^{5} f - c^{2} d^{4} e g + b c d^{3} e^{2} g - a b d e^{4} g + a^{2} e^{5} g\right )} x}{2 \, {\left (c d^{2} - b d e + a e^{2}\right )}^{3} {\left (e x + d\right )}^{2}} \] Input:

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

Output:

-1/2*(3*c^2*d^2*e*f - 3*b*c*d*e^2*f + b^2*e^3*f - a*c*e^3*f - c^2*d^3*g + 
3*a*c*d*e^2*g - a*b*e^3*g)*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) + (3*c^2*d^2*e^2*f 
 - 3*b*c*d*e^3*f + b^2*e^4*f - a*c*e^4*f - c^2*d^3*e*g + 3*a*c*d*e^3*g - a 
*b*e^4*g)*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) + (2*c^3*d^3*f - 3*b*c^2*d^2*e*f + 
 3*b^2*c*d*e^2*f - 6*a*c^2*d*e^2*f - b^3*e^3*f + 3*a*b*c*e^3*f - b*c^2*d^3 
*g + 6*a*c^2*d^2*e*g - 3*a*b*c*d*e^2*g + a*b^2*e^3*g - 2*a^2*c*e^3*g)*arct 
an((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((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)*sqrt(-b^2 + 4*a*c)) - 1/2*(5*c 
^2*d^4*e*f - 8*b*c*d^3*e^2*f + 3*b^2*d^2*e^3*f + 6*a*c*d^2*e^3*f - 4*a*b*d 
*e^4*f + a^2*e^5*f - 3*c^2*d^5*g + 4*b*c*d^4*e*g - b^2*d^3*e^2*g - 2*a*c*d 
^3*e^2*g + a^2*d*e^4*g + 2*(2*c^2*d^3*e^2*f - 3*b*c*d^2*e^3*f + b^2*d*e^4* 
f + 2*a*c*d*e^4*f - a*b*e^5*f - c^2*d^4*e*g + b*c*d^3*e^2*g - a*b*d*e^4*g 
+ a^2*e^5*g)*x)/((c*d^2 - b*d*e + a*e^2)^3*(e*x + d)^2)
                                                                                    
                                                                                    
 

Mupad [B] (verification not implemented)

Time = 41.29 (sec) , antiderivative size = 7045, normalized size of antiderivative = 18.06 \[ \int \frac {f+g x}{(d+e x)^3 \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

(log(((b^4*e^3*f + 4*a^2*c^2*e^3*f - b^2*c^2*d^3*g + (a*e^3*g*(b^2 - 4*a*c 
)^(3/2))/2 - (3*b*e^3*f*(b^2 - 4*a*c)^(3/2))/4 - a*b^3*e^3*g + 4*a*c^3*d^3 
*g - (b^3*e^3*f*(b^2 - 4*a*c)^(1/2))/4 + 2*c^3*d^3*f*(b^2 - 4*a*c)^(1/2) - 
 5*a*b^2*c*e^3*f + 4*a^2*b*c*e^3*g - 12*a*c^3*d^2*e*f - 3*b^3*c*d*e^2*f + 
(a*b^2*e^3*g*(b^2 - 4*a*c)^(1/2))/2 - b*c^2*d^3*g*(b^2 - 4*a*c)^(1/2) - (3 
*b^3*d*e^2*g*(b^2 - 4*a*c)^(1/2))/4 - 12*a^2*c^2*d*e^2*g + 3*b^2*c^2*d^2*e 
*f + (3*b*d*e^2*g*(b^2 - 4*a*c)^(3/2))/4 + (3*c*d*e^2*f*(b^2 - 4*a*c)^(3/2 
))/2 - (3*c*d^2*e*g*(b^2 - 4*a*c)^(3/2))/2 + 12*a*b*c^2*d*e^2*f + 3*a*b^2* 
c*d*e^2*g - 3*b*c^2*d^2*e*f*(b^2 - 4*a*c)^(1/2) + (3*b^2*c*d*e^2*f*(b^2 - 
4*a*c)^(1/2))/2 + (3*b^2*c*d^2*e*g*(b^2 - 4*a*c)^(1/2))/2)*(128*a^4*c^3*e^ 
6*g + 4*d*e^5*g*(b^2 - 4*a*c)^(7/2) + 3*e^6*g*x*(b^2 - 4*a*c)^(7/2) - 3*b^ 
2*e^6*f*(b^2 - 4*a*c)^(5/2) + 2*b^4*e^6*f*(b^2 - 4*a*c)^(3/2) + b^6*e^6*f* 
(b^2 - 4*a*c)^(1/2) - 48*a*b^5*c*e^6*f + 128*a*c^6*d^5*e*f + 16*b^6*c*d*e^ 
5*f - 32*b^6*c*e^6*f*x - 2*a*b^5*e^6*g*(b^2 - 4*a*c)^(1/2) + b^2*d*e^5*g*( 
b^2 - 4*a*c)^(5/2) - 10*b^4*d*e^5*g*(b^2 - 4*a*c)^(3/2) + 5*b^6*d*e^5*g*(b 
^2 - 4*a*c)^(1/2) + b^2*e^6*g*x*(b^2 - 4*a*c)^(5/2) - 3*b^4*e^6*g*x*(b^2 - 
 4*a*c)^(3/2) - b^6*e^6*g*x*(b^2 - 4*a*c)^(1/2) - 320*a^3*b*c^3*e^6*f + 48 
*a^2*b^4*c*e^6*g + 640*a^3*c^4*d*e^5*f - 32*b^2*c^5*d^5*e*f + 48*b^3*c^4*d 
^5*e*g - 48*c^2*d^2*e^4*f*(b^2 - 4*a*c)^(5/2) - 64*c^4*d^4*e^2*f*(b^2 - 4* 
a*c)^(3/2) + 48*c^2*d^3*e^3*g*(b^2 - 4*a*c)^(5/2) + 272*a^2*b^3*c^2*e^6...
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 4533, normalized size of antiderivative = 11.62 \[ \int \frac {f+g x}{(d+e x)^3 \left (a+b x+c x^2\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 4*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*c*d**3* 
e**3*g - 8*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*c* 
d**2*e**4*g*x - 4*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))* 
a**2*c*d*e**5*g*x**2 + 2*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - 
b**2))*a*b**2*d**3*e**3*g + 4*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a 
*c - b**2))*a*b**2*d**2*e**4*g*x + 2*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/s 
qrt(4*a*c - b**2))*a*b**2*d*e**5*g*x**2 - 6*sqrt(4*a*c - b**2)*atan((b + 2 
*c*x)/sqrt(4*a*c - b**2))*a*b*c*d**4*e**2*g + 6*sqrt(4*a*c - b**2)*atan((b 
 + 2*c*x)/sqrt(4*a*c - b**2))*a*b*c*d**3*e**3*f - 12*sqrt(4*a*c - b**2)*at 
an((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b*c*d**3*e**3*g*x + 12*sqrt(4*a*c - b 
**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b*c*d**2*e**4*f*x - 6*sqrt(4*a 
*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b*c*d**2*e**4*g*x**2 + 6 
*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b*c*d*e**5*f*x* 
*2 + 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*c**2*d** 
5*e*g - 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*c**2* 
d**4*e**2*f + 24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a 
*c**2*d**4*e**2*g*x - 24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - 
b**2))*a*c**2*d**3*e**3*f*x + 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt( 
4*a*c - b**2))*a*c**2*d**3*e**3*g*x**2 - 12*sqrt(4*a*c - b**2)*atan((b + 2 
*c*x)/sqrt(4*a*c - b**2))*a*c**2*d**2*e**4*f*x**2 - 2*sqrt(4*a*c - b**2...