\(\int \frac {(e+f x)^5}{(a c+(b c+a d) x+b d x^2)^3} \, dx\) [364]

Optimal result

Integrand size = 29, antiderivative size = 328 \[ \int \frac {(e+f x)^5}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=-\frac {(b e-a f)^5}{2 b^3 (b c-a d)^3 (a+b x)^2}+\frac {(b e-a f)^4 (3 b d e-5 b c f+2 a d f)}{b^3 (b c-a d)^4 (a+b x)}+\frac {(d e-c f)^5}{2 d^3 (b c-a d)^3 (c+d x)^2}+\frac {(d e-c f)^4 (3 b d e+2 b c f-5 a d f)}{d^3 (b c-a d)^4 (c+d x)}+\frac {(b e-a f)^3 \left (a^2 d^2 f^2+a b d f (3 d e-5 c f)+b^2 \left (6 d^2 e^2-15 c d e f+10 c^2 f^2\right )\right ) \log (a+b x)}{b^3 (b c-a d)^5}-\frac {(d e-c f)^3 \left (10 a^2 d^2 f^2-5 a b d f (3 d e+c f)+b^2 \left (6 d^2 e^2+3 c d e f+c^2 f^2\right )\right ) \log (c+d x)}{d^3 (b c-a d)^5} \] Output:

-1/2*(-a*f+b*e)^5/b^3/(-a*d+b*c)^3/(b*x+a)^2+(-a*f+b*e)^4*(2*a*d*f-5*b*c*f 
+3*b*d*e)/b^3/(-a*d+b*c)^4/(b*x+a)+1/2*(-c*f+d*e)^5/d^3/(-a*d+b*c)^3/(d*x+ 
c)^2+(-c*f+d*e)^4*(-5*a*d*f+2*b*c*f+3*b*d*e)/d^3/(-a*d+b*c)^4/(d*x+c)+(-a* 
f+b*e)^3*(a^2*d^2*f^2+a*b*d*f*(-5*c*f+3*d*e)+b^2*(10*c^2*f^2-15*c*d*e*f+6* 
d^2*e^2))*ln(b*x+a)/b^3/(-a*d+b*c)^5-(-c*f+d*e)^3*(10*a^2*d^2*f^2-5*a*b*d* 
f*(c*f+3*d*e)+b^2*(c^2*f^2+3*c*d*e*f+6*d^2*e^2))*ln(d*x+c)/d^3/(-a*d+b*c)^ 
5
 

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.00 \[ \int \frac {(e+f x)^5}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=-\frac {(b e-a f)^5}{2 b^3 (b c-a d)^3 (a+b x)^2}+\frac {(b e-a f)^4 (3 b d e-5 b c f+2 a d f)}{b^3 (b c-a d)^4 (a+b x)}-\frac {(d e-c f)^5}{2 d^3 (-b c+a d)^3 (c+d x)^2}+\frac {(d e-c f)^4 (3 b d e+2 b c f-5 a d f)}{d^3 (b c-a d)^4 (c+d x)}+\frac {(b e-a f)^3 \left (a^2 d^2 f^2+a b d f (3 d e-5 c f)+b^2 \left (6 d^2 e^2-15 c d e f+10 c^2 f^2\right )\right ) \log (a+b x)}{b^3 (b c-a d)^5}+\frac {(d e-c f)^3 \left (10 a^2 d^2 f^2-5 a b d f (3 d e+c f)+b^2 \left (6 d^2 e^2+3 c d e f+c^2 f^2\right )\right ) \log (c+d x)}{d^3 (-b c+a d)^5} \] Input:

Integrate[(e + f*x)^5/(a*c + (b*c + a*d)*x + b*d*x^2)^3,x]
 

Output:

-1/2*(b*e - a*f)^5/(b^3*(b*c - a*d)^3*(a + b*x)^2) + ((b*e - a*f)^4*(3*b*d 
*e - 5*b*c*f + 2*a*d*f))/(b^3*(b*c - a*d)^4*(a + b*x)) - (d*e - c*f)^5/(2* 
d^3*(-(b*c) + a*d)^3*(c + d*x)^2) + ((d*e - c*f)^4*(3*b*d*e + 2*b*c*f - 5* 
a*d*f))/(d^3*(b*c - a*d)^4*(c + d*x)) + ((b*e - a*f)^3*(a^2*d^2*f^2 + a*b* 
d*f*(3*d*e - 5*c*f) + b^2*(6*d^2*e^2 - 15*c*d*e*f + 10*c^2*f^2))*Log[a + b 
*x])/(b^3*(b*c - a*d)^5) + ((d*e - c*f)^3*(10*a^2*d^2*f^2 - 5*a*b*d*f*(3*d 
*e + c*f) + b^2*(6*d^2*e^2 + 3*c*d*e*f + c^2*f^2))*Log[c + d*x])/(d^3*(-(b 
*c) + a*d)^5)
 

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.08, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {1141, 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.

\(\displaystyle \int \frac {(e+f x)^5}{\left (x (a d+b c)+a c+b d x^2\right )^3} \, dx\)

\(\Big \downarrow \) 1141

\(\displaystyle b^3 d^3 \int \left (\frac {(b e-a f)^5}{b^5 d^3 (b c-a d)^3 (a+b x)^3}-\frac {(3 b d e-5 b c f+2 a d f) (b e-a f)^4}{b^5 d^3 (b c-a d)^4 (a+b x)^2}+\frac {\left (\left (6 d^2 e^2-15 c d f e+10 c^2 f^2\right ) b^2+a d f (3 d e-5 c f) b+a^2 d^2 f^2\right ) (b e-a f)^3}{b^5 d^3 (b c-a d)^5 (a+b x)}-\frac {(d e-c f)^3 \left (\left (6 d^2 e^2+3 c d f e+c^2 f^2\right ) b^2-5 a d f (3 d e+c f) b+10 a^2 d^2 f^2\right )}{b^3 d^5 (b c-a d)^5 (c+d x)}-\frac {(d e-c f)^4 (3 b d e+2 b c f-5 a d f)}{b^3 d^5 (b c-a d)^4 (c+d x)^2}-\frac {(d e-c f)^5}{b^3 d^5 (b c-a d)^3 (c+d x)^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle b^3 d^3 \left (\frac {(b e-a f)^3 \log (a+b x) \left (a^2 d^2 f^2+a b d f (3 d e-5 c f)+b^2 \left (10 c^2 f^2-15 c d e f+6 d^2 e^2\right )\right )}{b^6 d^3 (b c-a d)^5}-\frac {(d e-c f)^3 \log (c+d x) \left (10 a^2 d^2 f^2-5 a b d f (c f+3 d e)+b^2 \left (c^2 f^2+3 c d e f+6 d^2 e^2\right )\right )}{b^3 d^6 (b c-a d)^5}-\frac {(b e-a f)^5}{2 b^6 d^3 (a+b x)^2 (b c-a d)^3}+\frac {(b e-a f)^4 (2 a d f-5 b c f+3 b d e)}{b^6 d^3 (a+b x) (b c-a d)^4}+\frac {(d e-c f)^4 (-5 a d f+2 b c f+3 b d e)}{b^3 d^6 (c+d x) (b c-a d)^4}+\frac {(d e-c f)^5}{2 b^3 d^6 (c+d x)^2 (b c-a d)^3}\right )\)

Input:

Int[(e + f*x)^5/(a*c + (b*c + a*d)*x + b*d*x^2)^3,x]
 

Output:

b^3*d^3*(-1/2*(b*e - a*f)^5/(b^6*d^3*(b*c - a*d)^3*(a + b*x)^2) + ((b*e - 
a*f)^4*(3*b*d*e - 5*b*c*f + 2*a*d*f))/(b^6*d^3*(b*c - a*d)^4*(a + b*x)) + 
(d*e - c*f)^5/(2*b^3*d^6*(b*c - a*d)^3*(c + d*x)^2) + ((d*e - c*f)^4*(3*b* 
d*e + 2*b*c*f - 5*a*d*f))/(b^3*d^6*(b*c - a*d)^4*(c + d*x)) + ((b*e - a*f) 
^3*(a^2*d^2*f^2 + a*b*d*f*(3*d*e - 5*c*f) + b^2*(6*d^2*e^2 - 15*c*d*e*f + 
10*c^2*f^2))*Log[a + b*x])/(b^6*d^3*(b*c - a*d)^5) - ((d*e - c*f)^3*(10*a^ 
2*d^2*f^2 - 5*a*b*d*f*(3*d*e + c*f) + b^2*(6*d^2*e^2 + 3*c*d*e*f + c^2*f^2 
))*Log[c + d*x])/(b^3*d^6*(b*c - a*d)^5))
 

Defintions of rubi rules used

rule 1141

Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_ 
Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[1/c^p   Int[ExpandIntegrand[ 
(d + e*x)^m*(b/2 - q/2 + c*x)^p*(b/2 + q/2 + c*x)^p, x], x], x] /; EqQ[p, - 
1] ||  !FractionalPowerFactorQ[q]] /; FreeQ[{a, b, c, d, e}, x] && ILtQ[p, 
0] && IntegerQ[m] && NiceSqrtQ[b^2 - 4*a*c]
 

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. \(816\) vs. \(2(324)=648\).

Time = 1.11 (sec) , antiderivative size = 817, normalized size of antiderivative = 2.49

method	result	size
default	\(-\frac {a^{5} f^{5}-5 a^{4} b e \,f^{4}+10 e^{2} f^{3} a^{3} b^{2}-10 a^{2} b^{3} e^{3} f^{2}+5 a \,b^{4} e^{4} f -b^{5} e^{5}}{2 b^{3} \left (a d -b c \right )^{3} \left (b x +a \right )^{2}}-\frac {-2 a^{5} d \,f^{5}+5 a^{4} b c \,f^{5}+5 a^{4} b d e \,f^{4}-20 a^{3} b^{2} c e \,f^{4}+30 a^{2} b^{3} c \,e^{2} f^{3}-10 a^{2} b^{3} d \,e^{3} f^{2}-20 a \,b^{4} c \,e^{3} f^{2}+10 a \,b^{4} d \,e^{4} f +5 b^{5} c \,e^{4} f -3 b^{5} d \,e^{5}}{b^{3} \left (a d -b c \right )^{4} \left (b x +a \right )}+\frac {\left (a^{5} d^{2} f^{5}-5 a^{4} b c d \,f^{5}+10 a^{3} b^{2} c^{2} f^{5}-30 a^{2} b^{3} c^{2} e \,f^{4}+30 a^{2} b^{3} c d \,e^{2} f^{3}-10 a^{2} b^{3} d^{2} e^{3} f^{2}+30 a \,b^{4} c^{2} e^{2} f^{3}-40 a \,b^{4} c d \,e^{3} f^{2}+15 a \,b^{4} d^{2} e^{4} f -10 b^{5} c^{2} e^{3} f^{2}+15 b^{5} c d \,e^{4} f -6 b^{5} d^{2} e^{5}\right ) \ln \left (b x +a \right )}{\left (a d -b c \right )^{5} b^{3}}-\frac {5 a \,c^{4} d \,f^{5}-20 a \,c^{3} d^{2} e \,f^{4}+30 a \,c^{2} d^{3} e^{2} f^{3}-20 a c \,d^{4} e^{3} f^{2}+5 a \,d^{5} e^{4} f -2 b \,c^{5} f^{5}+5 b \,c^{4} d e \,f^{4}-10 b \,c^{2} d^{3} e^{3} f^{2}+10 b c \,d^{4} e^{4} f -3 b \,d^{5} e^{5}}{d^{3} \left (a d -b c \right )^{4} \left (d x +c \right )}-\frac {-c^{5} f^{5}+5 e \,f^{4} c^{4} d -10 c^{3} d^{2} e^{2} f^{3}+10 c^{2} d^{3} e^{3} f^{2}-5 c \,d^{4} e^{4} f +d^{5} e^{5}}{2 d^{3} \left (a d -b c \right )^{3} \left (d x +c \right )^{2}}+\frac {\left (-10 a^{2} c^{3} d^{2} f^{5}+30 a^{2} c^{2} d^{3} e \,f^{4}-30 a^{2} c \,d^{4} e^{2} f^{3}+10 a^{2} d^{5} e^{3} f^{2}+5 a b \,c^{4} d \,f^{5}-30 a b \,c^{2} d^{3} e^{2} f^{3}+40 a b c \,d^{4} e^{3} f^{2}-15 a b \,d^{5} e^{4} f -b^{2} c^{5} f^{5}+10 b^{2} c^{2} d^{3} e^{3} f^{2}-15 b^{2} c \,d^{4} e^{4} f +6 b^{2} d^{5} e^{5}\right ) \ln \left (d x +c \right )}{\left (a d -b c \right )^{5} d^{3}}\)	\(817\)
norman	\(\text {Expression too large to display}\)	\(2112\)
risch	\(\text {Expression too large to display}\)	\(3708\)
parallelrisch	\(\text {Expression too large to display}\)	\(6151\)

Input:

int((f*x+e)^5/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x,method=_RETURNVERBOSE)
 

Output:

-1/2*(a^5*f^5-5*a^4*b*e*f^4+10*a^3*b^2*e^2*f^3-10*a^2*b^3*e^3*f^2+5*a*b^4* 
e^4*f-b^5*e^5)/b^3/(a*d-b*c)^3/(b*x+a)^2-(-2*a^5*d*f^5+5*a^4*b*c*f^5+5*a^4 
*b*d*e*f^4-20*a^3*b^2*c*e*f^4+30*a^2*b^3*c*e^2*f^3-10*a^2*b^3*d*e^3*f^2-20 
*a*b^4*c*e^3*f^2+10*a*b^4*d*e^4*f+5*b^5*c*e^4*f-3*b^5*d*e^5)/b^3/(a*d-b*c) 
^4/(b*x+a)+(a^5*d^2*f^5-5*a^4*b*c*d*f^5+10*a^3*b^2*c^2*f^5-30*a^2*b^3*c^2* 
e*f^4+30*a^2*b^3*c*d*e^2*f^3-10*a^2*b^3*d^2*e^3*f^2+30*a*b^4*c^2*e^2*f^3-4 
0*a*b^4*c*d*e^3*f^2+15*a*b^4*d^2*e^4*f-10*b^5*c^2*e^3*f^2+15*b^5*c*d*e^4*f 
-6*b^5*d^2*e^5)/(a*d-b*c)^5/b^3*ln(b*x+a)-1/d^3*(5*a*c^4*d*f^5-20*a*c^3*d^ 
2*e*f^4+30*a*c^2*d^3*e^2*f^3-20*a*c*d^4*e^3*f^2+5*a*d^5*e^4*f-2*b*c^5*f^5+ 
5*b*c^4*d*e*f^4-10*b*c^2*d^3*e^3*f^2+10*b*c*d^4*e^4*f-3*b*d^5*e^5)/(a*d-b* 
c)^4/(d*x+c)-1/2*(-c^5*f^5+5*c^4*d*e*f^4-10*c^3*d^2*e^2*f^3+10*c^2*d^3*e^3 
*f^2-5*c*d^4*e^4*f+d^5*e^5)/d^3/(a*d-b*c)^3/(d*x+c)^2+1/(a*d-b*c)^5*(-10*a 
^2*c^3*d^2*f^5+30*a^2*c^2*d^3*e*f^4-30*a^2*c*d^4*e^2*f^3+10*a^2*d^5*e^3*f^ 
2+5*a*b*c^4*d*f^5-30*a*b*c^2*d^3*e^2*f^3+40*a*b*c*d^4*e^3*f^2-15*a*b*d^5*e 
^4*f-b^2*c^5*f^5+10*b^2*c^2*d^3*e^3*f^2-15*b^2*c*d^4*e^4*f+6*b^2*d^5*e^5)/ 
d^3*ln(d*x+c)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4299 vs. \(2 (324) = 648\).

Time = 1.22 (sec) , antiderivative size = 4299, normalized size of antiderivative = 13.11 \[ \int \frac {(e+f x)^5}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((f*x+e)^5/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(e+f x)^5}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate((f*x+e)**5/(a*c+(a*d+b*c)*x+b*d*x**2)**3,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1961 vs. \(2 (324) = 648\).

Time = 0.10 (sec) , antiderivative size = 1961, normalized size of antiderivative = 5.98 \[ \int \frac {(e+f x)^5}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((f*x+e)^5/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x, algorithm="maxima")
 

Output:

(6*b^5*d^2*e^5 + 30*a^2*b^3*c^2*e*f^4 - 15*(b^5*c*d + a*b^4*d^2)*e^4*f + 1 
0*(b^5*c^2 + 4*a*b^4*c*d + a^2*b^3*d^2)*e^3*f^2 - 30*(a*b^4*c^2 + a^2*b^3* 
c*d)*e^2*f^3 - (10*a^3*b^2*c^2 - 5*a^4*b*c*d + a^5*d^2)*f^5)*log(b*x + a)/ 
(b^8*c^5 - 5*a*b^7*c^4*d + 10*a^2*b^6*c^3*d^2 - 10*a^3*b^5*c^2*d^3 + 5*a^4 
*b^4*c*d^4 - a^5*b^3*d^5) - (6*b^2*d^5*e^5 + 30*a^2*c^2*d^3*e*f^4 - 15*(b^ 
2*c*d^4 + a*b*d^5)*e^4*f + 10*(b^2*c^2*d^3 + 4*a*b*c*d^4 + a^2*d^5)*e^3*f^ 
2 - 30*(a*b*c^2*d^3 + a^2*c*d^4)*e^2*f^3 - (b^2*c^5 - 5*a*b*c^4*d + 10*a^2 
*c^3*d^2)*f^5)*log(d*x + c)/(b^5*c^5*d^3 - 5*a*b^4*c^4*d^4 + 10*a^2*b^3*c^ 
3*d^5 - 10*a^3*b^2*c^2*d^6 + 5*a^4*b*c*d^7 - a^5*d^8) - 1/2*((b^6*c^3*d^3 
- 7*a*b^5*c^2*d^4 - 7*a^2*b^4*c*d^5 + a^3*b^3*d^6)*e^5 + 5*(a*b^5*c^3*d^3 
+ 10*a^2*b^4*c^2*d^4 + a^3*b^3*c*d^5)*e^4*f - 60*(a^2*b^4*c^3*d^3 + a^3*b^ 
3*c^2*d^4)*e^3*f^2 + 10*(a^2*b^4*c^4*d^2 + 10*a^3*b^3*c^3*d^3 + a^4*b^2*c^ 
2*d^4)*e^2*f^3 + 5*(a^2*b^4*c^5*d - 7*a^3*b^3*c^4*d^2 - 7*a^4*b^2*c^3*d^3 
+ a^5*b*c^2*d^4)*e*f^4 - 3*(a^2*b^4*c^6 - 3*a^3*b^3*c^5*d - 3*a^5*b*c^3*d^ 
3 + a^6*c^2*d^4)*f^5 - 2*(6*b^6*d^6*e^5 - 15*(b^6*c*d^5 + a*b^5*d^6)*e^4*f 
 + 10*(b^6*c^2*d^4 + 4*a*b^5*c*d^5 + a^2*b^4*d^6)*e^3*f^2 - 30*(a*b^5*c^2* 
d^4 + a^2*b^4*c*d^5)*e^2*f^3 - 5*(b^6*c^4*d^2 - 4*a*b^5*c^3*d^3 - 4*a^3*b^ 
3*c*d^5 + a^4*b^2*d^6)*e*f^4 + (2*b^6*c^5*d - 5*a*b^5*c^4*d^2 - 5*a^4*b^2* 
c*d^5 + 2*a^5*b*d^6)*f^5)*x^3 - (18*(b^6*c*d^5 + a*b^5*d^6)*e^5 - 45*(b^6* 
c^2*d^4 + 2*a*b^5*c*d^5 + a^2*b^4*d^6)*e^4*f + 30*(b^6*c^3*d^3 + 5*a*b^...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1915 vs. \(2 (324) = 648\).

Time = 0.29 (sec) , antiderivative size = 1915, normalized size of antiderivative = 5.84 \[ \int \frac {(e+f x)^5}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((f*x+e)^5/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x, algorithm="giac")
 

Output:

(6*b^5*d^2*e^5 - 15*b^5*c*d*e^4*f - 15*a*b^4*d^2*e^4*f + 10*b^5*c^2*e^3*f^ 
2 + 40*a*b^4*c*d*e^3*f^2 + 10*a^2*b^3*d^2*e^3*f^2 - 30*a*b^4*c^2*e^2*f^3 - 
 30*a^2*b^3*c*d*e^2*f^3 + 30*a^2*b^3*c^2*e*f^4 - 10*a^3*b^2*c^2*f^5 + 5*a^ 
4*b*c*d*f^5 - a^5*d^2*f^5)*log(abs(b*x + a))/(b^8*c^5 - 5*a*b^7*c^4*d + 10 
*a^2*b^6*c^3*d^2 - 10*a^3*b^5*c^2*d^3 + 5*a^4*b^4*c*d^4 - a^5*b^3*d^5) - ( 
6*b^2*d^5*e^5 - 15*b^2*c*d^4*e^4*f - 15*a*b*d^5*e^4*f + 10*b^2*c^2*d^3*e^3 
*f^2 + 40*a*b*c*d^4*e^3*f^2 + 10*a^2*d^5*e^3*f^2 - 30*a*b*c^2*d^3*e^2*f^3 
- 30*a^2*c*d^4*e^2*f^3 + 30*a^2*c^2*d^3*e*f^4 - b^2*c^5*f^5 + 5*a*b*c^4*d* 
f^5 - 10*a^2*c^3*d^2*f^5)*log(abs(d*x + c))/(b^5*c^5*d^3 - 5*a*b^4*c^4*d^4 
 + 10*a^2*b^3*c^3*d^5 - 10*a^3*b^2*c^2*d^6 + 5*a^4*b*c*d^7 - a^5*d^8) - 1/ 
2*(b^6*c^3*d^3*e^5 - 7*a*b^5*c^2*d^4*e^5 - 7*a^2*b^4*c*d^5*e^5 + a^3*b^3*d 
^6*e^5 + 5*a*b^5*c^3*d^3*e^4*f + 50*a^2*b^4*c^2*d^4*e^4*f + 5*a^3*b^3*c*d^ 
5*e^4*f - 60*a^2*b^4*c^3*d^3*e^3*f^2 - 60*a^3*b^3*c^2*d^4*e^3*f^2 + 10*a^2 
*b^4*c^4*d^2*e^2*f^3 + 100*a^3*b^3*c^3*d^3*e^2*f^3 + 10*a^4*b^2*c^2*d^4*e^ 
2*f^3 + 5*a^2*b^4*c^5*d*e*f^4 - 35*a^3*b^3*c^4*d^2*e*f^4 - 35*a^4*b^2*c^3* 
d^3*e*f^4 + 5*a^5*b*c^2*d^4*e*f^4 - 3*a^2*b^4*c^6*f^5 + 9*a^3*b^3*c^5*d*f^ 
5 + 9*a^5*b*c^3*d^3*f^5 - 3*a^6*c^2*d^4*f^5 - 2*(6*b^6*d^6*e^5 - 15*b^6*c* 
d^5*e^4*f - 15*a*b^5*d^6*e^4*f + 10*b^6*c^2*d^4*e^3*f^2 + 40*a*b^5*c*d^5*e 
^3*f^2 + 10*a^2*b^4*d^6*e^3*f^2 - 30*a*b^5*c^2*d^4*e^2*f^3 - 30*a^2*b^4*c* 
d^5*e^2*f^3 - 5*b^6*c^4*d^2*e*f^4 + 20*a*b^5*c^3*d^3*e*f^4 + 20*a^3*b^3...
 

Mupad [B] (verification not implemented)

Time = 8.01 (sec) , antiderivative size = 2161, normalized size of antiderivative = 6.59 \[ \int \frac {(e+f x)^5}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\text {Too large to display} \] Input:

int((e + f*x)^5/(a*c + x*(a*d + b*c) + b*d*x^2)^3,x)
 

Output:

- ((a^3*b^3*d^6*e^5 - 3*a^2*b^4*c^6*f^5 - 3*a^6*c^2*d^4*f^5 + b^6*c^3*d^3* 
e^5 - 7*a*b^5*c^2*d^4*e^5 - 7*a^2*b^4*c*d^5*e^5 + 9*a^3*b^3*c^5*d*f^5 + 9* 
a^5*b*c^3*d^3*f^5 + 5*a*b^5*c^3*d^3*e^4*f + 5*a^2*b^4*c^5*d*e*f^4 + 5*a^3* 
b^3*c*d^5*e^4*f + 5*a^5*b*c^2*d^4*e*f^4 + 50*a^2*b^4*c^2*d^4*e^4*f - 35*a^ 
3*b^3*c^4*d^2*e*f^4 - 35*a^4*b^2*c^3*d^3*e*f^4 - 60*a^2*b^4*c^3*d^3*e^3*f^ 
2 + 10*a^2*b^4*c^4*d^2*e^2*f^3 - 60*a^3*b^3*c^2*d^4*e^3*f^2 + 100*a^3*b^3* 
c^3*d^3*e^2*f^3 + 10*a^4*b^2*c^2*d^4*e^2*f^3)/(2*b^3*d^3*(a^4*d^4 + b^4*c^ 
4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)) - (x^3*(2*a^5*d^5* 
f^5 + 2*b^5*c^5*f^5 + 6*b^5*d^5*e^5 + 10*a^2*b^3*d^5*e^3*f^2 + 10*b^5*c^2* 
d^3*e^3*f^2 - 5*a*b^4*c^4*d*f^5 - 5*a^4*b*c*d^4*f^5 - 15*a*b^4*d^5*e^4*f - 
 5*a^4*b*d^5*e*f^4 - 15*b^5*c*d^4*e^4*f - 5*b^5*c^4*d*e*f^4 + 40*a*b^4*c*d 
^4*e^3*f^2 + 20*a*b^4*c^3*d^2*e*f^4 + 20*a^3*b^2*c*d^4*e*f^4 - 30*a*b^4*c^ 
2*d^3*e^2*f^3 - 30*a^2*b^3*c*d^4*e^2*f^3))/(b^2*d^2*(a^4*d^4 + b^4*c^4 + 6 
*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)) + (x*(5*a^3*b^3*c^4*d^2 
*f^5 - 3*a^6*c*d^5*f^5 - 2*a^2*b^4*d^6*e^5 - 2*b^6*c^2*d^4*e^5 - 3*a*b^5*c 
^6*f^5 + 5*a^4*b^2*c^3*d^3*f^5 - 14*a*b^5*c*d^5*e^5 + 7*a^2*b^4*c^5*d*f^5 
+ 7*a^5*b*c^2*d^4*f^5 + 5*a^3*b^3*d^6*e^4*f + 5*b^6*c^3*d^3*e^4*f + 40*a*b 
^5*c^2*d^4*e^4*f + 40*a^2*b^4*c*d^5*e^4*f - 50*a*b^5*c^3*d^3*e^3*f^2 + 10* 
a*b^5*c^4*d^2*e^2*f^3 - 30*a^2*b^4*c^4*d^2*e*f^4 - 50*a^3*b^3*c*d^5*e^3*f^ 
2 - 40*a^3*b^3*c^3*d^3*e*f^4 + 10*a^4*b^2*c*d^5*e^2*f^3 - 30*a^4*b^2*c^...
 

Reduce [B] (verification not implemented)

Time = 0.84 (sec) , antiderivative size = 8298, normalized size of antiderivative = 25.30 \[ \int \frac {(e+f x)^5}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int((f*x+e)^5/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x)
                                                                                    
                                                                                    
 

Output:

(2*log(a + b*x)*a**8*c**2*d**6*f**5 + 4*log(a + b*x)*a**8*c*d**7*f**5*x + 
2*log(a + b*x)*a**8*d**8*f**5*x**2 - 8*log(a + b*x)*a**7*b*c**3*d**5*f**5 
- 12*log(a + b*x)*a**7*b*c**2*d**6*f**5*x + 4*log(a + b*x)*a**7*b*d**8*f** 
5*x**3 + 10*log(a + b*x)*a**6*b**2*c**4*d**4*f**5 + 4*log(a + b*x)*a**6*b* 
*2*c**3*d**5*f**5*x - 20*log(a + b*x)*a**6*b**2*c**2*d**6*f**5*x**2 - 12*l 
og(a + b*x)*a**6*b**2*c*d**7*f**5*x**3 + 2*log(a + b*x)*a**6*b**2*d**8*f** 
5*x**4 + 20*log(a + b*x)*a**5*b**3*c**5*d**3*f**5 - 60*log(a + b*x)*a**5*b 
**3*c**4*d**4*e*f**4 + 60*log(a + b*x)*a**5*b**3*c**4*d**4*f**5*x + 60*log 
(a + b*x)*a**5*b**3*c**3*d**5*e**2*f**3 - 120*log(a + b*x)*a**5*b**3*c**3* 
d**5*e*f**4*x + 52*log(a + b*x)*a**5*b**3*c**3*d**5*f**5*x**2 - 20*log(a + 
 b*x)*a**5*b**3*c**2*d**6*e**3*f**2 + 120*log(a + b*x)*a**5*b**3*c**2*d**6 
*e**2*f**3*x - 60*log(a + b*x)*a**5*b**3*c**2*d**6*e*f**4*x**2 + 4*log(a + 
 b*x)*a**5*b**3*c**2*d**6*f**5*x**3 - 40*log(a + b*x)*a**5*b**3*c*d**7*e** 
3*f**2*x + 60*log(a + b*x)*a**5*b**3*c*d**7*e**2*f**3*x**2 - 8*log(a + b*x 
)*a**5*b**3*c*d**7*f**5*x**4 - 20*log(a + b*x)*a**5*b**3*d**8*e**3*f**2*x* 
*2 - 60*log(a + b*x)*a**4*b**4*c**5*d**3*e*f**4 + 40*log(a + b*x)*a**4*b** 
4*c**5*d**3*f**5*x + 120*log(a + b*x)*a**4*b**4*c**4*d**4*e**2*f**3 - 240* 
log(a + b*x)*a**4*b**4*c**4*d**4*e*f**4*x + 90*log(a + b*x)*a**4*b**4*c**4 
*d**4*f**5*x**2 - 100*log(a + b*x)*a**4*b**4*c**3*d**5*e**3*f**2 + 360*log 
(a + b*x)*a**4*b**4*c**3*d**5*e**2*f**3*x - 300*log(a + b*x)*a**4*b**4*...