\(\int \frac {(a+b x) (a^2+2 a b x+b^2 x^2)^2}{(d+e x)^4} \, dx\) [24]

Optimal result

Integrand size = 31, antiderivative size = 130 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^4} \, dx=-\frac {b^4 (4 b d-5 a e) x}{e^5}+\frac {b^5 x^2}{2 e^4}+\frac {(b d-a e)^5}{3 e^6 (d+e x)^3}-\frac {5 b (b d-a e)^4}{2 e^6 (d+e x)^2}+\frac {10 b^2 (b d-a e)^3}{e^6 (d+e x)}+\frac {10 b^3 (b d-a e)^2 \log (d+e x)}{e^6} \] Output:

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.76 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^4} \, dx=\frac {-2 a^5 e^5-5 a^4 b e^4 (d+3 e x)-20 a^3 b^2 e^3 \left (d^2+3 d e x+3 e^2 x^2\right )+10 a^2 b^3 d e^2 \left (11 d^2+27 d e x+18 e^2 x^2\right )+10 a b^4 e \left (-13 d^4-27 d^3 e x-9 d^2 e^2 x^2+9 d e^3 x^3+3 e^4 x^4\right )+b^5 \left (47 d^5+81 d^4 e x-9 d^3 e^2 x^2-63 d^2 e^3 x^3-15 d e^4 x^4+3 e^5 x^5\right )+60 b^3 (b d-a e)^2 (d+e x)^3 \log (d+e x)}{6 e^6 (d+e x)^3} \] Input:

Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/(d + e*x)^4,x]
 

Output:

(-2*a^5*e^5 - 5*a^4*b*e^4*(d + 3*e*x) - 20*a^3*b^2*e^3*(d^2 + 3*d*e*x + 3* 
e^2*x^2) + 10*a^2*b^3*d*e^2*(11*d^2 + 27*d*e*x + 18*e^2*x^2) + 10*a*b^4*e* 
(-13*d^4 - 27*d^3*e*x - 9*d^2*e^2*x^2 + 9*d*e^3*x^3 + 3*e^4*x^4) + b^5*(47 
*d^5 + 81*d^4*e*x - 9*d^3*e^2*x^2 - 63*d^2*e^3*x^3 - 15*d*e^4*x^4 + 3*e^5* 
x^5) + 60*b^3*(b*d - a*e)^2*(d + e*x)^3*Log[d + e*x])/(6*e^6*(d + e*x)^3)
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1184, 27, 49, 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.

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

\(\Big \downarrow \) 1184

\(\displaystyle \frac {\int \frac {b^4 (a+b x)^5}{(d+e x)^4}dx}{b^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \int \frac {(a+b x)^5}{(d+e x)^4}dx\)

\(\Big \downarrow \) 49

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

\(\Big \downarrow \) 2009

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

Input:

Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/(d + e*x)^4,x]
 

Output:

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

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 49

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] 
&& IGtQ[m, 0] && IGtQ[m + n + 2, 0]
 

rule 1184

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^p   Int[(d + e*x)^m*(f + g*x 
)^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && E 
qQ[b^2 - 4*a*c, 0] && IntegerQ[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. \(250\) vs. \(2(124)=248\).

Time = 1.19 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.93

method	result	size
default	\(\frac {b^{4} \left (\frac {1}{2} b e \,x^{2}+5 a e x -4 b d x \right )}{e^{5}}-\frac {e^{5} a^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}}{3 e^{6} \left (e x +d \right )^{3}}+\frac {10 b^{3} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \ln \left (e x +d \right )}{e^{6}}-\frac {10 b^{2} \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{e^{6} \left (e x +d \right )}-\frac {5 b \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}{2 e^{6} \left (e x +d \right )^{2}}\)	\(251\)
norman	\(\frac {-\frac {2 e^{5} a^{5}+5 a^{4} b d \,e^{4}+20 a^{3} b^{2} d^{2} e^{3}-110 a^{2} b^{3} d^{3} e^{2}+220 a \,b^{4} d^{4} e -110 b^{5} d^{5}}{6 e^{6}}+\frac {b^{5} x^{5}}{2 e}-\frac {\left (10 a^{3} b^{2} e^{3}-30 a^{2} b^{3} d \,e^{2}+60 a \,b^{4} d^{2} e -30 b^{5} d^{3}\right ) x^{2}}{e^{4}}-\frac {\left (5 a^{4} b \,e^{4}+20 a^{3} b^{2} d \,e^{3}-90 a^{2} b^{3} d^{2} e^{2}+180 a \,b^{4} d^{3} e -90 b^{5} d^{4}\right ) x}{2 e^{5}}+\frac {5 b^{4} \left (2 a e -b d \right ) x^{4}}{2 e^{2}}}{\left (e x +d \right )^{3}}+\frac {10 b^{3} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \ln \left (e x +d \right )}{e^{6}}\)	\(257\)
risch	\(\frac {b^{5} x^{2}}{2 e^{4}}+\frac {5 b^{4} a x}{e^{4}}-\frac {4 b^{5} d x}{e^{5}}+\frac {\left (-10 a^{3} b^{2} e^{4}+30 a^{2} b^{3} d \,e^{3}-30 a \,b^{4} d^{2} e^{2}+10 b^{5} d^{3} e \right ) x^{2}-\frac {5 b \left (a^{4} e^{4}+4 a^{3} b d \,e^{3}-18 a^{2} b^{2} d^{2} e^{2}+20 a \,b^{3} d^{3} e -7 b^{4} d^{4}\right ) x}{2}-\frac {2 e^{5} a^{5}+5 a^{4} b d \,e^{4}+20 a^{3} b^{2} d^{2} e^{3}-110 a^{2} b^{3} d^{3} e^{2}+130 a \,b^{4} d^{4} e -47 b^{5} d^{5}}{6 e}}{e^{5} \left (e x +d \right )^{3}}+\frac {10 b^{3} \ln \left (e x +d \right ) a^{2}}{e^{4}}-\frac {20 b^{4} \ln \left (e x +d \right ) a d}{e^{5}}+\frac {10 b^{5} \ln \left (e x +d \right ) d^{2}}{e^{6}}\)	\(268\)
parallelrisch	\(\frac {180 \ln \left (e x +d \right ) x \,a^{2} b^{3} d^{2} e^{3}-60 a^{3} b^{2} d \,e^{4} x +110 b^{5} d^{5}+60 \ln \left (e x +d \right ) a^{2} b^{3} e^{5} x^{3}+180 \ln \left (e x +d \right ) a^{2} b^{3} d \,e^{4} x^{2}-360 \ln \left (e x +d \right ) x a \,b^{4} d^{3} e^{2}-120 \ln \left (e x +d \right ) a \,b^{4} d \,e^{4} x^{3}-360 \ln \left (e x +d \right ) x^{2} a \,b^{4} d^{2} e^{3}+180 \ln \left (e x +d \right ) x^{2} b^{5} d^{3} e^{2}+60 \ln \left (e x +d \right ) x^{3} b^{5} d^{2} e^{3}+60 \ln \left (e x +d \right ) b^{5} d^{5}-5 a^{4} b d \,e^{4}-20 a^{3} b^{2} d^{2} e^{3}+60 \ln \left (e x +d \right ) a^{2} b^{3} d^{3} e^{2}-120 \ln \left (e x +d \right ) a \,b^{4} d^{4} e -15 a^{4} b \,e^{5} x +270 b^{5} d^{4} e x +110 a^{2} b^{3} d^{3} e^{2}-220 a \,b^{4} d^{4} e +270 x \,a^{2} b^{3} d^{2} e^{3}-540 x a \,b^{4} d^{3} e^{2}+180 x^{2} a^{2} b^{3} d \,e^{4}-360 x^{2} a \,b^{4} d^{2} e^{3}+30 x^{4} a \,b^{4} e^{5}-15 x^{4} b^{5} d \,e^{4}-60 x^{2} a^{3} b^{2} e^{5}+180 x^{2} b^{5} d^{3} e^{2}-2 e^{5} a^{5}+180 \ln \left (e x +d \right ) x \,b^{5} d^{4} e +3 x^{5} e^{5} b^{5}}{6 e^{6} \left (e x +d \right )^{3}}\)	\(459\)

Input:

int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^4,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 425 vs. \(2 (124) = 248\).

Time = 0.07 (sec) , antiderivative size = 425, normalized size of antiderivative = 3.27 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^4} \, dx=\frac {3 \, b^{5} e^{5} x^{5} + 47 \, b^{5} d^{5} - 130 \, a b^{4} d^{4} e + 110 \, a^{2} b^{3} d^{3} e^{2} - 20 \, a^{3} b^{2} d^{2} e^{3} - 5 \, a^{4} b d e^{4} - 2 \, a^{5} e^{5} - 15 \, {\left (b^{5} d e^{4} - 2 \, a b^{4} e^{5}\right )} x^{4} - 9 \, {\left (7 \, b^{5} d^{2} e^{3} - 10 \, a b^{4} d e^{4}\right )} x^{3} - 3 \, {\left (3 \, b^{5} d^{3} e^{2} + 30 \, a b^{4} d^{2} e^{3} - 60 \, a^{2} b^{3} d e^{4} + 20 \, a^{3} b^{2} e^{5}\right )} x^{2} + 3 \, {\left (27 \, b^{5} d^{4} e - 90 \, a b^{4} d^{3} e^{2} + 90 \, a^{2} b^{3} d^{2} e^{3} - 20 \, a^{3} b^{2} d e^{4} - 5 \, a^{4} b e^{5}\right )} x + 60 \, {\left (b^{5} d^{5} - 2 \, a b^{4} d^{4} e + a^{2} b^{3} d^{3} e^{2} + {\left (b^{5} d^{2} e^{3} - 2 \, a b^{4} d e^{4} + a^{2} b^{3} e^{5}\right )} x^{3} + 3 \, {\left (b^{5} d^{3} e^{2} - 2 \, a b^{4} d^{2} e^{3} + a^{2} b^{3} d e^{4}\right )} x^{2} + 3 \, {\left (b^{5} d^{4} e - 2 \, a b^{4} d^{3} e^{2} + a^{2} b^{3} d^{2} e^{3}\right )} x\right )} \log \left (e x + d\right )}{6 \, {\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} \] Input:

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^4,x, algorithm="fricas")
 

Output:

1/6*(3*b^5*e^5*x^5 + 47*b^5*d^5 - 130*a*b^4*d^4*e + 110*a^2*b^3*d^3*e^2 - 
20*a^3*b^2*d^2*e^3 - 5*a^4*b*d*e^4 - 2*a^5*e^5 - 15*(b^5*d*e^4 - 2*a*b^4*e 
^5)*x^4 - 9*(7*b^5*d^2*e^3 - 10*a*b^4*d*e^4)*x^3 - 3*(3*b^5*d^3*e^2 + 30*a 
*b^4*d^2*e^3 - 60*a^2*b^3*d*e^4 + 20*a^3*b^2*e^5)*x^2 + 3*(27*b^5*d^4*e - 
90*a*b^4*d^3*e^2 + 90*a^2*b^3*d^2*e^3 - 20*a^3*b^2*d*e^4 - 5*a^4*b*e^5)*x 
+ 60*(b^5*d^5 - 2*a*b^4*d^4*e + a^2*b^3*d^3*e^2 + (b^5*d^2*e^3 - 2*a*b^4*d 
*e^4 + a^2*b^3*e^5)*x^3 + 3*(b^5*d^3*e^2 - 2*a*b^4*d^2*e^3 + a^2*b^3*d*e^4 
)*x^2 + 3*(b^5*d^4*e - 2*a*b^4*d^3*e^2 + a^2*b^3*d^2*e^3)*x)*log(e*x + d)) 
/(e^9*x^3 + 3*d*e^8*x^2 + 3*d^2*e^7*x + d^3*e^6)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 284 vs. \(2 (119) = 238\).

Time = 2.06 (sec) , antiderivative size = 284, normalized size of antiderivative = 2.18 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^4} \, dx=\frac {b^{5} x^{2}}{2 e^{4}} + \frac {10 b^{3} \left (a e - b d\right )^{2} \log {\left (d + e x \right )}}{e^{6}} + x \left (\frac {5 a b^{4}}{e^{4}} - \frac {4 b^{5} d}{e^{5}}\right ) + \frac {- 2 a^{5} e^{5} - 5 a^{4} b d e^{4} - 20 a^{3} b^{2} d^{2} e^{3} + 110 a^{2} b^{3} d^{3} e^{2} - 130 a b^{4} d^{4} e + 47 b^{5} d^{5} + x^{2} \left (- 60 a^{3} b^{2} e^{5} + 180 a^{2} b^{3} d e^{4} - 180 a b^{4} d^{2} e^{3} + 60 b^{5} d^{3} e^{2}\right ) + x \left (- 15 a^{4} b e^{5} - 60 a^{3} b^{2} d e^{4} + 270 a^{2} b^{3} d^{2} e^{3} - 300 a b^{4} d^{3} e^{2} + 105 b^{5} d^{4} e\right )}{6 d^{3} e^{6} + 18 d^{2} e^{7} x + 18 d e^{8} x^{2} + 6 e^{9} x^{3}} \] Input:

integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**2/(e*x+d)**4,x)
 

Output:

b**5*x**2/(2*e**4) + 10*b**3*(a*e - b*d)**2*log(d + e*x)/e**6 + x*(5*a*b** 
4/e**4 - 4*b**5*d/e**5) + (-2*a**5*e**5 - 5*a**4*b*d*e**4 - 20*a**3*b**2*d 
**2*e**3 + 110*a**2*b**3*d**3*e**2 - 130*a*b**4*d**4*e + 47*b**5*d**5 + x* 
*2*(-60*a**3*b**2*e**5 + 180*a**2*b**3*d*e**4 - 180*a*b**4*d**2*e**3 + 60* 
b**5*d**3*e**2) + x*(-15*a**4*b*e**5 - 60*a**3*b**2*d*e**4 + 270*a**2*b**3 
*d**2*e**3 - 300*a*b**4*d**3*e**2 + 105*b**5*d**4*e))/(6*d**3*e**6 + 18*d* 
*2*e**7*x + 18*d*e**8*x**2 + 6*e**9*x**3)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 282 vs. \(2 (124) = 248\).

Time = 0.04 (sec) , antiderivative size = 282, normalized size of antiderivative = 2.17 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^4} \, dx=\frac {47 \, b^{5} d^{5} - 130 \, a b^{4} d^{4} e + 110 \, a^{2} b^{3} d^{3} e^{2} - 20 \, a^{3} b^{2} d^{2} e^{3} - 5 \, a^{4} b d e^{4} - 2 \, a^{5} e^{5} + 60 \, {\left (b^{5} d^{3} e^{2} - 3 \, a b^{4} d^{2} e^{3} + 3 \, a^{2} b^{3} d e^{4} - a^{3} b^{2} e^{5}\right )} x^{2} + 15 \, {\left (7 \, b^{5} d^{4} e - 20 \, a b^{4} d^{3} e^{2} + 18 \, a^{2} b^{3} d^{2} e^{3} - 4 \, a^{3} b^{2} d e^{4} - a^{4} b e^{5}\right )} x}{6 \, {\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} + \frac {b^{5} e x^{2} - 2 \, {\left (4 \, b^{5} d - 5 \, a b^{4} e\right )} x}{2 \, e^{5}} + \frac {10 \, {\left (b^{5} d^{2} - 2 \, a b^{4} d e + a^{2} b^{3} e^{2}\right )} \log \left (e x + d\right )}{e^{6}} \] Input:

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^4,x, algorithm="maxima")
 

Output:

1/6*(47*b^5*d^5 - 130*a*b^4*d^4*e + 110*a^2*b^3*d^3*e^2 - 20*a^3*b^2*d^2*e 
^3 - 5*a^4*b*d*e^4 - 2*a^5*e^5 + 60*(b^5*d^3*e^2 - 3*a*b^4*d^2*e^3 + 3*a^2 
*b^3*d*e^4 - a^3*b^2*e^5)*x^2 + 15*(7*b^5*d^4*e - 20*a*b^4*d^3*e^2 + 18*a^ 
2*b^3*d^2*e^3 - 4*a^3*b^2*d*e^4 - a^4*b*e^5)*x)/(e^9*x^3 + 3*d*e^8*x^2 + 3 
*d^2*e^7*x + d^3*e^6) + 1/2*(b^5*e*x^2 - 2*(4*b^5*d - 5*a*b^4*e)*x)/e^5 + 
10*(b^5*d^2 - 2*a*b^4*d*e + a^2*b^3*e^2)*log(e*x + d)/e^6
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 263 vs. \(2 (124) = 248\).

Time = 0.19 (sec) , antiderivative size = 263, normalized size of antiderivative = 2.02 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^4} \, dx=\frac {10 \, {\left (b^{5} d^{2} - 2 \, a b^{4} d e + a^{2} b^{3} e^{2}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{6}} + \frac {b^{5} e^{4} x^{2} - 8 \, b^{5} d e^{3} x + 10 \, a b^{4} e^{4} x}{2 \, e^{8}} + \frac {47 \, b^{5} d^{5} - 130 \, a b^{4} d^{4} e + 110 \, a^{2} b^{3} d^{3} e^{2} - 20 \, a^{3} b^{2} d^{2} e^{3} - 5 \, a^{4} b d e^{4} - 2 \, a^{5} e^{5} + 60 \, {\left (b^{5} d^{3} e^{2} - 3 \, a b^{4} d^{2} e^{3} + 3 \, a^{2} b^{3} d e^{4} - a^{3} b^{2} e^{5}\right )} x^{2} + 15 \, {\left (7 \, b^{5} d^{4} e - 20 \, a b^{4} d^{3} e^{2} + 18 \, a^{2} b^{3} d^{2} e^{3} - 4 \, a^{3} b^{2} d e^{4} - a^{4} b e^{5}\right )} x}{6 \, {\left (e x + d\right )}^{3} e^{6}} \] Input:

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^4,x, algorithm="giac")
 

Output:

10*(b^5*d^2 - 2*a*b^4*d*e + a^2*b^3*e^2)*log(abs(e*x + d))/e^6 + 1/2*(b^5* 
e^4*x^2 - 8*b^5*d*e^3*x + 10*a*b^4*e^4*x)/e^8 + 1/6*(47*b^5*d^5 - 130*a*b^ 
4*d^4*e + 110*a^2*b^3*d^3*e^2 - 20*a^3*b^2*d^2*e^3 - 5*a^4*b*d*e^4 - 2*a^5 
*e^5 + 60*(b^5*d^3*e^2 - 3*a*b^4*d^2*e^3 + 3*a^2*b^3*d*e^4 - a^3*b^2*e^5)* 
x^2 + 15*(7*b^5*d^4*e - 20*a*b^4*d^3*e^2 + 18*a^2*b^3*d^2*e^3 - 4*a^3*b^2* 
d*e^4 - a^4*b*e^5)*x)/((e*x + d)^3*e^6)
 

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 285, normalized size of antiderivative = 2.19 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^4} \, dx=x\,\left (\frac {5\,a\,b^4}{e^4}-\frac {4\,b^5\,d}{e^5}\right )-\frac {\frac {2\,a^5\,e^5+5\,a^4\,b\,d\,e^4+20\,a^3\,b^2\,d^2\,e^3-110\,a^2\,b^3\,d^3\,e^2+130\,a\,b^4\,d^4\,e-47\,b^5\,d^5}{6\,e}+x\,\left (\frac {5\,a^4\,b\,e^4}{2}+10\,a^3\,b^2\,d\,e^3-45\,a^2\,b^3\,d^2\,e^2+50\,a\,b^4\,d^3\,e-\frac {35\,b^5\,d^4}{2}\right )-x^2\,\left (-10\,a^3\,b^2\,e^4+30\,a^2\,b^3\,d\,e^3-30\,a\,b^4\,d^2\,e^2+10\,b^5\,d^3\,e\right )}{d^3\,e^5+3\,d^2\,e^6\,x+3\,d\,e^7\,x^2+e^8\,x^3}+\frac {b^5\,x^2}{2\,e^4}+\frac {\ln \left (d+e\,x\right )\,\left (10\,a^2\,b^3\,e^2-20\,a\,b^4\,d\,e+10\,b^5\,d^2\right )}{e^6} \] Input:

int(((a + b*x)*(a^2 + b^2*x^2 + 2*a*b*x)^2)/(d + e*x)^4,x)
 

Output:

x*((5*a*b^4)/e^4 - (4*b^5*d)/e^5) - ((2*a^5*e^5 - 47*b^5*d^5 - 110*a^2*b^3 
*d^3*e^2 + 20*a^3*b^2*d^2*e^3 + 130*a*b^4*d^4*e + 5*a^4*b*d*e^4)/(6*e) + x 
*((5*a^4*b*e^4)/2 - (35*b^5*d^4)/2 + 10*a^3*b^2*d*e^3 - 45*a^2*b^3*d^2*e^2 
 + 50*a*b^4*d^3*e) - x^2*(10*b^5*d^3*e - 10*a^3*b^2*e^4 - 30*a*b^4*d^2*e^2 
 + 30*a^2*b^3*d*e^3))/(d^3*e^5 + e^8*x^3 + 3*d^2*e^6*x + 3*d*e^7*x^2) + (b 
^5*x^2)/(2*e^4) + (log(d + e*x)*(10*b^5*d^2 + 10*a^2*b^3*e^2 - 20*a*b^4*d* 
e))/e^6
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 469, normalized size of antiderivative = 3.61 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^4} \, dx=\frac {50 b^{5} d^{6}+60 \,\mathrm {log}\left (e x +d \right ) a^{2} b^{3} d^{4} e^{2}-120 \,\mathrm {log}\left (e x +d \right ) a \,b^{4} d^{5} e +180 \,\mathrm {log}\left (e x +d \right ) b^{5} d^{5} e x -15 a^{4} b d \,e^{5} x +90 a^{2} b^{3} d^{3} e^{3} x -60 a^{2} b^{3} d \,e^{5} x^{3}-180 a \,b^{4} d^{4} e^{2} x +120 a \,b^{4} d^{2} e^{4} x^{3}+30 a \,b^{4} d \,e^{5} x^{4}+50 a^{2} b^{3} d^{4} e^{2}-100 a \,b^{4} d^{5} e +180 \,\mathrm {log}\left (e x +d \right ) b^{5} d^{4} e^{2} x^{2}+60 \,\mathrm {log}\left (e x +d \right ) a^{2} b^{3} d \,e^{5} x^{3}-120 \,\mathrm {log}\left (e x +d \right ) a \,b^{4} d^{2} e^{4} x^{3}-360 \,\mathrm {log}\left (e x +d \right ) a \,b^{4} d^{3} e^{3} x^{2}-2 a^{5} d \,e^{5}-5 a^{4} b \,d^{2} e^{4}+20 a^{3} b^{2} e^{6} x^{3}+180 \,\mathrm {log}\left (e x +d \right ) a^{2} b^{3} d^{3} e^{3} x -360 \,\mathrm {log}\left (e x +d \right ) a \,b^{4} d^{4} e^{2} x +180 \,\mathrm {log}\left (e x +d \right ) a^{2} b^{3} d^{2} e^{4} x^{2}+60 \,\mathrm {log}\left (e x +d \right ) b^{5} d^{6}+60 \,\mathrm {log}\left (e x +d \right ) b^{5} d^{3} e^{3} x^{3}+90 b^{5} d^{5} e x -60 b^{5} d^{3} e^{3} x^{3}-15 b^{5} d^{2} e^{4} x^{4}+3 b^{5} d \,e^{5} x^{5}}{6 d \,e^{6} \left (e^{3} x^{3}+3 d \,e^{2} x^{2}+3 d^{2} e x +d^{3}\right )} \] Input:

int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^4,x)
 

Output:

(60*log(d + e*x)*a**2*b**3*d**4*e**2 + 180*log(d + e*x)*a**2*b**3*d**3*e** 
3*x + 180*log(d + e*x)*a**2*b**3*d**2*e**4*x**2 + 60*log(d + e*x)*a**2*b** 
3*d*e**5*x**3 - 120*log(d + e*x)*a*b**4*d**5*e - 360*log(d + e*x)*a*b**4*d 
**4*e**2*x - 360*log(d + e*x)*a*b**4*d**3*e**3*x**2 - 120*log(d + e*x)*a*b 
**4*d**2*e**4*x**3 + 60*log(d + e*x)*b**5*d**6 + 180*log(d + e*x)*b**5*d** 
5*e*x + 180*log(d + e*x)*b**5*d**4*e**2*x**2 + 60*log(d + e*x)*b**5*d**3*e 
**3*x**3 - 2*a**5*d*e**5 - 5*a**4*b*d**2*e**4 - 15*a**4*b*d*e**5*x + 20*a* 
*3*b**2*e**6*x**3 + 50*a**2*b**3*d**4*e**2 + 90*a**2*b**3*d**3*e**3*x - 60 
*a**2*b**3*d*e**5*x**3 - 100*a*b**4*d**5*e - 180*a*b**4*d**4*e**2*x + 120* 
a*b**4*d**2*e**4*x**3 + 30*a*b**4*d*e**5*x**4 + 50*b**5*d**6 + 90*b**5*d** 
5*e*x - 60*b**5*d**3*e**3*x**3 - 15*b**5*d**2*e**4*x**4 + 3*b**5*d*e**5*x* 
*5)/(6*d*e**6*(d**3 + 3*d**2*e*x + 3*d*e**2*x**2 + e**3*x**3))