3.3.95 \(\int \frac {(3+2 x+5 x^2) (2+x+3 x^2-5 x^3+4 x^4)}{(d+e x)^3} \, dx\) [295]

3.3.95.1 Optimal result

Integrand size = 36, antiderivative size = 231 \[ \int \frac {\left (3+2 x+5 x^2\right ) \left (2+x+3 x^2-5 x^3+4 x^4\right )}{(d+e x)^3} \, dx=-\frac {\left (200 d^3+102 d^2 e+51 d e^2+4 e^3\right ) x}{e^6}+\frac {\left (120 d^2+51 d e+17 e^2\right ) x^2}{2 e^5}-\frac {(60 d+17 e) x^3}{3 e^4}+\frac {5 x^4}{e^3}-\frac {\left (5 d^2-2 d e+3 e^2\right ) \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right )}{2 e^7 (d+e x)^2}+\frac {120 d^5+85 d^4 e+68 d^3 e^2+12 d^2 e^3+42 d e^4-7 e^5}{e^7 (d+e x)}+\frac {\left (300 d^4+170 d^3 e+102 d^2 e^2+12 d e^3+21 e^4\right ) \log (d+e x)}{e^7} \]

output

-(200*d^3+102*d^2*e+51*d*e^2+4*e^3)*x/e^6+1/2*(120*d^2+51*d*e+17*e^2)*x^2/ 
e^5-1/3*(60*d+17*e)*x^3/e^4+5*x^4/e^3-1/2*(5*d^2-2*d*e+3*e^2)*(4*d^4+5*d^3 
*e+3*d^2*e^2-d*e^3+2*e^4)/e^7/(e*x+d)^2+(120*d^5+85*d^4*e+68*d^3*e^2+12*d^ 
2*e^3+42*d*e^4-7*e^5)/e^7/(e*x+d)+(300*d^4+170*d^3*e+102*d^2*e^2+12*d*e^3+ 
21*e^4)*ln(e*x+d)/e^7
 

3.3.95.2 Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.88 \[ \int \frac {\left (3+2 x+5 x^2\right ) \left (2+x+3 x^2-5 x^3+4 x^4\right )}{(d+e x)^3} \, dx=\frac {660 d^6+d^5 e (459-480 x)-51 d^4 e^2 \left (-7+2 x+40 x^2\right )-3 d^3 e^3 \left (-20-34 x+357 x^2+200 x^3\right )+d^2 e^4 \left (189+48 x-561 x^2-340 x^3+150 x^4\right )-d e^5 \left (21-252 x+48 x^2+204 x^3-85 x^4+60 x^5\right )+e^6 \left (-18-42 x-24 x^3+51 x^4-34 x^5+30 x^6\right )+6 \left (300 d^4+170 d^3 e+102 d^2 e^2+12 d e^3+21 e^4\right ) (d+e x)^2 \log (d+e x)}{6 e^7 (d+e x)^2} \]

input

Integrate[((3 + 2*x + 5*x^2)*(2 + x + 3*x^2 - 5*x^3 + 4*x^4))/(d + e*x)^3, 
x]
 

output

(660*d^6 + d^5*e*(459 - 480*x) - 51*d^4*e^2*(-7 + 2*x + 40*x^2) - 3*d^3*e^ 
3*(-20 - 34*x + 357*x^2 + 200*x^3) + d^2*e^4*(189 + 48*x - 561*x^2 - 340*x 
^3 + 150*x^4) - d*e^5*(21 - 252*x + 48*x^2 + 204*x^3 - 85*x^4 + 60*x^5) + 
e^6*(-18 - 42*x - 24*x^3 + 51*x^4 - 34*x^5 + 30*x^6) + 6*(300*d^4 + 170*d^ 
3*e + 102*d^2*e^2 + 12*d*e^3 + 21*e^4)*(d + e*x)^2*Log[d + e*x])/(6*e^7*(d 
 + e*x)^2)
 

3.3.95.3 Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2159, 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.

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

\(\Big \downarrow \) 2159

\(\displaystyle \int \left (\frac {x \left (120 d^2+51 d e+17 e^2\right )}{e^5}+\frac {-200 d^3-102 d^2 e-51 d e^2-4 e^3}{e^6}+\frac {300 d^4+170 d^3 e+102 d^2 e^2+12 d e^3+21 e^4}{e^6 (d+e x)}+\frac {-120 d^5-85 d^4 e-68 d^3 e^2-12 d^2 e^3-42 d e^4+7 e^5}{e^6 (d+e x)^2}+\frac {20 d^6+17 d^5 e+17 d^4 e^2+4 d^3 e^3+21 d^2 e^4-7 d e^5+6 e^6}{e^6 (d+e x)^3}-\frac {x^2 (60 d+17 e)}{e^4}+\frac {20 x^3}{e^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {x^2 \left (120 d^2+51 d e+17 e^2\right )}{2 e^5}-\frac {x \left (200 d^3+102 d^2 e+51 d e^2+4 e^3\right )}{e^6}-\frac {\left (5 d^2-2 d e+3 e^2\right ) \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right )}{2 e^7 (d+e x)^2}+\frac {\left (300 d^4+170 d^3 e+102 d^2 e^2+12 d e^3+21 e^4\right ) \log (d+e x)}{e^7}+\frac {120 d^5+85 d^4 e+68 d^3 e^2+12 d^2 e^3+42 d e^4-7 e^5}{e^7 (d+e x)}-\frac {x^3 (60 d+17 e)}{3 e^4}+\frac {5 x^4}{e^3}\)

input

Int[((3 + 2*x + 5*x^2)*(2 + x + 3*x^2 - 5*x^3 + 4*x^4))/(d + e*x)^3,x]
 

output

-(((200*d^3 + 102*d^2*e + 51*d*e^2 + 4*e^3)*x)/e^6) + ((120*d^2 + 51*d*e + 
 17*e^2)*x^2)/(2*e^5) - ((60*d + 17*e)*x^3)/(3*e^4) + (5*x^4)/e^3 - ((5*d^ 
2 - 2*d*e + 3*e^2)*(4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4))/(2*e^7*( 
d + e*x)^2) + (120*d^5 + 85*d^4*e + 68*d^3*e^2 + 12*d^2*e^3 + 42*d*e^4 - 7 
*e^5)/(e^7*(d + e*x)) + ((300*d^4 + 170*d^3*e + 102*d^2*e^2 + 12*d*e^3 + 2 
1*e^4)*Log[d + e*x])/e^7
 

3.3.95.3.1 Defintions of rubi rules used

rule 2009

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

rule 2159

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x 
], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
 

3.3.95.4 Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.97

method	result	size
norman	\(\frac {\frac {\left (600 d^{5}+340 d^{4} e +204 d^{3} e^{2}+24 d^{2} e^{3}+42 d \,e^{4}-7 e^{5}\right ) x}{e^{6}}+\frac {5 x^{6}}{e}+\frac {900 d^{6}+510 d^{5} e +306 d^{4} e^{2}+36 d^{3} e^{3}+63 d^{2} e^{4}-7 d \,e^{5}-6 e^{6}}{2 e^{7}}-\frac {\left (30 d +17 e \right ) x^{5}}{3 e^{2}}+\frac {\left (150 d^{2}+85 d e +51 e^{2}\right ) x^{4}}{6 e^{3}}-\frac {2 \left (150 d^{3}+85 d^{2} e +51 d \,e^{2}+6 e^{3}\right ) x^{3}}{3 e^{4}}}{\left (e x +d \right )^{2}}+\frac {\left (300 d^{4}+170 d^{3} e +102 d^{2} e^{2}+12 d \,e^{3}+21 e^{4}\right ) \ln \left (e x +d \right )}{e^{7}}\)	\(225\)
default	\(-\frac {-5 e^{3} x^{4}+20 d \,e^{2} x^{3}+\frac {17}{3} e^{3} x^{3}-60 d^{2} e \,x^{2}-\frac {51}{2} d \,e^{2} x^{2}-\frac {17}{2} e^{3} x^{2}+200 x \,d^{3}+102 d^{2} e x +51 d \,e^{2} x +4 e^{3} x}{e^{6}}-\frac {-120 d^{5}-85 d^{4} e -68 d^{3} e^{2}-12 d^{2} e^{3}-42 d \,e^{4}+7 e^{5}}{e^{7} \left (e x +d \right )}+\frac {\left (300 d^{4}+170 d^{3} e +102 d^{2} e^{2}+12 d \,e^{3}+21 e^{4}\right ) \ln \left (e x +d \right )}{e^{7}}-\frac {20 d^{6}+17 d^{5} e +17 d^{4} e^{2}+4 d^{3} e^{3}+21 d^{2} e^{4}-7 d \,e^{5}+6 e^{6}}{2 e^{7} \left (e x +d \right )^{2}}\)	\(236\)
risch	\(\frac {5 x^{4}}{e^{3}}-\frac {20 d \,x^{3}}{e^{4}}-\frac {17 x^{3}}{3 e^{3}}+\frac {60 d^{2} x^{2}}{e^{5}}+\frac {51 d \,x^{2}}{2 e^{4}}+\frac {17 x^{2}}{2 e^{3}}-\frac {200 x \,d^{3}}{e^{6}}-\frac {102 d^{2} x}{e^{5}}-\frac {51 d x}{e^{4}}-\frac {4 x}{e^{3}}+\frac {\left (120 d^{5}+85 d^{4} e +68 d^{3} e^{2}+12 d^{2} e^{3}+42 d \,e^{4}-7 e^{5}\right ) x +\frac {220 d^{6}+153 d^{5} e +119 d^{4} e^{2}+20 d^{3} e^{3}+63 d^{2} e^{4}-7 d \,e^{5}-6 e^{6}}{2 e}}{e^{6} \left (e x +d \right )^{2}}+\frac {300 \ln \left (e x +d \right ) d^{4}}{e^{7}}+\frac {170 \ln \left (e x +d \right ) d^{3}}{e^{6}}+\frac {102 \ln \left (e x +d \right ) d^{2}}{e^{5}}+\frac {12 \ln \left (e x +d \right ) d}{e^{4}}+\frac {21 \ln \left (e x +d \right )}{e^{3}}\)	\(256\)
parallelrisch	\(\frac {-34 e^{6} x^{5}-18 e^{6}+30 e^{6} x^{6}+1530 d^{5} e +1800 \ln \left (e x +d \right ) x^{2} d^{4} e^{2}+612 \ln \left (e x +d \right ) x^{2} d^{2} e^{4}+72 \ln \left (e x +d \right ) x^{2} d \,e^{5}+1020 \ln \left (e x +d \right ) x^{2} d^{3} e^{3}+252 \ln \left (e x +d \right ) x d \,e^{5}+3600 \ln \left (e x +d \right ) x \,d^{5} e +2040 \ln \left (e x +d \right ) x \,d^{4} e^{2}+1224 \ln \left (e x +d \right ) x \,d^{3} e^{3}+144 \ln \left (e x +d \right ) x \,d^{2} e^{4}-21 d \,e^{5}+918 d^{4} e^{2}+108 d^{3} e^{3}+189 d^{2} e^{4}-600 d^{3} x^{3} e^{3}+3600 d^{5} e x +150 d^{2} e^{4} x^{4}-60 d \,e^{5} x^{5}+51 x^{4} e^{6}-24 x^{3} e^{6}-42 x \,e^{6}+1800 \ln \left (e x +d \right ) d^{6}+85 d \,e^{5} x^{4}-340 d^{2} e^{4} x^{3}+2040 d^{4} e^{2} x -204 x^{3} d \,e^{5}+1224 x \,d^{3} e^{3}+144 x \,d^{2} e^{4}+252 x d \,e^{5}+1020 \ln \left (e x +d \right ) d^{5} e +612 \ln \left (e x +d \right ) d^{4} e^{2}+72 \ln \left (e x +d \right ) d^{3} e^{3}+126 \ln \left (e x +d \right ) d^{2} e^{4}+2700 d^{6}+126 \ln \left (e x +d \right ) x^{2} e^{6}}{6 e^{7} \left (e x +d \right )^{2}}\)	\(415\)

input

int((5*x^2+2*x+3)*(4*x^4-5*x^3+3*x^2+x+2)/(e*x+d)^3,x,method=_RETURNVERBOS 
E)
 

output

((600*d^5+340*d^4*e+204*d^3*e^2+24*d^2*e^3+42*d*e^4-7*e^5)/e^6*x+5*x^6/e+1 
/2*(900*d^6+510*d^5*e+306*d^4*e^2+36*d^3*e^3+63*d^2*e^4-7*d*e^5-6*e^6)/e^7 
-1/3*(30*d+17*e)/e^2*x^5+1/6*(150*d^2+85*d*e+51*e^2)/e^3*x^4-2/3*(150*d^3+ 
85*d^2*e+51*d*e^2+6*e^3)/e^4*x^3)/(e*x+d)^2+(300*d^4+170*d^3*e+102*d^2*e^2 
+12*d*e^3+21*e^4)*ln(e*x+d)/e^7
 

3.3.95.5 Fricas [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.56 \[ \int \frac {\left (3+2 x+5 x^2\right ) \left (2+x+3 x^2-5 x^3+4 x^4\right )}{(d+e x)^3} \, dx=\frac {30 \, e^{6} x^{6} + 660 \, d^{6} + 459 \, d^{5} e + 357 \, d^{4} e^{2} + 60 \, d^{3} e^{3} + 189 \, d^{2} e^{4} - 21 \, d e^{5} - 18 \, e^{6} - 2 \, {\left (30 \, d e^{5} + 17 \, e^{6}\right )} x^{5} + {\left (150 \, d^{2} e^{4} + 85 \, d e^{5} + 51 \, e^{6}\right )} x^{4} - 4 \, {\left (150 \, d^{3} e^{3} + 85 \, d^{2} e^{4} + 51 \, d e^{5} + 6 \, e^{6}\right )} x^{3} - 3 \, {\left (680 \, d^{4} e^{2} + 357 \, d^{3} e^{3} + 187 \, d^{2} e^{4} + 16 \, d e^{5}\right )} x^{2} - 6 \, {\left (80 \, d^{5} e + 17 \, d^{4} e^{2} - 17 \, d^{3} e^{3} - 8 \, d^{2} e^{4} - 42 \, d e^{5} + 7 \, e^{6}\right )} x + 6 \, {\left (300 \, d^{6} + 170 \, d^{5} e + 102 \, d^{4} e^{2} + 12 \, d^{3} e^{3} + 21 \, d^{2} e^{4} + {\left (300 \, d^{4} e^{2} + 170 \, d^{3} e^{3} + 102 \, d^{2} e^{4} + 12 \, d e^{5} + 21 \, e^{6}\right )} x^{2} + 2 \, {\left (300 \, d^{5} e + 170 \, d^{4} e^{2} + 102 \, d^{3} e^{3} + 12 \, d^{2} e^{4} + 21 \, d e^{5}\right )} x\right )} \log \left (e x + d\right )}{6 \, {\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} \]

input

integrate((5*x^2+2*x+3)*(4*x^4-5*x^3+3*x^2+x+2)/(e*x+d)^3,x, algorithm="fr 
icas")
 

output

1/6*(30*e^6*x^6 + 660*d^6 + 459*d^5*e + 357*d^4*e^2 + 60*d^3*e^3 + 189*d^2 
*e^4 - 21*d*e^5 - 18*e^6 - 2*(30*d*e^5 + 17*e^6)*x^5 + (150*d^2*e^4 + 85*d 
*e^5 + 51*e^6)*x^4 - 4*(150*d^3*e^3 + 85*d^2*e^4 + 51*d*e^5 + 6*e^6)*x^3 - 
 3*(680*d^4*e^2 + 357*d^3*e^3 + 187*d^2*e^4 + 16*d*e^5)*x^2 - 6*(80*d^5*e 
+ 17*d^4*e^2 - 17*d^3*e^3 - 8*d^2*e^4 - 42*d*e^5 + 7*e^6)*x + 6*(300*d^6 + 
 170*d^5*e + 102*d^4*e^2 + 12*d^3*e^3 + 21*d^2*e^4 + (300*d^4*e^2 + 170*d^ 
3*e^3 + 102*d^2*e^4 + 12*d*e^5 + 21*e^6)*x^2 + 2*(300*d^5*e + 170*d^4*e^2 
+ 102*d^3*e^3 + 12*d^2*e^4 + 21*d*e^5)*x)*log(e*x + d))/(e^9*x^2 + 2*d*e^8 
*x + d^2*e^7)
 

3.3.95.6 Sympy [A] (verification not implemented)

Time = 0.85 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.07 \[ \int \frac {\left (3+2 x+5 x^2\right ) \left (2+x+3 x^2-5 x^3+4 x^4\right )}{(d+e x)^3} \, dx=x^{3} \left (- \frac {20 d}{e^{4}} - \frac {17}{3 e^{3}}\right ) + x^{2} \cdot \left (\frac {60 d^{2}}{e^{5}} + \frac {51 d}{2 e^{4}} + \frac {17}{2 e^{3}}\right ) + x \left (- \frac {200 d^{3}}{e^{6}} - \frac {102 d^{2}}{e^{5}} - \frac {51 d}{e^{4}} - \frac {4}{e^{3}}\right ) + \frac {220 d^{6} + 153 d^{5} e + 119 d^{4} e^{2} + 20 d^{3} e^{3} + 63 d^{2} e^{4} - 7 d e^{5} - 6 e^{6} + x \left (240 d^{5} e + 170 d^{4} e^{2} + 136 d^{3} e^{3} + 24 d^{2} e^{4} + 84 d e^{5} - 14 e^{6}\right )}{2 d^{2} e^{7} + 4 d e^{8} x + 2 e^{9} x^{2}} + \frac {5 x^{4}}{e^{3}} + \frac {\left (300 d^{4} + 170 d^{3} e + 102 d^{2} e^{2} + 12 d e^{3} + 21 e^{4}\right ) \log {\left (d + e x \right )}}{e^{7}} \]

input

integrate((5*x**2+2*x+3)*(4*x**4-5*x**3+3*x**2+x+2)/(e*x+d)**3,x)
 

output

x**3*(-20*d/e**4 - 17/(3*e**3)) + x**2*(60*d**2/e**5 + 51*d/(2*e**4) + 17/ 
(2*e**3)) + x*(-200*d**3/e**6 - 102*d**2/e**5 - 51*d/e**4 - 4/e**3) + (220 
*d**6 + 153*d**5*e + 119*d**4*e**2 + 20*d**3*e**3 + 63*d**2*e**4 - 7*d*e** 
5 - 6*e**6 + x*(240*d**5*e + 170*d**4*e**2 + 136*d**3*e**3 + 24*d**2*e**4 
+ 84*d*e**5 - 14*e**6))/(2*d**2*e**7 + 4*d*e**8*x + 2*e**9*x**2) + 5*x**4/ 
e**3 + (300*d**4 + 170*d**3*e + 102*d**2*e**2 + 12*d*e**3 + 21*e**4)*log(d 
 + e*x)/e**7
 

3.3.95.7 Maxima [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.04 \[ \int \frac {\left (3+2 x+5 x^2\right ) \left (2+x+3 x^2-5 x^3+4 x^4\right )}{(d+e x)^3} \, dx=\frac {220 \, d^{6} + 153 \, d^{5} e + 119 \, d^{4} e^{2} + 20 \, d^{3} e^{3} + 63 \, d^{2} e^{4} - 7 \, d e^{5} - 6 \, e^{6} + 2 \, {\left (120 \, d^{5} e + 85 \, d^{4} e^{2} + 68 \, d^{3} e^{3} + 12 \, d^{2} e^{4} + 42 \, d e^{5} - 7 \, e^{6}\right )} x}{2 \, {\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} + \frac {30 \, e^{3} x^{4} - 2 \, {\left (60 \, d e^{2} + 17 \, e^{3}\right )} x^{3} + 3 \, {\left (120 \, d^{2} e + 51 \, d e^{2} + 17 \, e^{3}\right )} x^{2} - 6 \, {\left (200 \, d^{3} + 102 \, d^{2} e + 51 \, d e^{2} + 4 \, e^{3}\right )} x}{6 \, e^{6}} + \frac {{\left (300 \, d^{4} + 170 \, d^{3} e + 102 \, d^{2} e^{2} + 12 \, d e^{3} + 21 \, e^{4}\right )} \log \left (e x + d\right )}{e^{7}} \]

input

integrate((5*x^2+2*x+3)*(4*x^4-5*x^3+3*x^2+x+2)/(e*x+d)^3,x, algorithm="ma 
xima")
 

output

1/2*(220*d^6 + 153*d^5*e + 119*d^4*e^2 + 20*d^3*e^3 + 63*d^2*e^4 - 7*d*e^5 
 - 6*e^6 + 2*(120*d^5*e + 85*d^4*e^2 + 68*d^3*e^3 + 12*d^2*e^4 + 42*d*e^5 
- 7*e^6)*x)/(e^9*x^2 + 2*d*e^8*x + d^2*e^7) + 1/6*(30*e^3*x^4 - 2*(60*d*e^ 
2 + 17*e^3)*x^3 + 3*(120*d^2*e + 51*d*e^2 + 17*e^3)*x^2 - 6*(200*d^3 + 102 
*d^2*e + 51*d*e^2 + 4*e^3)*x)/e^6 + (300*d^4 + 170*d^3*e + 102*d^2*e^2 + 1 
2*d*e^3 + 21*e^4)*log(e*x + d)/e^7
 

3.3.95.8 Giac [A] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.03 \[ \int \frac {\left (3+2 x+5 x^2\right ) \left (2+x+3 x^2-5 x^3+4 x^4\right )}{(d+e x)^3} \, dx=\frac {{\left (300 \, d^{4} + 170 \, d^{3} e + 102 \, d^{2} e^{2} + 12 \, d e^{3} + 21 \, e^{4}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{7}} + \frac {220 \, d^{6} + 153 \, d^{5} e + 119 \, d^{4} e^{2} + 20 \, d^{3} e^{3} + 63 \, d^{2} e^{4} - 7 \, d e^{5} - 6 \, e^{6} + 2 \, {\left (120 \, d^{5} e + 85 \, d^{4} e^{2} + 68 \, d^{3} e^{3} + 12 \, d^{2} e^{4} + 42 \, d e^{5} - 7 \, e^{6}\right )} x}{2 \, {\left (e x + d\right )}^{2} e^{7}} + \frac {30 \, e^{9} x^{4} - 120 \, d e^{8} x^{3} - 34 \, e^{9} x^{3} + 360 \, d^{2} e^{7} x^{2} + 153 \, d e^{8} x^{2} + 51 \, e^{9} x^{2} - 1200 \, d^{3} e^{6} x - 612 \, d^{2} e^{7} x - 306 \, d e^{8} x - 24 \, e^{9} x}{6 \, e^{12}} \]

input

integrate((5*x^2+2*x+3)*(4*x^4-5*x^3+3*x^2+x+2)/(e*x+d)^3,x, algorithm="gi 
ac")
 

output

(300*d^4 + 170*d^3*e + 102*d^2*e^2 + 12*d*e^3 + 21*e^4)*log(abs(e*x + d))/ 
e^7 + 1/2*(220*d^6 + 153*d^5*e + 119*d^4*e^2 + 20*d^3*e^3 + 63*d^2*e^4 - 7 
*d*e^5 - 6*e^6 + 2*(120*d^5*e + 85*d^4*e^2 + 68*d^3*e^3 + 12*d^2*e^4 + 42* 
d*e^5 - 7*e^6)*x)/((e*x + d)^2*e^7) + 1/6*(30*e^9*x^4 - 120*d*e^8*x^3 - 34 
*e^9*x^3 + 360*d^2*e^7*x^2 + 153*d*e^8*x^2 + 51*e^9*x^2 - 1200*d^3*e^6*x - 
 612*d^2*e^7*x - 306*d*e^8*x - 24*e^9*x)/e^12
 

3.3.95.9 Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.29 \[ \int \frac {\left (3+2 x+5 x^2\right ) \left (2+x+3 x^2-5 x^3+4 x^4\right )}{(d+e x)^3} \, dx=x^2\,\left (\frac {17}{2\,e^3}-\frac {30\,d^2}{e^5}+\frac {3\,d\,\left (\frac {60\,d}{e^4}+\frac {17}{e^3}\right )}{2\,e}\right )-x^3\,\left (\frac {20\,d}{e^4}+\frac {17}{3\,e^3}\right )+\frac {x\,\left (120\,d^5+85\,d^4\,e+68\,d^3\,e^2+12\,d^2\,e^3+42\,d\,e^4-7\,e^5\right )+\frac {220\,d^6+153\,d^5\,e+119\,d^4\,e^2+20\,d^3\,e^3+63\,d^2\,e^4-7\,d\,e^5-6\,e^6}{2\,e}}{d^2\,e^6+2\,d\,e^7\,x+e^8\,x^2}-x\,\left (\frac {4}{e^3}+\frac {20\,d^3}{e^6}+\frac {3\,d\,\left (\frac {17}{e^3}-\frac {60\,d^2}{e^5}+\frac {3\,d\,\left (\frac {60\,d}{e^4}+\frac {17}{e^3}\right )}{e}\right )}{e}-\frac {3\,d^2\,\left (\frac {60\,d}{e^4}+\frac {17}{e^3}\right )}{e^2}\right )+\frac {5\,x^4}{e^3}+\frac {\ln \left (d+e\,x\right )\,\left (300\,d^4+170\,d^3\,e+102\,d^2\,e^2+12\,d\,e^3+21\,e^4\right )}{e^7} \]

input

int(((2*x + 5*x^2 + 3)*(x + 3*x^2 - 5*x^3 + 4*x^4 + 2))/(d + e*x)^3,x)
 

output

x^2*(17/(2*e^3) - (30*d^2)/e^5 + (3*d*((60*d)/e^4 + 17/e^3))/(2*e)) - x^3* 
((20*d)/e^4 + 17/(3*e^3)) + (x*(42*d*e^4 + 85*d^4*e + 120*d^5 - 7*e^5 + 12 
*d^2*e^3 + 68*d^3*e^2) + (153*d^5*e - 7*d*e^5 + 220*d^6 - 6*e^6 + 63*d^2*e 
^4 + 20*d^3*e^3 + 119*d^4*e^2)/(2*e))/(d^2*e^6 + e^8*x^2 + 2*d*e^7*x) - x* 
(4/e^3 + (20*d^3)/e^6 + (3*d*(17/e^3 - (60*d^2)/e^5 + (3*d*((60*d)/e^4 + 1 
7/e^3))/e))/e - (3*d^2*((60*d)/e^4 + 17/e^3))/e^2) + (5*x^4)/e^3 + (log(d 
+ e*x)*(12*d*e^3 + 170*d^3*e + 300*d^4 + 21*e^4 + 102*d^2*e^2))/e^7