3.2216 \(\int \frac{1}{(d+e x) \left (a+b x+c x^2\right )^5} \, dx\)

Optimal. Leaf size=1324 \[ \text{result too large to display} \]

[Out]

-(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)/(4*(b^2 - 4*a*c)*(c*d^2 - b*d*e +
a*e^2)*(a + b*x + c*x^2)^4) - (7*a*c*e*(2*c*d - b*e)^2 - (b*c*d - b^2*e + 2*a*c
*e)*(14*c^2*d^2 - 4*b^2*e^2 - c*e*(7*b*d - 16*a*e)) - 2*c*(2*c*d - b*e)*(7*c^2*d
^2 - 2*b^2*e^2 - c*e*(7*b*d - 15*a*e))*x)/(12*(b^2 - 4*a*c)^2*(c*d^2 - b*d*e + a
*e^2)^2*(a + b*x + c*x^2)^3) + (5*a*c*e*(2*c*d - b*e)^2*(7*c^2*d^2 - 2*b^2*e^2 -
c*e*(7*b*d - 15*a*e)) - (b*c*d - b^2*e + 2*a*c*e)*(70*c^4*d^4 + 6*b^4*e^4 + 2*b
^2*c*e^3*(5*b*d - 24*a*e) - 15*c^3*d^2*e*(7*b*d - 10*a*e) + 3*c^2*e^2*(5*b^2*d^2
- 25*a*b*d*e + 32*a^2*e^2)) - 2*c*(2*c*d - b*e)*(35*c^4*d^4 + 3*b^4*e^4 + 2*b^2
*c*e^3*(5*b*d - 17*a*e) - 10*c^3*d^2*e*(7*b*d - 11*a*e) + c^2*e^2*(25*b^2*d^2 -
110*a*b*d*e + 123*a^2*e^2))*x)/(12*(b^2 - 4*a*c)^3*(c*d^2 - b*d*e + a*e^2)^3*(a
+ b*x + c*x^2)^2) + (b^7*c*d*e^6 + 2*b^8*e^7 + 256*a^4*c^4*e^7 + b^6*c*e^5*(c*d^
2 - 31*a*e^2) + b^5*c^2*d*e^4*(c*d^2 - 14*a*e^2) - b^4*c^2*e^3*(125*c^2*d^4 + 13
*a*c*d^2*e^2 - 178*a^2*e^4) + b^3*c^3*d*e^2*(295*c^2*d^4 + 492*a*c*d^2*e^2 + 69*
a^2*e^4) + 2*b*c^4*d*(35*c^3*d^6 + 145*a*c^2*d^4*e^2 + 233*a^2*c*d^2*e^4 + 187*a
^3*e^6) - b^2*c^3*e*(245*c^3*d^6 + 725*a*c^2*d^4*e^2 + 699*a^2*c*d^2*e^4 + 443*a
^3*e^6) + 2*c*(2*c*d - b*e)*(35*c^6*d^6 - b^6*e^6 - 5*c^5*d^4*e*(21*b*d - 29*a*e
) - 3*b^4*c*e^5*(b*d - 5*a*e) - b^2*c^2*e^4*(7*b^2*d^2 - 44*a*b*d*e + 82*a^2*e^2
) + c^4*d^2*e^2*(95*b^2*d^2 - 290*a*b*d*e + 233*a^2*e^2) - c^3*e^3*(15*b^3*d^3 -
101*a*b^2*d^2*e + 233*a^2*b*d*e^2 - 187*a^3*e^3))*x)/(2*(b^2 - 4*a*c)^4*(c*d^2
- b*d*e + a*e^2)^4*(a + b*x + c*x^2)) - ((140*c^9*d^9 - b^9*e^9 + 18*a*b^7*c*e^9
- 126*a^2*b^5*c^2*e^9 + 420*a^3*b^3*c^3*e^9 - 630*a^4*b*c^4*e^9 - 90*c^8*d^7*e*
(7*b*d - 8*a*e) + 72*c^7*d^5*e^2*(15*b^2*d^2 - 35*a*b*d*e + 21*a^2*e^2) - 84*c^6
*d^3*e^3*(10*b^3*d^3 - 36*a*b^2*d^2*e + 45*a^2*b*d*e^2 - 20*a^3*e^3) + 252*c^5*d
*e^4*(b^4*d^4 - 5*a*b^3*d^3*e + 10*a^2*b^2*d^2*e^2 - 10*a^3*b*d*e^3 + 5*a^4*e^4)
)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/((b^2 - 4*a*c)^(9/2)*(c*d^2 - e*(b*d -
a*e))^5) + (e^9*Log[d + e*x])/(c*d^2 - b*d*e + a*e^2)^5 - (e^9*Log[a + b*x + c*
x^2])/(2*(c*d^2 - b*d*e + a*e^2)^5)

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Rubi [A]  time = 19.5993, antiderivative size = 1324, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ \frac{\log (d+e x) e^9}{\left (c d^2-b e d+a e^2\right )^5}-\frac{\log \left (c x^2+b x+a\right ) e^9}{2 \left (c d^2-b e d+a e^2\right )^5}-\frac{\left (140 c^9 d^9-90 c^8 e (7 b d-8 a e) d^7+72 c^7 e^2 \left (15 b^2 d^2-35 a b e d+21 a^2 e^2\right ) d^5-84 c^6 e^3 \left (10 b^3 d^3-36 a b^2 e d^2+45 a^2 b e^2 d-20 a^3 e^3\right ) d^3+252 c^5 e^4 \left (b^4 d^4-5 a b^3 e d^3+10 a^2 b^2 e^2 d^2-10 a^3 b e^3 d+5 a^4 e^4\right ) d-b^9 e^9-630 a^4 b c^4 e^9+420 a^3 b^3 c^3 e^9-126 a^2 b^5 c^2 e^9+18 a b^7 c e^9\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/2} \left (c d^2-e (b d-a e)\right )^5}+\frac{2 e^7 b^8+c d e^6 b^7+c e^5 \left (c d^2-31 a e^2\right ) b^6+c^2 d e^4 \left (c d^2-14 a e^2\right ) b^5-c^2 e^3 \left (125 c^2 d^4+13 a c e^2 d^2-178 a^2 e^4\right ) b^4+c^3 d e^2 \left (295 c^2 d^4+492 a c e^2 d^2+69 a^2 e^4\right ) b^3-c^3 e \left (245 c^3 d^6+725 a c^2 e^2 d^4+699 a^2 c e^4 d^2+443 a^3 e^6\right ) b^2+2 c^4 d \left (35 c^3 d^6+145 a c^2 e^2 d^4+233 a^2 c e^4 d^2+187 a^3 e^6\right ) b+256 a^4 c^4 e^7+2 c (2 c d-b e) \left (35 c^6 d^6-5 c^5 e (21 b d-29 a e) d^4+c^4 e^2 \left (95 b^2 d^2-290 a b e d+233 a^2 e^2\right ) d^2-b^6 e^6-3 b^4 c e^5 (b d-5 a e)-b^2 c^2 e^4 \left (7 b^2 d^2-44 a b e d+82 a^2 e^2\right )-c^3 e^3 \left (15 b^3 d^3-101 a b^2 e d^2+233 a^2 b e^2 d-187 a^3 e^3\right )\right ) x}{2 \left (b^2-4 a c\right )^4 \left (c d^2-b e d+a e^2\right )^4 \left (c x^2+b x+a\right )}+\frac{5 a c e \left (7 c^2 d^2-2 b^2 e^2-c e (7 b d-15 a e)\right ) (2 c d-b e)^2-2 c \left (35 c^4 d^4-10 c^3 e (7 b d-11 a e) d^2+3 b^4 e^4+2 b^2 c e^3 (5 b d-17 a e)+c^2 e^2 \left (25 b^2 d^2-110 a b e d+123 a^2 e^2\right )\right ) x (2 c d-b e)-\left (-e b^2+c d b+2 a c e\right ) \left (70 c^4 d^4-15 c^3 e (7 b d-10 a e) d^2+6 b^4 e^4+2 b^2 c e^3 (5 b d-24 a e)+3 c^2 e^2 \left (5 b^2 d^2-25 a b e d+32 a^2 e^2\right )\right )}{12 \left (b^2-4 a c\right )^3 \left (c d^2-b e d+a e^2\right )^3 \left (c x^2+b x+a\right )^2}-\frac{7 a c e (2 c d-b e)^2-2 c \left (7 c^2 d^2-2 b^2 e^2-c e (7 b d-15 a e)\right ) x (2 c d-b e)-\left (-e b^2+c d b+2 a c e\right ) \left (14 c^2 d^2-4 b^2 e^2-c e (7 b d-16 a e)\right )}{12 \left (b^2-4 a c\right )^2 \left (c d^2-b e d+a e^2\right )^2 \left (c x^2+b x+a\right )^3}-\frac{-e b^2+c d b+2 a c e+c (2 c d-b e) x}{4 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^4} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)*(a + b*x + c*x^2)^5),x]

[Out]

-(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)/(4*(b^2 - 4*a*c)*(c*d^2 - b*d*e +
a*e^2)*(a + b*x + c*x^2)^4) - (7*a*c*e*(2*c*d - b*e)^2 - (b*c*d - b^2*e + 2*a*c
*e)*(14*c^2*d^2 - 4*b^2*e^2 - c*e*(7*b*d - 16*a*e)) - 2*c*(2*c*d - b*e)*(7*c^2*d
^2 - 2*b^2*e^2 - c*e*(7*b*d - 15*a*e))*x)/(12*(b^2 - 4*a*c)^2*(c*d^2 - b*d*e + a
*e^2)^2*(a + b*x + c*x^2)^3) + (5*a*c*e*(2*c*d - b*e)^2*(7*c^2*d^2 - 2*b^2*e^2 -
c*e*(7*b*d - 15*a*e)) - (b*c*d - b^2*e + 2*a*c*e)*(70*c^4*d^4 + 6*b^4*e^4 + 2*b
^2*c*e^3*(5*b*d - 24*a*e) - 15*c^3*d^2*e*(7*b*d - 10*a*e) + 3*c^2*e^2*(5*b^2*d^2
- 25*a*b*d*e + 32*a^2*e^2)) - 2*c*(2*c*d - b*e)*(35*c^4*d^4 + 3*b^4*e^4 + 2*b^2
*c*e^3*(5*b*d - 17*a*e) - 10*c^3*d^2*e*(7*b*d - 11*a*e) + c^2*e^2*(25*b^2*d^2 -
110*a*b*d*e + 123*a^2*e^2))*x)/(12*(b^2 - 4*a*c)^3*(c*d^2 - b*d*e + a*e^2)^3*(a
+ b*x + c*x^2)^2) + (b^7*c*d*e^6 + 2*b^8*e^7 + 256*a^4*c^4*e^7 + b^6*c*e^5*(c*d^
2 - 31*a*e^2) + b^5*c^2*d*e^4*(c*d^2 - 14*a*e^2) - b^4*c^2*e^3*(125*c^2*d^4 + 13
*a*c*d^2*e^2 - 178*a^2*e^4) + b^3*c^3*d*e^2*(295*c^2*d^4 + 492*a*c*d^2*e^2 + 69*
a^2*e^4) + 2*b*c^4*d*(35*c^3*d^6 + 145*a*c^2*d^4*e^2 + 233*a^2*c*d^2*e^4 + 187*a
^3*e^6) - b^2*c^3*e*(245*c^3*d^6 + 725*a*c^2*d^4*e^2 + 699*a^2*c*d^2*e^4 + 443*a
^3*e^6) + 2*c*(2*c*d - b*e)*(35*c^6*d^6 - b^6*e^6 - 5*c^5*d^4*e*(21*b*d - 29*a*e
) - 3*b^4*c*e^5*(b*d - 5*a*e) - b^2*c^2*e^4*(7*b^2*d^2 - 44*a*b*d*e + 82*a^2*e^2
) + c^4*d^2*e^2*(95*b^2*d^2 - 290*a*b*d*e + 233*a^2*e^2) - c^3*e^3*(15*b^3*d^3 -
101*a*b^2*d^2*e + 233*a^2*b*d*e^2 - 187*a^3*e^3))*x)/(2*(b^2 - 4*a*c)^4*(c*d^2
- b*d*e + a*e^2)^4*(a + b*x + c*x^2)) - ((140*c^9*d^9 - b^9*e^9 + 18*a*b^7*c*e^9
- 126*a^2*b^5*c^2*e^9 + 420*a^3*b^3*c^3*e^9 - 630*a^4*b*c^4*e^9 - 90*c^8*d^7*e*
(7*b*d - 8*a*e) + 72*c^7*d^5*e^2*(15*b^2*d^2 - 35*a*b*d*e + 21*a^2*e^2) - 84*c^6
*d^3*e^3*(10*b^3*d^3 - 36*a*b^2*d^2*e + 45*a^2*b*d*e^2 - 20*a^3*e^3) + 252*c^5*d
*e^4*(b^4*d^4 - 5*a*b^3*d^3*e + 10*a^2*b^2*d^2*e^2 - 10*a^3*b*d*e^3 + 5*a^4*e^4)
)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/((b^2 - 4*a*c)^(9/2)*(c*d^2 - e*(b*d -
a*e))^5) + (e^9*Log[d + e*x])/(c*d^2 - b*d*e + a*e^2)^5 - (e^9*Log[a + b*x + c*
x^2])/(2*(c*d^2 - b*d*e + a*e^2)^5)

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)/(c*x**2+b*x+a)**5,x)

[Out]

Timed out

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Mathematica [A]  time = 7.03742, size = 1601, normalized size = 1.21 \[ \frac{\log (d+e x) e^9}{\left (c d^2-b e d+a e^2\right )^5}-\frac{\log \left (c x^2+b x+a\right ) e^9}{2 \left (c d^2-b e d+a e^2\right )^5}+\frac{\left (-140 c^9 d^9+630 b c^8 e d^8-720 a c^8 e^2 d^7-1080 b^2 c^7 e^2 d^7+2520 a b c^7 e^3 d^6+840 b^3 c^6 e^3 d^6-1512 a^2 c^7 e^4 d^5-3024 a b^2 c^6 e^4 d^5-252 b^4 c^5 e^4 d^5+3780 a^2 b c^6 e^5 d^4+1260 a b^3 c^5 e^5 d^4-1680 a^3 c^6 e^6 d^3-2520 a^2 b^2 c^5 e^6 d^3+2520 a^3 b c^5 e^7 d^2-1260 a^4 c^5 e^8 d+b^9 e^9+630 a^4 b c^4 e^9-420 a^3 b^3 c^3 e^9+126 a^2 b^5 c^2 e^9-18 a b^7 c e^9\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (b^2-4 a c\right )^4 \sqrt{4 a c-b^2} \left (-c d^2+b e d-a e^2\right )^5}+\frac{2 e^7 b^8+c d e^6 b^7+2 c e^7 x b^7-31 a c e^7 b^6+c^2 d^2 e^5 b^6+2 c^2 d e^6 x b^6-14 a c^2 d e^6 b^5+c^3 d^3 e^4 b^5-30 a c^2 e^7 x b^5+2 c^3 d^2 e^5 x b^5+178 a^2 c^2 e^7 b^4-13 a c^3 d^2 e^5 b^4-125 c^4 d^4 e^3 b^4-28 a c^3 d e^6 x b^4+2 c^4 d^3 e^4 x b^4+69 a^2 c^3 d e^6 b^3+492 a c^4 d^3 e^4 b^3+295 c^5 d^5 e^2 b^3+164 a^2 c^3 e^7 x b^3-26 a c^4 d^2 e^5 x b^3-250 c^5 d^4 e^3 x b^3-443 a^3 c^3 e^7 b^2-699 a^2 c^4 d^2 e^5 b^2-725 a c^5 d^4 e^3 b^2-245 c^6 d^6 e b^2+138 a^2 c^4 d e^6 x b^2+984 a c^5 d^3 e^4 x b^2+590 c^6 d^5 e^2 x b^2+70 c^7 d^7 b+374 a^3 c^4 d e^6 b+466 a^2 c^5 d^3 e^4 b+290 a c^6 d^5 e^2 b-374 a^3 c^4 e^7 x b-1398 a^2 c^5 d^2 e^5 x b-1450 a c^6 d^4 e^3 x b-490 c^7 d^6 e x b+256 a^4 c^4 e^7+140 c^8 d^7 x+748 a^3 c^5 d e^6 x+932 a^2 c^6 d^3 e^4 x+580 a c^7 d^5 e^2 x}{2 \left (4 a c-b^2\right )^4 \left (c d^2-b e d+a e^2\right )^4 \left (c x^2+b x+a\right )}+\frac{-6 e^5 b^6-4 c d e^4 b^5-6 c e^5 x b^5+70 a c e^5 b^4-5 c^2 d^2 e^3 b^4-8 c^2 d e^4 x b^4+42 a c^2 d e^4 b^3+120 c^3 d^3 e^2 b^3+68 a c^2 e^5 x b^3-10 c^3 d^2 e^3 x b^3-267 a^2 c^2 e^5 b^2-330 a c^3 d^2 e^3 b^2-175 c^4 d^4 e b^2+84 a c^3 d e^4 x b^2+240 c^4 d^3 e^2 x b^2+70 c^5 d^5 b+246 a^2 c^3 d e^4 b+220 a c^4 d^3 e^2 b-246 a^2 c^3 e^5 x b-660 a c^4 d^2 e^3 x b-350 c^5 d^4 e x b+192 a^3 c^3 e^5+140 c^6 d^5 x+492 a^2 c^4 d e^4 x+440 a c^5 d^3 e^2 x}{12 \left (4 a c-b^2\right )^3 \left (c d^2-b e d+a e^2\right )^3 \left (c x^2+b x+a\right )^2}+\frac{4 e^3 b^4+3 c d e^2 b^3+4 c e^3 x b^3-31 a c e^3 b^2-21 c^2 d^2 e b^2+6 c^2 d e^2 x b^2+14 c^3 d^3 b+30 a c^2 d e^2 b-30 a c^2 e^3 x b-42 c^3 d^2 e x b+32 a^2 c^2 e^3+28 c^4 d^3 x+60 a c^3 d e^2 x}{12 \left (4 a c-b^2\right )^2 \left (c d^2-b e d+a e^2\right )^2 \left (c x^2+b x+a\right )^3}+\frac{-e b^2+c d b-c e x b+2 a c e+2 c^2 d x}{4 \left (4 a c-b^2\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)*(a + b*x + c*x^2)^5),x]

[Out]

(b*c*d - b^2*e + 2*a*c*e + 2*c^2*d*x - b*c*e*x)/(4*(-b^2 + 4*a*c)*(c*d^2 - b*d*e
+ a*e^2)*(a + b*x + c*x^2)^4) + (14*b*c^3*d^3 - 21*b^2*c^2*d^2*e + 3*b^3*c*d*e^
2 + 30*a*b*c^2*d*e^2 + 4*b^4*e^3 - 31*a*b^2*c*e^3 + 32*a^2*c^2*e^3 + 28*c^4*d^3*
x - 42*b*c^3*d^2*e*x + 6*b^2*c^2*d*e^2*x + 60*a*c^3*d*e^2*x + 4*b^3*c*e^3*x - 30
*a*b*c^2*e^3*x)/(12*(-b^2 + 4*a*c)^2*(c*d^2 - b*d*e + a*e^2)^2*(a + b*x + c*x^2)
^3) + (70*b*c^5*d^5 - 175*b^2*c^4*d^4*e + 120*b^3*c^3*d^3*e^2 + 220*a*b*c^4*d^3*
e^2 - 5*b^4*c^2*d^2*e^3 - 330*a*b^2*c^3*d^2*e^3 - 4*b^5*c*d*e^4 + 42*a*b^3*c^2*d
*e^4 + 246*a^2*b*c^3*d*e^4 - 6*b^6*e^5 + 70*a*b^4*c*e^5 - 267*a^2*b^2*c^2*e^5 +
192*a^3*c^3*e^5 + 140*c^6*d^5*x - 350*b*c^5*d^4*e*x + 240*b^2*c^4*d^3*e^2*x + 44
0*a*c^5*d^3*e^2*x - 10*b^3*c^3*d^2*e^3*x - 660*a*b*c^4*d^2*e^3*x - 8*b^4*c^2*d*e
^4*x + 84*a*b^2*c^3*d*e^4*x + 492*a^2*c^4*d*e^4*x - 6*b^5*c*e^5*x + 68*a*b^3*c^2
*e^5*x - 246*a^2*b*c^3*e^5*x)/(12*(-b^2 + 4*a*c)^3*(c*d^2 - b*d*e + a*e^2)^3*(a
+ b*x + c*x^2)^2) + (70*b*c^7*d^7 - 245*b^2*c^6*d^6*e + 295*b^3*c^5*d^5*e^2 + 29
0*a*b*c^6*d^5*e^2 - 125*b^4*c^4*d^4*e^3 - 725*a*b^2*c^5*d^4*e^3 + b^5*c^3*d^3*e^
4 + 492*a*b^3*c^4*d^3*e^4 + 466*a^2*b*c^5*d^3*e^4 + b^6*c^2*d^2*e^5 - 13*a*b^4*c
^3*d^2*e^5 - 699*a^2*b^2*c^4*d^2*e^5 + b^7*c*d*e^6 - 14*a*b^5*c^2*d*e^6 + 69*a^2
*b^3*c^3*d*e^6 + 374*a^3*b*c^4*d*e^6 + 2*b^8*e^7 - 31*a*b^6*c*e^7 + 178*a^2*b^4*
c^2*e^7 - 443*a^3*b^2*c^3*e^7 + 256*a^4*c^4*e^7 + 140*c^8*d^7*x - 490*b*c^7*d^6*
e*x + 590*b^2*c^6*d^5*e^2*x + 580*a*c^7*d^5*e^2*x - 250*b^3*c^5*d^4*e^3*x - 1450
*a*b*c^6*d^4*e^3*x + 2*b^4*c^4*d^3*e^4*x + 984*a*b^2*c^5*d^3*e^4*x + 932*a^2*c^6
*d^3*e^4*x + 2*b^5*c^3*d^2*e^5*x - 26*a*b^3*c^4*d^2*e^5*x - 1398*a^2*b*c^5*d^2*e
^5*x + 2*b^6*c^2*d*e^6*x - 28*a*b^4*c^3*d*e^6*x + 138*a^2*b^2*c^4*d*e^6*x + 748*
a^3*c^5*d*e^6*x + 2*b^7*c*e^7*x - 30*a*b^5*c^2*e^7*x + 164*a^2*b^3*c^3*e^7*x - 3
74*a^3*b*c^4*e^7*x)/(2*(-b^2 + 4*a*c)^4*(c*d^2 - b*d*e + a*e^2)^4*(a + b*x + c*x
^2)) + ((-140*c^9*d^9 + 630*b*c^8*d^8*e - 1080*b^2*c^7*d^7*e^2 - 720*a*c^8*d^7*e
^2 + 840*b^3*c^6*d^6*e^3 + 2520*a*b*c^7*d^6*e^3 - 252*b^4*c^5*d^5*e^4 - 3024*a*b
^2*c^6*d^5*e^4 - 1512*a^2*c^7*d^5*e^4 + 1260*a*b^3*c^5*d^4*e^5 + 3780*a^2*b*c^6*
d^4*e^5 - 2520*a^2*b^2*c^5*d^3*e^6 - 1680*a^3*c^6*d^3*e^6 + 2520*a^3*b*c^5*d^2*e
^7 - 1260*a^4*c^5*d*e^8 + b^9*e^9 - 18*a*b^7*c*e^9 + 126*a^2*b^5*c^2*e^9 - 420*a
^3*b^3*c^3*e^9 + 630*a^4*b*c^4*e^9)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/((b^
2 - 4*a*c)^4*Sqrt[-b^2 + 4*a*c]*(-(c*d^2) + b*d*e - a*e^2)^5) + (e^9*Log[d + e*x
])/(c*d^2 - b*d*e + a*e^2)^5 - (e^9*Log[a + b*x + c*x^2])/(2*(c*d^2 - b*d*e + a*
e^2)^5)

_______________________________________________________________________________________

Maple [B]  time = 0.1, size = 37364, normalized size = 28.2 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)/(c*x^2+b*x+a)^5,x)

[Out]

result too large to display

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x + a)^5*(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x + a)^5*(e*x + d)),x, algorithm="fricas")

[Out]

Timed out

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)/(c*x**2+b*x+a)**5,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.280848, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x + a)^5*(e*x + d)),x, algorithm="giac")

[Out]

Done