3.2217 \(\int \frac{1}{(d+e x)^2 \left (a+b x+c x^2\right )^5} \, dx\)

Optimal. Leaf size=1761 \[ \text{result too large to display} \]

[Out]

(5*e*(14*c^8*d^8 - b^8*e^8 - 4*c^7*d^6*e*(14*b*d - 19*a*e) + b^6*c*e^7*(b*d + 15
*a*e) + b^4*c^2*e^6*(b^2*d^2 - 16*a*b*d*e - 82*a^2*e^2) + c^6*d^4*e^2*(79*b^2*d^
2 - 228*a*b*d*e + 176*a^2*e^2) - c^5*d^2*e^3*(41*b^3*d^3 - 197*a*b^2*d^2*e + 352
*a^2*b*d*e^2 - 244*a^3*e^3) + b^2*c^3*e^5*(b^3*d^3 - 15*a*b^2*d^2*e + 95*a^2*b*d
*e^2 + 187*a^3*e^3) + c^4*e^4*(b^4*d^4 - 14*a*b^3*d^3*e + 81*a^2*b^2*d^2*e^2 - 2
44*a^3*b*d*e^3 - 126*a^4*e^4)))/((b^2 - 4*a*c)^4*(c*d^2 - b*d*e + a*e^2)^5*(d +
e*x)) - (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)*(d + e*x)*(a + b*x + c*x^2)^4) - (8*a*c*e*(2*c*d - b*e)^2 - (b*c*
d - b^2*e + 2*a*c*e)*(14*c^2*d^2 - 5*b^2*e^2 - 6*c*e*(b*d - 3*a*e)) - c*(2*c*d -
b*e)*(14*c^2*d^2 - 5*b^2*e^2 - 2*c*e*(7*b*d - 17*a*e))*x)/(12*(b^2 - 4*a*c)^2*(
c*d^2 - b*d*e + a*e^2)^2*(d + e*x)*(a + b*x + c*x^2)^3) + (3*a*c*e*(2*c*d - b*e)
^2*(14*c^2*d^2 - 5*b^2*e^2 - 2*c*e*(7*b*d - 17*a*e)) - (b*c*d - b^2*e + 2*a*c*e)
*(70*c^4*d^4 + 10*b^4*e^4 - 2*c^3*d^2*e*(49*b*d - 78*a*e) + 5*b^2*c*e^3*(2*b*d -
15*a*e) + 3*c^2*e^2*(b^2*d^2 - 18*a*b*d*e + 42*a^2*e^2)) - 5*c*(2*c*d - b*e)*(1
4*c^4*d^4 + 2*b^4*e^4 + b^2*c*e^3*(5*b*d - 21*a*e) - 4*c^3*d^2*e*(7*b*d - 12*a*e
) + 3*c^2*e^2*(3*b^2*d^2 - 16*a*b*d*e + 22*a^2*e^2))*x)/(12*(b^2 - 4*a*c)^3*(c*d
^2 - b*d*e + a*e^2)^3*(d + e*x)*(a + b*x + c*x^2)^2) - (5*(2*a*c*e*(2*c*d - b*e)
^2*(14*c^4*d^4 + 2*b^4*e^4 + b^2*c*e^3*(5*b*d - 21*a*e) - 4*c^3*d^2*e*(7*b*d - 1
2*a*e) + 3*c^2*e^2*(3*b^2*d^2 - 16*a*b*d*e + 22*a^2*e^2)) - (b*c*d - b^2*e + 2*a
*c*e)*(42*c^6*d^6 - 3*b^6*e^6 - 2*c^5*d^4*e*(49*b*d - 65*a*e) - 2*b^4*c*e^5*(b*d
- 17*a*e) + b^2*c^2*e^4*(b^2*d^2 + 16*a*b*d*e - 123*a^2*e^2) + c^4*d^2*e^2*(55*
b^2*d^2 - 164*a*b*d*e + 150*a^2*e^2) + 6*c^3*e^3*(b^3*d^3 - 4*a*b^2*d^2*e - 3*a^
2*b*d*e^2 + 21*a^3*e^3)) - 3*c*(2*c*d - b*e)*(14*c^6*d^6 - b^6*e^6 - 2*c^5*d^4*e
*(21*b*d - 31*a*e) - 2*b^4*c*e^5*(b*d - 7*a*e) - b^2*c^2*e^4*(3*b^2*d^2 - 26*a*b
*d*e + 69*a^2*e^2) + c^4*d^2*e^2*(37*b^2*d^2 - 124*a*b*d*e + 114*a^2*e^2) - 2*c^
3*e^3*(2*b^3*d^3 - 18*a*b^2*d^2*e + 57*a^2*b*d*e^2 - 65*a^3*e^3))*x))/(6*(b^2 -
4*a*c)^4*(c*d^2 - b*d*e + a*e^2)^4*(d + e*x)*(a + b*x + c*x^2)) - (5*(28*c^10*d^
10 + b^10*e^10 - 20*c^9*d^8*e*(7*b*d - 9*a*e) - 252*a^4*c^5*e^9*(5*b*d + a*e) +
210*a^3*b^2*c^4*e^9*(4*b*d + 3*a*e) - 84*a^2*b^4*c^3*e^9*(3*b*d + 5*a*e) + 18*a*
b^6*c^2*e^9*(2*b*d + 7*a*e) - 2*b^8*c*e^9*(b*d + 9*a*e) + 18*c^8*d^6*e^2*(15*b^2
*d^2 - 40*a*b*d*e + 28*a^2*e^2) - 24*c^7*d^4*e^3*(10*b^3*d^3 - 42*a*b^2*d^2*e +
63*a^2*b*d*e^2 - 35*a^3*e^3) + 84*c^6*d^2*e^4*(b^4*d^4 - 6*a*b^3*d^3*e + 15*a^2*
b^2*d^2*e^2 - 20*a^3*b*d*e^3 + 15*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))^6) + (5*e^9*(2*c*d - b*e)*Log[d
+ e*x])/(c*d^2 - b*d*e + a*e^2)^6 - (5*e^9*(2*c*d - b*e)*Log[a + b*x + c*x^2])/
(2*(c*d^2 - b*d*e + a*e^2)^6)

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Rubi [A]  time = 25.3902, antiderivative size = 1761, 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 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

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

[Out]

(5*e*(14*c^8*d^8 - b^8*e^8 - 4*c^7*d^6*e*(14*b*d - 19*a*e) + b^6*c*e^7*(b*d + 15
*a*e) + b^4*c^2*e^6*(b^2*d^2 - 16*a*b*d*e - 82*a^2*e^2) + c^6*d^4*e^2*(79*b^2*d^
2 - 228*a*b*d*e + 176*a^2*e^2) - c^5*d^2*e^3*(41*b^3*d^3 - 197*a*b^2*d^2*e + 352
*a^2*b*d*e^2 - 244*a^3*e^3) + b^2*c^3*e^5*(b^3*d^3 - 15*a*b^2*d^2*e + 95*a^2*b*d
*e^2 + 187*a^3*e^3) + c^4*e^4*(b^4*d^4 - 14*a*b^3*d^3*e + 81*a^2*b^2*d^2*e^2 - 2
44*a^3*b*d*e^3 - 126*a^4*e^4)))/((b^2 - 4*a*c)^4*(c*d^2 - b*d*e + a*e^2)^5*(d +
e*x)) - (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)*(d + e*x)*(a + b*x + c*x^2)^4) - (8*a*c*e*(2*c*d - b*e)^2 - (b*c*
d - b^2*e + 2*a*c*e)*(14*c^2*d^2 - 5*b^2*e^2 - 6*c*e*(b*d - 3*a*e)) - c*(2*c*d -
b*e)*(14*c^2*d^2 - 5*b^2*e^2 - 2*c*e*(7*b*d - 17*a*e))*x)/(12*(b^2 - 4*a*c)^2*(
c*d^2 - b*d*e + a*e^2)^2*(d + e*x)*(a + b*x + c*x^2)^3) + (3*a*c*e*(2*c*d - b*e)
^2*(14*c^2*d^2 - 5*b^2*e^2 - 2*c*e*(7*b*d - 17*a*e)) - (b*c*d - b^2*e + 2*a*c*e)
*(70*c^4*d^4 + 10*b^4*e^4 - 2*c^3*d^2*e*(49*b*d - 78*a*e) + 5*b^2*c*e^3*(2*b*d -
15*a*e) + 3*c^2*e^2*(b^2*d^2 - 18*a*b*d*e + 42*a^2*e^2)) - 5*c*(2*c*d - b*e)*(1
4*c^4*d^4 + 2*b^4*e^4 + b^2*c*e^3*(5*b*d - 21*a*e) - 4*c^3*d^2*e*(7*b*d - 12*a*e
) + 3*c^2*e^2*(3*b^2*d^2 - 16*a*b*d*e + 22*a^2*e^2))*x)/(12*(b^2 - 4*a*c)^3*(c*d
^2 - b*d*e + a*e^2)^3*(d + e*x)*(a + b*x + c*x^2)^2) - (5*(2*a*c*e*(2*c*d - b*e)
^2*(14*c^4*d^4 + 2*b^4*e^4 + b^2*c*e^3*(5*b*d - 21*a*e) - 4*c^3*d^2*e*(7*b*d - 1
2*a*e) + 3*c^2*e^2*(3*b^2*d^2 - 16*a*b*d*e + 22*a^2*e^2)) - (b*c*d - b^2*e + 2*a
*c*e)*(42*c^6*d^6 - 3*b^6*e^6 - 2*c^5*d^4*e*(49*b*d - 65*a*e) - 2*b^4*c*e^5*(b*d
- 17*a*e) + b^2*c^2*e^4*(b^2*d^2 + 16*a*b*d*e - 123*a^2*e^2) + c^4*d^2*e^2*(55*
b^2*d^2 - 164*a*b*d*e + 150*a^2*e^2) + 6*c^3*e^3*(b^3*d^3 - 4*a*b^2*d^2*e - 3*a^
2*b*d*e^2 + 21*a^3*e^3)) - 3*c*(2*c*d - b*e)*(14*c^6*d^6 - b^6*e^6 - 2*c^5*d^4*e
*(21*b*d - 31*a*e) - 2*b^4*c*e^5*(b*d - 7*a*e) - b^2*c^2*e^4*(3*b^2*d^2 - 26*a*b
*d*e + 69*a^2*e^2) + c^4*d^2*e^2*(37*b^2*d^2 - 124*a*b*d*e + 114*a^2*e^2) - 2*c^
3*e^3*(2*b^3*d^3 - 18*a*b^2*d^2*e + 57*a^2*b*d*e^2 - 65*a^3*e^3))*x))/(6*(b^2 -
4*a*c)^4*(c*d^2 - b*d*e + a*e^2)^4*(d + e*x)*(a + b*x + c*x^2)) - (5*(28*c^10*d^
10 + b^10*e^10 - 20*c^9*d^8*e*(7*b*d - 9*a*e) - 252*a^4*c^5*e^9*(5*b*d + a*e) +
210*a^3*b^2*c^4*e^9*(4*b*d + 3*a*e) - 84*a^2*b^4*c^3*e^9*(3*b*d + 5*a*e) + 18*a*
b^6*c^2*e^9*(2*b*d + 7*a*e) - 2*b^8*c*e^9*(b*d + 9*a*e) + 18*c^8*d^6*e^2*(15*b^2
*d^2 - 40*a*b*d*e + 28*a^2*e^2) - 24*c^7*d^4*e^3*(10*b^3*d^3 - 42*a*b^2*d^2*e +
63*a^2*b*d*e^2 - 35*a^3*e^3) + 84*c^6*d^2*e^4*(b^4*d^4 - 6*a*b^3*d^3*e + 15*a^2*
b^2*d^2*e^2 - 20*a^3*b*d*e^3 + 15*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))^6) + (5*e^9*(2*c*d - b*e)*Log[d
+ e*x])/(c*d^2 - b*d*e + a*e^2)^6 - (5*e^9*(2*c*d - b*e)*Log[a + b*x + c*x^2])/
(2*(c*d^2 - b*d*e + a*e^2)^6)

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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)**2/(c*x**2+b*x+a)**5,x)

[Out]

Timed out

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Mathematica [A]  time = 7.96694, size = 2147, normalized size = 1.22 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

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

[Out]

-(e^9/((c*d^2 - b*d*e + a*e^2)^5*(d + e*x))) + (b*c^2*d^2 - 2*b^2*c*d*e + 4*a*c^
2*d*e + b^3*e^2 - 3*a*b*c*e^2 + 2*c^3*d^2*x - 2*b*c^2*d*e*x + b^2*c*e^2*x - 2*a*
c^2*e^2*x)/(4*(-b^2 + 4*a*c)*(c*d^2 - b*d*e + a*e^2)^2*(a + b*x + c*x^2)^4) + (1
4*b*c^4*d^4 - 28*b^2*c^3*d^3*e + 9*b^3*c^2*d^2*e^2 + 48*a*b*c^3*d^2*e^2 + 13*b^4
*c*d*e^3 - 112*a*b^2*c^2*d*e^3 + 128*a^2*c^3*d*e^3 - 8*b^5*e^4 + 59*a*b^3*c*e^4
- 94*a^2*b*c^2*e^4 + 28*c^5*d^4*x - 56*b*c^4*d^3*e*x + 18*b^2*c^3*d^2*e^2*x + 96
*a*c^4*d^2*e^2*x + 10*b^3*c^2*d*e^3*x - 96*a*b*c^3*d*e^3*x - 8*b^4*c*e^4*x + 54*
a*b^2*c^2*e^4*x - 60*a^2*c^3*e^4*x)/(12*(-b^2 + 4*a*c)^2*(c*d^2 - b*d*e + a*e^2)
^3*(a + b*x + c*x^2)^3) + (70*b*c^6*d^6 - 210*b^2*c^5*d^5*e + 185*b^3*c^4*d^4*e^
2 + 310*a*b*c^5*d^4*e^2 - 20*b^4*c^3*d^3*e^3 - 620*a*b^2*c^4*d^3*e^3 - 15*b^5*c^
2*d^2*e^4 + 180*a*b^3*c^3*d^2*e^4 + 570*a^2*b*c^4*d^2*e^4 - 28*b^6*c*d*e^5 + 346
*a*b^4*c^2*d*e^5 - 1434*a^2*b^2*c^3*d*e^5 + 1152*a^3*c^4*d*e^5 + 18*b^7*e^6 - 20
6*a*b^5*c*e^6 + 759*a^2*b^3*c^2*e^6 - 822*a^3*b*c^3*e^6 + 140*c^7*d^6*x - 420*b*
c^6*d^5*e*x + 370*b^2*c^5*d^4*e^2*x + 620*a*c^6*d^4*e^2*x - 40*b^3*c^4*d^3*e^3*x
- 1240*a*b*c^5*d^3*e^3*x - 30*b^4*c^3*d^2*e^4*x + 360*a*b^2*c^4*d^2*e^4*x + 114
0*a^2*c^5*d^2*e^4*x - 20*b^5*c^2*d*e^5*x + 260*a*b^3*c^3*d*e^5*x - 1140*a^2*b*c^
4*d*e^5*x + 18*b^6*c*e^6*x - 196*a*b^4*c^2*e^6*x + 654*a^2*b^2*c^3*e^6*x - 492*a
^3*c^4*e^6*x)/(12*(-b^2 + 4*a*c)^3*(c*d^2 - b*d*e + a*e^2)^4*(a + b*x + c*x^2)^2
) + (70*b*c^8*d^8 - 280*b^2*c^7*d^7*e + 395*b^3*c^6*d^6*e^2 + 380*a*b*c^7*d^6*e^
2 - 205*b^4*c^5*d^5*e^3 - 1140*a*b^2*c^6*d^5*e^3 + 5*b^5*c^4*d^4*e^4 + 985*a*b^3
*c^5*d^4*e^4 + 880*a^2*b*c^6*d^4*e^4 + 5*b^6*c^3*d^3*e^5 - 70*a*b^4*c^4*d^3*e^5
- 1760*a^2*b^2*c^5*d^3*e^5 + 5*b^7*c^2*d^2*e^6 - 75*a*b^5*c^3*d^2*e^6 + 405*a^2*
b^3*c^4*d^2*e^6 + 1220*a^3*b*c^5*d^2*e^6 + 13*b^8*c*d*e^7 - 208*a*b^6*c^2*d*e^7
+ 1243*a^2*b^4*c^3*d*e^7 - 3268*a^3*b^2*c^4*d*e^7 + 2048*a^4*c^5*d*e^7 - 8*b^9*e
^8 + 123*a*b^7*c*e^8 - 698*a^2*b^5*c^2*e^8 + 1703*a^3*b^3*c^3*e^8 - 1398*a^4*b*c
^4*e^8 + 140*c^9*d^8*x - 560*b*c^8*d^7*e*x + 790*b^2*c^7*d^6*e^2*x + 760*a*c^8*d
^6*e^2*x - 410*b^3*c^6*d^5*e^3*x - 2280*a*b*c^7*d^5*e^3*x + 10*b^4*c^5*d^4*e^4*x
+ 1970*a*b^2*c^6*d^4*e^4*x + 1760*a^2*c^7*d^4*e^4*x + 10*b^5*c^4*d^3*e^5*x - 14
0*a*b^3*c^5*d^3*e^5*x - 3520*a^2*b*c^6*d^3*e^5*x + 10*b^6*c^3*d^2*e^6*x - 150*a*
b^4*c^4*d^2*e^6*x + 810*a^2*b^2*c^5*d^2*e^6*x + 2440*a^3*c^6*d^2*e^6*x + 10*b^7*
c^2*d*e^7*x - 160*a*b^5*c^3*d*e^7*x + 950*a^2*b^3*c^4*d*e^7*x - 2440*a^3*b*c^5*d
*e^7*x - 8*b^8*c*e^8*x + 118*a*b^6*c^2*e^8*x - 628*a^2*b^4*c^3*e^8*x + 1358*a^3*
b^2*c^4*e^8*x - 748*a^4*c^5*e^8*x)/(2*(-b^2 + 4*a*c)^4*(c*d^2 - b*d*e + a*e^2)^5
*(a + b*x + c*x^2)) + (5*(28*c^10*d^10 - 140*b*c^9*d^9*e + 270*b^2*c^8*d^8*e^2 +
180*a*c^9*d^8*e^2 - 240*b^3*c^7*d^7*e^3 - 720*a*b*c^8*d^7*e^3 + 84*b^4*c^6*d^6*
e^4 + 1008*a*b^2*c^7*d^6*e^4 + 504*a^2*c^8*d^6*e^4 - 504*a*b^3*c^6*d^5*e^5 - 151
2*a^2*b*c^7*d^5*e^5 + 1260*a^2*b^2*c^6*d^4*e^6 + 840*a^3*c^7*d^4*e^6 - 1680*a^3*
b*c^6*d^3*e^7 + 1260*a^4*c^6*d^2*e^8 - 2*b^9*c*d*e^9 + 36*a*b^7*c^2*d*e^9 - 252*
a^2*b^5*c^3*d*e^9 + 840*a^3*b^3*c^4*d*e^9 - 1260*a^4*b*c^5*d*e^9 + b^10*e^10 - 1
8*a*b^8*c*e^10 + 126*a^2*b^6*c^2*e^10 - 420*a^3*b^4*c^3*e^10 + 630*a^4*b^2*c^4*e
^10 - 252*a^5*c^5*e^10)*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)^6) + (5*(2*c*d*e^9 - b*e^10)*Log[
d + e*x])/(c*d^2 - b*d*e + a*e^2)^6 - (5*(2*c*d*e^9 - b*e^10)*Log[a + b*x + c*x^
2])/(2*(c*d^2 - b*d*e + a*e^2)^6)

_______________________________________________________________________________________

Maple [B]  time = 0.099, size = 45635, normalized size = 25.9 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 1.57523, 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)^2),x, algorithm="giac")

[Out]

Done