3.2190 \(\int \frac{(d+e x)^5}{\left (a+b x+c x^2\right )^3} \, dx\)
Optimal. Leaf size=388 \[ -\frac{\left (20 c^3 d e^2 \left (3 a^2 e^2-3 a b d e+b^2 d^2\right )-30 a^2 b c^2 e^5+10 a b^3 c e^5-10 c^4 d^3 e (3 b d-4 a e)-b^5 e^5+12 c^5 d^5\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{5/2}}-\frac{e^2 x (2 c d-b e) \left (-c e (3 b d-7 a e)-b^2 e^2+3 c^2 d^2\right )}{c^2 \left (b^2-4 a c\right )^2}-\frac{(d+e x)^2 \left (-x (2 c d-b e) \left (-2 c e (3 b d-5 a e)-b^2 e^2+6 c^2 d^2\right )+b^2 \left (7 c d^2 e-a e^3\right )-6 b c d \left (3 a e^2+c d^2\right )+8 a c e \left (2 a e^2+c d^2\right )\right )}{2 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{(d+e x)^4 (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{e^5 \log \left (a+b x+c x^2\right )}{2 c^3} \]
[Out]
-((e^2*(2*c*d - b*e)*(3*c^2*d^2 - b^2*e^2 - c*e*(3*b*d - 7*a*e))*x)/(c^2*(b^2 -
4*a*c)^2)) - ((d + e*x)^4*(b*d - 2*a*e + (2*c*d - b*e)*x))/(2*(b^2 - 4*a*c)*(a +
b*x + c*x^2)^2) - ((d + e*x)^2*(8*a*c*e*(c*d^2 + 2*a*e^2) - 6*b*c*d*(c*d^2 + 3*
a*e^2) + b^2*(7*c*d^2*e - a*e^3) - (2*c*d - b*e)*(6*c^2*d^2 - b^2*e^2 - 2*c*e*(3
*b*d - 5*a*e))*x))/(2*c*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)) - ((12*c^5*d^5 - b^5*
e^5 + 10*a*b^3*c*e^5 - 30*a^2*b*c^2*e^5 - 10*c^4*d^3*e*(3*b*d - 4*a*e) + 20*c^3*
d*e^2*(b^2*d^2 - 3*a*b*d*e + 3*a^2*e^2))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])
/(c^3*(b^2 - 4*a*c)^(5/2)) + (e^5*Log[a + b*x + c*x^2])/(2*c^3)
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Rubi [A] time = 2.26937, antiderivative size = 388, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35
\[ -\frac{\left (20 c^3 d e^2 \left (3 a^2 e^2-3 a b d e+b^2 d^2\right )-30 a^2 b c^2 e^5+10 a b^3 c e^5-10 c^4 d^3 e (3 b d-4 a e)-b^5 e^5+12 c^5 d^5\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{5/2}}-\frac{e^2 x (2 c d-b e) \left (-c e (3 b d-7 a e)-b^2 e^2+3 c^2 d^2\right )}{c^2 \left (b^2-4 a c\right )^2}-\frac{(d+e x)^2 \left (-x (2 c d-b e) \left (-2 c e (3 b d-5 a e)-b^2 e^2+6 c^2 d^2\right )+b^2 \left (7 c d^2 e-a e^3\right )-6 b c d \left (3 a e^2+c d^2\right )+8 a c e \left (2 a e^2+c d^2\right )\right )}{2 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{(d+e x)^4 (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{e^5 \log \left (a+b x+c x^2\right )}{2 c^3} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^5/(a + b*x + c*x^2)^3,x]
[Out]
-((e^2*(2*c*d - b*e)*(3*c^2*d^2 - b^2*e^2 - c*e*(3*b*d - 7*a*e))*x)/(c^2*(b^2 -
4*a*c)^2)) - ((d + e*x)^4*(b*d - 2*a*e + (2*c*d - b*e)*x))/(2*(b^2 - 4*a*c)*(a +
b*x + c*x^2)^2) - ((d + e*x)^2*(8*a*c*e*(c*d^2 + 2*a*e^2) - 6*b*c*d*(c*d^2 + 3*
a*e^2) + b^2*(7*c*d^2*e - a*e^3) - (2*c*d - b*e)*(6*c^2*d^2 - b^2*e^2 - 2*c*e*(3
*b*d - 5*a*e))*x))/(2*c*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)) - ((12*c^5*d^5 - b^5*
e^5 + 10*a*b^3*c*e^5 - 30*a^2*b*c^2*e^5 - 10*c^4*d^3*e*(3*b*d - 4*a*e) + 20*c^3*
d*e^2*(b^2*d^2 - 3*a*b*d*e + 3*a^2*e^2))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])
/(c^3*(b^2 - 4*a*c)^(5/2)) + (e^5*Log[a + b*x + c*x^2])/(2*c^3)
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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((e*x+d)**5/(c*x**2+b*x+a)**3,x)
[Out]
Timed out
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Mathematica [A] time = 2.88549, size = 628, normalized size = 1.62 \[ \frac{\frac{2 c (2 c d-b e) \left (2 c^2 e^2 \left (15 a^2 e^2-10 a b d e+2 b^2 d^2\right )+2 b^2 c e^3 (b d-5 a e)-4 c^3 d^2 e (3 b d-5 a e)+b^4 e^4+6 c^4 d^4\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{5/2}}+\frac{-2 b^2 c e^2 \left (2 a^2 e^3-5 a c d e (d+2 e x)+5 c^2 d^3 x\right )+b c^2 \left (5 a^2 e^4 (3 d+e x)-10 a c d^2 e^2 (d+3 e x)-c^2 d^4 (d-5 e x)\right )+2 c^2 \left (a^3 e^5-5 a^2 c d e^3 (2 d+e x)+5 a c^2 d^3 e (d+2 e x)-c^3 d^5 x\right )+b^4 e^4 (a e-5 c d x)-5 b^3 c e^3 \left (a e (d+e x)-2 c d^2 x\right )+b^5 e^5 x}{\left (b^2-4 a c\right ) (a+x (b+c x))^2}+\frac{b^2 c^2 e \left (-39 a^2 e^4+10 a c d e^2 (5 d+8 e x)-5 c^2 d^3 (3 d-4 e x)\right )+2 b c^3 \left (5 a^2 e^4 (11 d+5 e x)+10 a c d^2 e^2 (d-3 e x)+3 c^2 d^4 (d-5 e x)\right )+4 c^3 \left (8 a^3 e^5-5 a^2 c d e^3 (8 d+5 e x)+10 a c^2 d^3 e^2 x+3 c^3 d^5 x\right )+b^4 c e^3 \left (11 a e^2-10 c d (d+e x)\right )+10 b^3 c^2 e^2 \left (c d^3-a e^2 (4 d+3 e x)\right )+b^6 \left (-e^5\right )+b^5 c e^4 (5 d+4 e x)}{\left (b^2-4 a c\right )^2 (a+x (b+c x))}+c e^5 \log (a+x (b+c x))}{2 c^4} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^5/(a + b*x + c*x^2)^3,x]
[Out]
((b^5*e^5*x + b^4*e^4*(a*e - 5*c*d*x) - 5*b^3*c*e^3*(-2*c*d^2*x + a*e*(d + e*x))
- 2*b^2*c*e^2*(2*a^2*e^3 + 5*c^2*d^3*x - 5*a*c*d*e*(d + 2*e*x)) + 2*c^2*(a^3*e^
5 - c^3*d^5*x - 5*a^2*c*d*e^3*(2*d + e*x) + 5*a*c^2*d^3*e*(d + 2*e*x)) + b*c^2*(
-(c^2*d^4*(d - 5*e*x)) + 5*a^2*e^4*(3*d + e*x) - 10*a*c*d^2*e^2*(d + 3*e*x)))/((
b^2 - 4*a*c)*(a + x*(b + c*x))^2) + (-(b^6*e^5) + b^5*c*e^4*(5*d + 4*e*x) + b^4*
c*e^3*(11*a*e^2 - 10*c*d*(d + e*x)) + 10*b^3*c^2*e^2*(c*d^3 - a*e^2*(4*d + 3*e*x
)) + 4*c^3*(8*a^3*e^5 + 3*c^3*d^5*x + 10*a*c^2*d^3*e^2*x - 5*a^2*c*d*e^3*(8*d +
5*e*x)) + 2*b*c^3*(3*c^2*d^4*(d - 5*e*x) + 10*a*c*d^2*e^2*(d - 3*e*x) + 5*a^2*e^
4*(11*d + 5*e*x)) + b^2*c^2*e*(-39*a^2*e^4 - 5*c^2*d^3*(3*d - 4*e*x) + 10*a*c*d*
e^2*(5*d + 8*e*x)))/((b^2 - 4*a*c)^2*(a + x*(b + c*x))) + (2*c*(2*c*d - b*e)*(6*
c^4*d^4 + b^4*e^4 + 2*b^2*c*e^3*(b*d - 5*a*e) - 4*c^3*d^2*e*(3*b*d - 5*a*e) + 2*
c^2*e^2*(2*b^2*d^2 - 10*a*b*d*e + 15*a^2*e^2))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*
a*c]])/(-b^2 + 4*a*c)^(5/2) + c*e^5*Log[a + x*(b + c*x)])/(2*c^4)
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Maple [B] time = 0.041, size = 2610, normalized size = 6.7 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^5/(c*x^2+b*x+a)^3,x)
[Out]
((25*a^2*b*c^2*e^5-50*a^2*c^3*d*e^4-15*a*b^3*c*e^5+40*a*b^2*c^2*d*e^4-30*a*b*c^3
*d^2*e^3+20*a*c^4*d^3*e^2+2*b^5*e^5-5*b^4*c*d*e^4+10*b^2*c^3*d^3*e^2-15*b*c^4*d^
4*e+6*c^5*d^5)/c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3+1/2*(32*a^3*c^3*e^5+11*a^2*b^2
*c^2*e^5+10*a^2*b*c^3*d*e^4-160*a^2*c^4*d^2*e^3-19*a*b^4*c*e^5+40*a*b^3*c^2*d*e^
4-10*a*b^2*c^3*d^2*e^3+60*a*b*c^4*d^3*e^2+3*b^6*e^5-5*b^5*c*d*e^4-10*b^4*c^2*d^2
*e^3+30*b^3*c^3*d^3*e^2-45*b^2*c^4*d^4*e+18*b*c^5*d^5)/(16*a^2*c^2-8*a*b^2*c+b^4
)/c^3*x^2+(31*a^3*b*c^2*e^5-30*a^3*c^3*d*e^4-22*a^2*b^3*c*e^5+50*a^2*b^2*c^2*d*e
^4-50*a^2*b*c^3*d^2*e^3-20*a^2*c^4*d^3*e^2+3*a*b^5*e^5-5*a*b^4*c*d*e^4-10*a*b^3*
c^2*d^2*e^3+50*a*b^2*c^3*d^3*e^2-25*a*b*c^4*d^4*e+10*a*c^5*d^5-5*b^3*c^3*d^4*e+2
*b^2*c^4*d^5)/(16*a^2*c^2-8*a*b^2*c+b^4)/c^3*x+1/2/c^3*(24*a^4*c^2*e^5-21*a^3*b^
2*c*e^5+50*a^3*b*c^2*d*e^4-80*a^3*c^3*d^2*e^3+3*a^2*b^4*e^5-5*a^2*b^3*c*d*e^4-10
*a^2*b^2*c^2*d^2*e^3+60*a^2*b*c^3*d^3*e^2-40*a^2*c^4*d^4*e-5*a*b^2*c^3*d^4*e+10*
a*b*c^4*d^5-b^3*c^3*d^5)/(16*a^2*c^2-8*a*b^2*c+b^4))/(c*x^2+b*x+a)^2+8/(16*a^2*c
^2-8*a*b^2*c+b^4)/c*ln(c^2*(16*a^2*c^2-8*a*b^2*c+b^4)*(c*x^2+b*x+a))*a^2*e^5-4/(
16*a^2*c^2-8*a*b^2*c+b^4)/c^2*ln(c^2*(16*a^2*c^2-8*a*b^2*c+b^4)*(c*x^2+b*x+a))*a
*b^2*e^5+1/2/(16*a^2*c^2-8*a*b^2*c+b^4)/c^3*ln(c^2*(16*a^2*c^2-8*a*b^2*c+b^4)*(c
*x^2+b*x+a))*b^4*e^5-30/(1024*a^5*c^9-1280*a^4*b^2*c^8+640*a^3*b^4*c^7-160*a^2*b
^6*c^6+20*a*b^8*c^5-b^10*c^4)^(1/2)*arctan((2*(16*a^2*c^2-8*a*b^2*c+b^4)*c^3*x+b
*c^2*(16*a^2*c^2-8*a*b^2*c+b^4))/(1024*a^5*c^9-1280*a^4*b^2*c^8+640*a^3*b^4*c^7-
160*a^2*b^6*c^6+20*a*b^8*c^5-b^10*c^4)^(1/2))*a^2*b*c*e^5+60/(1024*a^5*c^9-1280*
a^4*b^2*c^8+640*a^3*b^4*c^7-160*a^2*b^6*c^6+20*a*b^8*c^5-b^10*c^4)^(1/2)*arctan(
(2*(16*a^2*c^2-8*a*b^2*c+b^4)*c^3*x+b*c^2*(16*a^2*c^2-8*a*b^2*c+b^4))/(1024*a^5*
c^9-1280*a^4*b^2*c^8+640*a^3*b^4*c^7-160*a^2*b^6*c^6+20*a*b^8*c^5-b^10*c^4)^(1/2
))*a^2*d*e^4*c^2+10/(1024*a^5*c^9-1280*a^4*b^2*c^8+640*a^3*b^4*c^7-160*a^2*b^6*c
^6+20*a*b^8*c^5-b^10*c^4)^(1/2)*arctan((2*(16*a^2*c^2-8*a*b^2*c+b^4)*c^3*x+b*c^2
*(16*a^2*c^2-8*a*b^2*c+b^4))/(1024*a^5*c^9-1280*a^4*b^2*c^8+640*a^3*b^4*c^7-160*
a^2*b^6*c^6+20*a*b^8*c^5-b^10*c^4)^(1/2))*a*b^3*e^5-60/(1024*a^5*c^9-1280*a^4*b^
2*c^8+640*a^3*b^4*c^7-160*a^2*b^6*c^6+20*a*b^8*c^5-b^10*c^4)^(1/2)*arctan((2*(16
*a^2*c^2-8*a*b^2*c+b^4)*c^3*x+b*c^2*(16*a^2*c^2-8*a*b^2*c+b^4))/(1024*a^5*c^9-12
80*a^4*b^2*c^8+640*a^3*b^4*c^7-160*a^2*b^6*c^6+20*a*b^8*c^5-b^10*c^4)^(1/2))*d^2
*a*b*c^2*e^3+40/(1024*a^5*c^9-1280*a^4*b^2*c^8+640*a^3*b^4*c^7-160*a^2*b^6*c^6+2
0*a*b^8*c^5-b^10*c^4)^(1/2)*arctan((2*(16*a^2*c^2-8*a*b^2*c+b^4)*c^3*x+b*c^2*(16
*a^2*c^2-8*a*b^2*c+b^4))/(1024*a^5*c^9-1280*a^4*b^2*c^8+640*a^3*b^4*c^7-160*a^2*
b^6*c^6+20*a*b^8*c^5-b^10*c^4)^(1/2))*c^3*d^3*a*e^2+20/(1024*a^5*c^9-1280*a^4*b^
2*c^8+640*a^3*b^4*c^7-160*a^2*b^6*c^6+20*a*b^8*c^5-b^10*c^4)^(1/2)*arctan((2*(16
*a^2*c^2-8*a*b^2*c+b^4)*c^3*x+b*c^2*(16*a^2*c^2-8*a*b^2*c+b^4))/(1024*a^5*c^9-12
80*a^4*b^2*c^8+640*a^3*b^4*c^7-160*a^2*b^6*c^6+20*a*b^8*c^5-b^10*c^4)^(1/2))*b^2
*c^2*d^3*e^2-30/(1024*a^5*c^9-1280*a^4*b^2*c^8+640*a^3*b^4*c^7-160*a^2*b^6*c^6+2
0*a*b^8*c^5-b^10*c^4)^(1/2)*arctan((2*(16*a^2*c^2-8*a*b^2*c+b^4)*c^3*x+b*c^2*(16
*a^2*c^2-8*a*b^2*c+b^4))/(1024*a^5*c^9-1280*a^4*b^2*c^8+640*a^3*b^4*c^7-160*a^2*
b^6*c^6+20*a*b^8*c^5-b^10*c^4)^(1/2))*d^4*b*c^3*e+12/(1024*a^5*c^9-1280*a^4*b^2*
c^8+640*a^3*b^4*c^7-160*a^2*b^6*c^6+20*a*b^8*c^5-b^10*c^4)^(1/2)*arctan((2*(16*a
^2*c^2-8*a*b^2*c+b^4)*c^3*x+b*c^2*(16*a^2*c^2-8*a*b^2*c+b^4))/(1024*a^5*c^9-1280
*a^4*b^2*c^8+640*a^3*b^4*c^7-160*a^2*b^6*c^6+20*a*b^8*c^5-b^10*c^4)^(1/2))*c^4*d
^5-1/(1024*a^5*c^9-1280*a^4*b^2*c^8+640*a^3*b^4*c^7-160*a^2*b^6*c^6+20*a*b^8*c^5
-b^10*c^4)^(1/2)*arctan((2*(16*a^2*c^2-8*a*b^2*c+b^4)*c^3*x+b*c^2*(16*a^2*c^2-8*
a*b^2*c+b^4))/(1024*a^5*c^9-1280*a^4*b^2*c^8+640*a^3*b^4*c^7-160*a^2*b^6*c^6+20*
a*b^8*c^5-b^10*c^4)^(1/2))*b^5/c*e^5
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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((e*x + d)^5/(c*x^2 + b*x + a)^3,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.373873, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^5/(c*x^2 + b*x + a)^3,x, algorithm="fricas")
[Out]
[-1/2*((12*a^2*c^5*d^5 - 30*a^2*b*c^4*d^4*e - 60*a^3*b*c^3*d^2*e^3 + 60*a^4*c^3*
d*e^4 + 20*(a^2*b^2*c^3 + 2*a^3*c^4)*d^3*e^2 - (a^2*b^5 - 10*a^3*b^3*c + 30*a^4*
b*c^2)*e^5 + (12*c^7*d^5 - 30*b*c^6*d^4*e - 60*a*b*c^5*d^2*e^3 + 60*a^2*c^5*d*e^
4 + 20*(b^2*c^5 + 2*a*c^6)*d^3*e^2 - (b^5*c^2 - 10*a*b^3*c^3 + 30*a^2*b*c^4)*e^5
)*x^4 + 2*(12*b*c^6*d^5 - 30*b^2*c^5*d^4*e - 60*a*b^2*c^4*d^2*e^3 + 60*a^2*b*c^4
*d*e^4 + 20*(b^3*c^4 + 2*a*b*c^5)*d^3*e^2 - (b^6*c - 10*a*b^4*c^2 + 30*a^2*b^2*c
^3)*e^5)*x^3 + (12*(b^2*c^5 + 2*a*c^6)*d^5 - 30*(b^3*c^4 + 2*a*b*c^5)*d^4*e + 20
*(b^4*c^3 + 4*a*b^2*c^4 + 4*a^2*c^5)*d^3*e^2 - 60*(a*b^3*c^3 + 2*a^2*b*c^4)*d^2*
e^3 + 60*(a^2*b^2*c^3 + 2*a^3*c^4)*d*e^4 - (b^7 - 8*a*b^5*c + 10*a^2*b^3*c^2 + 6
0*a^3*b*c^3)*e^5)*x^2 + 2*(12*a*b*c^5*d^5 - 30*a*b^2*c^4*d^4*e - 60*a^2*b^2*c^3*
d^2*e^3 + 60*a^3*b*c^3*d*e^4 + 20*(a*b^3*c^3 + 2*a^2*b*c^4)*d^3*e^2 - (a*b^6 - 1
0*a^2*b^4*c + 30*a^3*b^2*c^2)*e^5)*x)*log((b^3 - 4*a*b*c + 2*(b^2*c - 4*a*c^2)*x
+ (2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c)*sqrt(b^2 - 4*a*c))/(c*x^2 + b*x + a)) - (
60*a^2*b*c^3*d^3*e^2 - (b^3*c^3 - 10*a*b*c^4)*d^5 - 5*(a*b^2*c^3 + 8*a^2*c^4)*d^
4*e - 10*(a^2*b^2*c^2 + 8*a^3*c^3)*d^2*e^3 - 5*(a^2*b^3*c - 10*a^3*b*c^2)*d*e^4
+ 3*(a^2*b^4 - 7*a^3*b^2*c + 8*a^4*c^2)*e^5 + 2*(6*c^6*d^5 - 15*b*c^5*d^4*e - 30
*a*b*c^4*d^2*e^3 + 10*(b^2*c^4 + 2*a*c^5)*d^3*e^2 - 5*(b^4*c^2 - 8*a*b^2*c^3 + 1
0*a^2*c^4)*d*e^4 + (2*b^5*c - 15*a*b^3*c^2 + 25*a^2*b*c^3)*e^5)*x^3 + (18*b*c^5*
d^5 - 45*b^2*c^4*d^4*e + 30*(b^3*c^3 + 2*a*b*c^4)*d^3*e^2 - 10*(b^4*c^2 + a*b^2*
c^3 + 16*a^2*c^4)*d^2*e^3 - 5*(b^5*c - 8*a*b^3*c^2 - 2*a^2*b*c^3)*d*e^4 + (3*b^6
- 19*a*b^4*c + 11*a^2*b^2*c^2 + 32*a^3*c^3)*e^5)*x^2 + 2*(2*(b^2*c^4 + 5*a*c^5)
*d^5 - 5*(b^3*c^3 + 5*a*b*c^4)*d^4*e + 10*(5*a*b^2*c^3 - 2*a^2*c^4)*d^3*e^2 - 10
*(a*b^3*c^2 + 5*a^2*b*c^3)*d^2*e^3 - 5*(a*b^4*c - 10*a^2*b^2*c^2 + 6*a^3*c^3)*d*
e^4 + (3*a*b^5 - 22*a^2*b^3*c + 31*a^3*b*c^2)*e^5)*x + ((b^4*c^2 - 8*a*b^2*c^3 +
16*a^2*c^4)*e^5*x^4 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*e^5*x^3 + (b^6 - 6
*a*b^4*c + 32*a^3*c^3)*e^5*x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*e^5*x +
(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2)*e^5)*log(c*x^2 + b*x + a))*sqrt(b^2 - 4*a*c
))/((a^2*b^4*c^3 - 8*a^3*b^2*c^4 + 16*a^4*c^5 + (b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*
c^7)*x^4 + 2*(b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6)*x^3 + (b^6*c^3 - 6*a*b^4*c^4
+ 32*a^3*c^6)*x^2 + 2*(a*b^5*c^3 - 8*a^2*b^3*c^4 + 16*a^3*b*c^5)*x)*sqrt(b^2 -
4*a*c)), 1/2*(2*(12*a^2*c^5*d^5 - 30*a^2*b*c^4*d^4*e - 60*a^3*b*c^3*d^2*e^3 + 60
*a^4*c^3*d*e^4 + 20*(a^2*b^2*c^3 + 2*a^3*c^4)*d^3*e^2 - (a^2*b^5 - 10*a^3*b^3*c
+ 30*a^4*b*c^2)*e^5 + (12*c^7*d^5 - 30*b*c^6*d^4*e - 60*a*b*c^5*d^2*e^3 + 60*a^2
*c^5*d*e^4 + 20*(b^2*c^5 + 2*a*c^6)*d^3*e^2 - (b^5*c^2 - 10*a*b^3*c^3 + 30*a^2*b
*c^4)*e^5)*x^4 + 2*(12*b*c^6*d^5 - 30*b^2*c^5*d^4*e - 60*a*b^2*c^4*d^2*e^3 + 60*
a^2*b*c^4*d*e^4 + 20*(b^3*c^4 + 2*a*b*c^5)*d^3*e^2 - (b^6*c - 10*a*b^4*c^2 + 30*
a^2*b^2*c^3)*e^5)*x^3 + (12*(b^2*c^5 + 2*a*c^6)*d^5 - 30*(b^3*c^4 + 2*a*b*c^5)*d
^4*e + 20*(b^4*c^3 + 4*a*b^2*c^4 + 4*a^2*c^5)*d^3*e^2 - 60*(a*b^3*c^3 + 2*a^2*b*
c^4)*d^2*e^3 + 60*(a^2*b^2*c^3 + 2*a^3*c^4)*d*e^4 - (b^7 - 8*a*b^5*c + 10*a^2*b^
3*c^2 + 60*a^3*b*c^3)*e^5)*x^2 + 2*(12*a*b*c^5*d^5 - 30*a*b^2*c^4*d^4*e - 60*a^2
*b^2*c^3*d^2*e^3 + 60*a^3*b*c^3*d*e^4 + 20*(a*b^3*c^3 + 2*a^2*b*c^4)*d^3*e^2 - (
a*b^6 - 10*a^2*b^4*c + 30*a^3*b^2*c^2)*e^5)*x)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x
+ b)/(b^2 - 4*a*c)) + (60*a^2*b*c^3*d^3*e^2 - (b^3*c^3 - 10*a*b*c^4)*d^5 - 5*(a
*b^2*c^3 + 8*a^2*c^4)*d^4*e - 10*(a^2*b^2*c^2 + 8*a^3*c^3)*d^2*e^3 - 5*(a^2*b^3*
c - 10*a^3*b*c^2)*d*e^4 + 3*(a^2*b^4 - 7*a^3*b^2*c + 8*a^4*c^2)*e^5 + 2*(6*c^6*d
^5 - 15*b*c^5*d^4*e - 30*a*b*c^4*d^2*e^3 + 10*(b^2*c^4 + 2*a*c^5)*d^3*e^2 - 5*(b
^4*c^2 - 8*a*b^2*c^3 + 10*a^2*c^4)*d*e^4 + (2*b^5*c - 15*a*b^3*c^2 + 25*a^2*b*c^
3)*e^5)*x^3 + (18*b*c^5*d^5 - 45*b^2*c^4*d^4*e + 30*(b^3*c^3 + 2*a*b*c^4)*d^3*e^
2 - 10*(b^4*c^2 + a*b^2*c^3 + 16*a^2*c^4)*d^2*e^3 - 5*(b^5*c - 8*a*b^3*c^2 - 2*a
^2*b*c^3)*d*e^4 + (3*b^6 - 19*a*b^4*c + 11*a^2*b^2*c^2 + 32*a^3*c^3)*e^5)*x^2 +
2*(2*(b^2*c^4 + 5*a*c^5)*d^5 - 5*(b^3*c^3 + 5*a*b*c^4)*d^4*e + 10*(5*a*b^2*c^3 -
2*a^2*c^4)*d^3*e^2 - 10*(a*b^3*c^2 + 5*a^2*b*c^3)*d^2*e^3 - 5*(a*b^4*c - 10*a^2
*b^2*c^2 + 6*a^3*c^3)*d*e^4 + (3*a*b^5 - 22*a^2*b^3*c + 31*a^3*b*c^2)*e^5)*x + (
(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*e^5*x^4 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b
*c^3)*e^5*x^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*e^5*x^2 + 2*(a*b^5 - 8*a^2*b^3*c
+ 16*a^3*b*c^2)*e^5*x + (a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2)*e^5)*log(c*x^2 + b*
x + a))*sqrt(-b^2 + 4*a*c))/((a^2*b^4*c^3 - 8*a^3*b^2*c^4 + 16*a^4*c^5 + (b^4*c^
5 - 8*a*b^2*c^6 + 16*a^2*c^7)*x^4 + 2*(b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6)*x^3
+ (b^6*c^3 - 6*a*b^4*c^4 + 32*a^3*c^6)*x^2 + 2*(a*b^5*c^3 - 8*a^2*b^3*c^4 + 16*
a^3*b*c^5)*x)*sqrt(-b^2 + 4*a*c))]
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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((e*x+d)**5/(c*x**2+b*x+a)**3,x)
[Out]
Timed out
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GIAC/XCAS [A] time = 0.212197, size = 1087, normalized size = 2.8 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^5/(c*x^2 + b*x + a)^3,x, algorithm="giac")
[Out]
(12*c^5*d^5 - 30*b*c^4*d^4*e + 20*b^2*c^3*d^3*e^2 + 40*a*c^4*d^3*e^2 - 60*a*b*c^
3*d^2*e^3 + 60*a^2*c^3*d*e^4 - b^5*e^5 + 10*a*b^3*c*e^5 - 30*a^2*b*c^2*e^5)*arct
an((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*sqrt(-b
^2 + 4*a*c)) + 1/2*e^5*ln(c*x^2 + b*x + a)/c^3 - 1/2*(b^3*c^3*d^5 - 10*a*b*c^4*d
^5 + 5*a*b^2*c^3*d^4*e + 40*a^2*c^4*d^4*e - 60*a^2*b*c^3*d^3*e^2 + 10*a^2*b^2*c^
2*d^2*e^3 + 80*a^3*c^3*d^2*e^3 + 5*a^2*b^3*c*d*e^4 - 50*a^3*b*c^2*d*e^4 - 3*a^2*
b^4*e^5 + 21*a^3*b^2*c*e^5 - 24*a^4*c^2*e^5 - 2*(6*c^6*d^5 - 15*b*c^5*d^4*e + 10
*b^2*c^4*d^3*e^2 + 20*a*c^5*d^3*e^2 - 30*a*b*c^4*d^2*e^3 - 5*b^4*c^2*d*e^4 + 40*
a*b^2*c^3*d*e^4 - 50*a^2*c^4*d*e^4 + 2*b^5*c*e^5 - 15*a*b^3*c^2*e^5 + 25*a^2*b*c
^3*e^5)*x^3 - (18*b*c^5*d^5 - 45*b^2*c^4*d^4*e + 30*b^3*c^3*d^3*e^2 + 60*a*b*c^4
*d^3*e^2 - 10*b^4*c^2*d^2*e^3 - 10*a*b^2*c^3*d^2*e^3 - 160*a^2*c^4*d^2*e^3 - 5*b
^5*c*d*e^4 + 40*a*b^3*c^2*d*e^4 + 10*a^2*b*c^3*d*e^4 + 3*b^6*e^5 - 19*a*b^4*c*e^
5 + 11*a^2*b^2*c^2*e^5 + 32*a^3*c^3*e^5)*x^2 - 2*(2*b^2*c^4*d^5 + 10*a*c^5*d^5 -
5*b^3*c^3*d^4*e - 25*a*b*c^4*d^4*e + 50*a*b^2*c^3*d^3*e^2 - 20*a^2*c^4*d^3*e^2
- 10*a*b^3*c^2*d^2*e^3 - 50*a^2*b*c^3*d^2*e^3 - 5*a*b^4*c*d*e^4 + 50*a^2*b^2*c^2
*d*e^4 - 30*a^3*c^3*d*e^4 + 3*a*b^5*e^5 - 22*a^2*b^3*c*e^5 + 31*a^3*b*c^2*e^5)*x
)/((c*x^2 + b*x + a)^2*(b^2 - 4*a*c)^2*c^3)