3.2179 \(\int \frac{(d+e x)^5}{\left (a+b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=374 \[ \frac{(2 c d-b e) \left (-2 c^2 e^2 \left (15 a^2 e^2+10 a b d e+b^2 d^2\right )+4 b^2 c e^3 (5 a e+b d)-4 c^3 d^2 e (b d-5 a e)-3 b^4 e^4+2 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^4 \left (b^2-4 a c\right )^{3/2}}+\frac{e^3 x^2 \left (-2 c e (4 a e+5 b d)+3 b^2 e^2+16 c^2 d^2\right )}{2 c^2 \left (b^2-4 a c\right )}+\frac{e^3 \left (-2 c e (a e+5 b d)+3 b^2 e^2+10 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{2 c^4}+\frac{e^4 x^3 (2 c d-b e)}{c \left (b^2-4 a c\right )}-\frac{(d+e x)^4 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{e^2 x \left (-10 c^2 d e (3 a e+b d)+b c e^2 (11 a e+10 b d)-3 b^3 e^3+12 c^3 d^3\right )}{c^3 \left (b^2-4 a c\right )} \]

[Out]

(e^2*(12*c^3*d^3 - 3*b^3*e^3 - 10*c^2*d*e*(b*d + 3*a*e) + b*c*e^2*(10*b*d + 11*a
*e))*x)/(c^3*(b^2 - 4*a*c)) + (e^3*(16*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(5*b*d + 4*a*
e))*x^2)/(2*c^2*(b^2 - 4*a*c)) + (e^4*(2*c*d - b*e)*x^3)/(c*(b^2 - 4*a*c)) - ((d
+ e*x)^4*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)*(a + b*x + c*x^2)) + (
(2*c*d - b*e)*(2*c^4*d^4 - 3*b^4*e^4 - 4*c^3*d^2*e*(b*d - 5*a*e) + 4*b^2*c*e^3*(
b*d + 5*a*e) - 2*c^2*e^2*(b^2*d^2 + 10*a*b*d*e + 15*a^2*e^2))*ArcTanh[(b + 2*c*x
)/Sqrt[b^2 - 4*a*c]])/(c^4*(b^2 - 4*a*c)^(3/2)) + (e^3*(10*c^2*d^2 + 3*b^2*e^2 -
2*c*e*(5*b*d + a*e))*Log[a + b*x + c*x^2])/(2*c^4)

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Rubi [A]  time = 1.56848, antiderivative size = 374, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{(2 c d-b e) \left (-2 c^2 e^2 \left (15 a^2 e^2+10 a b d e+b^2 d^2\right )+4 b^2 c e^3 (5 a e+b d)-4 c^3 d^2 e (b d-5 a e)-3 b^4 e^4+2 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^4 \left (b^2-4 a c\right )^{3/2}}+\frac{e^3 x^2 \left (-2 c e (4 a e+5 b d)+3 b^2 e^2+16 c^2 d^2\right )}{2 c^2 \left (b^2-4 a c\right )}+\frac{e^3 \left (-2 c e (a e+5 b d)+3 b^2 e^2+10 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{2 c^4}+\frac{e^4 x^3 (2 c d-b e)}{c \left (b^2-4 a c\right )}-\frac{(d+e x)^4 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{e^2 x \left (-10 c^2 d e (3 a e+b d)+b c e^2 (11 a e+10 b d)-3 b^3 e^3+12 c^3 d^3\right )}{c^3 \left (b^2-4 a c\right )} \]

Antiderivative was successfully verified.

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

[Out]

(e^2*(12*c^3*d^3 - 3*b^3*e^3 - 10*c^2*d*e*(b*d + 3*a*e) + b*c*e^2*(10*b*d + 11*a
*e))*x)/(c^3*(b^2 - 4*a*c)) + (e^3*(16*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(5*b*d + 4*a*
e))*x^2)/(2*c^2*(b^2 - 4*a*c)) + (e^4*(2*c*d - b*e)*x^3)/(c*(b^2 - 4*a*c)) - ((d
+ e*x)^4*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)*(a + b*x + c*x^2)) + (
(2*c*d - b*e)*(2*c^4*d^4 - 3*b^4*e^4 - 4*c^3*d^2*e*(b*d - 5*a*e) + 4*b^2*c*e^3*(
b*d + 5*a*e) - 2*c^2*e^2*(b^2*d^2 + 10*a*b*d*e + 15*a^2*e^2))*ArcTanh[(b + 2*c*x
)/Sqrt[b^2 - 4*a*c]])/(c^4*(b^2 - 4*a*c)^(3/2)) + (e^3*(10*c^2*d^2 + 3*b^2*e^2 -
2*c*e*(5*b*d + a*e))*Log[a + b*x + c*x^2])/(2*c^4)

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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)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 1.63531, size = 422, normalized size = 1.13 \[ \frac{\frac{2 (b e-2 c d) \left (2 c^2 e^2 \left (15 a^2 e^2+10 a b d e+b^2 d^2\right )-4 b^2 c e^3 (5 a e+b d)+4 c^3 d^2 e (b d-5 a e)+3 b^4 e^4-2 c^4 d^4\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}+\frac{2 \left (-2 b^2 c e^2 \left (2 a^2 e^3-5 a c d e (d+2 e x)+5 c^2 d^3 x\right )+b c^2 \left (5 a^2 e^4 (3 d+e x)-10 a c d^2 e^2 (d+3 e x)-c^2 d^4 (d-5 e x)\right )+2 c^2 \left (a^3 e^5-5 a^2 c d e^3 (2 d+e x)+5 a c^2 d^3 e (d+2 e x)-c^3 d^5 x\right )+b^4 e^4 (a e-5 c d x)-5 b^3 c e^3 \left (a e (d+e x)-2 c d^2 x\right )+b^5 e^5 x\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}+e^3 \left (-2 c e (a e+5 b d)+3 b^2 e^2+10 c^2 d^2\right ) \log (a+x (b+c x))+2 c e^4 x (5 c d-2 b e)+c^2 e^5 x^2}{2 c^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.032, size = 2334, normalized size = 6.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

20/c/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^2+b*x+a))*a*e^3*d^2-5/c^2/(4*a*c-b^2)*ln((4
*a*c-b^2)*(c*x^2+b*x+a))*b^2*d^2*e^3-60/c/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-
b^6)^(1/2)*arctan((2*c*(4*a*c-b^2)*x+(4*a*c-b^2)*b)/(64*a^3*c^3-48*a^2*b^2*c^2+1
2*a*b^4*c-b^6)^(1/2))*d^2*a*b*e^3-20/c^2/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^2+b*x+a
))*a*b*d*e^4-15/c^2/(c*x^2+b*x+a)/(4*a*c-b^2)*a^2*b*d*e^4-10/c^2/(c*x^2+b*x+a)/(
4*a*c-b^2)*x*b^3*d^2*e^3+10/c/(c*x^2+b*x+a)/(4*a*c-b^2)*x*b^2*d^3*e^2-5/c^2/(c*x
^2+b*x+a)/(4*a*c-b^2)*x*a^2*b*e^5-10/c^2/(c*x^2+b*x+a)/(4*a*c-b^2)*a*b^2*d^2*e^3
+10/c/(c*x^2+b*x+a)/(4*a*c-b^2)*a*b*d^3*e^2+5/c^3/(c*x^2+b*x+a)/(4*a*c-b^2)*a*b^
3*d*e^4-2*e^5/c^3*b*x+5*e^4/c^2*d*x-10/(c*x^2+b*x+a)/(4*a*c-b^2)*a*d^4*e-10/(64*
a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*arctan((2*c*(4*a*c-b^2)*x+(4*a*c-b^
2)*b)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))*d^4*b*e+40/(64*a^3*c^3-4
8*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*arctan((2*c*(4*a*c-b^2)*x+(4*a*c-b^2)*b)/(64
*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))*d^3*a*e^2+1/2*e^5*x^2/c^2+30/c^2/
(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*arctan((2*c*(4*a*c-b^2)*x+(4*a*
c-b^2)*b)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))*a^2*b*e^5-60/c/(64*a
^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*arctan((2*c*(4*a*c-b^2)*x+(4*a*c-b^2
)*b)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))*a^2*d*e^4-20/c^3/(64*a^3*
c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*arctan((2*c*(4*a*c-b^2)*x+(4*a*c-b^2)*b
)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))*a*b^3*e^5-10/c^3/(64*a^3*c^3
-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*arctan((2*c*(4*a*c-b^2)*x+(4*a*c-b^2)*b)/(
64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))*d*b^4*e^4+7/c^3/(4*a*c-b^2)*ln(
(4*a*c-b^2)*(c*x^2+b*x+a))*a*b^2*e^5+2*c/(c*x^2+b*x+a)/(4*a*c-b^2)*x*d^5-2/c^2/(
c*x^2+b*x+a)/(4*a*c-b^2)*a^3*e^5-3/2/c^4/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^2+b*x+a
))*b^4*e^5+3/c^4/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*arctan((2*c*(4
*a*c-b^2)*x+(4*a*c-b^2)*b)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))*b^5
*e^5-4/c^2/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^2+b*x+a))*a^2*e^5+30/c/(c*x^2+b*x+a)/
(4*a*c-b^2)*x*a*b*d^2*e^3+4*c/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*a
rctan((2*c*(4*a*c-b^2)*x+(4*a*c-b^2)*b)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^
6)^(1/2))*d^5+1/(c*x^2+b*x+a)/(4*a*c-b^2)*b*d^5-20/c^2/(c*x^2+b*x+a)/(4*a*c-b^2)
*x*a*b^2*d*e^4+10/c/(c*x^2+b*x+a)/(4*a*c-b^2)*x*e^4*a^2*d+20/c/(c*x^2+b*x+a)/(4*
a*c-b^2)*a^2*d^2*e^3-1/c^4/(c*x^2+b*x+a)/(4*a*c-b^2)*a*b^4*e^5+10/c^2/(64*a^3*c^
3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*arctan((2*c*(4*a*c-b^2)*x+(4*a*c-b^2)*b)/
(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))*b^3*d^2*e^3+5/c^3/(4*a*c-b^2)*
ln((4*a*c-b^2)*(c*x^2+b*x+a))*b^3*d*e^4-5/(c*x^2+b*x+a)/(4*a*c-b^2)*x*b*d^4*e+4/
c^3/(c*x^2+b*x+a)/(4*a*c-b^2)*a^2*b^2*e^5-20/(c*x^2+b*x+a)/(4*a*c-b^2)*x*a*d^3*e
^2-1/c^4/(c*x^2+b*x+a)/(4*a*c-b^2)*x*b^5*e^5+5/c^3/(c*x^2+b*x+a)/(4*a*c-b^2)*x*a
*b^3*e^5+5/c^3/(c*x^2+b*x+a)/(4*a*c-b^2)*x*b^4*d*e^4+60/c^2/(64*a^3*c^3-48*a^2*b
^2*c^2+12*a*b^4*c-b^6)^(1/2)*arctan((2*c*(4*a*c-b^2)*x+(4*a*c-b^2)*b)/(64*a^3*c^
3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))*d*a*b^2*e^4

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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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.335293, 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)^2,x, algorithm="fricas")

[Out]

[-1/2*((4*a*c^5*d^5 - 10*a*b*c^4*d^4*e + 40*a^2*c^4*d^3*e^2 + 10*(a*b^3*c^2 - 6*
a^2*b*c^3)*d^2*e^3 - 10*(a*b^4*c - 6*a^2*b^2*c^2 + 6*a^3*c^3)*d*e^4 + (3*a*b^5 -
20*a^2*b^3*c + 30*a^3*b*c^2)*e^5 + (4*c^6*d^5 - 10*b*c^5*d^4*e + 40*a*c^5*d^3*e
^2 + 10*(b^3*c^3 - 6*a*b*c^4)*d^2*e^3 - 10*(b^4*c^2 - 6*a*b^2*c^3 + 6*a^2*c^4)*d
*e^4 + (3*b^5*c - 20*a*b^3*c^2 + 30*a^2*b*c^3)*e^5)*x^2 + (4*b*c^5*d^5 - 10*b^2*
c^4*d^4*e + 40*a*b*c^4*d^3*e^2 + 10*(b^4*c^2 - 6*a*b^2*c^3)*d^2*e^3 - 10*(b^5*c
- 6*a*b^3*c^2 + 6*a^2*b*c^3)*d*e^4 + (3*b^6 - 20*a*b^4*c + 30*a^2*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)) + (2*b*c^4*d^5 - 20*a*c^4*d^4*e + 20*
a*b*c^3*d^3*e^2 - (b^2*c^3 - 4*a*c^4)*e^5*x^4 - 20*(a*b^2*c^2 - 2*a^2*c^3)*d^2*e
^3 + 10*(a*b^3*c - 3*a^2*b*c^2)*d*e^4 - 2*(a*b^4 - 4*a^2*b^2*c + 2*a^3*c^2)*e^5
- (10*(b^2*c^3 - 4*a*c^4)*d*e^4 - 3*(b^3*c^2 - 4*a*b*c^3)*e^5)*x^3 - (10*(b^3*c^
2 - 4*a*b*c^3)*d*e^4 - (4*b^4*c - 17*a*b^2*c^2 + 4*a^2*c^3)*e^5)*x^2 + 2*(2*c^5*
d^5 - 5*b*c^4*d^4*e + 10*(b^2*c^3 - 2*a*c^4)*d^3*e^2 - 10*(b^3*c^2 - 3*a*b*c^3)*
d^2*e^3 + 5*(b^4*c - 5*a*b^2*c^2 + 6*a^2*c^3)*d*e^4 - (b^5 - 7*a*b^3*c + 13*a^2*
b*c^2)*e^5)*x - (10*(a*b^2*c^2 - 4*a^2*c^3)*d^2*e^3 - 10*(a*b^3*c - 4*a^2*b*c^2)
*d*e^4 + (3*a*b^4 - 14*a^2*b^2*c + 8*a^3*c^2)*e^5 + (10*(b^2*c^3 - 4*a*c^4)*d^2*
e^3 - 10*(b^3*c^2 - 4*a*b*c^3)*d*e^4 + (3*b^4*c - 14*a*b^2*c^2 + 8*a^2*c^3)*e^5)
*x^2 + (10*(b^3*c^2 - 4*a*b*c^3)*d^2*e^3 - 10*(b^4*c - 4*a*b^2*c^2)*d*e^4 + (3*b
^5 - 14*a*b^3*c + 8*a^2*b*c^2)*e^5)*x)*log(c*x^2 + b*x + a))*sqrt(b^2 - 4*a*c))/
((a*b^2*c^4 - 4*a^2*c^5 + (b^2*c^5 - 4*a*c^6)*x^2 + (b^3*c^4 - 4*a*b*c^5)*x)*sqr
t(b^2 - 4*a*c)), -1/2*(2*(4*a*c^5*d^5 - 10*a*b*c^4*d^4*e + 40*a^2*c^4*d^3*e^2 +
10*(a*b^3*c^2 - 6*a^2*b*c^3)*d^2*e^3 - 10*(a*b^4*c - 6*a^2*b^2*c^2 + 6*a^3*c^3)*
d*e^4 + (3*a*b^5 - 20*a^2*b^3*c + 30*a^3*b*c^2)*e^5 + (4*c^6*d^5 - 10*b*c^5*d^4*
e + 40*a*c^5*d^3*e^2 + 10*(b^3*c^3 - 6*a*b*c^4)*d^2*e^3 - 10*(b^4*c^2 - 6*a*b^2*
c^3 + 6*a^2*c^4)*d*e^4 + (3*b^5*c - 20*a*b^3*c^2 + 30*a^2*b*c^3)*e^5)*x^2 + (4*b
*c^5*d^5 - 10*b^2*c^4*d^4*e + 40*a*b*c^4*d^3*e^2 + 10*(b^4*c^2 - 6*a*b^2*c^3)*d^
2*e^3 - 10*(b^5*c - 6*a*b^3*c^2 + 6*a^2*b*c^3)*d*e^4 + (3*b^6 - 20*a*b^4*c + 30*
a^2*b^2*c^2)*e^5)*x)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + (2*
b*c^4*d^5 - 20*a*c^4*d^4*e + 20*a*b*c^3*d^3*e^2 - (b^2*c^3 - 4*a*c^4)*e^5*x^4 -
20*(a*b^2*c^2 - 2*a^2*c^3)*d^2*e^3 + 10*(a*b^3*c - 3*a^2*b*c^2)*d*e^4 - 2*(a*b^4
- 4*a^2*b^2*c + 2*a^3*c^2)*e^5 - (10*(b^2*c^3 - 4*a*c^4)*d*e^4 - 3*(b^3*c^2 - 4
*a*b*c^3)*e^5)*x^3 - (10*(b^3*c^2 - 4*a*b*c^3)*d*e^4 - (4*b^4*c - 17*a*b^2*c^2 +
4*a^2*c^3)*e^5)*x^2 + 2*(2*c^5*d^5 - 5*b*c^4*d^4*e + 10*(b^2*c^3 - 2*a*c^4)*d^3
*e^2 - 10*(b^3*c^2 - 3*a*b*c^3)*d^2*e^3 + 5*(b^4*c - 5*a*b^2*c^2 + 6*a^2*c^3)*d*
e^4 - (b^5 - 7*a*b^3*c + 13*a^2*b*c^2)*e^5)*x - (10*(a*b^2*c^2 - 4*a^2*c^3)*d^2*
e^3 - 10*(a*b^3*c - 4*a^2*b*c^2)*d*e^4 + (3*a*b^4 - 14*a^2*b^2*c + 8*a^3*c^2)*e^
5 + (10*(b^2*c^3 - 4*a*c^4)*d^2*e^3 - 10*(b^3*c^2 - 4*a*b*c^3)*d*e^4 + (3*b^4*c
- 14*a*b^2*c^2 + 8*a^2*c^3)*e^5)*x^2 + (10*(b^3*c^2 - 4*a*b*c^3)*d^2*e^3 - 10*(b
^4*c - 4*a*b^2*c^2)*d*e^4 + (3*b^5 - 14*a*b^3*c + 8*a^2*b*c^2)*e^5)*x)*log(c*x^2
+ b*x + a))*sqrt(-b^2 + 4*a*c))/((a*b^2*c^4 - 4*a^2*c^5 + (b^2*c^5 - 4*a*c^6)*x
^2 + (b^3*c^4 - 4*a*b*c^5)*x)*sqrt(-b^2 + 4*a*c))]

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Sympy [A]  time = 84.2198, size = 2669, normalized size = 7.14 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-e**3*(2*a*c*e**2 - 3*b**2*e**2 + 10*b*c*d*e - 10*c**2*d**2)/(2*c**4) - sqrt(-(
4*a*c - b**2)**3)*(b*e - 2*c*d)*(30*a**2*c**2*e**4 - 20*a*b**2*c*e**4 + 20*a*b*c
**2*d*e**3 - 20*a*c**3*d**2*e**2 + 3*b**4*e**4 - 4*b**3*c*d*e**3 + 2*b**2*c**2*d
**2*e**2 + 4*b*c**3*d**3*e - 2*c**4*d**4)/(2*c**4*(64*a**3*c**3 - 48*a**2*b**2*c
**2 + 12*a*b**4*c - b**6)))*log(x + (16*a**3*c**2*e**5 - 17*a**2*b**2*c*e**5 + 5
0*a**2*b*c**2*d*e**4 + 16*a**2*c**5*(-e**3*(2*a*c*e**2 - 3*b**2*e**2 + 10*b*c*d*
e - 10*c**2*d**2)/(2*c**4) - sqrt(-(4*a*c - b**2)**3)*(b*e - 2*c*d)*(30*a**2*c**
2*e**4 - 20*a*b**2*c*e**4 + 20*a*b*c**2*d*e**3 - 20*a*c**3*d**2*e**2 + 3*b**4*e*
*4 - 4*b**3*c*d*e**3 + 2*b**2*c**2*d**2*e**2 + 4*b*c**3*d**3*e - 2*c**4*d**4)/(2
*c**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) - 80*a**2*c**3*d
**2*e**3 + 3*a*b**4*e**5 - 10*a*b**3*c*d*e**4 - 8*a*b**2*c**4*(-e**3*(2*a*c*e**2
- 3*b**2*e**2 + 10*b*c*d*e - 10*c**2*d**2)/(2*c**4) - sqrt(-(4*a*c - b**2)**3)*
(b*e - 2*c*d)*(30*a**2*c**2*e**4 - 20*a*b**2*c*e**4 + 20*a*b*c**2*d*e**3 - 20*a*
c**3*d**2*e**2 + 3*b**4*e**4 - 4*b**3*c*d*e**3 + 2*b**2*c**2*d**2*e**2 + 4*b*c**
3*d**3*e - 2*c**4*d**4)/(2*c**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c
- b**6))) + 10*a*b**2*c**2*d**2*e**3 + 20*a*b*c**3*d**3*e**2 + b**4*c**3*(-e**3*
(2*a*c*e**2 - 3*b**2*e**2 + 10*b*c*d*e - 10*c**2*d**2)/(2*c**4) - sqrt(-(4*a*c -
b**2)**3)*(b*e - 2*c*d)*(30*a**2*c**2*e**4 - 20*a*b**2*c*e**4 + 20*a*b*c**2*d*e
**3 - 20*a*c**3*d**2*e**2 + 3*b**4*e**4 - 4*b**3*c*d*e**3 + 2*b**2*c**2*d**2*e**
2 + 4*b*c**3*d**3*e - 2*c**4*d**4)/(2*c**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 1
2*a*b**4*c - b**6))) - 5*b**2*c**3*d**4*e + 2*b*c**4*d**5)/(30*a**2*b*c**2*e**5
- 60*a**2*c**3*d*e**4 - 20*a*b**3*c*e**5 + 60*a*b**2*c**2*d*e**4 - 60*a*b*c**3*d
**2*e**3 + 40*a*c**4*d**3*e**2 + 3*b**5*e**5 - 10*b**4*c*d*e**4 + 10*b**3*c**2*d
**2*e**3 - 10*b*c**4*d**4*e + 4*c**5*d**5)) + (-e**3*(2*a*c*e**2 - 3*b**2*e**2 +
10*b*c*d*e - 10*c**2*d**2)/(2*c**4) + sqrt(-(4*a*c - b**2)**3)*(b*e - 2*c*d)*(3
0*a**2*c**2*e**4 - 20*a*b**2*c*e**4 + 20*a*b*c**2*d*e**3 - 20*a*c**3*d**2*e**2 +
3*b**4*e**4 - 4*b**3*c*d*e**3 + 2*b**2*c**2*d**2*e**2 + 4*b*c**3*d**3*e - 2*c**
4*d**4)/(2*c**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))*log(x
+ (16*a**3*c**2*e**5 - 17*a**2*b**2*c*e**5 + 50*a**2*b*c**2*d*e**4 + 16*a**2*c**
5*(-e**3*(2*a*c*e**2 - 3*b**2*e**2 + 10*b*c*d*e - 10*c**2*d**2)/(2*c**4) + sqrt(
-(4*a*c - b**2)**3)*(b*e - 2*c*d)*(30*a**2*c**2*e**4 - 20*a*b**2*c*e**4 + 20*a*b
*c**2*d*e**3 - 20*a*c**3*d**2*e**2 + 3*b**4*e**4 - 4*b**3*c*d*e**3 + 2*b**2*c**2
*d**2*e**2 + 4*b*c**3*d**3*e - 2*c**4*d**4)/(2*c**4*(64*a**3*c**3 - 48*a**2*b**2
*c**2 + 12*a*b**4*c - b**6))) - 80*a**2*c**3*d**2*e**3 + 3*a*b**4*e**5 - 10*a*b*
*3*c*d*e**4 - 8*a*b**2*c**4*(-e**3*(2*a*c*e**2 - 3*b**2*e**2 + 10*b*c*d*e - 10*c
**2*d**2)/(2*c**4) + sqrt(-(4*a*c - b**2)**3)*(b*e - 2*c*d)*(30*a**2*c**2*e**4 -
20*a*b**2*c*e**4 + 20*a*b*c**2*d*e**3 - 20*a*c**3*d**2*e**2 + 3*b**4*e**4 - 4*b
**3*c*d*e**3 + 2*b**2*c**2*d**2*e**2 + 4*b*c**3*d**3*e - 2*c**4*d**4)/(2*c**4*(6
4*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) + 10*a*b**2*c**2*d**2*e*
*3 + 20*a*b*c**3*d**3*e**2 + b**4*c**3*(-e**3*(2*a*c*e**2 - 3*b**2*e**2 + 10*b*c
*d*e - 10*c**2*d**2)/(2*c**4) + sqrt(-(4*a*c - b**2)**3)*(b*e - 2*c*d)*(30*a**2*
c**2*e**4 - 20*a*b**2*c*e**4 + 20*a*b*c**2*d*e**3 - 20*a*c**3*d**2*e**2 + 3*b**4
*e**4 - 4*b**3*c*d*e**3 + 2*b**2*c**2*d**2*e**2 + 4*b*c**3*d**3*e - 2*c**4*d**4)
/(2*c**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) - 5*b**2*c**3
*d**4*e + 2*b*c**4*d**5)/(30*a**2*b*c**2*e**5 - 60*a**2*c**3*d*e**4 - 20*a*b**3*
c*e**5 + 60*a*b**2*c**2*d*e**4 - 60*a*b*c**3*d**2*e**3 + 40*a*c**4*d**3*e**2 + 3
*b**5*e**5 - 10*b**4*c*d*e**4 + 10*b**3*c**2*d**2*e**3 - 10*b*c**4*d**4*e + 4*c*
*5*d**5)) - (2*a**3*c**2*e**5 - 4*a**2*b**2*c*e**5 + 15*a**2*b*c**2*d*e**4 - 20*
a**2*c**3*d**2*e**3 + a*b**4*e**5 - 5*a*b**3*c*d*e**4 + 10*a*b**2*c**2*d**2*e**3
- 10*a*b*c**3*d**3*e**2 + 10*a*c**4*d**4*e - b*c**4*d**5 + x*(5*a**2*b*c**2*e**
5 - 10*a**2*c**3*d*e**4 - 5*a*b**3*c*e**5 + 20*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 + b**5*e**5 - 5*b**4*c*d*e**4 + 10*b**3*c**2*d**
2*e**3 - 10*b**2*c**3*d**3*e**2 + 5*b*c**4*d**4*e - 2*c**5*d**5))/(4*a**2*c**5 -
a*b**2*c**4 + x**2*(4*a*c**6 - b**2*c**5) + x*(4*a*b*c**5 - b**3*c**4)) + e**5*
x**2/(2*c**2) - x*(2*b*e**5 - 5*c*d*e**4)/c**3

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GIAC/XCAS [A]  time = 0.208868, size = 690, normalized size = 1.84 \[ -\frac{{\left (4 \, c^{5} d^{5} - 10 \, b c^{4} d^{4} e + 40 \, a c^{4} d^{3} e^{2} + 10 \, b^{3} c^{2} d^{2} e^{3} - 60 \, a b c^{3} d^{2} e^{3} - 10 \, b^{4} c d e^{4} + 60 \, a b^{2} c^{2} d e^{4} - 60 \, a^{2} c^{3} d e^{4} + 3 \, b^{5} e^{5} - 20 \, a b^{3} c e^{5} + 30 \, a^{2} b c^{2} e^{5}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{4} - 4 \, a c^{5}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{{\left (10 \, c^{2} d^{2} e^{3} - 10 \, b c d e^{4} + 3 \, b^{2} e^{5} - 2 \, a c e^{5}\right )}{\rm ln}\left (c x^{2} + b x + a\right )}{2 \, c^{4}} + \frac{c^{2} x^{2} e^{5} + 10 \, c^{2} d x e^{4} - 4 \, b c x e^{5}}{2 \, c^{4}} - \frac{b c^{4} d^{5} - 10 \, a c^{4} d^{4} e + 10 \, a b c^{3} d^{3} e^{2} - 10 \, a b^{2} c^{2} d^{2} e^{3} + 20 \, a^{2} c^{3} d^{2} e^{3} + 5 \, a b^{3} c d e^{4} - 15 \, a^{2} b c^{2} d e^{4} - a b^{4} e^{5} + 4 \, a^{2} b^{2} c e^{5} - 2 \, a^{3} c^{2} e^{5} +{\left (2 \, c^{5} d^{5} - 5 \, b c^{4} d^{4} e + 10 \, b^{2} c^{3} d^{3} e^{2} - 20 \, a c^{4} d^{3} e^{2} - 10 \, b^{3} c^{2} d^{2} e^{3} + 30 \, a b c^{3} d^{2} e^{3} + 5 \, b^{4} c d e^{4} - 20 \, a b^{2} c^{2} d e^{4} + 10 \, a^{2} c^{3} d e^{4} - b^{5} e^{5} + 5 \, a b^{3} c e^{5} - 5 \, a^{2} b c^{2} e^{5}\right )} x}{{\left (c x^{2} + b x + a\right )}{\left (b^{2} - 4 \, a c\right )} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(4*c^5*d^5 - 10*b*c^4*d^4*e + 40*a*c^4*d^3*e^2 + 10*b^3*c^2*d^2*e^3 - 60*a*b*c^
3*d^2*e^3 - 10*b^4*c*d*e^4 + 60*a*b^2*c^2*d*e^4 - 60*a^2*c^3*d*e^4 + 3*b^5*e^5 -
20*a*b^3*c*e^5 + 30*a^2*b*c^2*e^5)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^2
*c^4 - 4*a*c^5)*sqrt(-b^2 + 4*a*c)) + 1/2*(10*c^2*d^2*e^3 - 10*b*c*d*e^4 + 3*b^2
*e^5 - 2*a*c*e^5)*ln(c*x^2 + b*x + a)/c^4 + 1/2*(c^2*x^2*e^5 + 10*c^2*d*x*e^4 -
4*b*c*x*e^5)/c^4 - (b*c^4*d^5 - 10*a*c^4*d^4*e + 10*a*b*c^3*d^3*e^2 - 10*a*b^2*c
^2*d^2*e^3 + 20*a^2*c^3*d^2*e^3 + 5*a*b^3*c*d*e^4 - 15*a^2*b*c^2*d*e^4 - a*b^4*e
^5 + 4*a^2*b^2*c*e^5 - 2*a^3*c^2*e^5 + (2*c^5*d^5 - 5*b*c^4*d^4*e + 10*b^2*c^3*d
^3*e^2 - 20*a*c^4*d^3*e^2 - 10*b^3*c^2*d^2*e^3 + 30*a*b*c^3*d^2*e^3 + 5*b^4*c*d*
e^4 - 20*a*b^2*c^2*d*e^4 + 10*a^2*c^3*d*e^4 - b^5*e^5 + 5*a*b^3*c*e^5 - 5*a^2*b*
c^2*e^5)*x)/((c*x^2 + b*x + a)*(b^2 - 4*a*c)*c^4)