3.1539 \(\int \frac{(b+2 c x) (d+e x)^5}{\left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=293 \[ \frac{5 e^3 x \left (-c e (3 a e+2 b d)+b^2 e^2+3 c^2 d^2\right )}{c^2 \left (b^2-4 a c\right )}+\frac{5 e^4 x^2 (2 c d-b e)}{2 c \left (b^2-4 a c\right )}-\frac{5 e (d+e x)^3 (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{5 e \left (2 b^2 c e^3 (3 a e+b d)-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (a e+2 b d)-b^4 e^4+2 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{3/2}}+\frac{5 e^4 (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 c^3}-\frac{(d+e x)^5}{2 \left (a+b x+c x^2\right )^2} \]

[Out]

(5*e^3*(3*c^2*d^2 + b^2*e^2 - c*e*(2*b*d + 3*a*e))*x)/(c^2*(b^2 - 4*a*c)) + (5*e
^4*(2*c*d - b*e)*x^2)/(2*c*(b^2 - 4*a*c)) - (d + e*x)^5/(2*(a + b*x + c*x^2)^2)
- (5*e*(d + e*x)^3*(b*d - 2*a*e + (2*c*d - b*e)*x))/(2*(b^2 - 4*a*c)*(a + b*x +
c*x^2)) + (5*e*(2*c^4*d^4 - b^4*e^4 - 4*c^3*d^2*e*(b*d - 3*a*e) - 6*a*c^2*e^3*(2
*b*d + a*e) + 2*b^2*c*e^3*(b*d + 3*a*e))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])
/(c^3*(b^2 - 4*a*c)^(3/2)) + (5*e^4*(2*c*d - b*e)*Log[a + b*x + c*x^2])/(2*c^3)

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Rubi [A]  time = 1.33642, antiderivative size = 293, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ \frac{5 e^3 x \left (-c e (3 a e+2 b d)+b^2 e^2+3 c^2 d^2\right )}{c^2 \left (b^2-4 a c\right )}+\frac{5 e^4 x^2 (2 c d-b e)}{2 c \left (b^2-4 a c\right )}-\frac{5 e (d+e x)^3 (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{5 e \left (2 b^2 c e^3 (3 a e+b d)-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (a e+2 b d)-b^4 e^4+2 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{3/2}}+\frac{5 e^4 (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 c^3}-\frac{(d+e x)^5}{2 \left (a+b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[((b + 2*c*x)*(d + e*x)^5)/(a + b*x + c*x^2)^3,x]

[Out]

(5*e^3*(3*c^2*d^2 + b^2*e^2 - c*e*(2*b*d + 3*a*e))*x)/(c^2*(b^2 - 4*a*c)) + (5*e
^4*(2*c*d - b*e)*x^2)/(2*c*(b^2 - 4*a*c)) - (d + e*x)^5/(2*(a + b*x + c*x^2)^2)
- (5*e*(d + e*x)^3*(b*d - 2*a*e + (2*c*d - b*e)*x))/(2*(b^2 - 4*a*c)*(a + b*x +
c*x^2)) + (5*e*(2*c^4*d^4 - b^4*e^4 - 4*c^3*d^2*e*(b*d - 3*a*e) - 6*a*c^2*e^3*(2
*b*d + a*e) + 2*b^2*c*e^3*(b*d + 3*a*e))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])
/(c^3*(b^2 - 4*a*c)^(3/2)) + (5*e^4*(2*c*d - b*e)*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((2*c*x+b)*(e*x+d)**5/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 2.01721, size = 480, normalized size = 1.64 \[ \frac{-\frac{c^2 e^3 \left (a^2 e (5 d+e x)+10 a b d (d+e x)+10 b^2 d^2 x\right )-b c e^4 \left (2 a^2 e+a b (5 d+3 e x)+5 b^2 d x\right )+b^3 e^5 (a+b x)-10 c^3 d^2 e^2 (a (d+e x)+b d x)+c^4 d^4 (d+5 e x)}{(a+x (b+c x))^2}+\frac{e \left (b c^2 \left (31 a^2 e^4-10 a c d e^2 (7 d+10 e x)-5 c^2 d^3 (d-4 e x)\right )-2 c^3 \left (a^2 e^3 (40 d+9 e x)-10 a c d^2 e (4 d+5 e x)+5 c^2 d^4 x\right )+b^3 c e^2 \left (10 c d (d+3 e x)-13 a e^2\right )+2 b^2 c^2 e \left (a e^2 (25 d+17 e x)-5 c d^2 (d+4 e x)\right )+b^5 e^4-b^4 c e^3 (5 d+8 e x)\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}-\frac{10 c e \left (-2 b^2 c e^3 (3 a e+b d)+4 c^3 d^2 e (b d-3 a e)+6 a c^2 e^3 (a e+2 b d)+b^4 e^4-2 c^4 d^4\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}+5 c e^4 (2 c d-b e) \log (a+x (b+c x))+4 c^2 e^5 x}{2 c^4} \]

Antiderivative was successfully verified.

[In]  Integrate[((b + 2*c*x)*(d + e*x)^5)/(a + b*x + c*x^2)^3,x]

[Out]

(4*c^2*e^5*x - (b^3*e^5*(a + b*x) + c^4*d^4*(d + 5*e*x) - 10*c^3*d^2*e^2*(b*d*x
+ a*(d + e*x)) + c^2*e^3*(10*b^2*d^2*x + 10*a*b*d*(d + e*x) + a^2*e*(5*d + e*x))
- b*c*e^4*(2*a^2*e + 5*b^2*d*x + a*b*(5*d + 3*e*x)))/(a + x*(b + c*x))^2 + (e*(
b^5*e^4 - b^4*c*e^3*(5*d + 8*e*x) + b^3*c*e^2*(-13*a*e^2 + 10*c*d*(d + 3*e*x)) -
2*c^3*(5*c^2*d^4*x - 10*a*c*d^2*e*(4*d + 5*e*x) + a^2*e^3*(40*d + 9*e*x)) + b*c
^2*(31*a^2*e^4 - 5*c^2*d^3*(d - 4*e*x) - 10*a*c*d*e^2*(7*d + 10*e*x)) + 2*b^2*c^
2*e*(-5*c*d^2*(d + 4*e*x) + a*e^2*(25*d + 17*e*x))))/((b^2 - 4*a*c)*(a + x*(b +
c*x))) - (10*c*e*(-2*c^4*d^4 + b^4*e^4 + 4*c^3*d^2*e*(b*d - 3*a*e) + 6*a*c^2*e^3
*(2*b*d + a*e) - 2*b^2*c*e^3*(b*d + 3*a*e))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c
]])/(-b^2 + 4*a*c)^(3/2) + 5*c*e^4*(2*c*d - b*e)*Log[a + x*(b + c*x)])/(2*c^4)

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Maple [B]  time = 0.027, size = 2485, normalized size = 8.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*x+b)*(e*x+d)^5/(c*x^2+b*x+a)^3,x)

[Out]

15/c/(c*x^2+b*x+a)^2/(4*a*c-b^2)*a^2*b*d^2*e^3-17/c/(c*x^2+b*x+a)^2*e^5/(4*a*c-b
^2)*x^3*a*b^2-40*c/(c*x^2+b*x+a)^2*e^2/(4*a*c-b^2)*x^2*a*d^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+12*a*b^4*c-b^6)^(1/2))*b*e^4*a*d-21/2/c^2/(c*x^2+b*x+a)^2
*e^5/(4*a*c-b^2)*x^2*a*b^3-25/2/c^2/(c*x^2+b*x+a)^2*e^4/(4*a*c-b^2)*x^2*b^4*d+7/
c^3/(c*x^2+b*x+a)^2*e^5/(4*a*c-b^2)*x*a*b^4+50/(c*x^2+b*x+a)^2*e^4/(4*a*c-b^2)*x
^3*a*b*d+60/(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*e^3*d^
2-20/(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*d^3*e^2-30/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*e^5+5/2/c^3*e^5/
(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^2+b*x+a))*b^3+9/(c*x^2+b*x+a)^2*e^5/(4*a*c-b^2)*
x^3*a^2-20/(c*x^2+b*x+a)^2/(4*a*c-b^2)*a^2*d^3*e^2-2*c/(c*x^2+b*x+a)^2/(4*a*c-b^
2)*a*d^5+2*e^5/c^2*x-30/(c*x^2+b*x+a)^2*e^2/(4*a*c-b^2)*x*a*b*d^3-25/2/c^2/(c*x^
2+b*x+a)^2/(4*a*c-b^2)*a^2*b^2*d*e^4-10*c/(c*x^2+b*x+a)^2*e^2/(4*a*c-b^2)*x^3*b*
d^3-13/2/c/(c*x^2+b*x+a)^2*e^5/(4*a*c-b^2)*x^2*a^2*b-15/(c*x^2+b*x+a)^2*e^3/(4*a
*c-b^2)*x^2*a*b*d^2-5*c/(c*x^2+b*x+a)^2*e/(4*a*c-b^2)*x*a*d^4-50*c/(c*x^2+b*x+a)
^2*e^3/(4*a*c-b^2)*x^3*a*d^2-15/c/(c*x^2+b*x+a)^2*e^4/(4*a*c-b^2)*x^3*b^3*d-26/c
^2/(c*x^2+b*x+a)^2*e^5/(4*a*c-b^2)*x*a^2*b^2+15/c/(c*x^2+b*x+a)^2*e^3/(4*a*c-b^2
)*x^2*b^3*d^2+15/2*c/(c*x^2+b*x+a)^2*e/(4*a*c-b^2)*x^2*b*d^4+25/c/(c*x^2+b*x+a)^
2*e^4/(4*a*c-b^2)*x^2*a*b^2*d+10*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))*d^4*e-5/c^3/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*arct
an((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^4*e^5+1/2/(c*x^2+b*x+a)^2/(4*a*c-b^2)*b^2*d^5+30/c/(c*x^2+b*x+a)^2*e^3/
(4*a*c-b^2)*x*a*b^2*d^2-25/c^2/(c*x^2+b*x+a)^2*e^4/(4*a*c-b^2)*x*a*b^3*d+70/c/(c
*x^2+b*x+a)^2*e^4/(4*a*c-b^2)*x*a^2*b*d+5/2/(c*x^2+b*x+a)^2/(4*a*c-b^2)*a*b*d^4*
e-30/(c*x^2+b*x+a)^2*e^3/(4*a*c-b^2)*x*a^2*d^2+40/(c*x^2+b*x+a)^2*e^4/(4*a*c-b^2
)*x^2*a^2*d-10/c^2*e^5/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^2+b*x+a))*a*b+20/c*e^4/(4
*a*c-b^2)*ln((4*a*c-b^2)*(c*x^2+b*x+a))*a*d-5/c^2*e^4/(4*a*c-b^2)*ln((4*a*c-b^2)
*(c*x^2+b*x+a))*b^2*d+10/c^2/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*ar
ctan((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*e^4*d-23/2/c^2/(c*x^2+b*x+a)^2/(4*a*c-b^2)*a^3*b*e^5+4/c^2/(c*x^2+b
*x+a)^2*e^5/(4*a*c-b^2)*x^3*b^4+5*c^2/(c*x^2+b*x+a)^2*e/(4*a*c-b^2)*x^3*d^4+7/2/
c^3/(c*x^2+b*x+a)^2*e^5/(4*a*c-b^2)*x^2*b^5+20/(c*x^2+b*x+a)^2*e^3/(4*a*c-b^2)*x
^3*b^2*d^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*
b^2*e^5+7/c/(c*x^2+b*x+a)^2*e^5/(4*a*c-b^2)*x*a^3+30/c/(c*x^2+b*x+a)^2/(4*a*c-b^
2)*a^3*d*e^4+7/2/c^3/(c*x^2+b*x+a)^2/(4*a*c-b^2)*a^2*b^3*e^5-5/(c*x^2+b*x+a)^2*e
^2/(4*a*c-b^2)*x^2*b^2*d^3+5/(c*x^2+b*x+a)^2*e/(4*a*c-b^2)*x*b^2*d^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((2*c*x + b)*(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 = 0.374622, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x + b)*(e*x + d)^5/(c*x^2 + b*x + a)^3,x, algorithm="fricas")

[Out]

[1/2*(5*(2*a^2*c^4*d^4*e - 4*a^2*b*c^3*d^3*e^2 + 12*a^3*c^3*d^2*e^3 + 2*(a^2*b^3
*c - 6*a^3*b*c^2)*d*e^4 - (a^2*b^4 - 6*a^3*b^2*c + 6*a^4*c^2)*e^5 + (2*c^6*d^4*e
- 4*b*c^5*d^3*e^2 + 12*a*c^5*d^2*e^3 + 2*(b^3*c^3 - 6*a*b*c^4)*d*e^4 - (b^4*c^2
- 6*a*b^2*c^3 + 6*a^2*c^4)*e^5)*x^4 + 2*(2*b*c^5*d^4*e - 4*b^2*c^4*d^3*e^2 + 12
*a*b*c^4*d^2*e^3 + 2*(b^4*c^2 - 6*a*b^2*c^3)*d*e^4 - (b^5*c - 6*a*b^3*c^2 + 6*a^
2*b*c^3)*e^5)*x^3 + (2*(b^2*c^4 + 2*a*c^5)*d^4*e - 4*(b^3*c^3 + 2*a*b*c^4)*d^3*e
^2 + 12*(a*b^2*c^3 + 2*a^2*c^4)*d^2*e^3 + 2*(b^5*c - 4*a*b^3*c^2 - 12*a^2*b*c^3)
*d*e^4 - (b^6 - 4*a*b^4*c - 6*a^2*b^2*c^2 + 12*a^3*c^3)*e^5)*x^2 + 2*(2*a*b*c^4*
d^4*e - 4*a*b^2*c^3*d^3*e^2 + 12*a^2*b*c^3*d^2*e^3 + 2*(a*b^4*c - 6*a^2*b^2*c^2)
*d*e^4 - (a*b^5 - 6*a^2*b^3*c + 6*a^3*b*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)) + (4*(b^2*c^3 - 4*a*c^4)*e^5*x^5 - 5*a*b*c^3*d^4*e + 40*a^2*c^3*d^3*
e^2 - 30*a^2*b*c^2*d^2*e^3 + 8*(b^3*c^2 - 4*a*b*c^3)*e^5*x^4 - (b^2*c^3 - 4*a*c^
4)*d^5 + 5*(5*a^2*b^2*c - 12*a^3*c^2)*d*e^4 - (7*a^2*b^3 - 23*a^3*b*c)*e^5 - 2*(
5*c^5*d^4*e - 10*b*c^4*d^3*e^2 + 10*(2*b^2*c^3 - 5*a*c^4)*d^2*e^3 - 5*(3*b^3*c^2
- 10*a*b*c^3)*d*e^4 + (2*b^4*c - 13*a*b^2*c^2 + 25*a^2*c^3)*e^5)*x^3 - (15*b*c^
4*d^4*e - 10*(b^2*c^3 + 8*a*c^4)*d^3*e^2 + 30*(b^3*c^2 - a*b*c^3)*d^2*e^3 - 5*(5
*b^4*c - 10*a*b^2*c^2 - 16*a^2*c^3)*d*e^4 + (7*b^5 - 29*a*b^3*c + 19*a^2*b*c^2)*
e^5)*x^2 + 2*(30*a*b*c^3*d^3*e^2 - 5*(b^2*c^3 - a*c^4)*d^4*e - 30*(a*b^2*c^2 - a
^2*c^3)*d^2*e^3 + 5*(5*a*b^3*c - 14*a^2*b*c^2)*d*e^4 - (7*a*b^4 - 28*a^2*b^2*c +
15*a^3*c^2)*e^5)*x + 5*(2*(a^2*b^2*c - 4*a^3*c^2)*d*e^4 - (a^2*b^3 - 4*a^3*b*c)
*e^5 + (2*(b^2*c^3 - 4*a*c^4)*d*e^4 - (b^3*c^2 - 4*a*b*c^3)*e^5)*x^4 + 2*(2*(b^3
*c^2 - 4*a*b*c^3)*d*e^4 - (b^4*c - 4*a*b^2*c^2)*e^5)*x^3 + (2*(b^4*c - 2*a*b^2*c
^2 - 8*a^2*c^3)*d*e^4 - (b^5 - 2*a*b^3*c - 8*a^2*b*c^2)*e^5)*x^2 + 2*(2*(a*b^3*c
- 4*a^2*b*c^2)*d*e^4 - (a*b^4 - 4*a^2*b^2*c)*e^5)*x)*log(c*x^2 + b*x + a))*sqrt
(b^2 - 4*a*c))/((a^2*b^2*c^3 - 4*a^3*c^4 + (b^2*c^5 - 4*a*c^6)*x^4 + 2*(b^3*c^4
- 4*a*b*c^5)*x^3 + (b^4*c^3 - 2*a*b^2*c^4 - 8*a^2*c^5)*x^2 + 2*(a*b^3*c^3 - 4*a^
2*b*c^4)*x)*sqrt(b^2 - 4*a*c)), -1/2*(10*(2*a^2*c^4*d^4*e - 4*a^2*b*c^3*d^3*e^2
+ 12*a^3*c^3*d^2*e^3 + 2*(a^2*b^3*c - 6*a^3*b*c^2)*d*e^4 - (a^2*b^4 - 6*a^3*b^2*
c + 6*a^4*c^2)*e^5 + (2*c^6*d^4*e - 4*b*c^5*d^3*e^2 + 12*a*c^5*d^2*e^3 + 2*(b^3*
c^3 - 6*a*b*c^4)*d*e^4 - (b^4*c^2 - 6*a*b^2*c^3 + 6*a^2*c^4)*e^5)*x^4 + 2*(2*b*c
^5*d^4*e - 4*b^2*c^4*d^3*e^2 + 12*a*b*c^4*d^2*e^3 + 2*(b^4*c^2 - 6*a*b^2*c^3)*d*
e^4 - (b^5*c - 6*a*b^3*c^2 + 6*a^2*b*c^3)*e^5)*x^3 + (2*(b^2*c^4 + 2*a*c^5)*d^4*
e - 4*(b^3*c^3 + 2*a*b*c^4)*d^3*e^2 + 12*(a*b^2*c^3 + 2*a^2*c^4)*d^2*e^3 + 2*(b^
5*c - 4*a*b^3*c^2 - 12*a^2*b*c^3)*d*e^4 - (b^6 - 4*a*b^4*c - 6*a^2*b^2*c^2 + 12*
a^3*c^3)*e^5)*x^2 + 2*(2*a*b*c^4*d^4*e - 4*a*b^2*c^3*d^3*e^2 + 12*a^2*b*c^3*d^2*
e^3 + 2*(a*b^4*c - 6*a^2*b^2*c^2)*d*e^4 - (a*b^5 - 6*a^2*b^3*c + 6*a^3*b*c^2)*e^
5)*x)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) - (4*(b^2*c^3 - 4*a*
c^4)*e^5*x^5 - 5*a*b*c^3*d^4*e + 40*a^2*c^3*d^3*e^2 - 30*a^2*b*c^2*d^2*e^3 + 8*(
b^3*c^2 - 4*a*b*c^3)*e^5*x^4 - (b^2*c^3 - 4*a*c^4)*d^5 + 5*(5*a^2*b^2*c - 12*a^3
*c^2)*d*e^4 - (7*a^2*b^3 - 23*a^3*b*c)*e^5 - 2*(5*c^5*d^4*e - 10*b*c^4*d^3*e^2 +
10*(2*b^2*c^3 - 5*a*c^4)*d^2*e^3 - 5*(3*b^3*c^2 - 10*a*b*c^3)*d*e^4 + (2*b^4*c
- 13*a*b^2*c^2 + 25*a^2*c^3)*e^5)*x^3 - (15*b*c^4*d^4*e - 10*(b^2*c^3 + 8*a*c^4)
*d^3*e^2 + 30*(b^3*c^2 - a*b*c^3)*d^2*e^3 - 5*(5*b^4*c - 10*a*b^2*c^2 - 16*a^2*c
^3)*d*e^4 + (7*b^5 - 29*a*b^3*c + 19*a^2*b*c^2)*e^5)*x^2 + 2*(30*a*b*c^3*d^3*e^2
- 5*(b^2*c^3 - a*c^4)*d^4*e - 30*(a*b^2*c^2 - a^2*c^3)*d^2*e^3 + 5*(5*a*b^3*c -
14*a^2*b*c^2)*d*e^4 - (7*a*b^4 - 28*a^2*b^2*c + 15*a^3*c^2)*e^5)*x + 5*(2*(a^2*
b^2*c - 4*a^3*c^2)*d*e^4 - (a^2*b^3 - 4*a^3*b*c)*e^5 + (2*(b^2*c^3 - 4*a*c^4)*d*
e^4 - (b^3*c^2 - 4*a*b*c^3)*e^5)*x^4 + 2*(2*(b^3*c^2 - 4*a*b*c^3)*d*e^4 - (b^4*c
- 4*a*b^2*c^2)*e^5)*x^3 + (2*(b^4*c - 2*a*b^2*c^2 - 8*a^2*c^3)*d*e^4 - (b^5 - 2
*a*b^3*c - 8*a^2*b*c^2)*e^5)*x^2 + 2*(2*(a*b^3*c - 4*a^2*b*c^2)*d*e^4 - (a*b^4 -
4*a^2*b^2*c)*e^5)*x)*log(c*x^2 + b*x + a))*sqrt(-b^2 + 4*a*c))/((a^2*b^2*c^3 -
4*a^3*c^4 + (b^2*c^5 - 4*a*c^6)*x^4 + 2*(b^3*c^4 - 4*a*b*c^5)*x^3 + (b^4*c^3 - 2
*a*b^2*c^4 - 8*a^2*c^5)*x^2 + 2*(a*b^3*c^3 - 4*a^2*b*c^4)*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((2*c*x+b)*(e*x+d)**5/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.283432, size = 859, normalized size = 2.93 \[ -\frac{5 \,{\left (2 \, c^{4} d^{4} e - 4 \, b c^{3} d^{3} e^{2} + 12 \, a c^{3} d^{2} e^{3} + 2 \, b^{3} c d e^{4} - 12 \, a b c^{2} d e^{4} - b^{4} e^{5} + 6 \, a b^{2} c e^{5} - 6 \, a^{2} c^{2} e^{5}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{2 \, x e^{5}}{c^{2}} + \frac{5 \,{\left (2 \, c d e^{4} - b e^{5}\right )}{\rm ln}\left (c x^{2} + b x + a\right )}{2 \, c^{3}} - \frac{b^{2} c^{3} d^{5} - 4 \, a c^{4} d^{5} + 5 \, a b c^{3} d^{4} e - 40 \, a^{2} c^{3} d^{3} e^{2} + 30 \, a^{2} b c^{2} d^{2} e^{3} - 25 \, a^{2} b^{2} c d e^{4} + 60 \, a^{3} c^{2} d e^{4} + 7 \, a^{2} b^{3} e^{5} - 23 \, a^{3} b c e^{5} + 2 \,{\left (5 \, c^{5} d^{4} e - 10 \, b c^{4} d^{3} e^{2} + 20 \, b^{2} c^{3} d^{2} e^{3} - 50 \, a c^{4} d^{2} e^{3} - 15 \, b^{3} c^{2} d e^{4} + 50 \, a b c^{3} d e^{4} + 4 \, b^{4} c e^{5} - 17 \, a b^{2} c^{2} e^{5} + 9 \, a^{2} c^{3} e^{5}\right )} x^{3} +{\left (15 \, b c^{4} d^{4} e - 10 \, b^{2} c^{3} d^{3} e^{2} - 80 \, a c^{4} d^{3} e^{2} + 30 \, b^{3} c^{2} d^{2} e^{3} - 30 \, a b c^{3} d^{2} e^{3} - 25 \, b^{4} c d e^{4} + 50 \, a b^{2} c^{2} d e^{4} + 80 \, a^{2} c^{3} d e^{4} + 7 \, b^{5} e^{5} - 21 \, a b^{3} c e^{5} - 13 \, a^{2} b c^{2} e^{5}\right )} x^{2} + 2 \,{\left (5 \, b^{2} c^{3} d^{4} e - 5 \, a c^{4} d^{4} e - 30 \, a b c^{3} d^{3} e^{2} + 30 \, a b^{2} c^{2} d^{2} e^{3} - 30 \, a^{2} c^{3} d^{2} e^{3} - 25 \, a b^{3} c d e^{4} + 70 \, a^{2} b c^{2} d e^{4} + 7 \, a b^{4} e^{5} - 26 \, a^{2} b^{2} c e^{5} + 7 \, a^{3} c^{2} e^{5}\right )} x}{2 \,{\left (c x^{2} + b x + a\right )}^{2}{\left (b^{2} - 4 \, a c\right )} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x + b)*(e*x + d)^5/(c*x^2 + b*x + a)^3,x, algorithm="giac")

[Out]

-5*(2*c^4*d^4*e - 4*b*c^3*d^3*e^2 + 12*a*c^3*d^2*e^3 + 2*b^3*c*d*e^4 - 12*a*b*c^
2*d*e^4 - b^4*e^5 + 6*a*b^2*c*e^5 - 6*a^2*c^2*e^5)*arctan((2*c*x + b)/sqrt(-b^2
+ 4*a*c))/((b^2*c^3 - 4*a*c^4)*sqrt(-b^2 + 4*a*c)) + 2*x*e^5/c^2 + 5/2*(2*c*d*e^
4 - b*e^5)*ln(c*x^2 + b*x + a)/c^3 - 1/2*(b^2*c^3*d^5 - 4*a*c^4*d^5 + 5*a*b*c^3*
d^4*e - 40*a^2*c^3*d^3*e^2 + 30*a^2*b*c^2*d^2*e^3 - 25*a^2*b^2*c*d*e^4 + 60*a^3*
c^2*d*e^4 + 7*a^2*b^3*e^5 - 23*a^3*b*c*e^5 + 2*(5*c^5*d^4*e - 10*b*c^4*d^3*e^2 +
20*b^2*c^3*d^2*e^3 - 50*a*c^4*d^2*e^3 - 15*b^3*c^2*d*e^4 + 50*a*b*c^3*d*e^4 + 4
*b^4*c*e^5 - 17*a*b^2*c^2*e^5 + 9*a^2*c^3*e^5)*x^3 + (15*b*c^4*d^4*e - 10*b^2*c^
3*d^3*e^2 - 80*a*c^4*d^3*e^2 + 30*b^3*c^2*d^2*e^3 - 30*a*b*c^3*d^2*e^3 - 25*b^4*
c*d*e^4 + 50*a*b^2*c^2*d*e^4 + 80*a^2*c^3*d*e^4 + 7*b^5*e^5 - 21*a*b^3*c*e^5 - 1
3*a^2*b*c^2*e^5)*x^2 + 2*(5*b^2*c^3*d^4*e - 5*a*c^4*d^4*e - 30*a*b*c^3*d^3*e^2 +
30*a*b^2*c^2*d^2*e^3 - 30*a^2*c^3*d^2*e^3 - 25*a*b^3*c*d*e^4 + 70*a^2*b*c^2*d*e
^4 + 7*a*b^4*e^5 - 26*a^2*b^2*c*e^5 + 7*a^3*c^2*e^5)*x)/((c*x^2 + b*x + a)^2*(b^
2 - 4*a*c)*c^3)