3.1545 \(\int \frac{b+2 c x}{(d+e x) \left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=397 \[ -\frac{\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}-\frac{e \left (2 b^2 c e^3 (3 a e+b d)-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (a e+2 b d)-b^4 e^4+2 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} \left (a e^2-b d e+c d^2\right )^3}-\frac{e \left (-2 c x \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )-b c \left (c d^2-7 a e^2\right )-8 a c^2 d e-2 b^3 e^2+3 b^2 c d e\right )}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )^2}+\frac{e^4 (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^3}-\frac{e^4 (2 c d-b e) \log (d+e x)}{\left (a e^2-b d e+c d^2\right )^3} \]

[Out]

-((b^2 - 4*a*c)*(c*d - b*e) - c*(b^2 - 4*a*c)*e*x)/(2*(b^2 - 4*a*c)*(c*d^2 - b*d
*e + a*e^2)*(a + b*x + c*x^2)^2) - (e*(3*b^2*c*d*e - 8*a*c^2*d*e - 2*b^3*e^2 - b
*c*(c*d^2 - 7*a*e^2) - 2*c*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*x))/(2*(b^2 -
4*a*c)*(c*d^2 - b*d*e + a*e^2)^2*(a + b*x + c*x^2)) - (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]])/((b^2 - 4*a*c)^(3/2)*(c*d^2 - b*d*e
+ a*e^2)^3) - (e^4*(2*c*d - b*e)*Log[d + e*x])/(c*d^2 - b*d*e + a*e^2)^3 + (e^4*
(2*c*d - b*e)*Log[a + b*x + c*x^2])/(2*(c*d^2 - b*d*e + a*e^2)^3)

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Rubi [A]  time = 1.73046, antiderivative size = 397, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{e \left (2 b^2 c e^3 (3 a e+b d)-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (a e+2 b d)-b^4 e^4+2 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} \left (a e^2-b d e+c d^2\right )^3}-\frac{e \left (-2 c x \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )-b c \left (c d^2-7 a e^2\right )-8 a c^2 d e-2 b^3 e^2+3 b^2 c d e\right )}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )^2}-\frac{-b e+c d-c e x}{2 \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}+\frac{e^4 (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^3}-\frac{e^4 (2 c d-b e) \log (d+e x)}{\left (a e^2-b d e+c d^2\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

-(c*d - b*e - c*e*x)/(2*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)^2) - (e*(3*b^2
*c*d*e - 8*a*c^2*d*e - 2*b^3*e^2 - b*c*(c*d^2 - 7*a*e^2) - 2*c*(c^2*d^2 + b^2*e^
2 - c*e*(b*d + 3*a*e))*x))/(2*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^2*(a + b*x +
c*x^2)) - (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]])/
((b^2 - 4*a*c)^(3/2)*(c*d^2 - b*d*e + a*e^2)^3) - (e^4*(2*c*d - b*e)*Log[d + e*x
])/(c*d^2 - b*d*e + a*e^2)^3 + (e^4*(2*c*d - b*e)*Log[a + b*x + c*x^2])/(2*(c*d^
2 - b*d*e + a*e^2)^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)/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 2.35302, size = 356, normalized size = 0.9 \[ \frac{1}{2} \left (-\frac{2 e \left (-2 b^2 c e^3 (3 a e+b d)+4 c^3 d^2 e (b d-3 a e)+6 a c^2 e^3 (a e+2 b d)+b^4 e^4-2 c^4 d^4\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2} \left (e (b d-a e)-c d^2\right )^3}+\frac{e \left (b c \left (c d (d-2 e x)-7 a e^2\right )+2 c^2 \left (a e (4 d-3 e x)+c d^2 x\right )+2 b^3 e^2+b^2 c e (2 e x-3 d)\right )}{\left (b^2-4 a c\right ) (a+x (b+c x)) \left (e (a e-b d)+c d^2\right )^2}+\frac{2 e^4 (b e-2 c d) \log (d+e x)}{\left (e (a e-b d)+c d^2\right )^3}+\frac{e^4 (2 c d-b e) \log (a+x (b+c x))}{\left (e (a e-b d)+c d^2\right )^3}+\frac{b e-c d+c e x}{(a+x (b+c x))^2 \left (e (a e-b d)+c d^2\right )}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.038, size = 3797, normalized size = 9.6 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(4*a*c-b^2)*a*b^2*c^2*d^3*e^2-12/(a*e^2
-b*d*e+c*d^2)^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*c^
3*d^2*e^3-1/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^5*e/(4*a*c-b^2)*x^3*d^4+5/(a
*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*e^5/(4*a*c-b^2)*x*a^3*c^2+11/2/(a*e^2-b*d*e+
c*d^2)^3/(c*x^2+b*x+a)^2/(4*a*c-b^2)*a^3*b*c*e^5-6/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+
b*x+a)^2/(4*a*c-b^2)*a^3*c^2*d*e^4-8/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(4*a*
c-b^2)*a^2*c^3*d^3*e^2+2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(4*a*c-b^2)*a*b^4
*d*e^4+3/2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(4*a*c-b^2)*b^4*c*d^3*e^2-3/2/(
a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(4*a*c-b^2)*b^3*c^2*d^4*e+3/(a*e^2-b*d*e+c*
d^2)^3/(c*x^2+b*x+a)^2*c^3*e^5/(4*a*c-b^2)*x^3*a^2-1/(a*e^2-b*d*e+c*d^2)^3/(c*x^
2+b*x+a)^2*e^5/(4*a*c-b^2)*x*a*b^4+1/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*e^4/(
4*a*c-b^2)*x*b^5*d-2/(a*e^2-b*d*e+c*d^2)^3*e^5*c/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x
^2+b*x+a))*a*b+4/(a*e^2-b*d*e+c*d^2)^3*e^4*c^2/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^2
+b*x+a))*a*d-1/(a*e^2-b*d*e+c*d^2)^3*e^4*c/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^2+b*x
+a))*b^2*d-2/(a*e^2-b*d*e+c*d^2)^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))*b^3*c*d*e^4+4/(a*e^2-b*d*e+c*d^2)^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))*b*c^3*d^3*e^2-6/(a*e^2-b*d*e+c*d^2)^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^2*c*e^5+3/(a*e^2-b*d*e+c*d^
2)^3/(c*x^2+b*x+a)^2*e^2/(4*a*c-b^2)*x*b^3*c^2*d^3-1/(a*e^2-b*d*e+c*d^2)^3/(c*x^
2+b*x+a)^2*e/(4*a*c-b^2)*x*b^2*c^3*d^4-11/2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^
2/(4*a*c-b^2)*a^2*b^2*c*d*e^4+15/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(4*a*c-b^
2)*a^2*b*c^2*d^2*e^3+4/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^3*e^2/(4*a*c-b^2)
*x^2*b^2*d^3-1/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^2*e^5/(4*a*c-b^2)*x^3*a*b
^2-9/2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^2*e^3/(4*a*c-b^2)*x^2*b^3*d^2+2/(
a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^4*e^3/(4*a*c-b^2)*x^3*a*d^2-4/(a*e^2-b*d*
e+c*d^2)^3/(c*x^2+b*x+a)^2*c^4*e^2/(4*a*c-b^2)*x^2*a*d^3+2/(a*e^2-b*d*e+c*d^2)^3
/(c*x^2+b*x+a)^2*c*e^4/(4*a*c-b^2)*x^2*b^4*d+1/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+
a)^2*e/(4*a*c-b^2)*x*a*c^4*d^4-3/2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^4*e/(
4*a*c-b^2)*x^2*b*d^4+13/2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^2*e^5/(4*a*c-b
^2)*x^2*a^2*b-4/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^3*e^4/(4*a*c-b^2)*x^2*a^
2*d-2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c*e^5/(4*a*c-b^2)*x^2*a*b^3+11/2/(a*
e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(4*a*c-b^2)*a*b*c^3*d^4*e+12/(a*e^2-b*d*e+c*d
^2)^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*c^2*d*e^4-
5/2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(4*a*c-b^2)*a*b^3*c*d^2*e^3+2/(a*e^2-b
*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^4*e^2/(4*a*c-b^2)*x^3*d^3*b+2/(a*e^2-b*d*e+c*d^2
)^3/(c*x^2+b*x+a)^2*e^5/(4*a*c-b^2)*x*a^2*b^2*c+6/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b
*x+a)^2*e^3/(4*a*c-b^2)*x*a^2*c^3*d^2+1/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^
2*e^4/(4*a*c-b^2)*x^3*b^3*d-2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^3*e^3/(4*a
*c-b^2)*x^3*b^2*d^2-3/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*e^3/(4*a*c-b^2)*x*b^
4*c*d^2+e^5/(a*e^2-b*d*e+c*d^2)^3*ln(e*x+d)*b-2*e^4/(a*e^2-b*d*e+c*d^2)^3*ln(e*x
+d)*c*d-10/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*e^4/(4*a*c-b^2)*x*a^2*b*c^2*d+1
/2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(4*a*c-b^2)*b^2*c^3*d^5-3/2/(a*e^2-b*d*
e+c*d^2)^3/(c*x^2+b*x+a)^2/(4*a*c-b^2)*a^2*b^3*e^5-2/(a*e^2-b*d*e+c*d^2)^3/(c*x^
2+b*x+a)^2/(4*a*c-b^2)*a*c^4*d^5-1/2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(4*a*
c-b^2)*b^5*d^2*e^3+6/(a*e^2-b*d*e+c*d^2)^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^2*c^2*e^5-2/(a*e^2-b*d*e+c*d^2)^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))*c^4*d^4*e+1/2/(a*e^2-b*d*e+c*d^2)^3*e^5/(
4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^2+b*x+a))*b^3+1/(a*e^2-b*d*e+c*d^2)^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))*b^4*e^5-2/(a*e^2-b*d*e+c*d^2)^3
/(c*x^2+b*x+a)^2*c^3*e^4/(4*a*c-b^2)*x^3*a*b*d-4/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*
x+a)^2*c^2*e^4/(4*a*c-b^2)*x^2*a*b^2*d+9/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c
^3*e^3/(4*a*c-b^2)*x^2*a*b*d^2+6/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*e^3/(4*a*
c-b^2)*x*a*b^2*c^2*d^2-6/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*e^2/(4*a*c-b^2)*x
*a*b*c^3*d^3

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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)/((c*x^2 + b*x + a)^3*(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 93.1755, 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)/((c*x^2 + b*x + a)^3*(e*x + d)),x, algorithm="fricas")

[Out]

[1/2*((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)) - ((b^2*c^3 - 4*a*c^4)*d^5 - (3*b^3*c^2 - 11*a*b*c^3)*d^4*e + (3*b^4*
c - 6*a*b^2*c^2 - 16*a^2*c^3)*d^3*e^2 - (b^5 + 5*a*b^3*c - 30*a^2*b*c^2)*d^2*e^3
+ (4*a*b^4 - 11*a^2*b^2*c - 12*a^3*c^2)*d*e^4 - (3*a^2*b^3 - 11*a^3*b*c)*e^5 -
2*(c^5*d^4*e - 2*b*c^4*d^3*e^2 + 2*(b^2*c^3 - a*c^4)*d^2*e^3 - (b^3*c^2 - 2*a*b*
c^3)*d*e^4 + (a*b^2*c^2 - 3*a^2*c^3)*e^5)*x^3 - (3*b*c^4*d^4*e - 8*(b^2*c^3 - a*
c^4)*d^3*e^2 + 9*(b^3*c^2 - 2*a*b*c^3)*d^2*e^3 - 4*(b^4*c - 2*a*b^2*c^2 - 2*a^2*
c^3)*d*e^4 + (4*a*b^3*c - 13*a^2*b*c^2)*e^5)*x^2 - 2*((b^2*c^3 - a*c^4)*d^4*e -
3*(b^3*c^2 - 2*a*b*c^3)*d^3*e^2 + 3*(b^4*c - 2*a*b^2*c^2 - 2*a^2*c^3)*d^2*e^3 -
(b^5 - 10*a^2*b*c^2)*d*e^4 + (a*b^4 - 2*a^2*b^2*c - 5*a^3*c^2)*e^5)*x - (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) + 2*(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
(e*x + d))*sqrt(b^2 - 4*a*c))/(((a^2*b^2*c^3 - 4*a^3*c^4)*d^6 - 3*(a^2*b^3*c^2 -
4*a^3*b*c^3)*d^5*e + 3*(a^2*b^4*c - 3*a^3*b^2*c^2 - 4*a^4*c^3)*d^4*e^2 - (a^2*b
^5 + 2*a^3*b^3*c - 24*a^4*b*c^2)*d^3*e^3 + 3*(a^3*b^4 - 3*a^4*b^2*c - 4*a^5*c^2)
*d^2*e^4 - 3*(a^4*b^3 - 4*a^5*b*c)*d*e^5 + (a^5*b^2 - 4*a^6*c)*e^6 + ((b^2*c^5 -
4*a*c^6)*d^6 - 3*(b^3*c^4 - 4*a*b*c^5)*d^5*e + 3*(b^4*c^3 - 3*a*b^2*c^4 - 4*a^2
*c^5)*d^4*e^2 - (b^5*c^2 + 2*a*b^3*c^3 - 24*a^2*b*c^4)*d^3*e^3 + 3*(a*b^4*c^2 -
3*a^2*b^2*c^3 - 4*a^3*c^4)*d^2*e^4 - 3*(a^2*b^3*c^2 - 4*a^3*b*c^3)*d*e^5 + (a^3*
b^2*c^2 - 4*a^4*c^3)*e^6)*x^4 + 2*((b^3*c^4 - 4*a*b*c^5)*d^6 - 3*(b^4*c^3 - 4*a*
b^2*c^4)*d^5*e + 3*(b^5*c^2 - 3*a*b^3*c^3 - 4*a^2*b*c^4)*d^4*e^2 - (b^6*c + 2*a*
b^4*c^2 - 24*a^2*b^2*c^3)*d^3*e^3 + 3*(a*b^5*c - 3*a^2*b^3*c^2 - 4*a^3*b*c^3)*d^
2*e^4 - 3*(a^2*b^4*c - 4*a^3*b^2*c^2)*d*e^5 + (a^3*b^3*c - 4*a^4*b*c^2)*e^6)*x^3
+ ((b^4*c^3 - 2*a*b^2*c^4 - 8*a^2*c^5)*d^6 - 3*(b^5*c^2 - 2*a*b^3*c^3 - 8*a^2*b
*c^4)*d^5*e + 3*(b^6*c - a*b^4*c^2 - 10*a^2*b^2*c^3 - 8*a^3*c^4)*d^4*e^2 - (b^7
+ 4*a*b^5*c - 20*a^2*b^3*c^2 - 48*a^3*b*c^3)*d^3*e^3 + 3*(a*b^6 - a^2*b^4*c - 10
*a^3*b^2*c^2 - 8*a^4*c^3)*d^2*e^4 - 3*(a^2*b^5 - 2*a^3*b^3*c - 8*a^4*b*c^2)*d*e^
5 + (a^3*b^4 - 2*a^4*b^2*c - 8*a^5*c^2)*e^6)*x^2 + 2*((a*b^3*c^3 - 4*a^2*b*c^4)*
d^6 - 3*(a*b^4*c^2 - 4*a^2*b^2*c^3)*d^5*e + 3*(a*b^5*c - 3*a^2*b^3*c^2 - 4*a^3*b
*c^3)*d^4*e^2 - (a*b^6 + 2*a^2*b^4*c - 24*a^3*b^2*c^2)*d^3*e^3 + 3*(a^2*b^5 - 3*
a^3*b^3*c - 4*a^4*b*c^2)*d^2*e^4 - 3*(a^3*b^4 - 4*a^4*b^2*c)*d*e^5 + (a^4*b^3 -
4*a^5*b*c)*e^6)*x)*sqrt(b^2 - 4*a*c)), 1/2*(2*(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)) - ((b^2*c^3 - 4
*a*c^4)*d^5 - (3*b^3*c^2 - 11*a*b*c^3)*d^4*e + (3*b^4*c - 6*a*b^2*c^2 - 16*a^2*c
^3)*d^3*e^2 - (b^5 + 5*a*b^3*c - 30*a^2*b*c^2)*d^2*e^3 + (4*a*b^4 - 11*a^2*b^2*c
- 12*a^3*c^2)*d*e^4 - (3*a^2*b^3 - 11*a^3*b*c)*e^5 - 2*(c^5*d^4*e - 2*b*c^4*d^3
*e^2 + 2*(b^2*c^3 - a*c^4)*d^2*e^3 - (b^3*c^2 - 2*a*b*c^3)*d*e^4 + (a*b^2*c^2 -
3*a^2*c^3)*e^5)*x^3 - (3*b*c^4*d^4*e - 8*(b^2*c^3 - a*c^4)*d^3*e^2 + 9*(b^3*c^2
- 2*a*b*c^3)*d^2*e^3 - 4*(b^4*c - 2*a*b^2*c^2 - 2*a^2*c^3)*d*e^4 + (4*a*b^3*c -
13*a^2*b*c^2)*e^5)*x^2 - 2*((b^2*c^3 - a*c^4)*d^4*e - 3*(b^3*c^2 - 2*a*b*c^3)*d^
3*e^2 + 3*(b^4*c - 2*a*b^2*c^2 - 2*a^2*c^3)*d^2*e^3 - (b^5 - 10*a^2*b*c^2)*d*e^4
+ (a*b^4 - 2*a^2*b^2*c - 5*a^3*c^2)*e^5)*x - (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) + 2*(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(e*x + d))*sqrt(-b^2 + 4*a
*c))/(((a^2*b^2*c^3 - 4*a^3*c^4)*d^6 - 3*(a^2*b^3*c^2 - 4*a^3*b*c^3)*d^5*e + 3*(
a^2*b^4*c - 3*a^3*b^2*c^2 - 4*a^4*c^3)*d^4*e^2 - (a^2*b^5 + 2*a^3*b^3*c - 24*a^4
*b*c^2)*d^3*e^3 + 3*(a^3*b^4 - 3*a^4*b^2*c - 4*a^5*c^2)*d^2*e^4 - 3*(a^4*b^3 - 4
*a^5*b*c)*d*e^5 + (a^5*b^2 - 4*a^6*c)*e^6 + ((b^2*c^5 - 4*a*c^6)*d^6 - 3*(b^3*c^
4 - 4*a*b*c^5)*d^5*e + 3*(b^4*c^3 - 3*a*b^2*c^4 - 4*a^2*c^5)*d^4*e^2 - (b^5*c^2
+ 2*a*b^3*c^3 - 24*a^2*b*c^4)*d^3*e^3 + 3*(a*b^4*c^2 - 3*a^2*b^2*c^3 - 4*a^3*c^4
)*d^2*e^4 - 3*(a^2*b^3*c^2 - 4*a^3*b*c^3)*d*e^5 + (a^3*b^2*c^2 - 4*a^4*c^3)*e^6)
*x^4 + 2*((b^3*c^4 - 4*a*b*c^5)*d^6 - 3*(b^4*c^3 - 4*a*b^2*c^4)*d^5*e + 3*(b^5*c
^2 - 3*a*b^3*c^3 - 4*a^2*b*c^4)*d^4*e^2 - (b^6*c + 2*a*b^4*c^2 - 24*a^2*b^2*c^3)
*d^3*e^3 + 3*(a*b^5*c - 3*a^2*b^3*c^2 - 4*a^3*b*c^3)*d^2*e^4 - 3*(a^2*b^4*c - 4*
a^3*b^2*c^2)*d*e^5 + (a^3*b^3*c - 4*a^4*b*c^2)*e^6)*x^3 + ((b^4*c^3 - 2*a*b^2*c^
4 - 8*a^2*c^5)*d^6 - 3*(b^5*c^2 - 2*a*b^3*c^3 - 8*a^2*b*c^4)*d^5*e + 3*(b^6*c -
a*b^4*c^2 - 10*a^2*b^2*c^3 - 8*a^3*c^4)*d^4*e^2 - (b^7 + 4*a*b^5*c - 20*a^2*b^3*
c^2 - 48*a^3*b*c^3)*d^3*e^3 + 3*(a*b^6 - a^2*b^4*c - 10*a^3*b^2*c^2 - 8*a^4*c^3)
*d^2*e^4 - 3*(a^2*b^5 - 2*a^3*b^3*c - 8*a^4*b*c^2)*d*e^5 + (a^3*b^4 - 2*a^4*b^2*
c - 8*a^5*c^2)*e^6)*x^2 + 2*((a*b^3*c^3 - 4*a^2*b*c^4)*d^6 - 3*(a*b^4*c^2 - 4*a^
2*b^2*c^3)*d^5*e + 3*(a*b^5*c - 3*a^2*b^3*c^2 - 4*a^3*b*c^3)*d^4*e^2 - (a*b^6 +
2*a^2*b^4*c - 24*a^3*b^2*c^2)*d^3*e^3 + 3*(a^2*b^5 - 3*a^3*b^3*c - 4*a^4*b*c^2)*
d^2*e^4 - 3*(a^3*b^4 - 4*a^4*b^2*c)*d*e^5 + (a^4*b^3 - 4*a^5*b*c)*e^6)*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)/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.284393, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done