3.2178 \(\int \frac{1}{(d+e x)^3 \left (a+b x+c x^2\right )} \, dx\)

Optimal. Leaf size=272 \[ -\frac{e \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^3}+\frac{e \log (d+e x) \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )}{\left (a e^2-b d e+c d^2\right )^3}-\frac{(2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c} \left (a e^2-b d e+c d^2\right )^3}-\frac{e (2 c d-b e)}{(d+e x) \left (a e^2-b d e+c d^2\right )^2}-\frac{e}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )} \]

[Out]

-e/(2*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^2) - (e*(2*c*d - b*e))/((c*d^2 - b*d*e +
a*e^2)^2*(d + e*x)) - ((2*c*d - b*e)*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*Ar
cTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)
^3) + (e*(3*c^2*d^2 + b^2*e^2 - c*e*(3*b*d + a*e))*Log[d + e*x])/(c*d^2 - b*d*e
+ a*e^2)^3 - (e*(3*c^2*d^2 + b^2*e^2 - c*e*(3*b*d + a*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 = 1.01897, antiderivative size = 272, 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 \[ -\frac{e \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^3}+\frac{e \log (d+e x) \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )}{\left (a e^2-b d e+c d^2\right )^3}-\frac{(2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c} \left (a e^2-b d e+c d^2\right )^3}-\frac{e (2 c d-b e)}{(d+e x) \left (a e^2-b d e+c d^2\right )^2}-\frac{e}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^3*(a + b*x + c*x^2)),x]

[Out]

-e/(2*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^2) - (e*(2*c*d - b*e))/((c*d^2 - b*d*e +
a*e^2)^2*(d + e*x)) - ((2*c*d - b*e)*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*Ar
cTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)
^3) + (e*(3*c^2*d^2 + b^2*e^2 - c*e*(3*b*d + a*e))*Log[d + e*x])/(c*d^2 - b*d*e
+ a*e^2)^3 - (e*(3*c^2*d^2 + b^2*e^2 - c*e*(3*b*d + a*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(1/(e*x+d)**3/(c*x**2+b*x+a),x)

[Out]

Timed out

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Mathematica [A]  time = 0.668545, size = 272, normalized size = 1. \[ \frac{e \log (d+e x) \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )}{\left (e (a e-b d)+c d^2\right )^3}+\frac{e \left (c e (a e+3 b d)-b^2 e^2-3 c^2 d^2\right ) \log (a+x (b+c x))}{2 \left (e (a e-b d)+c d^2\right )^3}+\frac{(b e-2 c d) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2} \left (e (b d-a e)-c d^2\right )^3}+\frac{e (b e-2 c d)}{(d+e x) \left (e (a e-b d)+c d^2\right )^2}-\frac{e}{2 (d+e x)^2 \left (e (a e-b d)+c d^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^3*(a + b*x + c*x^2)),x]

[Out]

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

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Maple [B]  time = 0.018, size = 719, normalized size = 2.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^3/(c*x^2+b*x+a),x)

[Out]

-1/2*e/(a*e^2-b*d*e+c*d^2)/(e*x+d)^2+e^2/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)*b-2*e/(a*
e^2-b*d*e+c*d^2)^2/(e*x+d)*c*d-e^3/(a*e^2-b*d*e+c*d^2)^3*ln(e*x+d)*a*c+e^3/(a*e^
2-b*d*e+c*d^2)^3*ln(e*x+d)*b^2-3*e^2/(a*e^2-b*d*e+c*d^2)^3*ln(e*x+d)*b*c*d+3*e/(
a*e^2-b*d*e+c*d^2)^3*ln(e*x+d)*c^2*d^2+1/2/(a*e^2-b*d*e+c*d^2)^3*c*ln(c*x^2+b*x+
a)*a*e^3-1/2/(a*e^2-b*d*e+c*d^2)^3*ln(c*x^2+b*x+a)*b^2*e^3+3/2/(a*e^2-b*d*e+c*d^
2)^3*c*ln(c*x^2+b*x+a)*d*e^2*b-3/2/(a*e^2-b*d*e+c*d^2)^3*c^2*ln(c*x^2+b*x+a)*d^2
*e+3/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))
*a*b*c*e^3-6/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2
)^(1/2))*a*c^2*e^2*d-1/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/
(4*a*c-b^2)^(1/2))*b^3*e^3+3/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(1/2)*arctan((2*c
*x+b)/(4*a*c-b^2)^(1/2))*b^2*c*d*e^2-3/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(1/2)*a
rctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*c^2*d^2*e+2/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^
2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*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(1/((c*x^2 + b*x + a)*(e*x + d)^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 50.317, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x + a)*(e*x + d)^3),x, algorithm="fricas")

[Out]

[1/2*((2*c^3*d^5 - 3*b*c^2*d^4*e + 3*(b^2*c - 2*a*c^2)*d^3*e^2 - (b^3 - 3*a*b*c)
*d^2*e^3 + (2*c^3*d^3*e^2 - 3*b*c^2*d^2*e^3 + 3*(b^2*c - 2*a*c^2)*d*e^4 - (b^3 -
3*a*b*c)*e^5)*x^2 + 2*(2*c^3*d^4*e - 3*b*c^2*d^3*e^2 + 3*(b^2*c - 2*a*c^2)*d^2*
e^3 - (b^3 - 3*a*b*c)*d*e^4)*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)) - (5*c^2
*d^4*e - 8*b*c*d^3*e^2 - 4*a*b*d*e^4 + a^2*e^5 + 3*(b^2 + 2*a*c)*d^2*e^3 + 2*(2*
c^2*d^3*e^2 - 3*b*c*d^2*e^3 - a*b*e^5 + (b^2 + 2*a*c)*d*e^4)*x + (3*c^2*d^4*e -
3*b*c*d^3*e^2 + (b^2 - a*c)*d^2*e^3 + (3*c^2*d^2*e^3 - 3*b*c*d*e^4 + (b^2 - a*c)
*e^5)*x^2 + 2*(3*c^2*d^3*e^2 - 3*b*c*d^2*e^3 + (b^2 - a*c)*d*e^4)*x)*log(c*x^2 +
b*x + a) - 2*(3*c^2*d^4*e - 3*b*c*d^3*e^2 + (b^2 - a*c)*d^2*e^3 + (3*c^2*d^2*e^
3 - 3*b*c*d*e^4 + (b^2 - a*c)*e^5)*x^2 + 2*(3*c^2*d^3*e^2 - 3*b*c*d^2*e^3 + (b^2
- a*c)*d*e^4)*x)*log(e*x + d))*sqrt(b^2 - 4*a*c))/((c^3*d^8 - 3*b*c^2*d^7*e - 3
*a^2*b*d^3*e^5 + a^3*d^2*e^6 + 3*(b^2*c + a*c^2)*d^6*e^2 - (b^3 + 6*a*b*c)*d^5*e
^3 + 3*(a*b^2 + a^2*c)*d^4*e^4 + (c^3*d^6*e^2 - 3*b*c^2*d^5*e^3 - 3*a^2*b*d*e^7
+ a^3*e^8 + 3*(b^2*c + a*c^2)*d^4*e^4 - (b^3 + 6*a*b*c)*d^3*e^5 + 3*(a*b^2 + a^2
*c)*d^2*e^6)*x^2 + 2*(c^3*d^7*e - 3*b*c^2*d^6*e^2 - 3*a^2*b*d^2*e^6 + a^3*d*e^7
+ 3*(b^2*c + a*c^2)*d^5*e^3 - (b^3 + 6*a*b*c)*d^4*e^4 + 3*(a*b^2 + a^2*c)*d^3*e^
5)*x)*sqrt(b^2 - 4*a*c)), 1/2*(2*(2*c^3*d^5 - 3*b*c^2*d^4*e + 3*(b^2*c - 2*a*c^2
)*d^3*e^2 - (b^3 - 3*a*b*c)*d^2*e^3 + (2*c^3*d^3*e^2 - 3*b*c^2*d^2*e^3 + 3*(b^2*
c - 2*a*c^2)*d*e^4 - (b^3 - 3*a*b*c)*e^5)*x^2 + 2*(2*c^3*d^4*e - 3*b*c^2*d^3*e^2
+ 3*(b^2*c - 2*a*c^2)*d^2*e^3 - (b^3 - 3*a*b*c)*d*e^4)*x)*arctan(-sqrt(-b^2 + 4
*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) - (5*c^2*d^4*e - 8*b*c*d^3*e^2 - 4*a*b*d*e^4 +
a^2*e^5 + 3*(b^2 + 2*a*c)*d^2*e^3 + 2*(2*c^2*d^3*e^2 - 3*b*c*d^2*e^3 - a*b*e^5 +
(b^2 + 2*a*c)*d*e^4)*x + (3*c^2*d^4*e - 3*b*c*d^3*e^2 + (b^2 - a*c)*d^2*e^3 + (
3*c^2*d^2*e^3 - 3*b*c*d*e^4 + (b^2 - a*c)*e^5)*x^2 + 2*(3*c^2*d^3*e^2 - 3*b*c*d^
2*e^3 + (b^2 - a*c)*d*e^4)*x)*log(c*x^2 + b*x + a) - 2*(3*c^2*d^4*e - 3*b*c*d^3*
e^2 + (b^2 - a*c)*d^2*e^3 + (3*c^2*d^2*e^3 - 3*b*c*d*e^4 + (b^2 - a*c)*e^5)*x^2
+ 2*(3*c^2*d^3*e^2 - 3*b*c*d^2*e^3 + (b^2 - a*c)*d*e^4)*x)*log(e*x + d))*sqrt(-b
^2 + 4*a*c))/((c^3*d^8 - 3*b*c^2*d^7*e - 3*a^2*b*d^3*e^5 + a^3*d^2*e^6 + 3*(b^2*
c + a*c^2)*d^6*e^2 - (b^3 + 6*a*b*c)*d^5*e^3 + 3*(a*b^2 + a^2*c)*d^4*e^4 + (c^3*
d^6*e^2 - 3*b*c^2*d^5*e^3 - 3*a^2*b*d*e^7 + a^3*e^8 + 3*(b^2*c + a*c^2)*d^4*e^4
- (b^3 + 6*a*b*c)*d^3*e^5 + 3*(a*b^2 + a^2*c)*d^2*e^6)*x^2 + 2*(c^3*d^7*e - 3*b*
c^2*d^6*e^2 - 3*a^2*b*d^2*e^6 + a^3*d*e^7 + 3*(b^2*c + a*c^2)*d^5*e^3 - (b^3 + 6
*a*b*c)*d^4*e^4 + 3*(a*b^2 + a^2*c)*d^3*e^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(1/(e*x+d)**3/(c*x**2+b*x+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.208373, size = 801, normalized size = 2.94 \[ -\frac{{\left (3 \, c^{2} d^{2} e - 3 \, b c d e^{2} + b^{2} e^{3} - a c e^{3}\right )}{\rm ln}\left (c x^{2} + b x + a\right )}{2 \,{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} + 3 \, a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} - 6 \, a b c d^{3} e^{3} + 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} c d^{2} e^{4} - 3 \, a^{2} b d e^{5} + a^{3} e^{6}\right )}} + \frac{{\left (3 \, c^{2} d^{2} e^{2} - 3 \, b c d e^{3} + b^{2} e^{4} - a c e^{4}\right )}{\rm ln}\left ({\left | x e + d \right |}\right )}{c^{3} d^{6} e - 3 \, b c^{2} d^{5} e^{2} + 3 \, b^{2} c d^{4} e^{3} + 3 \, a c^{2} d^{4} e^{3} - b^{3} d^{3} e^{4} - 6 \, a b c d^{3} e^{4} + 3 \, a b^{2} d^{2} e^{5} + 3 \, a^{2} c d^{2} e^{5} - 3 \, a^{2} b d e^{6} + a^{3} e^{7}} + \frac{{\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \, b^{2} c d e^{2} - 6 \, a c^{2} d e^{2} - b^{3} e^{3} + 3 \, a b c e^{3}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} + 3 \, a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} - 6 \, a b c d^{3} e^{3} + 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} c d^{2} e^{4} - 3 \, a^{2} b d e^{5} + a^{3} e^{6}\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{5 \, c^{2} d^{4} e - 8 \, b c d^{3} e^{2} + 3 \, b^{2} d^{2} e^{3} + 6 \, a c d^{2} e^{3} - 4 \, a b d e^{4} + a^{2} e^{5} + 2 \,{\left (2 \, c^{2} d^{3} e^{2} - 3 \, b c d^{2} e^{3} + b^{2} d e^{4} + 2 \, a c d e^{4} - a b e^{5}\right )} x}{2 \,{\left (c d^{2} - b d e + a e^{2}\right )}^{3}{\left (x e + d\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x + a)*(e*x + d)^3),x, algorithm="giac")

[Out]

-1/2*(3*c^2*d^2*e - 3*b*c*d*e^2 + b^2*e^3 - a*c*e^3)*ln(c*x^2 + b*x + a)/(c^3*d^
6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 + 3*a*c^2*d^4*e^2 - b^3*d^3*e^3 - 6*a*b*c*d^
3*e^3 + 3*a*b^2*d^2*e^4 + 3*a^2*c*d^2*e^4 - 3*a^2*b*d*e^5 + a^3*e^6) + (3*c^2*d^
2*e^2 - 3*b*c*d*e^3 + b^2*e^4 - a*c*e^4)*ln(abs(x*e + d))/(c^3*d^6*e - 3*b*c^2*d
^5*e^2 + 3*b^2*c*d^4*e^3 + 3*a*c^2*d^4*e^3 - b^3*d^3*e^4 - 6*a*b*c*d^3*e^4 + 3*a
*b^2*d^2*e^5 + 3*a^2*c*d^2*e^5 - 3*a^2*b*d*e^6 + a^3*e^7) + (2*c^3*d^3 - 3*b*c^2
*d^2*e + 3*b^2*c*d*e^2 - 6*a*c^2*d*e^2 - b^3*e^3 + 3*a*b*c*e^3)*arctan((2*c*x +
b)/sqrt(-b^2 + 4*a*c))/((c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 + 3*a*c^2*d^4
*e^2 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a^2*c*d^2*e^4 - 3*a^2
*b*d*e^5 + a^3*e^6)*sqrt(-b^2 + 4*a*c)) - 1/2*(5*c^2*d^4*e - 8*b*c*d^3*e^2 + 3*b
^2*d^2*e^3 + 6*a*c*d^2*e^3 - 4*a*b*d*e^4 + a^2*e^5 + 2*(2*c^2*d^3*e^2 - 3*b*c*d^
2*e^3 + b^2*d*e^4 + 2*a*c*d*e^4 - a*b*e^5)*x)/((c*d^2 - b*d*e + a*e^2)^3*(x*e +
d)^2)