3.2177 \(\int \frac{1}{(d+e x)^2 \left (a+b x+c x^2\right )} \, dx\)

Optimal. Leaf size=186 \[ -\frac{\left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c} \left (a e^2-b d e+c d^2\right )^2}-\frac{e (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^2}-\frac{e}{(d+e x) \left (a e^2-b d e+c d^2\right )}+\frac{e (2 c d-b e) \log (d+e x)}{\left (a e^2-b d e+c d^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.645475, antiderivative size = 186, 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 \[ -\frac{\left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c} \left (a e^2-b d e+c d^2\right )^2}-\frac{e (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^2}-\frac{e}{(d+e x) \left (a e^2-b d e+c d^2\right )}+\frac{e (2 c d-b e) \log (d+e x)}{\left (a e^2-b d e+c d^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 125.244, size = 177, normalized size = 0.95 \[ - \frac{e \left (b e - 2 c d\right ) \log{\left (d + e x \right )}}{\left (a e^{2} - b d e + c d^{2}\right )^{2}} + \frac{e \left (b e - 2 c d\right ) \log{\left (a + b x + c x^{2} \right )}}{2 \left (a e^{2} - b d e + c d^{2}\right )^{2}} - \frac{e}{\left (d + e x\right ) \left (a e^{2} - b d e + c d^{2}\right )} - \frac{\left (- 2 a c e^{2} + b^{2} e^{2} - 2 b c d e + 2 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{\sqrt{- 4 a c + b^{2}} \left (a e^{2} - b d e + c d^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-e*(b*e - 2*c*d)*log(d + e*x)/(a*e**2 - b*d*e + c*d**2)**2 + e*(b*e - 2*c*d)*log
(a + b*x + c*x**2)/(2*(a*e**2 - b*d*e + c*d**2)**2) - e/((d + e*x)*(a*e**2 - b*d
*e + c*d**2)) - (-2*a*c*e**2 + b**2*e**2 - 2*b*c*d*e + 2*c**2*d**2)*atanh((b + 2
*c*x)/sqrt(-4*a*c + b**2))/(sqrt(-4*a*c + b**2)*(a*e**2 - b*d*e + c*d**2)**2)

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Mathematica [A]  time = 0.412365, size = 151, normalized size = 0.81 \[ \frac{\frac{2 \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}-\frac{2 e \left (e (a e-b d)+c d^2\right )}{d+e x}+e (b e-2 c d) \log (a+x (b+c x))-2 e (b e-2 c d) \log (d+e x)}{2 \left (e (a e-b d)+c d^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.012, size = 386, normalized size = 2.1 \[ -{\frac{e}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) \left ( ex+d \right ) }}-{\frac{{e}^{2}\ln \left ( ex+d \right ) b}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) ^{2}}}+2\,{\frac{e\ln \left ( ex+d \right ) cd}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) ^{2}}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ) b{e}^{2}}{2\, \left ( a{e}^{2}-bde+c{d}^{2} \right ) ^{2}}}-{\frac{c\ln \left ( c{x}^{2}+bx+a \right ) de}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) ^{2}}}-2\,{\frac{ac{e}^{2}}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) ^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{{b}^{2}{e}^{2}}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) ^{2}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-2\,{\frac{bcde}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) ^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+2\,{\frac{{c}^{2}{d}^{2}}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) ^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-e/(a*e^2-b*d*e+c*d^2)/(e*x+d)-e^2/(a*e^2-b*d*e+c*d^2)^2*ln(e*x+d)*b+2*e/(a*e^2-
b*d*e+c*d^2)^2*ln(e*x+d)*c*d+1/2/(a*e^2-b*d*e+c*d^2)^2*ln(c*x^2+b*x+a)*b*e^2-1/(
a*e^2-b*d*e+c*d^2)^2*c*ln(c*x^2+b*x+a)*d*e-2/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)^(
1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*c*e^2+1/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-
b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^2*e^2-2/(a*e^2-b*d*e+c*d^2)^2/(
4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*c*d*e+2/(a*e^2-b*d*e+c*d^
2)^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*c^2*d^2

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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)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.88816, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (2 \, c^{2} d^{3} - 2 \, b c d^{2} e +{\left (b^{2} - 2 \, a c\right )} d e^{2} +{\left (2 \, c^{2} d^{2} e - 2 \, b c d e^{2} +{\left (b^{2} - 2 \, a c\right )} e^{3}\right )} x\right )} \log \left (\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x +{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right ) +{\left (2 \, c d^{2} e - 2 \, b d e^{2} + 2 \, a e^{3} +{\left (2 \, c d^{2} e - b d e^{2} +{\left (2 \, c d e^{2} - b e^{3}\right )} x\right )} \log \left (c x^{2} + b x + a\right ) - 2 \,{\left (2 \, c d^{2} e - b d e^{2} +{\left (2 \, c d e^{2} - b e^{3}\right )} x\right )} \log \left (e x + d\right )\right )} \sqrt{b^{2} - 4 \, a c}}{2 \,{\left (c^{2} d^{5} - 2 \, b c d^{4} e - 2 \, a b d^{2} e^{3} + a^{2} d e^{4} +{\left (b^{2} + 2 \, a c\right )} d^{3} e^{2} +{\left (c^{2} d^{4} e - 2 \, b c d^{3} e^{2} - 2 \, a b d e^{4} + a^{2} e^{5} +{\left (b^{2} + 2 \, a c\right )} d^{2} e^{3}\right )} x\right )} \sqrt{b^{2} - 4 \, a c}}, \frac{2 \,{\left (2 \, c^{2} d^{3} - 2 \, b c d^{2} e +{\left (b^{2} - 2 \, a c\right )} d e^{2} +{\left (2 \, c^{2} d^{2} e - 2 \, b c d e^{2} +{\left (b^{2} - 2 \, a c\right )} e^{3}\right )} x\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) -{\left (2 \, c d^{2} e - 2 \, b d e^{2} + 2 \, a e^{3} +{\left (2 \, c d^{2} e - b d e^{2} +{\left (2 \, c d e^{2} - b e^{3}\right )} x\right )} \log \left (c x^{2} + b x + a\right ) - 2 \,{\left (2 \, c d^{2} e - b d e^{2} +{\left (2 \, c d e^{2} - b e^{3}\right )} x\right )} \log \left (e x + d\right )\right )} \sqrt{-b^{2} + 4 \, a c}}{2 \,{\left (c^{2} d^{5} - 2 \, b c d^{4} e - 2 \, a b d^{2} e^{3} + a^{2} d e^{4} +{\left (b^{2} + 2 \, a c\right )} d^{3} e^{2} +{\left (c^{2} d^{4} e - 2 \, b c d^{3} e^{2} - 2 \, a b d e^{4} + a^{2} e^{5} +{\left (b^{2} + 2 \, a c\right )} d^{2} e^{3}\right )} x\right )} \sqrt{-b^{2} + 4 \, a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.208739, size = 447, normalized size = 2.4 \[ -\frac{{\left (2 \, c^{2} d^{2} e^{2} - 2 \, b c d e^{3} + b^{2} e^{4} - 2 \, a c e^{4}\right )} \arctan \left (-\frac{{\left (2 \, c d - \frac{2 \, c d^{2}}{x e + d} - b e + \frac{2 \, b d e}{x e + d} - \frac{2 \, a e^{2}}{x e + d}\right )} e^{\left (-1\right )}}{\sqrt{-b^{2} + 4 \, a c}}\right ) e^{\left (-2\right )}}{{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{{\left (2 \, c d e - b e^{2}\right )}{\rm ln}\left (-c + \frac{2 \, c d}{x e + d} - \frac{c d^{2}}{{\left (x e + d\right )}^{2}} - \frac{b e}{x e + d} + \frac{b d e}{{\left (x e + d\right )}^{2}} - \frac{a e^{2}}{{\left (x e + d\right )}^{2}}\right )}{2 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )}} - \frac{e^{3}}{{\left (c d^{2} e^{2} - b d e^{3} + a e^{4}\right )}{\left (x e + d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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