3.1530 \(\int \frac{b+2 c x}{(d+e x)^2 \left (a+b x+c x^2\right )} \, dx\)

Optimal. Leaf size=210 \[ \frac{\left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^2}-\frac{\log (d+e x) \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right )}{\left (a e^2-b d e+c d^2\right )^2}+\frac{e \sqrt{b^2-4 a c} (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (a e^2-b d e+c d^2\right )^2}+\frac{2 c d-b e}{(d+e x) \left (a e^2-b d e+c d^2\right )} \]

[Out]

(2*c*d - b*e)/((c*d^2 - b*d*e + a*e^2)*(d + e*x)) + (Sqrt[b^2 - 4*a*c]*e*(2*c*d
- b*e)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c*d^2 - b*d*e + a*e^2)^2 - ((2*c
^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e))*Log[d + e*x])/(c*d^2 - b*d*e + a*e^2)^2 +
((2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*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 = 0.616566, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{\left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^2}-\frac{\log (d+e x) \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right )}{\left (a e^2-b d e+c d^2\right )^2}+\frac{e \sqrt{b^2-4 a c} (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (a e^2-b d e+c d^2\right )^2}+\frac{2 c d-b e}{(d+e x) \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(2*c*d - b*e)/((c*d^2 - b*d*e + a*e^2)*(d + e*x)) + (Sqrt[b^2 - 4*a*c]*e*(2*c*d
- b*e)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c*d^2 - b*d*e + a*e^2)^2 - ((2*c
^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e))*Log[d + e*x])/(c*d^2 - b*d*e + a*e^2)^2 +
((2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*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 = 158.661, size = 201, normalized size = 0.96 \[ - \frac{e \sqrt{- 4 a c + b^{2}} \left (b e - 2 c d\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{\left (a e^{2} - b d e + c d^{2}\right )^{2}} - \frac{\left (- 2 a c e^{2} + b^{2} e^{2} - 2 b c d e + 2 c^{2} d^{2}\right ) \log{\left (d + e x \right )}}{\left (a e^{2} - b d e + c d^{2}\right )^{2}} + \frac{\left (- a c e^{2} + \frac{b^{2} e^{2}}{2} - b c d e + c^{2} d^{2}\right ) \log{\left (a + b x + c x^{2} \right )}}{\left (a e^{2} - b d e + c d^{2}\right )^{2}} - \frac{b e - 2 c d}{\left (d + e x\right ) \left (a e^{2} - b d e + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.579624, size = 177, normalized size = 0.84 \[ \frac{\log (d+e x) \left (4 c e (a e+b d)-2 b^2 e^2-4 c^2 d^2\right )+\left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \log (a+x (b+c x))-2 e \sqrt{4 a c-b^2} (b e-2 c d) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )+\frac{2 (2 c d-b e) \left (e (a e-b d)+c d^2\right )}{d+e x}}{2 \left (e (a e-b d)+c d^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.019, size = 560, normalized size = 2.7 \[ 2\,{\frac{\ln \left ( ex+d \right ) ac{e}^{2}}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) ^{2}}}-{\frac{\ln \left ( ex+d \right ){b}^{2}{e}^{2}}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) ^{2}}}+2\,{\frac{\ln \left ( ex+d \right ) bcde}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) ^{2}}}-2\,{\frac{\ln \left ( ex+d \right ){c}^{2}{d}^{2}}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) ^{2}}}-{\frac{be}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) \left ( ex+d \right ) }}+2\,{\frac{cd}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) \left ( ex+d \right ) }}-{\frac{c\ln \left ( c{x}^{2}+bx+a \right ) a{e}^{2}}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) ^{2}}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ){b}^{2}{e}^{2}}{2\, \left ( a{e}^{2}-bde+c{d}^{2} \right ) ^{2}}}-{\frac{c\ln \left ( c{x}^{2}+bx+a \right ) bde}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) ^{2}}}+{\frac{{c}^{2}\ln \left ( c{x}^{2}+bx+a \right ){d}^{2}}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) ^{2}}}-4\,{\frac{c{e}^{2}ab}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) ^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+8\,{\frac{a{c}^{2}de}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) ^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{{b}^{3}{e}^{2}}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) ^{2}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-2\,{\frac{{b}^{2}cde}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) ^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/(a*e^2-b*d*e+c*d^2)^2*ln(e*x+d)*a*c*e^2-1/(a*e^2-b*d*e+c*d^2)^2*ln(e*x+d)*b^2*
e^2+2/(a*e^2-b*d*e+c*d^2)^2*ln(e*x+d)*b*c*d*e-2/(a*e^2-b*d*e+c*d^2)^2*ln(e*x+d)*
c^2*d^2-1/(a*e^2-b*d*e+c*d^2)/(e*x+d)*b*e+2/(a*e^2-b*d*e+c*d^2)/(e*x+d)*c*d-1/(a
*e^2-b*d*e+c*d^2)^2*c*ln(c*x^2+b*x+a)*a*e^2+1/2/(a*e^2-b*d*e+c*d^2)^2*ln(c*x^2+b
*x+a)*b^2*e^2-1/(a*e^2-b*d*e+c*d^2)^2*c*ln(c*x^2+b*x+a)*b*d*e+1/(a*e^2-b*d*e+c*d
^2)^2*c^2*ln(c*x^2+b*x+a)*d^2-4/(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*e^2*a*b+8/(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^2*d*e+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^3*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^2*c*d*e

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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)*(e*x + d)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.11427, 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)*(e*x + d)^2),x, algorithm="fricas")

[Out]

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

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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)**2/(c*x**2+b*x+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.278633, size = 494, normalized size = 2.35 \[ \frac{{\left (2 \, b^{2} c d e^{3} - 8 \, a c^{2} d e^{3} - b^{3} e^{4} + 4 \, a b c e^{4}\right )} \arctan \left (-\frac{{\left (2 \, c d - \frac{2 \, c d^{2}}{x e + d} - b e + \frac{2 \, b d e}{x e + d} - \frac{2 \, a e^{2}}{x e + d}\right )} e^{\left (-1\right )}}{\sqrt{-b^{2} + 4 \, a c}}\right ) e^{\left (-2\right )}}{{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{{\left (2 \, c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2} - 2 \, a c e^{2}\right )}{\rm ln}\left (-c + \frac{2 \, c d}{x e + d} - \frac{c d^{2}}{{\left (x e + d\right )}^{2}} - \frac{b e}{x e + d} + \frac{b d e}{{\left (x e + d\right )}^{2}} - \frac{a e^{2}}{{\left (x e + d\right )}^{2}}\right )}{2 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )}} + \frac{\frac{2 \, c d e^{2}}{x e + d} - \frac{b e^{3}}{x e + d}}{c d^{2} e^{2} - b d e^{3} + a e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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