3.1537 \(\int \frac{b+2 c x}{(d+e x) \left (a+b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=236 \[ \frac{e \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c} \left (a e^2-b d e+c d^2\right )^2}-\frac{\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}+\frac{e^2 (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^2}-\frac{e^2 (2 c d-b e) \log (d+e x)}{\left (a e^2-b d e+c d^2\right )^2} \]

[Out]

-(((b^2 - 4*a*c)*(c*d - b*e) - c*(b^2 - 4*a*c)*e*x)/((b^2 - 4*a*c)*(c*d^2 - b*d*
e + a*e^2)*(a + b*x + c*x^2))) + (e*(2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + 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)
^2) - (e^2*(2*c*d - b*e)*Log[d + e*x])/(c*d^2 - b*d*e + a*e^2)^2 + (e^2*(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 = 0.798519, antiderivative size = 236, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{e \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c} \left (a e^2-b d e+c d^2\right )^2}-\frac{-b e+c d-c e x}{\left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}+\frac{e^2 (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^2}-\frac{e^2 (2 c d-b e) \log (d+e x)}{\left (a e^2-b d e+c d^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-((c*d - b*e - c*e*x)/((c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2))) + (e*(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*(2*c*d - b*e)*Log[d + e*x])/(c*d^
2 - b*d*e + a*e^2)^2 + (e^2*(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 = 156.089, size = 197, normalized size = 0.83 \[ \frac{e^{2} \left (b e - 2 c d\right ) \log{\left (d + e x \right )}}{\left (a e^{2} - b d e + c d^{2}\right )^{2}} - \frac{e^{2} \left (b e - 2 c d\right ) \log{\left (a + b x + c x^{2} \right )}}{2 \left (a e^{2} - b d e + c d^{2}\right )^{2}} + \frac{e \left (- 2 a c e^{2} + b^{2} e^{2} - 2 b c d e + 2 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{\sqrt{- 4 a c + b^{2}} \left (a e^{2} - b d e + c d^{2}\right )^{2}} + \frac{b e - c d + c e x}{\left (a + b x + c x^{2}\right ) \left (a e^{2} - b d e + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.464648, size = 173, normalized size = 0.73 \[ \frac{\frac{2 e \left (2 c e (a e+b d)-b^2 e^2-2 c^2 d^2\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+\frac{2 \left (e (a e-b d)+c d^2\right ) (b e-c d+c e x)}{a+x (b+c x)}-e^2 (b e-2 c d) \log (a+x (b+c x))+2 e^2 (b e-2 c d) \log (d+e x)}{2 \left (e (a e-b d)+c d^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

((2*(c*d^2 + e*(-(b*d) + a*e))*(-(c*d) + b*e + c*e*x))/(a + x*(b + c*x)) + (2*e*
(-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*(-2*c*d + b*e)*Log[d + e*x] - e^2*(-2*c*d + b*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 = 0.025, size = 668, normalized size = 2.8 \[{\frac{{e}^{3}\ln \left ( ex+d \right ) b}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) ^{2}}}-2\,{\frac{{e}^{2}\ln \left ( ex+d \right ) cd}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) ^{2}}}+{\frac{axc{e}^{3}}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) }}-{\frac{bxcd{e}^{2}}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) }}+{\frac{x{c}^{2}{d}^{2}e}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) }}+{\frac{ab{e}^{3}}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) }}-{\frac{ad{e}^{2}c}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) }}-{\frac{{b}^{2}d{e}^{2}}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) }}+2\,{\frac{bc{d}^{2}e}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) }}-{\frac{{c}^{2}{d}^{3}}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) }}-{\frac{{e}^{3}\ln \left ( c{x}^{2}+bx+a \right ) b}{2\, \left ( a{e}^{2}-bde+c{d}^{2} \right ) ^{2}}}+{\frac{c{e}^{2}\ln \left ( c{x}^{2}+bx+a \right ) d}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) ^{2}}}+2\,{\frac{ac{e}^{3}}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) ^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{{b}^{2}{e}^{3}}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) ^{2}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+2\,{\frac{d{e}^{2}bc}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) ^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-2\,{\frac{{d}^{2}e{c}^{2}}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) ^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.04565, 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)^2*(e*x + d)),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.275748, size = 482, normalized size = 2.04 \[ \frac{{\left (2 \, c d e^{2} - b e^{3}\right )}{\rm ln}\left (c x^{2} + b x + a\right )}{2 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )}} - \frac{{\left (2 \, c d e^{3} - b e^{4}\right )}{\rm ln}\left ({\left | x e + d \right |}\right )}{c^{2} d^{4} e - 2 \, b c d^{3} e^{2} + b^{2} d^{2} e^{3} + 2 \, a c d^{2} e^{3} - 2 \, a b d e^{4} + a^{2} e^{5}} - \frac{{\left (2 \, c^{2} d^{2} e - 2 \, b c d e^{2} + b^{2} e^{3} - 2 \, a c e^{3}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{c^{2} d^{3} - 2 \, b c d^{2} e + b^{2} d e^{2} + a c d e^{2} - a b e^{3} -{\left (c^{2} d^{2} e - b c d e^{2} + a c e^{3}\right )} x}{{\left (c d^{2} - b d e + a e^{2}\right )}^{2}{\left (c x^{2} + b x + a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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