3.616 \(\int \frac{d+e x}{a+b (d+e x)^2+c (d+e x)^4} \, dx\)

Optimal. Leaf size=43 \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right )}{e \sqrt{b^2-4 a c}} \]

[Out]

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

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Rubi [A]  time = 0.123995, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right )}{e \sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)/(a + b*(d + e*x)^2 + c*(d + e*x)^4),x]

[Out]

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

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Rubi in Sympy [A]  time = 27.0955, size = 39, normalized size = 0.91 \[ - \frac{\operatorname{atanh}{\left (\frac{b + 2 c \left (d + e x\right )^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{e \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)/(a+b*(e*x+d)**2+c*(e*x+d)**4),x)

[Out]

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

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Mathematica [A]  time = 0.0234109, size = 46, normalized size = 1.07 \[ \frac{\tan ^{-1}\left (\frac{b+2 c (d+e x)^2}{\sqrt{4 a c-b^2}}\right )}{e \sqrt{4 a c-b^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)/(a + b*(d + e*x)^2 + c*(d + e*x)^4),x]

[Out]

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

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Maple [C]  time = 0.008, size = 129, normalized size = 3. \[{\frac{1}{2\,e}\sum _{{\it \_R}={\it RootOf} \left ( c{e}^{4}{{\it \_Z}}^{4}+4\,cd{e}^{3}{{\it \_Z}}^{3}+ \left ( 6\,c{d}^{2}{e}^{2}+b{e}^{2} \right ){{\it \_Z}}^{2}+ \left ( 4\,c{d}^{3}e+2\,bde \right ){\it \_Z}+c{d}^{4}+b{d}^{2}+a \right ) }{\frac{ \left ({\it \_R}\,e+d \right ) \ln \left ( x-{\it \_R} \right ) }{2\,c{e}^{3}{{\it \_R}}^{3}+6\,cd{e}^{2}{{\it \_R}}^{2}+6\,{d}^{2}ec{\it \_R}+2\,c{d}^{3}+be{\it \_R}+bd}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)/(a+b*(e*x+d)^2+c*(e*x+d)^4),x)

[Out]

1/2/e*sum((_R*e+d)/(2*_R^3*c*e^3+6*_R^2*c*d*e^2+6*_R*c*d^2*e+2*c*d^3+_R*b*e+b*d)
*ln(x-_R),_R=RootOf(c*e^4*_Z^4+4*c*d*e^3*_Z^3+(6*c*d^2*e^2+b*e^2)*_Z^2+(4*c*d^3*
e+2*b*d*e)*_Z+c*d^4+b*d^2+a))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{e x + d}{{\left (e x + d\right )}^{4} c +{\left (e x + d\right )}^{2} b + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)/((e*x + d)^4*c + (e*x + d)^2*b + a),x, algorithm="maxima")

[Out]

integrate((e*x + d)/((e*x + d)^4*c + (e*x + d)^2*b + a), x)

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Fricas [A]  time = 0.277257, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (-\frac{2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} e^{2} x^{2} + 4 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d e x + b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} -{\left (2 \, c^{2} e^{4} x^{4} + 8 \, c^{2} d e^{3} x^{3} + 2 \, c^{2} d^{4} + 2 \,{\left (6 \, c^{2} d^{2} + b c\right )} e^{2} x^{2} + 2 \, b c d^{2} + 4 \,{\left (2 \, c^{2} d^{3} + b c d\right )} e x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c e^{4} x^{4} + 4 \, c d e^{3} x^{3} + c d^{4} +{\left (6 \, c d^{2} + b\right )} e^{2} x^{2} + b d^{2} + 2 \,{\left (2 \, c d^{3} + b d\right )} e x + a}\right )}{2 \, \sqrt{b^{2} - 4 \, a c} e}, \frac{\arctan \left (-\frac{{\left (2 \, c e^{2} x^{2} + 4 \, c d e x + 2 \, c d^{2} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right )}{\sqrt{-b^{2} + 4 \, a c} e}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)/((e*x + d)^4*c + (e*x + d)^2*b + a),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 3.57723, size = 168, normalized size = 3.91 \[ - \frac{\sqrt{- \frac{1}{4 a c - b^{2}}} \log{\left (\frac{2 d x}{e} + x^{2} + \frac{- 4 a c \sqrt{- \frac{1}{4 a c - b^{2}}} + b^{2} \sqrt{- \frac{1}{4 a c - b^{2}}} + b + 2 c d^{2}}{2 c e^{2}} \right )}}{2 e} + \frac{\sqrt{- \frac{1}{4 a c - b^{2}}} \log{\left (\frac{2 d x}{e} + x^{2} + \frac{4 a c \sqrt{- \frac{1}{4 a c - b^{2}}} - b^{2} \sqrt{- \frac{1}{4 a c - b^{2}}} + b + 2 c d^{2}}{2 c e^{2}} \right )}}{2 e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)/(a+b*(e*x+d)**2+c*(e*x+d)**4),x)

[Out]

-sqrt(-1/(4*a*c - b**2))*log(2*d*x/e + x**2 + (-4*a*c*sqrt(-1/(4*a*c - b**2)) +
b**2*sqrt(-1/(4*a*c - b**2)) + b + 2*c*d**2)/(2*c*e**2))/(2*e) + sqrt(-1/(4*a*c
- b**2))*log(2*d*x/e + x**2 + (4*a*c*sqrt(-1/(4*a*c - b**2)) - b**2*sqrt(-1/(4*a
*c - b**2)) + b + 2*c*d**2)/(2*c*e**2))/(2*e)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{e x + d}{{\left (e x + d\right )}^{4} c +{\left (e x + d\right )}^{2} b + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)/((e*x + d)^4*c + (e*x + d)^2*b + a),x, algorithm="giac")

[Out]

integrate((e*x + d)/((e*x + d)^4*c + (e*x + d)^2*b + a), x)