3.619 \(\int \frac{1}{(d+e x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right )} \, dx\)

Optimal. Leaf size=121 \[ -\frac{\left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right )}{2 a^2 e \sqrt{b^2-4 a c}}+\frac{b \log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^2 e}-\frac{b \log (d+e x)}{a^2 e}-\frac{1}{2 a e (d+e x)^2} \]

[Out]

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

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Rubi [A]  time = 0.401787, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{\left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right )}{2 a^2 e \sqrt{b^2-4 a c}}+\frac{b \log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^2 e}-\frac{b \log (d+e x)}{a^2 e}-\frac{1}{2 a e (d+e x)^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4)),x]

[Out]

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

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Rubi in Sympy [A]  time = 64.7209, size = 110, normalized size = 0.91 \[ - \frac{1}{2 a e \left (d + e x\right )^{2}} - \frac{b \log{\left (\left (d + e x\right )^{2} \right )}}{2 a^{2} e} + \frac{b \log{\left (a + b \left (d + e x\right )^{2} + c \left (d + e x\right )^{4} \right )}}{4 a^{2} e} - \frac{\left (- 2 a c + b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c \left (d + e x\right )^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{2 a^{2} e \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**3/(a+b*(e*x+d)**2+c*(e*x+d)**4),x)

[Out]

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

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Mathematica [A]  time = 0.249792, size = 154, normalized size = 1.27 \[ \frac{\frac{\left (b \sqrt{b^2-4 a c}-2 a c+b^2\right ) \log \left (-\sqrt{b^2-4 a c}+b+2 c (d+e x)^2\right )}{\sqrt{b^2-4 a c}}+\frac{\left (b \sqrt{b^2-4 a c}+2 a c-b^2\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c (d+e x)^2\right )}{\sqrt{b^2-4 a c}}-\frac{2 a}{(d+e x)^2}-4 b \log (d+e x)}{4 a^2 e} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4)),x]

[Out]

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

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Maple [C]  time = 0.018, size = 213, normalized size = 1.8 \[{\frac{1}{2\,{a}^{2}e}\sum _{{\it \_R}={\it RootOf} \left ( c{e}^{4}{{\it \_Z}}^{4}+4\,cd{e}^{3}{{\it \_Z}}^{3}+ \left ( 6\,c{d}^{2}{e}^{2}+b{e}^{2} \right ){{\it \_Z}}^{2}+ \left ( 4\,c{d}^{3}e+2\,bde \right ){\it \_Z}+c{d}^{4}+b{d}^{2}+a \right ) }{\frac{ \left ({{\it \_R}}^{3}bc{e}^{3}+3\,{{\it \_R}}^{2}bcd{e}^{2}+e \left ( 3\,bc{d}^{2}-ac+{b}^{2} \right ){\it \_R}+bc{d}^{3}-acd+{b}^{2}d \right ) \ln \left ( x-{\it \_R} \right ) }{2\,c{e}^{3}{{\it \_R}}^{3}+6\,cd{e}^{2}{{\it \_R}}^{2}+6\,{d}^{2}ec{\it \_R}+2\,c{d}^{3}+be{\it \_R}+bd}}}-{\frac{1}{2\,ae \left ( ex+d \right ) ^{2}}}-{\frac{b\ln \left ( ex+d \right ) }{{a}^{2}e}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^3/(a+b*(e*x+d)^2+c*(e*x+d)^4),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{1}{2 \,{\left (a e^{3} x^{2} + 2 \, a d e^{2} x + a d^{2} e\right )}} + \frac{\int \frac{b c e^{3} x^{3} + 3 \, b c d e^{2} x^{2} + b c d^{3} +{\left (3 \, b c d^{2} + b^{2} - a c\right )} e x +{\left (b^{2} - a c\right )} d}{c e^{4} x^{4} + 4 \, c d e^{3} x^{3} + c d^{4} +{\left (6 \, c d^{2} + b\right )} e^{2} x^{2} + b d^{2} + 2 \,{\left (2 \, c d^{3} + b d\right )} e x + a}\,{d x}}{a^{2}} - \frac{b \log \left (e x + d\right )}{a^{2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(((e*x + d)^4*c + (e*x + d)^2*b + a)*(e*x + d)^3),x, algorithm="maxima")

[Out]

-1/2/(a*e^3*x^2 + 2*a*d*e^2*x + a*d^2*e) + integrate((b*c*e^3*x^3 + 3*b*c*d*e^2*
x^2 + b*c*d^3 + (3*b*c*d^2 + b^2 - a*c)*e*x + (b^2 - a*c)*d)/(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), x)
/a^2 - b*log(e*x + d)/(a^2*e)

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Fricas [A]  time = 0.30177, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left ({\left (b^{2} - 2 \, a c\right )} e^{2} x^{2} + 2 \,{\left (b^{2} - 2 \, a c\right )} d e x +{\left (b^{2} - 2 \, a c\right )} d^{2}\right )} \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 ) - \sqrt{b^{2} - 4 \, a c}{\left ({\left (b e^{2} x^{2} + 2 \, b d e x + b d^{2}\right )} \log \left (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 ) - 4 \,{\left (b e^{2} x^{2} + 2 \, b d e x + b d^{2}\right )} \log \left (e x + d\right ) - 2 \, a\right )}}{4 \,{\left (a^{2} e^{3} x^{2} + 2 \, a^{2} d e^{2} x + a^{2} d^{2} e\right )} \sqrt{b^{2} - 4 \, a c}}, \frac{2 \,{\left ({\left (b^{2} - 2 \, a c\right )} e^{2} x^{2} + 2 \,{\left (b^{2} - 2 \, a c\right )} d e x +{\left (b^{2} - 2 \, a c\right )} d^{2}\right )} \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}{\left ({\left (b e^{2} x^{2} + 2 \, b d e x + b d^{2}\right )} \log \left (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 ) - 4 \,{\left (b e^{2} x^{2} + 2 \, b d e x + b d^{2}\right )} \log \left (e x + d\right ) - 2 \, a\right )}}{4 \,{\left (a^{2} e^{3} x^{2} + 2 \, a^{2} d e^{2} x + a^{2} d^{2} e\right )} \sqrt{-b^{2} + 4 \, a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(((e*x + d)^4*c + (e*x + d)^2*b + a)*(e*x + d)^3),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 46.3686, size = 464, normalized size = 3.83 \[ \left (\frac{b}{4 a^{2} e} - \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 a^{2} e \left (4 a c - b^{2}\right )}\right ) \log{\left (\frac{2 d x}{e} + x^{2} + \frac{- 8 a^{3} c e \left (\frac{b}{4 a^{2} e} - \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 a^{2} e \left (4 a c - b^{2}\right )}\right ) + 2 a^{2} b^{2} e \left (\frac{b}{4 a^{2} e} - \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 a^{2} e \left (4 a c - b^{2}\right )}\right ) + 3 a b c + 2 a c^{2} d^{2} - b^{3} - b^{2} c d^{2}}{2 a c^{2} e^{2} - b^{2} c e^{2}} \right )} + \left (\frac{b}{4 a^{2} e} + \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 a^{2} e \left (4 a c - b^{2}\right )}\right ) \log{\left (\frac{2 d x}{e} + x^{2} + \frac{- 8 a^{3} c e \left (\frac{b}{4 a^{2} e} + \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 a^{2} e \left (4 a c - b^{2}\right )}\right ) + 2 a^{2} b^{2} e \left (\frac{b}{4 a^{2} e} + \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 a^{2} e \left (4 a c - b^{2}\right )}\right ) + 3 a b c + 2 a c^{2} d^{2} - b^{3} - b^{2} c d^{2}}{2 a c^{2} e^{2} - b^{2} c e^{2}} \right )} - \frac{1}{2 a d^{2} e + 4 a d e^{2} x + 2 a e^{3} x^{2}} - \frac{b \log{\left (\frac{d}{e} + x \right )}}{a^{2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**3/(a+b*(e*x+d)**2+c*(e*x+d)**4),x)

[Out]

(b/(4*a**2*e) - sqrt(-4*a*c + b**2)*(2*a*c - b**2)/(4*a**2*e*(4*a*c - b**2)))*lo
g(2*d*x/e + x**2 + (-8*a**3*c*e*(b/(4*a**2*e) - sqrt(-4*a*c + b**2)*(2*a*c - b**
2)/(4*a**2*e*(4*a*c - b**2))) + 2*a**2*b**2*e*(b/(4*a**2*e) - sqrt(-4*a*c + b**2
)*(2*a*c - b**2)/(4*a**2*e*(4*a*c - b**2))) + 3*a*b*c + 2*a*c**2*d**2 - b**3 - b
**2*c*d**2)/(2*a*c**2*e**2 - b**2*c*e**2)) + (b/(4*a**2*e) + sqrt(-4*a*c + b**2)
*(2*a*c - b**2)/(4*a**2*e*(4*a*c - b**2)))*log(2*d*x/e + x**2 + (-8*a**3*c*e*(b/
(4*a**2*e) + sqrt(-4*a*c + b**2)*(2*a*c - b**2)/(4*a**2*e*(4*a*c - b**2))) + 2*a
**2*b**2*e*(b/(4*a**2*e) + sqrt(-4*a*c + b**2)*(2*a*c - b**2)/(4*a**2*e*(4*a*c -
b**2))) + 3*a*b*c + 2*a*c**2*d**2 - b**3 - b**2*c*d**2)/(2*a*c**2*e**2 - b**2*c
*e**2)) - 1/(2*a*d**2*e + 4*a*d*e**2*x + 2*a*e**3*x**2) - b*log(d/e + x)/(a**2*e
)

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GIAC/XCAS [A]  time = 0.296676, size = 138, normalized size = 1.14 \[ \frac{b e^{\left (-1\right )}{\rm ln}\left (c + \frac{b}{{\left (x e + d\right )}^{2}} + \frac{a}{{\left (x e + d\right )}^{4}}\right )}{4 \, a^{2}} + \frac{{\left (b^{2} - 2 \, a c\right )} \arctan \left (-\frac{b + \frac{2 \, a}{{\left (x e + d\right )}^{2}}}{\sqrt{-b^{2} + 4 \, a c}}\right ) e^{\left (-1\right )}}{2 \, \sqrt{-b^{2} + 4 \, a c} a^{2}} - \frac{e^{\left (-1\right )}}{2 \,{\left (x e + d\right )}^{2} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(((e*x + d)^4*c + (e*x + d)^2*b + a)*(e*x + d)^3),x, algorithm="giac")

[Out]

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