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

Optimal. Leaf size=103 \[ \frac{b \tanh ^{-1}\left (\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right )}{2 a e f \sqrt{b^2-4 a c}}-\frac{\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a e f}+\frac{\log (d+e x)}{a e f} \]

[Out]

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

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Rubi [A]  time = 0.283488, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.242 \[ \frac{b \tanh ^{-1}\left (\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right )}{2 a e f \sqrt{b^2-4 a c}}-\frac{\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a e f}+\frac{\log (d+e x)}{a e f} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 39.0735, size = 87, normalized size = 0.84 \[ \frac{b \operatorname{atanh}{\left (\frac{b + 2 c \left (d + e x\right )^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{2 a e f \sqrt{- 4 a c + b^{2}}} + \frac{\log{\left (\left (d + e x\right )^{2} \right )}}{2 a e f} - \frac{\log{\left (a + b \left (d + e x\right )^{2} + c \left (d + e x\right )^{4} \right )}}{4 a e f} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.126468, size = 131, normalized size = 1.27 \[ \frac{4 \sqrt{b^2-4 a c} \log (d+e x)-\left (\sqrt{b^2-4 a c}+b\right ) \log \left (-\sqrt{b^2-4 a c}+b+2 c (d+e x)^2\right )+\left (b-\sqrt{b^2-4 a c}\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c (d+e x)^2\right )}{4 a e f \sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.007, size = 190, normalized size = 1.8 \[{\frac{1}{2\,aef}\sum _{{\it \_R}={\it RootOf} \left ( c{e}^{4}{{\it \_Z}}^{4}+4\,cd{e}^{3}{{\it \_Z}}^{3}+ \left ( 6\,c{d}^{2}{e}^{2}+b{e}^{2} \right ){{\it \_Z}}^{2}+ \left ( 4\,c{d}^{3}e+2\,bde \right ){\it \_Z}+c{d}^{4}+b{d}^{2}+a \right ) }{\frac{ \left ( -c{e}^{3}{{\it \_R}}^{3}-3\,cd{e}^{2}{{\it \_R}}^{2}+e \left ( -3\,c{d}^{2}-b \right ){\it \_R}-c{d}^{3}-bd \right ) \ln \left ( x-{\it \_R} \right ) }{2\,c{e}^{3}{{\it \_R}}^{3}+6\,cd{e}^{2}{{\it \_R}}^{2}+6\,{\it \_R}\,c{d}^{2}e+2\,c{d}^{3}+be{\it \_R}+bd}}}+{\frac{\ln \left ( ex+d \right ) }{aef}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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*f*x + d*f)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.294892, size = 1, normalized size = 0.01 \[ \left [\frac{b \log \left (\frac{2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} e^{2} x^{2} + 4 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d e x + b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} +{\left (2 \, c^{2} e^{4} x^{4} + 8 \, c^{2} d e^{3} x^{3} + 2 \, c^{2} d^{4} + 2 \,{\left (6 \, c^{2} d^{2} + b c\right )} e^{2} x^{2} + 2 \, b c d^{2} + 4 \,{\left (2 \, c^{2} d^{3} + b c d\right )} e x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c e^{4} x^{4} + 4 \, c d e^{3} x^{3} + c d^{4} +{\left (6 \, c d^{2} + b\right )} e^{2} x^{2} + b d^{2} + 2 \,{\left (2 \, c d^{3} + b d\right )} e x + a}\right ) - \sqrt{b^{2} - 4 \, a c}{\left (\log \left (c e^{4} x^{4} + 4 \, c d e^{3} x^{3} + c d^{4} +{\left (6 \, c d^{2} + b\right )} e^{2} x^{2} + b d^{2} + 2 \,{\left (2 \, c d^{3} + b d\right )} e x + a\right ) - 4 \, \log \left (e x + d\right )\right )}}{4 \, \sqrt{b^{2} - 4 \, a c} a e f}, -\frac{2 \, b \arctan \left (-\frac{{\left (2 \, c e^{2} x^{2} + 4 \, c d e x + 2 \, c d^{2} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) + \sqrt{-b^{2} + 4 \, a c}{\left (\log \left (c e^{4} x^{4} + 4 \, c d e^{3} x^{3} + c d^{4} +{\left (6 \, c d^{2} + b\right )} e^{2} x^{2} + b d^{2} + 2 \,{\left (2 \, c d^{3} + b d\right )} e x + a\right ) - 4 \, \log \left (e x + d\right )\right )}}{4 \, \sqrt{-b^{2} + 4 \, a c} a e f}\right ] \]

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*f*x + d*f)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 20.4612, size = 348, normalized size = 3.38 \[ \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{4 a e f \left (4 a c - b^{2}\right )} - \frac{1}{4 a e f}\right ) \log{\left (\frac{2 d x}{e} + x^{2} + \frac{- 8 a^{2} c e f \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{4 a e f \left (4 a c - b^{2}\right )} - \frac{1}{4 a e f}\right ) + 2 a b^{2} e f \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{4 a e f \left (4 a c - b^{2}\right )} - \frac{1}{4 a e f}\right ) - 2 a c + b^{2} + b c d^{2}}{b c e^{2}} \right )} + \left (\frac{b \sqrt{- 4 a c + b^{2}}}{4 a e f \left (4 a c - b^{2}\right )} - \frac{1}{4 a e f}\right ) \log{\left (\frac{2 d x}{e} + x^{2} + \frac{- 8 a^{2} c e f \left (\frac{b \sqrt{- 4 a c + b^{2}}}{4 a e f \left (4 a c - b^{2}\right )} - \frac{1}{4 a e f}\right ) + 2 a b^{2} e f \left (\frac{b \sqrt{- 4 a c + b^{2}}}{4 a e f \left (4 a c - b^{2}\right )} - \frac{1}{4 a e f}\right ) - 2 a c + b^{2} + b c d^{2}}{b c e^{2}} \right )} + \frac{\log{\left (\frac{d}{e} + x \right )}}{a e f} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left ({\left (e x + d\right )}^{4} c +{\left (e x + d\right )}^{2} b + a\right )}{\left (e f x + d f\right )}}\,{d x} \]

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*f*x + d*f)),x, algorithm="giac")

[Out]

integrate(1/(((e*x + d)^4*c + (e*x + d)^2*b + a)*(e*f*x + d*f)), x)