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

Optimal. Leaf size=204 \[ -\frac{\sqrt{c} \left (\frac{b}{\sqrt{b^2-4 a c}}+1\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} a e f^2 \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} a e f^2 \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{1}{a e f^2 (d+e x)} \]

[Out]

-(1/(a*e*f^2*(d + e*x))) - (Sqrt[c]*(1 + b/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sq
rt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*a*Sqrt[b - Sqrt[b^2 - 4*
a*c]]*e*f^2) - (Sqrt[c]*(1 - b/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e
*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*a*Sqrt[b + Sqrt[b^2 - 4*a*c]]*e*f^2)

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Rubi [A]  time = 0.694051, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121 \[ -\frac{\sqrt{c} \left (\frac{b}{\sqrt{b^2-4 a c}}+1\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} a e f^2 \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} a e f^2 \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{1}{a e f^2 (d+e x)} \]

Antiderivative was successfully verified.

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

[Out]

-(1/(a*e*f^2*(d + e*x))) - (Sqrt[c]*(1 + b/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sq
rt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*a*Sqrt[b - Sqrt[b^2 - 4*
a*c]]*e*f^2) - (Sqrt[c]*(1 - b/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e
*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*a*Sqrt[b + Sqrt[b^2 - 4*a*c]]*e*f^2)

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Rubi in Sympy [A]  time = 64.9485, size = 202, normalized size = 0.99 \[ \frac{\sqrt{2} \sqrt{c} \left (b - \sqrt{- 4 a c + b^{2}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} \left (d + e x\right )}{\sqrt{b + \sqrt{- 4 a c + b^{2}}}} \right )}}{2 a e f^{2} \sqrt{b + \sqrt{- 4 a c + b^{2}}} \sqrt{- 4 a c + b^{2}}} - \frac{\sqrt{2} \sqrt{c} \left (b + \sqrt{- 4 a c + b^{2}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} \left (d + e x\right )}{\sqrt{b - \sqrt{- 4 a c + b^{2}}}} \right )}}{2 a e f^{2} \sqrt{b - \sqrt{- 4 a c + b^{2}}} \sqrt{- 4 a c + b^{2}}} - \frac{1}{a e f^{2} \left (d + e x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.643669, size = 209, normalized size = 1.02 \[ -\frac{\frac{\sqrt{2} \sqrt{c} \left (\sqrt{b^2-4 a c}+b\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \sqrt{c} \left (\sqrt{b^2-4 a c}-b\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2}{d+e x}}{2 a e f^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.007, size = 174, normalized size = 0.9 \[ -{\frac{1}{ae{f}^{2} \left ( ex+d \right ) }}+{\frac{1}{2\,ae{f}^{2}}\sum _{{\it \_R}={\it RootOf} \left ( c{e}^{4}{{\it \_Z}}^{4}+4\,cd{e}^{3}{{\it \_Z}}^{3}+ \left ( 6\,c{d}^{2}{e}^{2}+b{e}^{2} \right ){{\it \_Z}}^{2}+ \left ( 4\,c{d}^{3}e+2\,bde \right ){\it \_Z}+c{d}^{4}+b{d}^{2}+a \right ) }{\frac{ \left ( -{{\it \_R}}^{2}c{e}^{2}-2\,{\it \_R}\,cde-c{d}^{2}-b \right ) \ln \left ( x-{\it \_R} \right ) }{2\,c{e}^{3}{{\it \_R}}^{3}+6\,cd{e}^{2}{{\it \_R}}^{2}+6\,{\it \_R}\,c{d}^{2}e+2\,c{d}^{3}+be{\it \_R}+bd}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/a/e/f^2/(e*x+d)+1/2/f^2/a/e*sum((-_R^2*c*e^2-2*_R*c*d*e-c*d^2-b)/(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. \[ -\frac{1}{a e^{2} f^{2} x + a d e f^{2}} - \frac{\int \frac{c e^{2} x^{2} + 2 \, c d e x + c d^{2} + b}{c e^{4} x^{4} + 4 \, c d e^{3} x^{3} + c d^{4} +{\left (6 \, c d^{2} + b\right )} e^{2} x^{2} + b d^{2} + 2 \,{\left (2 \, c d^{3} + b d\right )} e x + a}\,{d x}}{a f^{2}} \]

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)^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.289381, size = 1994, normalized size = 9.77 \[ \text{result too large to display} \]

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)^2),x, algorithm="fricas")

[Out]

1/2*(sqrt(1/2)*(a*e^2*f^2*x + a*d*e*f^2)*sqrt(-((a^3*b^2 - 4*a^4*c)*e^2*f^4*sqrt
((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^2 - 4*a^7*c)*e^4*f^8)) + b^3 - 3*a*b*c)/((a
^3*b^2 - 4*a^4*c)*e^2*f^4))*log(-2*(b^2*c^2 - a*c^3)*e*x - 2*(b^2*c^2 - a*c^3)*d
+ sqrt(1/2)*((a^3*b^4 - 6*a^4*b^2*c + 8*a^5*c^2)*e^3*f^6*sqrt((b^4 - 2*a*b^2*c
+ a^2*c^2)/((a^6*b^2 - 4*a^7*c)*e^4*f^8)) - (b^5 - 5*a*b^3*c + 4*a^2*b*c^2)*e*f^
2)*sqrt(-((a^3*b^2 - 4*a^4*c)*e^2*f^4*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^2
- 4*a^7*c)*e^4*f^8)) + b^3 - 3*a*b*c)/((a^3*b^2 - 4*a^4*c)*e^2*f^4))) - sqrt(1/
2)*(a*e^2*f^2*x + a*d*e*f^2)*sqrt(-((a^3*b^2 - 4*a^4*c)*e^2*f^4*sqrt((b^4 - 2*a*
b^2*c + a^2*c^2)/((a^6*b^2 - 4*a^7*c)*e^4*f^8)) + b^3 - 3*a*b*c)/((a^3*b^2 - 4*a
^4*c)*e^2*f^4))*log(-2*(b^2*c^2 - a*c^3)*e*x - 2*(b^2*c^2 - a*c^3)*d - sqrt(1/2)
*((a^3*b^4 - 6*a^4*b^2*c + 8*a^5*c^2)*e^3*f^6*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/(
(a^6*b^2 - 4*a^7*c)*e^4*f^8)) - (b^5 - 5*a*b^3*c + 4*a^2*b*c^2)*e*f^2)*sqrt(-((a
^3*b^2 - 4*a^4*c)*e^2*f^4*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^2 - 4*a^7*c)*
e^4*f^8)) + b^3 - 3*a*b*c)/((a^3*b^2 - 4*a^4*c)*e^2*f^4))) - sqrt(1/2)*(a*e^2*f^
2*x + a*d*e*f^2)*sqrt(((a^3*b^2 - 4*a^4*c)*e^2*f^4*sqrt((b^4 - 2*a*b^2*c + a^2*c
^2)/((a^6*b^2 - 4*a^7*c)*e^4*f^8)) - b^3 + 3*a*b*c)/((a^3*b^2 - 4*a^4*c)*e^2*f^4
))*log(-2*(b^2*c^2 - a*c^3)*e*x - 2*(b^2*c^2 - a*c^3)*d + sqrt(1/2)*((a^3*b^4 -
6*a^4*b^2*c + 8*a^5*c^2)*e^3*f^6*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^2 - 4*
a^7*c)*e^4*f^8)) + (b^5 - 5*a*b^3*c + 4*a^2*b*c^2)*e*f^2)*sqrt(((a^3*b^2 - 4*a^4
*c)*e^2*f^4*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^2 - 4*a^7*c)*e^4*f^8)) - b^
3 + 3*a*b*c)/((a^3*b^2 - 4*a^4*c)*e^2*f^4))) + sqrt(1/2)*(a*e^2*f^2*x + a*d*e*f^
2)*sqrt(((a^3*b^2 - 4*a^4*c)*e^2*f^4*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^2
- 4*a^7*c)*e^4*f^8)) - b^3 + 3*a*b*c)/((a^3*b^2 - 4*a^4*c)*e^2*f^4))*log(-2*(b^2
*c^2 - a*c^3)*e*x - 2*(b^2*c^2 - a*c^3)*d - sqrt(1/2)*((a^3*b^4 - 6*a^4*b^2*c +
8*a^5*c^2)*e^3*f^6*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^2 - 4*a^7*c)*e^4*f^8
)) + (b^5 - 5*a*b^3*c + 4*a^2*b*c^2)*e*f^2)*sqrt(((a^3*b^2 - 4*a^4*c)*e^2*f^4*sq
rt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^2 - 4*a^7*c)*e^4*f^8)) - b^3 + 3*a*b*c)/(
(a^3*b^2 - 4*a^4*c)*e^2*f^4))) - 2)/(a*e^2*f^2*x + a*d*e*f^2)

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Sympy [A]  time = 16.9901, size = 258, normalized size = 1.26 \[ \operatorname{RootSum}{\left (t^{4} \left (256 a^{5} c^{2} e^{4} f^{8} - 128 a^{4} b^{2} c e^{4} f^{8} + 16 a^{3} b^{4} e^{4} f^{8}\right ) + t^{2} \left (48 a^{2} b c^{2} e^{2} f^{4} - 28 a b^{3} c e^{2} f^{4} + 4 b^{5} e^{2} f^{4}\right ) + c^{3}, \left ( t \mapsto t \log{\left (x + \frac{- 64 t^{3} a^{5} c^{2} e^{3} f^{6} + 48 t^{3} a^{4} b^{2} c e^{3} f^{6} - 8 t^{3} a^{3} b^{4} e^{3} f^{6} - 10 t a^{2} b c^{2} e f^{2} + 10 t a b^{3} c e f^{2} - 2 t b^{5} e f^{2} + a c^{3} d - b^{2} c^{2} d}{a c^{3} e - b^{2} c^{2} e} \right )} \right )\right )} - \frac{1}{a d e f^{2} + a e^{2} f^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(_t**4*(256*a**5*c**2*e**4*f**8 - 128*a**4*b**2*c*e**4*f**8 + 16*a**3*b**
4*e**4*f**8) + _t**2*(48*a**2*b*c**2*e**2*f**4 - 28*a*b**3*c*e**2*f**4 + 4*b**5*
e**2*f**4) + c**3, Lambda(_t, _t*log(x + (-64*_t**3*a**5*c**2*e**3*f**6 + 48*_t*
*3*a**4*b**2*c*e**3*f**6 - 8*_t**3*a**3*b**4*e**3*f**6 - 10*_t*a**2*b*c**2*e*f**
2 + 10*_t*a*b**3*c*e*f**2 - 2*_t*b**5*e*f**2 + a*c**3*d - b**2*c**2*d)/(a*c**3*e
- b**2*c**2*e)))) - 1/(a*d*e*f**2 + a*e**2*f**2*x)

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GIAC/XCAS [A]  time = 1.01916, size = 1, normalized size = 0. \[ \mathit{Done} \]

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)^2),x, algorithm="giac")

[Out]

Done