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

Optimal. Leaf size=193 \[ -\frac{\left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} c^{3/2} e \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} c^{3/2} e \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{x}{c} \]

[Out]

x/c - ((b - (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/
Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]*e) -
((b + (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b
+ Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]*e)

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Rubi [A]  time = 0.874143, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{\left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} c^{3/2} e \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} c^{3/2} e \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{x}{c} \]

Antiderivative was successfully verified.

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

[Out]

x/c - ((b - (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/
Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]*e) -
((b + (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b
+ Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]*e)

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Rubi in Sympy [A]  time = 85.4182, size = 204, normalized size = 1.06 \[ \frac{d + e x}{c e} - \frac{\sqrt{2} \left (- 2 a c + b^{2} + b \sqrt{- 4 a c + b^{2}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} \left (d + e x\right )}{\sqrt{b + \sqrt{- 4 a c + b^{2}}}} \right )}}{2 c^{\frac{3}{2}} e \sqrt{b + \sqrt{- 4 a c + b^{2}}} \sqrt{- 4 a c + b^{2}}} + \frac{\sqrt{2} \left (- 2 a c + b^{2} - b \sqrt{- 4 a c + b^{2}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} \left (d + e x\right )}{\sqrt{b - \sqrt{- 4 a c + b^{2}}}} \right )}}{2 c^{\frac{3}{2}} e \sqrt{b - \sqrt{- 4 a c + b^{2}}} \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(d + e*x)/(c*e) - sqrt(2)*(-2*a*c + b**2 + b*sqrt(-4*a*c + b**2))*atan(sqrt(2)*s
qrt(c)*(d + e*x)/sqrt(b + sqrt(-4*a*c + b**2)))/(2*c**(3/2)*e*sqrt(b + sqrt(-4*a
*c + b**2))*sqrt(-4*a*c + b**2)) + sqrt(2)*(-2*a*c + b**2 - b*sqrt(-4*a*c + b**2
))*atan(sqrt(2)*sqrt(c)*(d + e*x)/sqrt(b - sqrt(-4*a*c + b**2)))/(2*c**(3/2)*e*s
qrt(b - sqrt(-4*a*c + b**2))*sqrt(-4*a*c + b**2))

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Mathematica [A]  time = 0.256574, size = 219, normalized size = 1.13 \[ \frac{-\frac{\sqrt{2} \left (b \sqrt{b^2-4 a c}+2 a c-b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} \left (b \sqrt{b^2-4 a c}-2 a c+b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}+2 \sqrt{c} (d+e x)}{2 c^{3/2} e} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[c]*(d + e*x) - (Sqrt[2]*(-b^2 + 2*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqr
t[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]*(b^2 - 2*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqr
t[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*c^(3/2)*e)

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Maple [C]  time = 0.095, size = 158, normalized size = 0.8 \[{\frac{x}{c}}+{\frac{1}{2\,ce}\sum _{{\it \_R}={\it RootOf} \left ( c{e}^{4}{{\it \_Z}}^{4}+4\,cd{e}^{3}{{\it \_Z}}^{3}+ \left ( 6\,c{d}^{2}{e}^{2}+b{e}^{2} \right ){{\it \_Z}}^{2}+ \left ( 4\,c{d}^{3}e+2\,bde \right ){\it \_Z}+c{d}^{4}+b{d}^{2}+a \right ) }{\frac{ \left ( -{{\it \_R}}^{2}b{e}^{2}-2\,{\it \_R}\,bde-b{d}^{2}-a \right ) \ln \left ( x-{\it \_R} \right ) }{2\,c{e}^{3}{{\it \_R}}^{3}+6\,cd{e}^{2}{{\it \_R}}^{2}+6\,{d}^{2}ec{\it \_R}+2\,c{d}^{3}+be{\it \_R}+bd}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.297898, size = 1662, normalized size = 8.61 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 9.92905, size = 178, normalized size = 0.92 \[ \operatorname{RootSum}{\left (t^{4} \left (256 a^{2} c^{5} e^{4} - 128 a b^{2} c^{4} e^{4} + 16 b^{4} c^{3} e^{4}\right ) + t^{2} \left (48 a^{2} b c^{2} e^{2} - 28 a b^{3} c e^{2} + 4 b^{5} e^{2}\right ) + a^{3}, \left ( t \mapsto t \log{\left (x + \frac{32 t^{3} a b c^{4} e^{3} - 8 t^{3} b^{3} c^{3} e^{3} - 4 t a^{2} c^{2} e + 8 t a b^{2} c e - 2 t b^{4} e + a^{2} c d - a b^{2} d}{a^{2} c e - a b^{2} e} \right )} \right )\right )} + \frac{x}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(_t**4*(256*a**2*c**5*e**4 - 128*a*b**2*c**4*e**4 + 16*b**4*c**3*e**4) +
_t**2*(48*a**2*b*c**2*e**2 - 28*a*b**3*c*e**2 + 4*b**5*e**2) + a**3, Lambda(_t,
_t*log(x + (32*_t**3*a*b*c**4*e**3 - 8*_t**3*b**3*c**3*e**3 - 4*_t*a**2*c**2*e +
8*_t*a*b**2*c*e - 2*_t*b**4*e + a**2*c*d - a*b**2*d)/(a**2*c*e - a*b**2*e)))) +
x/c

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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