3.37 \(\int \frac{1}{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4} \, dx\)

Optimal. Leaf size=529 \[ -\frac{d \log \left (\sqrt{c} \sqrt{4 a d^2+c^3}-\sqrt{2} \sqrt [4]{c} d \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}} \left (\frac{c}{d}+x\right )+d^2 \left (\frac{c}{d}+x\right )^2\right )}{4 \sqrt{2} c^{3/4} \sqrt{4 a d^2+c^3} \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}}}+\frac{d \log \left (\sqrt{c} \sqrt{4 a d^2+c^3}+\sqrt{2} \sqrt [4]{c} d \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}} \left (\frac{c}{d}+x\right )+d^2 \left (\frac{c}{d}+x\right )^2\right )}{4 \sqrt{2} c^{3/4} \sqrt{4 a d^2+c^3} \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}}}-\frac{d \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}}+\sqrt{2} c+\sqrt{2} d x}{\sqrt [4]{c} \sqrt{c^{3/2}-\sqrt{4 a d^2+c^3}}}\right )}{2 \sqrt{2} c^{3/4} \sqrt{4 a d^2+c^3} \sqrt{c^{3/2}-\sqrt{4 a d^2+c^3}}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}}-\sqrt{2} (c+d x)}{\sqrt [4]{c} \sqrt{c^{3/2}-\sqrt{4 a d^2+c^3}}}\right )}{2 \sqrt{2} c^{3/4} \sqrt{4 a d^2+c^3} \sqrt{c^{3/2}-\sqrt{4 a d^2+c^3}}} \]

[Out]

-(d*ArcTanh[(Sqrt[2]*c + c^(1/4)*Sqrt[c^(3/2) + Sqrt[c^3 + 4*a*d^2]] + Sqrt[2]*d
*x)/(c^(1/4)*Sqrt[c^(3/2) - Sqrt[c^3 + 4*a*d^2]])])/(2*Sqrt[2]*c^(3/4)*Sqrt[c^3
+ 4*a*d^2]*Sqrt[c^(3/2) - Sqrt[c^3 + 4*a*d^2]]) + (d*ArcTanh[(c^(1/4)*Sqrt[c^(3/
2) + Sqrt[c^3 + 4*a*d^2]] - Sqrt[2]*(c + d*x))/(c^(1/4)*Sqrt[c^(3/2) - Sqrt[c^3
+ 4*a*d^2]])])/(2*Sqrt[2]*c^(3/4)*Sqrt[c^3 + 4*a*d^2]*Sqrt[c^(3/2) - Sqrt[c^3 +
4*a*d^2]]) - (d*Log[Sqrt[c]*Sqrt[c^3 + 4*a*d^2] - Sqrt[2]*c^(1/4)*d*Sqrt[c^(3/2)
+ Sqrt[c^3 + 4*a*d^2]]*(c/d + x) + d^2*(c/d + x)^2])/(4*Sqrt[2]*c^(3/4)*Sqrt[c^
3 + 4*a*d^2]*Sqrt[c^(3/2) + Sqrt[c^3 + 4*a*d^2]]) + (d*Log[Sqrt[c]*Sqrt[c^3 + 4*
a*d^2] + Sqrt[2]*c^(1/4)*d*Sqrt[c^(3/2) + Sqrt[c^3 + 4*a*d^2]]*(c/d + x) + d^2*(
c/d + x)^2])/(4*Sqrt[2]*c^(3/4)*Sqrt[c^3 + 4*a*d^2]*Sqrt[c^(3/2) + Sqrt[c^3 + 4*
a*d^2]])

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Rubi [A]  time = 1.99819, antiderivative size = 529, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ -\frac{d \log \left (\sqrt{c} \sqrt{4 a d^2+c^3}-\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}} (c+d x)+(c+d x)^2\right )}{4 \sqrt{2} c^{3/4} \sqrt{4 a d^2+c^3} \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}}}+\frac{d \log \left (\sqrt{c} \sqrt{4 a d^2+c^3}+\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}} (c+d x)+(c+d x)^2\right )}{4 \sqrt{2} c^{3/4} \sqrt{4 a d^2+c^3} \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}}}-\frac{d \tanh ^{-1}\left (\frac{\sqrt [4]{c} \left (\sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}}+\sqrt{2} c^{3/4}\right )+\sqrt{2} d x}{\sqrt [4]{c} \sqrt{c^{3/2}-\sqrt{4 a d^2+c^3}}}\right )}{2 \sqrt{2} c^{3/4} \sqrt{4 a d^2+c^3} \sqrt{c^{3/2}-\sqrt{4 a d^2+c^3}}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}}-\sqrt{2} (c+d x)}{\sqrt [4]{c} \sqrt{c^{3/2}-\sqrt{4 a d^2+c^3}}}\right )}{2 \sqrt{2} c^{3/4} \sqrt{4 a d^2+c^3} \sqrt{c^{3/2}-\sqrt{4 a d^2+c^3}}} \]

Antiderivative was successfully verified.

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

[Out]

-(d*ArcTanh[(c^(1/4)*(Sqrt[2]*c^(3/4) + Sqrt[c^(3/2) + Sqrt[c^3 + 4*a*d^2]]) + S
qrt[2]*d*x)/(c^(1/4)*Sqrt[c^(3/2) - Sqrt[c^3 + 4*a*d^2]])])/(2*Sqrt[2]*c^(3/4)*S
qrt[c^3 + 4*a*d^2]*Sqrt[c^(3/2) - Sqrt[c^3 + 4*a*d^2]]) + (d*ArcTanh[(c^(1/4)*Sq
rt[c^(3/2) + Sqrt[c^3 + 4*a*d^2]] - Sqrt[2]*(c + d*x))/(c^(1/4)*Sqrt[c^(3/2) - S
qrt[c^3 + 4*a*d^2]])])/(2*Sqrt[2]*c^(3/4)*Sqrt[c^3 + 4*a*d^2]*Sqrt[c^(3/2) - Sqr
t[c^3 + 4*a*d^2]]) - (d*Log[Sqrt[c]*Sqrt[c^3 + 4*a*d^2] - Sqrt[2]*c^(1/4)*Sqrt[c
^(3/2) + Sqrt[c^3 + 4*a*d^2]]*(c + d*x) + (c + d*x)^2])/(4*Sqrt[2]*c^(3/4)*Sqrt[
c^3 + 4*a*d^2]*Sqrt[c^(3/2) + Sqrt[c^3 + 4*a*d^2]]) + (d*Log[Sqrt[c]*Sqrt[c^3 +
4*a*d^2] + Sqrt[2]*c^(1/4)*Sqrt[c^(3/2) + Sqrt[c^3 + 4*a*d^2]]*(c + d*x) + (c +
d*x)^2])/(4*Sqrt[2]*c^(3/4)*Sqrt[c^3 + 4*a*d^2]*Sqrt[c^(3/2) + Sqrt[c^3 + 4*a*d^
2]])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [C]  time = 0.0451106, size = 71, normalized size = 0.13 \[ \frac{1}{4} \text{RootSum}\left [\text{$\#$1}^4 d^2+4 \text{$\#$1}^3 c d+4 \text{$\#$1}^2 c^2+4 a c\&,\frac{\log (x-\text{$\#$1})}{\text{$\#$1}^3 d^2+3 \text{$\#$1}^2 c d+2 \text{$\#$1} c^2}\&\right ] \]

Antiderivative was successfully verified.

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

[Out]

RootSum[4*a*c + 4*c^2*#1^2 + 4*c*d*#1^3 + d^2*#1^4 & , Log[x - #1]/(2*c^2*#1 + 3
*c*d*#1^2 + d^2*#1^3) & ]/4

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Maple [C]  time = 0.056, size = 64, normalized size = 0.1 \[{\frac{1}{4}\sum _{{\it \_R}={\it RootOf} \left ({d}^{2}{{\it \_Z}}^{4}+4\,cd{{\it \_Z}}^{3}+4\,{c}^{2}{{\it \_Z}}^{2}+4\,ac \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{3}{d}^{2}+3\,{{\it \_R}}^{2}cd+2\,{\it \_R}\,{c}^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{d^{2} x^{4} + 4 \, c d x^{3} + 4 \, c^{2} x^{2} + 4 \, a c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(d^2*x^4 + 4*c*d*x^3 + 4*c^2*x^2 + 4*a*c), x)

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Fricas [A]  time = 0.299792, size = 1222, normalized size = 2.31 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*sqrt(-(2*(a*c^3 + 4*a^2*d^2)*sqrt(-d^2/(a*c^9 + 8*a^2*c^6*d^2 + 16*a^3*c^3*d
^4)) + 1)/(a*c^3 + 4*a^2*d^2))*log(d^2*x + c*d + (2*a*c*d^2 + (a*c^7 + 4*a^2*c^4
*d^2)*sqrt(-d^2/(a*c^9 + 8*a^2*c^6*d^2 + 16*a^3*c^3*d^4)))*sqrt(-(2*(a*c^3 + 4*a
^2*d^2)*sqrt(-d^2/(a*c^9 + 8*a^2*c^6*d^2 + 16*a^3*c^3*d^4)) + 1)/(a*c^3 + 4*a^2*
d^2))) - 1/8*sqrt(-(2*(a*c^3 + 4*a^2*d^2)*sqrt(-d^2/(a*c^9 + 8*a^2*c^6*d^2 + 16*
a^3*c^3*d^4)) + 1)/(a*c^3 + 4*a^2*d^2))*log(d^2*x + c*d - (2*a*c*d^2 + (a*c^7 +
4*a^2*c^4*d^2)*sqrt(-d^2/(a*c^9 + 8*a^2*c^6*d^2 + 16*a^3*c^3*d^4)))*sqrt(-(2*(a*
c^3 + 4*a^2*d^2)*sqrt(-d^2/(a*c^9 + 8*a^2*c^6*d^2 + 16*a^3*c^3*d^4)) + 1)/(a*c^3
+ 4*a^2*d^2))) + 1/8*sqrt((2*(a*c^3 + 4*a^2*d^2)*sqrt(-d^2/(a*c^9 + 8*a^2*c^6*d
^2 + 16*a^3*c^3*d^4)) - 1)/(a*c^3 + 4*a^2*d^2))*log(d^2*x + c*d + (2*a*c*d^2 - (
a*c^7 + 4*a^2*c^4*d^2)*sqrt(-d^2/(a*c^9 + 8*a^2*c^6*d^2 + 16*a^3*c^3*d^4)))*sqrt
((2*(a*c^3 + 4*a^2*d^2)*sqrt(-d^2/(a*c^9 + 8*a^2*c^6*d^2 + 16*a^3*c^3*d^4)) - 1)
/(a*c^3 + 4*a^2*d^2))) - 1/8*sqrt((2*(a*c^3 + 4*a^2*d^2)*sqrt(-d^2/(a*c^9 + 8*a^
2*c^6*d^2 + 16*a^3*c^3*d^4)) - 1)/(a*c^3 + 4*a^2*d^2))*log(d^2*x + c*d - (2*a*c*
d^2 - (a*c^7 + 4*a^2*c^4*d^2)*sqrt(-d^2/(a*c^9 + 8*a^2*c^6*d^2 + 16*a^3*c^3*d^4)
))*sqrt((2*(a*c^3 + 4*a^2*d^2)*sqrt(-d^2/(a*c^9 + 8*a^2*c^6*d^2 + 16*a^3*c^3*d^4
)) - 1)/(a*c^3 + 4*a^2*d^2)))

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Sympy [A]  time = 3.1435, size = 88, normalized size = 0.17 \[ \operatorname{RootSum}{\left (t^{4} \left (16384 a^{3} c^{3} d^{2} + 4096 a^{2} c^{6}\right ) + 128 t^{2} a c^{3} + 1, \left ( t \mapsto t \log{\left (x + \frac{- 1024 t^{3} a^{2} c^{4} d^{2} - 256 t^{3} a c^{7} + 16 t a c d^{2} - 4 t c^{4} + c d}{d^{2}} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(_t**4*(16384*a**3*c**3*d**2 + 4096*a**2*c**6) + 128*_t**2*a*c**3 + 1, La
mbda(_t, _t*log(x + (-1024*_t**3*a**2*c**4*d**2 - 256*_t**3*a*c**7 + 16*_t*a*c*d
**2 - 4*_t*c**4 + c*d)/d**2)))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{d^{2} x^{4} + 4 \, c d x^{3} + 4 \, c^{2} x^{2} + 4 \, a c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(d^2*x^4 + 4*c*d*x^3 + 4*c^2*x^2 + 4*a*c), x)