3.38 \(\int \frac{1}{\left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^2} \, dx\)
Optimal. Leaf size=746 \[ -\frac{d \left (-c^{3/2} \sqrt{4 a d^2+c^3}+12 a d^2+c^3\right ) \log \left (\sqrt{c} \sqrt{4 a d^2+c^3}-\sqrt{2} \sqrt [4]{c} d \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}} \left (\frac{c}{d}+x\right )+d^2 \left (\frac{c}{d}+x\right )^2\right )}{64 \sqrt{2} a c^{7/4} \left (4 a d^2+c^3\right )^{3/2} \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}}}+\frac{d \left (-c^{3/2} \sqrt{4 a d^2+c^3}+12 a d^2+c^3\right ) \log \left (\sqrt{c} \sqrt{4 a d^2+c^3}+\sqrt{2} \sqrt [4]{c} d \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}} \left (\frac{c}{d}+x\right )+d^2 \left (\frac{c}{d}+x\right )^2\right )}{64 \sqrt{2} a c^{7/4} \left (4 a d^2+c^3\right )^{3/2} \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}}}-\frac{d \left (c^{3/2} \sqrt{4 a d^2+c^3}+12 a d^2+c^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}}+\sqrt{2} c+\sqrt{2} d x}{\sqrt [4]{c} \sqrt{c^{3/2}-\sqrt{4 a d^2+c^3}}}\right )}{32 \sqrt{2} a c^{7/4} \left (4 a d^2+c^3\right )^{3/2} \sqrt{c^{3/2}-\sqrt{4 a d^2+c^3}}}+\frac{d \left (c^{3/2} \sqrt{4 a d^2+c^3}+12 a d^2+c^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}}-\sqrt{2} (c+d x)}{\sqrt [4]{c} \sqrt{c^{3/2}-\sqrt{4 a d^2+c^3}}}\right )}{32 \sqrt{2} a c^{7/4} \left (4 a d^2+c^3\right )^{3/2} \sqrt{c^{3/2}-\sqrt{4 a d^2+c^3}}}-\frac{\left (\frac{c}{d}+x\right ) \left (-4 a d^2+c^3-c d^2 \left (\frac{c}{d}+x\right )^2\right )}{16 a c \left (4 a d^2+c^3\right ) \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )} \]
[Out]
-((c/d + x)*(c^3 - 4*a*d^2 - c*d^2*(c/d + x)^2))/(16*a*c*(c^3 + 4*a*d^2)*(4*a*c
+ 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4)) - (d*(c^3 + 12*a*d^2 + c^(3/2)*Sqrt[c^3 + 4*
a*d^2])*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]])])/(32*Sqrt[2]*a*c^(7/4)*(c
^3 + 4*a*d^2)^(3/2)*Sqrt[c^(3/2) - Sqrt[c^3 + 4*a*d^2]]) + (d*(c^3 + 12*a*d^2 +
c^(3/2)*Sqrt[c^3 + 4*a*d^2])*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]])])/(32*Sqrt[
2]*a*c^(7/4)*(c^3 + 4*a*d^2)^(3/2)*Sqrt[c^(3/2) - Sqrt[c^3 + 4*a*d^2]]) - (d*(c^
3 + 12*a*d^2 - c^(3/2)*Sqrt[c^3 + 4*a*d^2])*Log[Sqrt[c]*Sqrt[c^3 + 4*a*d^2] - Sq
rt[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]
)/(64*Sqrt[2]*a*c^(7/4)*(c^3 + 4*a*d^2)^(3/2)*Sqrt[c^(3/2) + Sqrt[c^3 + 4*a*d^2]
]) + (d*(c^3 + 12*a*d^2 - c^(3/2)*Sqrt[c^3 + 4*a*d^2])*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])/(64*Sqrt[2]*a*c^(7/4)*(c^3 + 4*a*d^2)^(3/2)*Sqrt[c^(3/2) + Sqrt[c^3
+ 4*a*d^2]])
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Rubi [A] time = 4.06436, antiderivative size = 746, normalized size of antiderivative = 1.,
number of steps used = 11, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241
\[ -\frac{d \left (-c^{3/2} \sqrt{4 a d^2+c^3}+12 a d^2+c^3\right ) \log \left (\sqrt{c} \sqrt{4 a d^2+c^3}-\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}} (c+d x)+(c+d x)^2\right )}{64 \sqrt{2} a c^{7/4} \left (4 a d^2+c^3\right )^{3/2} \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}}}+\frac{d \left (-c^{3/2} \sqrt{4 a d^2+c^3}+12 a d^2+c^3\right ) \log \left (\sqrt{c} \sqrt{4 a d^2+c^3}+\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}} (c+d x)+(c+d x)^2\right )}{64 \sqrt{2} a c^{7/4} \left (4 a d^2+c^3\right )^{3/2} \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}}}-\frac{d \left (c^{3/2} \sqrt{4 a d^2+c^3}+12 a d^2+c^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \left (\sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}}+\sqrt{2} c^{3/4}\right )+\sqrt{2} d x}{\sqrt [4]{c} \sqrt{c^{3/2}-\sqrt{4 a d^2+c^3}}}\right )}{32 \sqrt{2} a c^{7/4} \left (4 a d^2+c^3\right )^{3/2} \sqrt{c^{3/2}-\sqrt{4 a d^2+c^3}}}+\frac{d \left (c^{3/2} \sqrt{4 a d^2+c^3}+12 a d^2+c^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}}-\sqrt{2} (c+d x)}{\sqrt [4]{c} \sqrt{c^{3/2}-\sqrt{4 a d^2+c^3}}}\right )}{32 \sqrt{2} a c^{7/4} \left (4 a d^2+c^3\right )^{3/2} \sqrt{c^{3/2}-\sqrt{4 a d^2+c^3}}}-\frac{(c+d x) \left (-4 a d^2+c^3-c (c+d x)^2\right )}{16 a c d \left (4 a d^2+c^3\right ) \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )} \]
Antiderivative was successfully verified.
[In] Int[(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4)^(-2),x]
[Out]
-((c + d*x)*(c^3 - 4*a*d^2 - c*(c + d*x)^2))/(16*a*c*d*(c^3 + 4*a*d^2)*(4*a*c +
4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4)) - (d*(c^3 + 12*a*d^2 + c^(3/2)*Sqrt[c^3 + 4*a*
d^2])*ArcTanh[(c^(1/4)*(Sqrt[2]*c^(3/4) + 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]])])/(32*Sqrt[2]*a*c^(7
/4)*(c^3 + 4*a*d^2)^(3/2)*Sqrt[c^(3/2) - Sqrt[c^3 + 4*a*d^2]]) + (d*(c^3 + 12*a*
d^2 + c^(3/2)*Sqrt[c^3 + 4*a*d^2])*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]])])/(32
*Sqrt[2]*a*c^(7/4)*(c^3 + 4*a*d^2)^(3/2)*Sqrt[c^(3/2) - Sqrt[c^3 + 4*a*d^2]]) -
(d*(c^3 + 12*a*d^2 - c^(3/2)*Sqrt[c^3 + 4*a*d^2])*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]
)/(64*Sqrt[2]*a*c^(7/4)*(c^3 + 4*a*d^2)^(3/2)*Sqrt[c^(3/2) + Sqrt[c^3 + 4*a*d^2]
]) + (d*(c^3 + 12*a*d^2 - c^(3/2)*Sqrt[c^3 + 4*a*d^2])*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])/(64*Sqrt[2]*a*c^(7/4)*(c^3 + 4*a*d^2)^(3/2)*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)**2,x)
[Out]
Timed out
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Mathematica [C] time = 0.173404, size = 182, normalized size = 0.24 \[ \frac{\text{RootSum}\left [\text{$\#$1}^4 d^2+4 \text{$\#$1}^3 c d+4 \text{$\#$1}^2 c^2+4 a c\&,\frac{\text{$\#$1}^2 c d^2 \log (x-\text{$\#$1})+12 a d^2 \log (x-\text{$\#$1})+2 c^3 \log (x-\text{$\#$1})+2 \text{$\#$1} c^2 d \log (x-\text{$\#$1})}{\text{$\#$1}^3 d^2+3 \text{$\#$1}^2 c d+2 \text{$\#$1} c^2}\&\right ]+\frac{4 (c+d x) (4 a d+c x (2 c+d x))}{4 a c+x^2 (2 c+d x)^2}}{64 a c \left (4 a d^2+c^3\right )} \]
Antiderivative was successfully verified.
[In] Integrate[(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4)^(-2),x]
[Out]
((4*(c + d*x)*(4*a*d + c*x*(2*c + d*x)))/(4*a*c + x^2*(2*c + d*x)^2) + RootSum[4
*a*c + 4*c^2*#1^2 + 4*c*d*#1^3 + d^2*#1^4 & , (2*c^3*Log[x - #1] + 12*a*d^2*Log[
x - #1] + 2*c^2*d*Log[x - #1]*#1 + c*d^2*Log[x - #1]*#1^2)/(2*c^2*#1 + 3*c*d*#1^
2 + d^2*#1^3) & ])/(64*a*c*(c^3 + 4*a*d^2))
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Maple [C] time = 0.045, size = 232, normalized size = 0.3 \[{\frac{1}{{d}^{2}{x}^{4}+4\,cd{x}^{3}+4\,{c}^{2}{x}^{2}+4\,ac} \left ({\frac{{d}^{2}{x}^{3}}{16\,a \left ( 4\,a{d}^{2}+{c}^{3} \right ) }}+{\frac{3\,cd{x}^{2}}{16\,a \left ( 4\,a{d}^{2}+{c}^{3} \right ) }}+{\frac{ \left ( 2\,a{d}^{2}+{c}^{3} \right ) x}{8\, \left ( 4\,a{d}^{2}+{c}^{3} \right ) ac}}+{\frac{d}{16\,a{d}^{2}+4\,{c}^{3}}} \right ) }+{\frac{1}{64\,ac}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{4}{d}^{2}+4\,{{\it \_Z}}^{3}cd+4\,{{\it \_Z}}^{2}{c}^{2}+4\,ac \right ) }{\frac{ \left ({{\it \_R}}^{2}c{d}^{2}+2\,{\it \_R}\,{c}^{2}d+12\,a{d}^{2}+2\,{c}^{3} \right ) \ln \left ( x-{\it \_R} \right ) }{ \left ( 4\,a{d}^{2}+{c}^{3} \right ) \left ({{\it \_R}}^{3}{d}^{2}+3\,{{\it \_R}}^{2}cd+2\,{\it \_R}\,{c}^{2} \right ) }}} \]
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)^2,x)
[Out]
(1/16*d^2/a/(4*a*d^2+c^3)*x^3+3/16/a*c*d/(4*a*d^2+c^3)*x^2+1/8/c*(2*a*d^2+c^3)/(
4*a*d^2+c^3)/a*x+1/4*d/(4*a*d^2+c^3))/(d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c)+1/64/a
/c*sum((_R^2*c*d^2+2*_R*c^2*d+12*a*d^2+2*c^3)/(4*a*d^2+c^3)/(_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. \[ \frac{c d^{2} x^{3} + 3 \, c^{2} d x^{2} + 4 \, a c d + 2 \,{\left (c^{3} + 2 \, a d^{2}\right )} x}{16 \,{\left (4 \, a^{2} c^{5} + 16 \, a^{3} c^{2} d^{2} +{\left (a c^{4} d^{2} + 4 \, a^{2} c d^{4}\right )} x^{4} + 4 \,{\left (a c^{5} d + 4 \, a^{2} c^{2} d^{3}\right )} x^{3} + 4 \,{\left (a c^{6} + 4 \, a^{2} c^{3} d^{2}\right )} x^{2}\right )}} + \frac{\int \frac{c d^{2} x^{2} + 2 \, c^{2} d x + 2 \, c^{3} + 12 \, a d^{2}}{d^{2} x^{4} + 4 \, c d x^{3} + 4 \, c^{2} x^{2} + 4 \, a c}\,{d x}}{16 \,{\left (a c^{4} + 4 \, a^{2} c d^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d^2*x^4 + 4*c*d*x^3 + 4*c^2*x^2 + 4*a*c)^(-2),x, algorithm="maxima")
[Out]
1/16*(c*d^2*x^3 + 3*c^2*d*x^2 + 4*a*c*d + 2*(c^3 + 2*a*d^2)*x)/(4*a^2*c^5 + 16*a
^3*c^2*d^2 + (a*c^4*d^2 + 4*a^2*c*d^4)*x^4 + 4*(a*c^5*d + 4*a^2*c^2*d^3)*x^3 + 4
*(a*c^6 + 4*a^2*c^3*d^2)*x^2) + 1/16*integrate((c*d^2*x^2 + 2*c^2*d*x + 2*c^3 +
12*a*d^2)/(d^2*x^4 + 4*c*d*x^3 + 4*c^2*x^2 + 4*a*c), x)/(a*c^4 + 4*a^2*c*d^2)
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Fricas [A] time = 0.347202, size = 4350, normalized size = 5.83 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d^2*x^4 + 4*c*d*x^3 + 4*c^2*x^2 + 4*a*c)^(-2),x, algorithm="fricas")
[Out]
1/64*(4*c*d^2*x^3 + 12*c^2*d*x^2 + 16*a*c*d + (4*a^2*c^5 + 16*a^3*c^2*d^2 + (a*c
^4*d^2 + 4*a^2*c*d^4)*x^4 + 4*(a*c^5*d + 4*a^2*c^2*d^3)*x^3 + 4*(a*c^6 + 4*a^2*c
^3*d^2)*x^2)*sqrt(-(c^6 + 15*a*c^3*d^2 + 60*a^2*d^4 + 2*(a^3*c^11 + 12*a^4*c^8*d
^2 + 48*a^5*c^5*d^4 + 64*a^6*c^2*d^6)*sqrt(-(25*c^6*d^6 + 360*a*c^3*d^8 + 1296*a
^2*d^10)/(a^3*c^25 + 24*a^4*c^22*d^2 + 240*a^5*c^19*d^4 + 1280*a^6*c^16*d^6 + 38
40*a^7*c^13*d^8 + 6144*a^8*c^10*d^10 + 4096*a^9*c^7*d^12)))/(a^3*c^11 + 12*a^4*c
^8*d^2 + 48*a^5*c^5*d^4 + 64*a^6*c^2*d^6))*log(5*c^7*d^3 + 81*a*c^4*d^5 + 324*a^
2*c*d^7 + (5*c^6*d^4 + 81*a*c^3*d^6 + 324*a^2*d^8)*x + (5*a^2*c^8*d^4 + 96*a^3*c
^5*d^6 + 432*a^4*c^2*d^8 + (a^3*c^19 + 20*a^4*c^16*d^2 + 144*a^5*c^13*d^4 + 448*
a^6*c^10*d^6 + 512*a^7*c^7*d^8)*sqrt(-(25*c^6*d^6 + 360*a*c^3*d^8 + 1296*a^2*d^1
0)/(a^3*c^25 + 24*a^4*c^22*d^2 + 240*a^5*c^19*d^4 + 1280*a^6*c^16*d^6 + 3840*a^7
*c^13*d^8 + 6144*a^8*c^10*d^10 + 4096*a^9*c^7*d^12)))*sqrt(-(c^6 + 15*a*c^3*d^2
+ 60*a^2*d^4 + 2*(a^3*c^11 + 12*a^4*c^8*d^2 + 48*a^5*c^5*d^4 + 64*a^6*c^2*d^6)*s
qrt(-(25*c^6*d^6 + 360*a*c^3*d^8 + 1296*a^2*d^10)/(a^3*c^25 + 24*a^4*c^22*d^2 +
240*a^5*c^19*d^4 + 1280*a^6*c^16*d^6 + 3840*a^7*c^13*d^8 + 6144*a^8*c^10*d^10 +
4096*a^9*c^7*d^12)))/(a^3*c^11 + 12*a^4*c^8*d^2 + 48*a^5*c^5*d^4 + 64*a^6*c^2*d^
6))) - (4*a^2*c^5 + 16*a^3*c^2*d^2 + (a*c^4*d^2 + 4*a^2*c*d^4)*x^4 + 4*(a*c^5*d
+ 4*a^2*c^2*d^3)*x^3 + 4*(a*c^6 + 4*a^2*c^3*d^2)*x^2)*sqrt(-(c^6 + 15*a*c^3*d^2
+ 60*a^2*d^4 + 2*(a^3*c^11 + 12*a^4*c^8*d^2 + 48*a^5*c^5*d^4 + 64*a^6*c^2*d^6)*s
qrt(-(25*c^6*d^6 + 360*a*c^3*d^8 + 1296*a^2*d^10)/(a^3*c^25 + 24*a^4*c^22*d^2 +
240*a^5*c^19*d^4 + 1280*a^6*c^16*d^6 + 3840*a^7*c^13*d^8 + 6144*a^8*c^10*d^10 +
4096*a^9*c^7*d^12)))/(a^3*c^11 + 12*a^4*c^8*d^2 + 48*a^5*c^5*d^4 + 64*a^6*c^2*d^
6))*log(5*c^7*d^3 + 81*a*c^4*d^5 + 324*a^2*c*d^7 + (5*c^6*d^4 + 81*a*c^3*d^6 + 3
24*a^2*d^8)*x - (5*a^2*c^8*d^4 + 96*a^3*c^5*d^6 + 432*a^4*c^2*d^8 + (a^3*c^19 +
20*a^4*c^16*d^2 + 144*a^5*c^13*d^4 + 448*a^6*c^10*d^6 + 512*a^7*c^7*d^8)*sqrt(-(
25*c^6*d^6 + 360*a*c^3*d^8 + 1296*a^2*d^10)/(a^3*c^25 + 24*a^4*c^22*d^2 + 240*a^
5*c^19*d^4 + 1280*a^6*c^16*d^6 + 3840*a^7*c^13*d^8 + 6144*a^8*c^10*d^10 + 4096*a
^9*c^7*d^12)))*sqrt(-(c^6 + 15*a*c^3*d^2 + 60*a^2*d^4 + 2*(a^3*c^11 + 12*a^4*c^8
*d^2 + 48*a^5*c^5*d^4 + 64*a^6*c^2*d^6)*sqrt(-(25*c^6*d^6 + 360*a*c^3*d^8 + 1296
*a^2*d^10)/(a^3*c^25 + 24*a^4*c^22*d^2 + 240*a^5*c^19*d^4 + 1280*a^6*c^16*d^6 +
3840*a^7*c^13*d^8 + 6144*a^8*c^10*d^10 + 4096*a^9*c^7*d^12)))/(a^3*c^11 + 12*a^4
*c^8*d^2 + 48*a^5*c^5*d^4 + 64*a^6*c^2*d^6))) + (4*a^2*c^5 + 16*a^3*c^2*d^2 + (a
*c^4*d^2 + 4*a^2*c*d^4)*x^4 + 4*(a*c^5*d + 4*a^2*c^2*d^3)*x^3 + 4*(a*c^6 + 4*a^2
*c^3*d^2)*x^2)*sqrt(-(c^6 + 15*a*c^3*d^2 + 60*a^2*d^4 - 2*(a^3*c^11 + 12*a^4*c^8
*d^2 + 48*a^5*c^5*d^4 + 64*a^6*c^2*d^6)*sqrt(-(25*c^6*d^6 + 360*a*c^3*d^8 + 1296
*a^2*d^10)/(a^3*c^25 + 24*a^4*c^22*d^2 + 240*a^5*c^19*d^4 + 1280*a^6*c^16*d^6 +
3840*a^7*c^13*d^8 + 6144*a^8*c^10*d^10 + 4096*a^9*c^7*d^12)))/(a^3*c^11 + 12*a^4
*c^8*d^2 + 48*a^5*c^5*d^4 + 64*a^6*c^2*d^6))*log(5*c^7*d^3 + 81*a*c^4*d^5 + 324*
a^2*c*d^7 + (5*c^6*d^4 + 81*a*c^3*d^6 + 324*a^2*d^8)*x + (5*a^2*c^8*d^4 + 96*a^3
*c^5*d^6 + 432*a^4*c^2*d^8 - (a^3*c^19 + 20*a^4*c^16*d^2 + 144*a^5*c^13*d^4 + 44
8*a^6*c^10*d^6 + 512*a^7*c^7*d^8)*sqrt(-(25*c^6*d^6 + 360*a*c^3*d^8 + 1296*a^2*d
^10)/(a^3*c^25 + 24*a^4*c^22*d^2 + 240*a^5*c^19*d^4 + 1280*a^6*c^16*d^6 + 3840*a
^7*c^13*d^8 + 6144*a^8*c^10*d^10 + 4096*a^9*c^7*d^12)))*sqrt(-(c^6 + 15*a*c^3*d^
2 + 60*a^2*d^4 - 2*(a^3*c^11 + 12*a^4*c^8*d^2 + 48*a^5*c^5*d^4 + 64*a^6*c^2*d^6)
*sqrt(-(25*c^6*d^6 + 360*a*c^3*d^8 + 1296*a^2*d^10)/(a^3*c^25 + 24*a^4*c^22*d^2
+ 240*a^5*c^19*d^4 + 1280*a^6*c^16*d^6 + 3840*a^7*c^13*d^8 + 6144*a^8*c^10*d^10
+ 4096*a^9*c^7*d^12)))/(a^3*c^11 + 12*a^4*c^8*d^2 + 48*a^5*c^5*d^4 + 64*a^6*c^2*
d^6))) - (4*a^2*c^5 + 16*a^3*c^2*d^2 + (a*c^4*d^2 + 4*a^2*c*d^4)*x^4 + 4*(a*c^5*
d + 4*a^2*c^2*d^3)*x^3 + 4*(a*c^6 + 4*a^2*c^3*d^2)*x^2)*sqrt(-(c^6 + 15*a*c^3*d^
2 + 60*a^2*d^4 - 2*(a^3*c^11 + 12*a^4*c^8*d^2 + 48*a^5*c^5*d^4 + 64*a^6*c^2*d^6)
*sqrt(-(25*c^6*d^6 + 360*a*c^3*d^8 + 1296*a^2*d^10)/(a^3*c^25 + 24*a^4*c^22*d^2
+ 240*a^5*c^19*d^4 + 1280*a^6*c^16*d^6 + 3840*a^7*c^13*d^8 + 6144*a^8*c^10*d^10
+ 4096*a^9*c^7*d^12)))/(a^3*c^11 + 12*a^4*c^8*d^2 + 48*a^5*c^5*d^4 + 64*a^6*c^2*
d^6))*log(5*c^7*d^3 + 81*a*c^4*d^5 + 324*a^2*c*d^7 + (5*c^6*d^4 + 81*a*c^3*d^6 +
324*a^2*d^8)*x - (5*a^2*c^8*d^4 + 96*a^3*c^5*d^6 + 432*a^4*c^2*d^8 - (a^3*c^19
+ 20*a^4*c^16*d^2 + 144*a^5*c^13*d^4 + 448*a^6*c^10*d^6 + 512*a^7*c^7*d^8)*sqrt(
-(25*c^6*d^6 + 360*a*c^3*d^8 + 1296*a^2*d^10)/(a^3*c^25 + 24*a^4*c^22*d^2 + 240*
a^5*c^19*d^4 + 1280*a^6*c^16*d^6 + 3840*a^7*c^13*d^8 + 6144*a^8*c^10*d^10 + 4096
*a^9*c^7*d^12)))*sqrt(-(c^6 + 15*a*c^3*d^2 + 60*a^2*d^4 - 2*(a^3*c^11 + 12*a^4*c
^8*d^2 + 48*a^5*c^5*d^4 + 64*a^6*c^2*d^6)*sqrt(-(25*c^6*d^6 + 360*a*c^3*d^8 + 12
96*a^2*d^10)/(a^3*c^25 + 24*a^4*c^22*d^2 + 240*a^5*c^19*d^4 + 1280*a^6*c^16*d^6
+ 3840*a^7*c^13*d^8 + 6144*a^8*c^10*d^10 + 4096*a^9*c^7*d^12)))/(a^3*c^11 + 12*a
^4*c^8*d^2 + 48*a^5*c^5*d^4 + 64*a^6*c^2*d^6))) + 8*(c^3 + 2*a*d^2)*x)/(4*a^2*c^
5 + 16*a^3*c^2*d^2 + (a*c^4*d^2 + 4*a^2*c*d^4)*x^4 + 4*(a*c^5*d + 4*a^2*c^2*d^3)
*x^3 + 4*(a*c^6 + 4*a^2*c^3*d^2)*x^2)
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Sympy [A] time = 20.2243, size = 427, normalized size = 0.57 \[ \frac{4 a c d + 3 c^{2} d x^{2} + c d^{2} x^{3} + x \left (4 a d^{2} + 2 c^{3}\right )}{256 a^{3} c^{2} d^{2} + 64 a^{2} c^{5} + x^{4} \left (64 a^{2} c d^{4} + 16 a c^{4} d^{2}\right ) + x^{3} \left (256 a^{2} c^{2} d^{3} + 64 a c^{5} d\right ) + x^{2} \left (256 a^{2} c^{3} d^{2} + 64 a c^{6}\right )} + \operatorname{RootSum}{\left (t^{4} \left (1073741824 a^{9} c^{7} d^{6} + 805306368 a^{8} c^{10} d^{4} + 201326592 a^{7} c^{13} d^{2} + 16777216 a^{6} c^{16}\right ) + t^{2} \left (491520 a^{5} c^{5} d^{4} + 122880 a^{4} c^{8} d^{2} + 8192 a^{3} c^{11}\right ) + 81 a^{2} d^{4} + 18 a c^{3} d^{2} + c^{6}, \left ( t \mapsto t \log{\left (x + \frac{- 67108864 t^{3} a^{7} c^{7} d^{8} - 58720256 t^{3} a^{6} c^{10} d^{6} - 18874368 t^{3} a^{5} c^{13} d^{4} - 2621440 t^{3} a^{4} c^{16} d^{2} - 131072 t^{3} a^{3} c^{19} + 27648 t a^{4} c^{2} d^{8} - 9216 t a^{3} c^{5} d^{6} - 5440 t a^{2} c^{8} d^{4} - 736 t a c^{11} d^{2} - 32 t c^{14} + 324 a^{2} c d^{7} + 81 a c^{4} d^{5} + 5 c^{7} d^{3}}{324 a^{2} d^{8} + 81 a c^{3} d^{6} + 5 c^{6} d^{4}} \right )} \right )\right )} \]
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)**2,x)
[Out]
(4*a*c*d + 3*c**2*d*x**2 + c*d**2*x**3 + x*(4*a*d**2 + 2*c**3))/(256*a**3*c**2*d
**2 + 64*a**2*c**5 + x**4*(64*a**2*c*d**4 + 16*a*c**4*d**2) + x**3*(256*a**2*c**
2*d**3 + 64*a*c**5*d) + x**2*(256*a**2*c**3*d**2 + 64*a*c**6)) + RootSum(_t**4*(
1073741824*a**9*c**7*d**6 + 805306368*a**8*c**10*d**4 + 201326592*a**7*c**13*d**
2 + 16777216*a**6*c**16) + _t**2*(491520*a**5*c**5*d**4 + 122880*a**4*c**8*d**2
+ 8192*a**3*c**11) + 81*a**2*d**4 + 18*a*c**3*d**2 + c**6, Lambda(_t, _t*log(x +
(-67108864*_t**3*a**7*c**7*d**8 - 58720256*_t**3*a**6*c**10*d**6 - 18874368*_t*
*3*a**5*c**13*d**4 - 2621440*_t**3*a**4*c**16*d**2 - 131072*_t**3*a**3*c**19 + 2
7648*_t*a**4*c**2*d**8 - 9216*_t*a**3*c**5*d**6 - 5440*_t*a**2*c**8*d**4 - 736*_
t*a*c**11*d**2 - 32*_t*c**14 + 324*a**2*c*d**7 + 81*a*c**4*d**5 + 5*c**7*d**3)/(
324*a**2*d**8 + 81*a*c**3*d**6 + 5*c**6*d**4))))
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (d^{2} x^{4} + 4 \, c d x^{3} + 4 \, c^{2} x^{2} + 4 \, a c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d^2*x^4 + 4*c*d*x^3 + 4*c^2*x^2 + 4*a*c)^(-2),x, algorithm="giac")
[Out]
integrate((d^2*x^4 + 4*c*d*x^3 + 4*c^2*x^2 + 4*a*c)^(-2), x)