3.1082 \(\int \frac{x^{11/2}}{\left (a+b x^2+c x^4\right )^3} \, dx\)

Optimal. Leaf size=569 \[ \frac{\sqrt{x} \left (x^2 \left (20 a c+7 b^2\right )+24 a b\right )}{16 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{x^{5/2} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac{3 \left (\sqrt{b^2-4 a c} \left (20 a c+7 b^2\right )+36 a b c+7 b^3\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} \sqrt [4]{c} \left (b^2-4 a c\right )^{5/2} \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{3 \left (-\frac{36 a b c+7 b^3}{\sqrt{b^2-4 a c}}+20 a c+7 b^2\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} \sqrt [4]{c} \left (b^2-4 a c\right )^2 \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{3 \left (\sqrt{b^2-4 a c} \left (20 a c+7 b^2\right )+36 a b c+7 b^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} \sqrt [4]{c} \left (b^2-4 a c\right )^{5/2} \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{3 \left (-\frac{36 a b c+7 b^3}{\sqrt{b^2-4 a c}}+20 a c+7 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} \sqrt [4]{c} \left (b^2-4 a c\right )^2 \left (\sqrt{b^2-4 a c}-b\right )^{3/4}} \]

[Out]

(x^(5/2)*(2*a + b*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (Sqrt[x]*(24*a
*b + (7*b^2 + 20*a*c)*x^2))/(16*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) - (3*(7*b^3
+ 36*a*b*c + Sqrt[b^2 - 4*a*c]*(7*b^2 + 20*a*c))*ArcTan[(2^(1/4)*c^(1/4)*Sqrt[x
])/(-b - Sqrt[b^2 - 4*a*c])^(1/4)])/(32*2^(1/4)*c^(1/4)*(b^2 - 4*a*c)^(5/2)*(-b
- Sqrt[b^2 - 4*a*c])^(3/4)) - (3*(7*b^2 + 20*a*c - (7*b^3 + 36*a*b*c)/Sqrt[b^2 -
4*a*c])*ArcTan[(2^(1/4)*c^(1/4)*Sqrt[x])/(-b + Sqrt[b^2 - 4*a*c])^(1/4)])/(32*2
^(1/4)*c^(1/4)*(b^2 - 4*a*c)^2*(-b + Sqrt[b^2 - 4*a*c])^(3/4)) - (3*(7*b^3 + 36*
a*b*c + Sqrt[b^2 - 4*a*c]*(7*b^2 + 20*a*c))*ArcTanh[(2^(1/4)*c^(1/4)*Sqrt[x])/(-
b - Sqrt[b^2 - 4*a*c])^(1/4)])/(32*2^(1/4)*c^(1/4)*(b^2 - 4*a*c)^(5/2)*(-b - Sqr
t[b^2 - 4*a*c])^(3/4)) - (3*(7*b^2 + 20*a*c - (7*b^3 + 36*a*b*c)/Sqrt[b^2 - 4*a*
c])*ArcTanh[(2^(1/4)*c^(1/4)*Sqrt[x])/(-b + Sqrt[b^2 - 4*a*c])^(1/4)])/(32*2^(1/
4)*c^(1/4)*(b^2 - 4*a*c)^2*(-b + Sqrt[b^2 - 4*a*c])^(3/4))

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Rubi [A]  time = 3.49101, antiderivative size = 569, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ \frac{\sqrt{x} \left (x^2 \left (20 a c+7 b^2\right )+24 a b\right )}{16 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{x^{5/2} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac{3 \left (\sqrt{b^2-4 a c} \left (20 a c+7 b^2\right )+36 a b c+7 b^3\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} \sqrt [4]{c} \left (b^2-4 a c\right )^{5/2} \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{3 \left (-\frac{36 a b c+7 b^3}{\sqrt{b^2-4 a c}}+20 a c+7 b^2\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} \sqrt [4]{c} \left (b^2-4 a c\right )^2 \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{3 \left (\sqrt{b^2-4 a c} \left (20 a c+7 b^2\right )+36 a b c+7 b^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} \sqrt [4]{c} \left (b^2-4 a c\right )^{5/2} \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{3 \left (-\frac{36 a b c+7 b^3}{\sqrt{b^2-4 a c}}+20 a c+7 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} \sqrt [4]{c} \left (b^2-4 a c\right )^2 \left (\sqrt{b^2-4 a c}-b\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Int[x^(11/2)/(a + b*x^2 + c*x^4)^3,x]

[Out]

(x^(5/2)*(2*a + b*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (Sqrt[x]*(24*a
*b + (7*b^2 + 20*a*c)*x^2))/(16*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) - (3*(7*b^3
+ 36*a*b*c + Sqrt[b^2 - 4*a*c]*(7*b^2 + 20*a*c))*ArcTan[(2^(1/4)*c^(1/4)*Sqrt[x
])/(-b - Sqrt[b^2 - 4*a*c])^(1/4)])/(32*2^(1/4)*c^(1/4)*(b^2 - 4*a*c)^(5/2)*(-b
- Sqrt[b^2 - 4*a*c])^(3/4)) - (3*(7*b^2 + 20*a*c - (7*b^3 + 36*a*b*c)/Sqrt[b^2 -
4*a*c])*ArcTan[(2^(1/4)*c^(1/4)*Sqrt[x])/(-b + Sqrt[b^2 - 4*a*c])^(1/4)])/(32*2
^(1/4)*c^(1/4)*(b^2 - 4*a*c)^2*(-b + Sqrt[b^2 - 4*a*c])^(3/4)) - (3*(7*b^3 + 36*
a*b*c + Sqrt[b^2 - 4*a*c]*(7*b^2 + 20*a*c))*ArcTanh[(2^(1/4)*c^(1/4)*Sqrt[x])/(-
b - Sqrt[b^2 - 4*a*c])^(1/4)])/(32*2^(1/4)*c^(1/4)*(b^2 - 4*a*c)^(5/2)*(-b - Sqr
t[b^2 - 4*a*c])^(3/4)) - (3*(7*b^2 + 20*a*c - (7*b^3 + 36*a*b*c)/Sqrt[b^2 - 4*a*
c])*ArcTanh[(2^(1/4)*c^(1/4)*Sqrt[x])/(-b + Sqrt[b^2 - 4*a*c])^(1/4)])/(32*2^(1/
4)*c^(1/4)*(b^2 - 4*a*c)^2*(-b + Sqrt[b^2 - 4*a*c])^(3/4))

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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(x**(11/2)/(c*x**4+b*x**2+a)**3,x)

[Out]

Timed out

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Mathematica [C]  time = 0.594087, size = 219, normalized size = 0.38 \[ \frac{3 c \left (a+b x^2+c x^4\right )^2 \text{RootSum}\left [\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{20 \text{$\#$1}^4 a c \log \left (\sqrt{x}-\text{$\#$1}\right )+7 \text{$\#$1}^4 b^2 \log \left (\sqrt{x}-\text{$\#$1}\right )-8 a b \log \left (\sqrt{x}-\text{$\#$1}\right )}{2 \text{$\#$1}^7 c+\text{$\#$1}^3 b}\&\right ]-16 \sqrt{x} \left (b^2-4 a c\right ) \left (a \left (b-2 c x^2\right )+b^2 x^2\right )+4 \sqrt{x} \left (8 a b c+20 a c^2 x^2+4 b^3+7 b^2 c x^2\right ) \left (a+b x^2+c x^4\right )}{64 c \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(11/2)/(a + b*x^2 + c*x^4)^3,x]

[Out]

(4*Sqrt[x]*(4*b^3 + 8*a*b*c + 7*b^2*c*x^2 + 20*a*c^2*x^2)*(a + b*x^2 + c*x^4) -
16*(b^2 - 4*a*c)*Sqrt[x]*(b^2*x^2 + a*(b - 2*c*x^2)) + 3*c*(a + b*x^2 + c*x^4)^2
*RootSum[a + b*#1^4 + c*#1^8 & , (-8*a*b*Log[Sqrt[x] - #1] + 7*b^2*Log[Sqrt[x] -
#1]*#1^4 + 20*a*c*Log[Sqrt[x] - #1]*#1^4)/(b*#1^3 + 2*c*#1^7) & ])/(64*c*(b^2 -
4*a*c)^2*(a + b*x^2 + c*x^4)^2)

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Maple [C]  time = 0.046, size = 241, normalized size = 0.4 \[ 2\,{\frac{1}{ \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{2}} \left ( 3/4\,{\frac{{a}^{2}b\sqrt{x}}{16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4}}}-{\frac{ \left ( 12\,ac-39\,{b}^{2} \right ) a{x}^{5/2}}{512\,{a}^{2}{c}^{2}-256\,a{b}^{2}c+32\,{b}^{4}}}+1/32\,{\frac{b \left ( 28\,ac+11\,{b}^{2} \right ){x}^{9/2}}{16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4}}}+1/32\,{\frac{c \left ( 20\,ac+7\,{b}^{2} \right ){x}^{13/2}}{16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4}}} \right ) }+{\frac{3}{64}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}c+{{\it \_Z}}^{4}b+a \right ) }{\frac{ \left ( 20\,ac+7\,{b}^{2} \right ){{\it \_R}}^{4}-8\,ab}{ \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) \left ( 2\,{{\it \_R}}^{7}c+{{\it \_R}}^{3}b \right ) }\ln \left ( \sqrt{x}-{\it \_R} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(11/2)/(c*x^4+b*x^2+a)^3,x)

[Out]

2*(3/4*a^2*b/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(1/2)-3/32*(4*a*c-13*b^2)*a/(16*a^2*c^
2-8*a*b^2*c+b^4)*x^(5/2)+1/32*b*(28*a*c+11*b^2)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(9/
2)+1/32*c*(20*a*c+7*b^2)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(13/2))/(c*x^4+b*x^2+a)^2+
3/64*sum(((20*a*c+7*b^2)*_R^4-8*a*b)/(16*a^2*c^2-8*a*b^2*c+b^4)/(2*_R^7*c+_R^3*b
)*ln(x^(1/2)-_R),_R=RootOf(_Z^8*c+_Z^4*b+a))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{24 \, b c^{2} x^{\frac{17}{2}} +{\left (41 \, b^{2} c - 20 \, a c^{2}\right )} x^{\frac{13}{2}} +{\left (13 \, b^{3} + 20 \, a b c\right )} x^{\frac{9}{2}} + 3 \,{\left (3 \, a b^{2} + 4 \, a^{2} c\right )} x^{\frac{5}{2}}}{16 \,{\left ({\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{8} + 2 \,{\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{6} + a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} +{\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} x^{4} + 2 \,{\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x^{2}\right )}} + \int \frac{3 \,{\left (8 \, b c x^{\frac{7}{2}} + 5 \,{\left (3 \, b^{2} + 4 \, a c\right )} x^{\frac{3}{2}}\right )}}{32 \,{\left (a b^{4} - 8 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} +{\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{4} +{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{2}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(11/2)/(c*x^4 + b*x^2 + a)^3,x, algorithm="maxima")

[Out]

-1/16*(24*b*c^2*x^(17/2) + (41*b^2*c - 20*a*c^2)*x^(13/2) + (13*b^3 + 20*a*b*c)*
x^(9/2) + 3*(3*a*b^2 + 4*a^2*c)*x^(5/2))/((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x
^8 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^6 + a^2*b^4 - 8*a^3*b^2*c + 16*a^4
*c^2 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^4 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^
2)*x^2) + integrate(3/32*(8*b*c*x^(7/2) + 5*(3*b^2 + 4*a*c)*x^(3/2))/(a*b^4 - 8*
a^2*b^2*c + 16*a^3*c^2 + (b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*x^4 + (b^5 - 8*a*b^3
*c + 16*a^2*b*c^2)*x^2), x)

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Fricas [A]  time = 4.5942, size = 16012, normalized size = 28.14 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(11/2)/(c*x^4 + b*x^2 + a)^3,x, algorithm="fricas")

[Out]

-1/64*(12*((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^8 + 2*(b^5*c - 8*a*b^3*c^2 + 1
6*a^2*b*c^3)*x^6 + a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^6 - 6*a*b^4*c + 32*a^
3*c^3)*x^4 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x^2)*sqrt(sqrt(1/2)*sqrt(-(2
401*b^9 + 86640*a*b^7*c + 413280*a^2*b^5*c^2 + 833280*a^3*b^3*c^3 + 672000*a^4*b
*c^4 + (b^20*c - 40*a*b^18*c^2 + 720*a^2*b^16*c^3 - 7680*a^3*b^14*c^4 + 53760*a^
4*b^12*c^5 - 258048*a^5*b^10*c^6 + 860160*a^6*b^8*c^7 - 1966080*a^7*b^6*c^8 + 29
49120*a^8*b^4*c^9 - 2621440*a^9*b^2*c^10 + 1048576*a^10*c^11)*sqrt((5764801*b^8
+ 45138800*a*b^6*c + 136380000*a^2*b^4*c^2 + 188000000*a^3*b^2*c^3 + 100000000*a
^4*c^4)/(b^30*c^2 - 60*a*b^28*c^3 + 1680*a^2*b^26*c^4 - 29120*a^3*b^24*c^5 + 349
440*a^4*b^22*c^6 - 3075072*a^5*b^20*c^7 + 20500480*a^6*b^18*c^8 - 105431040*a^7*
b^16*c^9 + 421724160*a^8*b^14*c^10 - 1312030720*a^9*b^12*c^11 + 3148873728*a^10*
b^10*c^12 - 5725224960*a^11*b^8*c^13 + 7633633280*a^12*b^6*c^14 - 7046430720*a^1
3*b^4*c^15 + 4026531840*a^14*b^2*c^16 - 1073741824*a^15*c^17)))/(b^20*c - 40*a*b
^18*c^2 + 720*a^2*b^16*c^3 - 7680*a^3*b^14*c^4 + 53760*a^4*b^12*c^5 - 258048*a^5
*b^10*c^6 + 860160*a^6*b^8*c^7 - 1966080*a^7*b^6*c^8 + 2949120*a^8*b^4*c^9 - 262
1440*a^9*b^2*c^10 + 1048576*a^10*c^11)))*arctan((9604*b^11 - 77648*a*b^9*c + 497
92*a^2*b^7*c^2 + 710144*a^3*b^5*c^3 - 486400*a^4*b^3*c^4 - 2560000*a^5*b*c^5 + (
11*b^22*c - 420*a*b^20*c^2 + 7120*a^2*b^18*c^3 - 70080*a^3*b^16*c^4 + 437760*a^4
*b^14*c^5 - 1763328*a^5*b^12*c^6 + 4300800*a^6*b^10*c^7 - 4423680*a^7*b^8*c^8 -
6881280*a^8*b^6*c^9 + 30146560*a^9*b^4*c^10 - 40894464*a^10*b^2*c^11 + 20971520*
a^11*c^12)*sqrt((5764801*b^8 + 45138800*a*b^6*c + 136380000*a^2*b^4*c^2 + 188000
000*a^3*b^2*c^3 + 100000000*a^4*c^4)/(b^30*c^2 - 60*a*b^28*c^3 + 1680*a^2*b^26*c
^4 - 29120*a^3*b^24*c^5 + 349440*a^4*b^22*c^6 - 3075072*a^5*b^20*c^7 + 20500480*
a^6*b^18*c^8 - 105431040*a^7*b^16*c^9 + 421724160*a^8*b^14*c^10 - 1312030720*a^9
*b^12*c^11 + 3148873728*a^10*b^10*c^12 - 5725224960*a^11*b^8*c^13 + 7633633280*a
^12*b^6*c^14 - 7046430720*a^13*b^4*c^15 + 4026531840*a^14*b^2*c^16 - 1073741824*
a^15*c^17)))*sqrt(sqrt(1/2)*sqrt(-(2401*b^9 + 86640*a*b^7*c + 413280*a^2*b^5*c^2
+ 833280*a^3*b^3*c^3 + 672000*a^4*b*c^4 + (b^20*c - 40*a*b^18*c^2 + 720*a^2*b^1
6*c^3 - 7680*a^3*b^14*c^4 + 53760*a^4*b^12*c^5 - 258048*a^5*b^10*c^6 + 860160*a^
6*b^8*c^7 - 1966080*a^7*b^6*c^8 + 2949120*a^8*b^4*c^9 - 2621440*a^9*b^2*c^10 + 1
048576*a^10*c^11)*sqrt((5764801*b^8 + 45138800*a*b^6*c + 136380000*a^2*b^4*c^2 +
188000000*a^3*b^2*c^3 + 100000000*a^4*c^4)/(b^30*c^2 - 60*a*b^28*c^3 + 1680*a^2
*b^26*c^4 - 29120*a^3*b^24*c^5 + 349440*a^4*b^22*c^6 - 3075072*a^5*b^20*c^7 + 20
500480*a^6*b^18*c^8 - 105431040*a^7*b^16*c^9 + 421724160*a^8*b^14*c^10 - 1312030
720*a^9*b^12*c^11 + 3148873728*a^10*b^10*c^12 - 5725224960*a^11*b^8*c^13 + 76336
33280*a^12*b^6*c^14 - 7046430720*a^13*b^4*c^15 + 4026531840*a^14*b^2*c^16 - 1073
741824*a^15*c^17)))/(b^20*c - 40*a*b^18*c^2 + 720*a^2*b^16*c^3 - 7680*a^3*b^14*c
^4 + 53760*a^4*b^12*c^5 - 258048*a^5*b^10*c^6 + 860160*a^6*b^8*c^7 - 1966080*a^7
*b^6*c^8 + 2949120*a^8*b^4*c^9 - 2621440*a^9*b^2*c^10 + 1048576*a^10*c^11)))/((2
52105*b^8 + 2197104*a*b^6*c + 6748000*a^2*b^4*c^2 + 8800000*a^3*b^2*c^3 + 400000
0*a^4*c^4)*sqrt(x) + sqrt((63556931025*b^16 + 1107801807840*a*b^14*c + 822967506
6816*a^2*b^12*c^2 + 34089163584000*a^3*b^10*c^3 + 86221374400000*a^4*b^8*c^4 + 1
36341632000000*a^5*b^6*c^5 + 131424000000000*a^6*b^4*c^6 + 70400000000000*a^7*b^
2*c^7 + 16000000000000*a^8*c^8)*x + sqrt(1/2)*(789777737*b^22 - 7443973964*a*b^2
0*c - 2705008400*a^2*b^18*c^2 + 166642188480*a^3*b^16*c^3 - 23017121280*a^4*b^14
*c^4 - 1866033297408*a^5*b^12*c^5 - 803898138624*a^6*b^10*c^6 + 11168850739200*a
^7*b^8*c^7 + 14678630400000*a^8*b^6*c^8 - 23490560000000*a^9*b^4*c^9 - 643072000
00000*a^10*b^2*c^10 - 40960000000000*a^11*c^11 + 8*(26411*b^33*c - 1221952*a*b^3
1*c^2 + 25385088*a^2*b^29*c^3 - 309750784*a^3*b^27*c^4 + 2424181760*a^4*b^25*c^5
- 12295815168*a^5*b^23*c^6 + 36966465536*a^6*b^21*c^7 - 34375204864*a^7*b^19*c^
8 - 198547734528*a^8*b^17*c^9 + 848696442880*a^9*b^15*c^10 - 948860616704*a^10*b
^13*c^11 - 2216807104512*a^11*b^11*c^12 + 8103865090048*a^12*b^9*c^13 - 62609885
75744*a^13*b^7*c^14 - 10597831802880*a^14*b^5*c^15 + 23622320128000*a^15*b^3*c^1
6 - 13421772800000*a^16*b*c^17)*sqrt((5764801*b^8 + 45138800*a*b^6*c + 136380000
*a^2*b^4*c^2 + 188000000*a^3*b^2*c^3 + 100000000*a^4*c^4)/(b^30*c^2 - 60*a*b^28*
c^3 + 1680*a^2*b^26*c^4 - 29120*a^3*b^24*c^5 + 349440*a^4*b^22*c^6 - 3075072*a^5
*b^20*c^7 + 20500480*a^6*b^18*c^8 - 105431040*a^7*b^16*c^9 + 421724160*a^8*b^14*
c^10 - 1312030720*a^9*b^12*c^11 + 3148873728*a^10*b^10*c^12 - 5725224960*a^11*b^
8*c^13 + 7633633280*a^12*b^6*c^14 - 7046430720*a^13*b^4*c^15 + 4026531840*a^14*b
^2*c^16 - 1073741824*a^15*c^17)))*sqrt(-(2401*b^9 + 86640*a*b^7*c + 413280*a^2*b
^5*c^2 + 833280*a^3*b^3*c^3 + 672000*a^4*b*c^4 + (b^20*c - 40*a*b^18*c^2 + 720*a
^2*b^16*c^3 - 7680*a^3*b^14*c^4 + 53760*a^4*b^12*c^5 - 258048*a^5*b^10*c^6 + 860
160*a^6*b^8*c^7 - 1966080*a^7*b^6*c^8 + 2949120*a^8*b^4*c^9 - 2621440*a^9*b^2*c^
10 + 1048576*a^10*c^11)*sqrt((5764801*b^8 + 45138800*a*b^6*c + 136380000*a^2*b^4
*c^2 + 188000000*a^3*b^2*c^3 + 100000000*a^4*c^4)/(b^30*c^2 - 60*a*b^28*c^3 + 16
80*a^2*b^26*c^4 - 29120*a^3*b^24*c^5 + 349440*a^4*b^22*c^6 - 3075072*a^5*b^20*c^
7 + 20500480*a^6*b^18*c^8 - 105431040*a^7*b^16*c^9 + 421724160*a^8*b^14*c^10 - 1
312030720*a^9*b^12*c^11 + 3148873728*a^10*b^10*c^12 - 5725224960*a^11*b^8*c^13 +
7633633280*a^12*b^6*c^14 - 7046430720*a^13*b^4*c^15 + 4026531840*a^14*b^2*c^16
- 1073741824*a^15*c^17)))/(b^20*c - 40*a*b^18*c^2 + 720*a^2*b^16*c^3 - 7680*a^3*
b^14*c^4 + 53760*a^4*b^12*c^5 - 258048*a^5*b^10*c^6 + 860160*a^6*b^8*c^7 - 19660
80*a^7*b^6*c^8 + 2949120*a^8*b^4*c^9 - 2621440*a^9*b^2*c^10 + 1048576*a^10*c^11)
)))) - 12*((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^8 + 2*(b^5*c - 8*a*b^3*c^2 + 1
6*a^2*b*c^3)*x^6 + a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^6 - 6*a*b^4*c + 32*a^
3*c^3)*x^4 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x^2)*sqrt(sqrt(1/2)*sqrt(-(2
401*b^9 + 86640*a*b^7*c + 413280*a^2*b^5*c^2 + 833280*a^3*b^3*c^3 + 672000*a^4*b
*c^4 - (b^20*c - 40*a*b^18*c^2 + 720*a^2*b^16*c^3 - 7680*a^3*b^14*c^4 + 53760*a^
4*b^12*c^5 - 258048*a^5*b^10*c^6 + 860160*a^6*b^8*c^7 - 1966080*a^7*b^6*c^8 + 29
49120*a^8*b^4*c^9 - 2621440*a^9*b^2*c^10 + 1048576*a^10*c^11)*sqrt((5764801*b^8
+ 45138800*a*b^6*c + 136380000*a^2*b^4*c^2 + 188000000*a^3*b^2*c^3 + 100000000*a
^4*c^4)/(b^30*c^2 - 60*a*b^28*c^3 + 1680*a^2*b^26*c^4 - 29120*a^3*b^24*c^5 + 349
440*a^4*b^22*c^6 - 3075072*a^5*b^20*c^7 + 20500480*a^6*b^18*c^8 - 105431040*a^7*
b^16*c^9 + 421724160*a^8*b^14*c^10 - 1312030720*a^9*b^12*c^11 + 3148873728*a^10*
b^10*c^12 - 5725224960*a^11*b^8*c^13 + 7633633280*a^12*b^6*c^14 - 7046430720*a^1
3*b^4*c^15 + 4026531840*a^14*b^2*c^16 - 1073741824*a^15*c^17)))/(b^20*c - 40*a*b
^18*c^2 + 720*a^2*b^16*c^3 - 7680*a^3*b^14*c^4 + 53760*a^4*b^12*c^5 - 258048*a^5
*b^10*c^6 + 860160*a^6*b^8*c^7 - 1966080*a^7*b^6*c^8 + 2949120*a^8*b^4*c^9 - 262
1440*a^9*b^2*c^10 + 1048576*a^10*c^11)))*arctan(-(9604*b^11 - 77648*a*b^9*c + 49
792*a^2*b^7*c^2 + 710144*a^3*b^5*c^3 - 486400*a^4*b^3*c^4 - 2560000*a^5*b*c^5 -
(11*b^22*c - 420*a*b^20*c^2 + 7120*a^2*b^18*c^3 - 70080*a^3*b^16*c^4 + 437760*a^
4*b^14*c^5 - 1763328*a^5*b^12*c^6 + 4300800*a^6*b^10*c^7 - 4423680*a^7*b^8*c^8 -
6881280*a^8*b^6*c^9 + 30146560*a^9*b^4*c^10 - 40894464*a^10*b^2*c^11 + 20971520
*a^11*c^12)*sqrt((5764801*b^8 + 45138800*a*b^6*c + 136380000*a^2*b^4*c^2 + 18800
0000*a^3*b^2*c^3 + 100000000*a^4*c^4)/(b^30*c^2 - 60*a*b^28*c^3 + 1680*a^2*b^26*
c^4 - 29120*a^3*b^24*c^5 + 349440*a^4*b^22*c^6 - 3075072*a^5*b^20*c^7 + 20500480
*a^6*b^18*c^8 - 105431040*a^7*b^16*c^9 + 421724160*a^8*b^14*c^10 - 1312030720*a^
9*b^12*c^11 + 3148873728*a^10*b^10*c^12 - 5725224960*a^11*b^8*c^13 + 7633633280*
a^12*b^6*c^14 - 7046430720*a^13*b^4*c^15 + 4026531840*a^14*b^2*c^16 - 1073741824
*a^15*c^17)))*sqrt(sqrt(1/2)*sqrt(-(2401*b^9 + 86640*a*b^7*c + 413280*a^2*b^5*c^
2 + 833280*a^3*b^3*c^3 + 672000*a^4*b*c^4 - (b^20*c - 40*a*b^18*c^2 + 720*a^2*b^
16*c^3 - 7680*a^3*b^14*c^4 + 53760*a^4*b^12*c^5 - 258048*a^5*b^10*c^6 + 860160*a
^6*b^8*c^7 - 1966080*a^7*b^6*c^8 + 2949120*a^8*b^4*c^9 - 2621440*a^9*b^2*c^10 +
1048576*a^10*c^11)*sqrt((5764801*b^8 + 45138800*a*b^6*c + 136380000*a^2*b^4*c^2
+ 188000000*a^3*b^2*c^3 + 100000000*a^4*c^4)/(b^30*c^2 - 60*a*b^28*c^3 + 1680*a^
2*b^26*c^4 - 29120*a^3*b^24*c^5 + 349440*a^4*b^22*c^6 - 3075072*a^5*b^20*c^7 + 2
0500480*a^6*b^18*c^8 - 105431040*a^7*b^16*c^9 + 421724160*a^8*b^14*c^10 - 131203
0720*a^9*b^12*c^11 + 3148873728*a^10*b^10*c^12 - 5725224960*a^11*b^8*c^13 + 7633
633280*a^12*b^6*c^14 - 7046430720*a^13*b^4*c^15 + 4026531840*a^14*b^2*c^16 - 107
3741824*a^15*c^17)))/(b^20*c - 40*a*b^18*c^2 + 720*a^2*b^16*c^3 - 7680*a^3*b^14*
c^4 + 53760*a^4*b^12*c^5 - 258048*a^5*b^10*c^6 + 860160*a^6*b^8*c^7 - 1966080*a^
7*b^6*c^8 + 2949120*a^8*b^4*c^9 - 2621440*a^9*b^2*c^10 + 1048576*a^10*c^11)))/((
252105*b^8 + 2197104*a*b^6*c + 6748000*a^2*b^4*c^2 + 8800000*a^3*b^2*c^3 + 40000
00*a^4*c^4)*sqrt(x) + sqrt((63556931025*b^16 + 1107801807840*a*b^14*c + 82296750
66816*a^2*b^12*c^2 + 34089163584000*a^3*b^10*c^3 + 86221374400000*a^4*b^8*c^4 +
136341632000000*a^5*b^6*c^5 + 131424000000000*a^6*b^4*c^6 + 70400000000000*a^7*b
^2*c^7 + 16000000000000*a^8*c^8)*x + sqrt(1/2)*(789777737*b^22 - 7443973964*a*b^
20*c - 2705008400*a^2*b^18*c^2 + 166642188480*a^3*b^16*c^3 - 23017121280*a^4*b^1
4*c^4 - 1866033297408*a^5*b^12*c^5 - 803898138624*a^6*b^10*c^6 + 11168850739200*
a^7*b^8*c^7 + 14678630400000*a^8*b^6*c^8 - 23490560000000*a^9*b^4*c^9 - 64307200
000000*a^10*b^2*c^10 - 40960000000000*a^11*c^11 - 8*(26411*b^33*c - 1221952*a*b^
31*c^2 + 25385088*a^2*b^29*c^3 - 309750784*a^3*b^27*c^4 + 2424181760*a^4*b^25*c^
5 - 12295815168*a^5*b^23*c^6 + 36966465536*a^6*b^21*c^7 - 34375204864*a^7*b^19*c
^8 - 198547734528*a^8*b^17*c^9 + 848696442880*a^9*b^15*c^10 - 948860616704*a^10*
b^13*c^11 - 2216807104512*a^11*b^11*c^12 + 8103865090048*a^12*b^9*c^13 - 6260988
575744*a^13*b^7*c^14 - 10597831802880*a^14*b^5*c^15 + 23622320128000*a^15*b^3*c^
16 - 13421772800000*a^16*b*c^17)*sqrt((5764801*b^8 + 45138800*a*b^6*c + 13638000
0*a^2*b^4*c^2 + 188000000*a^3*b^2*c^3 + 100000000*a^4*c^4)/(b^30*c^2 - 60*a*b^28
*c^3 + 1680*a^2*b^26*c^4 - 29120*a^3*b^24*c^5 + 349440*a^4*b^22*c^6 - 3075072*a^
5*b^20*c^7 + 20500480*a^6*b^18*c^8 - 105431040*a^7*b^16*c^9 + 421724160*a^8*b^14
*c^10 - 1312030720*a^9*b^12*c^11 + 3148873728*a^10*b^10*c^12 - 5725224960*a^11*b
^8*c^13 + 7633633280*a^12*b^6*c^14 - 7046430720*a^13*b^4*c^15 + 4026531840*a^14*
b^2*c^16 - 1073741824*a^15*c^17)))*sqrt(-(2401*b^9 + 86640*a*b^7*c + 413280*a^2*
b^5*c^2 + 833280*a^3*b^3*c^3 + 672000*a^4*b*c^4 - (b^20*c - 40*a*b^18*c^2 + 720*
a^2*b^16*c^3 - 7680*a^3*b^14*c^4 + 53760*a^4*b^12*c^5 - 258048*a^5*b^10*c^6 + 86
0160*a^6*b^8*c^7 - 1966080*a^7*b^6*c^8 + 2949120*a^8*b^4*c^9 - 2621440*a^9*b^2*c
^10 + 1048576*a^10*c^11)*sqrt((5764801*b^8 + 45138800*a*b^6*c + 136380000*a^2*b^
4*c^2 + 188000000*a^3*b^2*c^3 + 100000000*a^4*c^4)/(b^30*c^2 - 60*a*b^28*c^3 + 1
680*a^2*b^26*c^4 - 29120*a^3*b^24*c^5 + 349440*a^4*b^22*c^6 - 3075072*a^5*b^20*c
^7 + 20500480*a^6*b^18*c^8 - 105431040*a^7*b^16*c^9 + 421724160*a^8*b^14*c^10 -
1312030720*a^9*b^12*c^11 + 3148873728*a^10*b^10*c^12 - 5725224960*a^11*b^8*c^13
+ 7633633280*a^12*b^6*c^14 - 7046430720*a^13*b^4*c^15 + 4026531840*a^14*b^2*c^16
- 1073741824*a^15*c^17)))/(b^20*c - 40*a*b^18*c^2 + 720*a^2*b^16*c^3 - 7680*a^3
*b^14*c^4 + 53760*a^4*b^12*c^5 - 258048*a^5*b^10*c^6 + 860160*a^6*b^8*c^7 - 1966
080*a^7*b^6*c^8 + 2949120*a^8*b^4*c^9 - 2621440*a^9*b^2*c^10 + 1048576*a^10*c^11
))))) - 3*((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^8 + 2*(b^5*c - 8*a*b^3*c^2 + 1
6*a^2*b*c^3)*x^6 + a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^6 - 6*a*b^4*c + 32*a^
3*c^3)*x^4 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x^2)*sqrt(sqrt(1/2)*sqrt(-(2
401*b^9 + 86640*a*b^7*c + 413280*a^2*b^5*c^2 + 833280*a^3*b^3*c^3 + 672000*a^4*b
*c^4 + (b^20*c - 40*a*b^18*c^2 + 720*a^2*b^16*c^3 - 7680*a^3*b^14*c^4 + 53760*a^
4*b^12*c^5 - 258048*a^5*b^10*c^6 + 860160*a^6*b^8*c^7 - 1966080*a^7*b^6*c^8 + 29
49120*a^8*b^4*c^9 - 2621440*a^9*b^2*c^10 + 1048576*a^10*c^11)*sqrt((5764801*b^8
+ 45138800*a*b^6*c + 136380000*a^2*b^4*c^2 + 188000000*a^3*b^2*c^3 + 100000000*a
^4*c^4)/(b^30*c^2 - 60*a*b^28*c^3 + 1680*a^2*b^26*c^4 - 29120*a^3*b^24*c^5 + 349
440*a^4*b^22*c^6 - 3075072*a^5*b^20*c^7 + 20500480*a^6*b^18*c^8 - 105431040*a^7*
b^16*c^9 + 421724160*a^8*b^14*c^10 - 1312030720*a^9*b^12*c^11 + 3148873728*a^10*
b^10*c^12 - 5725224960*a^11*b^8*c^13 + 7633633280*a^12*b^6*c^14 - 7046430720*a^1
3*b^4*c^15 + 4026531840*a^14*b^2*c^16 - 1073741824*a^15*c^17)))/(b^20*c - 40*a*b
^18*c^2 + 720*a^2*b^16*c^3 - 7680*a^3*b^14*c^4 + 53760*a^4*b^12*c^5 - 258048*a^5
*b^10*c^6 + 860160*a^6*b^8*c^7 - 1966080*a^7*b^6*c^8 + 2949120*a^8*b^4*c^9 - 262
1440*a^9*b^2*c^10 + 1048576*a^10*c^11)))*log(243*(252105*b^8 + 2197104*a*b^6*c +
6748000*a^2*b^4*c^2 + 8800000*a^3*b^2*c^3 + 4000000*a^4*c^4)*sqrt(x) + 243*(960
4*b^11 - 77648*a*b^9*c + 49792*a^2*b^7*c^2 + 710144*a^3*b^5*c^3 - 486400*a^4*b^3
*c^4 - 2560000*a^5*b*c^5 + (11*b^22*c - 420*a*b^20*c^2 + 7120*a^2*b^18*c^3 - 700
80*a^3*b^16*c^4 + 437760*a^4*b^14*c^5 - 1763328*a^5*b^12*c^6 + 4300800*a^6*b^10*
c^7 - 4423680*a^7*b^8*c^8 - 6881280*a^8*b^6*c^9 + 30146560*a^9*b^4*c^10 - 408944
64*a^10*b^2*c^11 + 20971520*a^11*c^12)*sqrt((5764801*b^8 + 45138800*a*b^6*c + 13
6380000*a^2*b^4*c^2 + 188000000*a^3*b^2*c^3 + 100000000*a^4*c^4)/(b^30*c^2 - 60*
a*b^28*c^3 + 1680*a^2*b^26*c^4 - 29120*a^3*b^24*c^5 + 349440*a^4*b^22*c^6 - 3075
072*a^5*b^20*c^7 + 20500480*a^6*b^18*c^8 - 105431040*a^7*b^16*c^9 + 421724160*a^
8*b^14*c^10 - 1312030720*a^9*b^12*c^11 + 3148873728*a^10*b^10*c^12 - 5725224960*
a^11*b^8*c^13 + 7633633280*a^12*b^6*c^14 - 7046430720*a^13*b^4*c^15 + 4026531840
*a^14*b^2*c^16 - 1073741824*a^15*c^17)))*sqrt(sqrt(1/2)*sqrt(-(2401*b^9 + 86640*
a*b^7*c + 413280*a^2*b^5*c^2 + 833280*a^3*b^3*c^3 + 672000*a^4*b*c^4 + (b^20*c -
40*a*b^18*c^2 + 720*a^2*b^16*c^3 - 7680*a^3*b^14*c^4 + 53760*a^4*b^12*c^5 - 258
048*a^5*b^10*c^6 + 860160*a^6*b^8*c^7 - 1966080*a^7*b^6*c^8 + 2949120*a^8*b^4*c^
9 - 2621440*a^9*b^2*c^10 + 1048576*a^10*c^11)*sqrt((5764801*b^8 + 45138800*a*b^6
*c + 136380000*a^2*b^4*c^2 + 188000000*a^3*b^2*c^3 + 100000000*a^4*c^4)/(b^30*c^
2 - 60*a*b^28*c^3 + 1680*a^2*b^26*c^4 - 29120*a^3*b^24*c^5 + 349440*a^4*b^22*c^6
- 3075072*a^5*b^20*c^7 + 20500480*a^6*b^18*c^8 - 105431040*a^7*b^16*c^9 + 42172
4160*a^8*b^14*c^10 - 1312030720*a^9*b^12*c^11 + 3148873728*a^10*b^10*c^12 - 5725
224960*a^11*b^8*c^13 + 7633633280*a^12*b^6*c^14 - 7046430720*a^13*b^4*c^15 + 402
6531840*a^14*b^2*c^16 - 1073741824*a^15*c^17)))/(b^20*c - 40*a*b^18*c^2 + 720*a^
2*b^16*c^3 - 7680*a^3*b^14*c^4 + 53760*a^4*b^12*c^5 - 258048*a^5*b^10*c^6 + 8601
60*a^6*b^8*c^7 - 1966080*a^7*b^6*c^8 + 2949120*a^8*b^4*c^9 - 2621440*a^9*b^2*c^1
0 + 1048576*a^10*c^11)))) + 3*((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^8 + 2*(b^5
*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^6 + a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^6
- 6*a*b^4*c + 32*a^3*c^3)*x^4 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x^2)*sqr
t(sqrt(1/2)*sqrt(-(2401*b^9 + 86640*a*b^7*c + 413280*a^2*b^5*c^2 + 833280*a^3*b^
3*c^3 + 672000*a^4*b*c^4 + (b^20*c - 40*a*b^18*c^2 + 720*a^2*b^16*c^3 - 7680*a^3
*b^14*c^4 + 53760*a^4*b^12*c^5 - 258048*a^5*b^10*c^6 + 860160*a^6*b^8*c^7 - 1966
080*a^7*b^6*c^8 + 2949120*a^8*b^4*c^9 - 2621440*a^9*b^2*c^10 + 1048576*a^10*c^11
)*sqrt((5764801*b^8 + 45138800*a*b^6*c + 136380000*a^2*b^4*c^2 + 188000000*a^3*b
^2*c^3 + 100000000*a^4*c^4)/(b^30*c^2 - 60*a*b^28*c^3 + 1680*a^2*b^26*c^4 - 2912
0*a^3*b^24*c^5 + 349440*a^4*b^22*c^6 - 3075072*a^5*b^20*c^7 + 20500480*a^6*b^18*
c^8 - 105431040*a^7*b^16*c^9 + 421724160*a^8*b^14*c^10 - 1312030720*a^9*b^12*c^1
1 + 3148873728*a^10*b^10*c^12 - 5725224960*a^11*b^8*c^13 + 7633633280*a^12*b^6*c
^14 - 7046430720*a^13*b^4*c^15 + 4026531840*a^14*b^2*c^16 - 1073741824*a^15*c^17
)))/(b^20*c - 40*a*b^18*c^2 + 720*a^2*b^16*c^3 - 7680*a^3*b^14*c^4 + 53760*a^4*b
^12*c^5 - 258048*a^5*b^10*c^6 + 860160*a^6*b^8*c^7 - 1966080*a^7*b^6*c^8 + 29491
20*a^8*b^4*c^9 - 2621440*a^9*b^2*c^10 + 1048576*a^10*c^11)))*log(243*(252105*b^8
+ 2197104*a*b^6*c + 6748000*a^2*b^4*c^2 + 8800000*a^3*b^2*c^3 + 4000000*a^4*c^4
)*sqrt(x) - 243*(9604*b^11 - 77648*a*b^9*c + 49792*a^2*b^7*c^2 + 710144*a^3*b^5*
c^3 - 486400*a^4*b^3*c^4 - 2560000*a^5*b*c^5 + (11*b^22*c - 420*a*b^20*c^2 + 712
0*a^2*b^18*c^3 - 70080*a^3*b^16*c^4 + 437760*a^4*b^14*c^5 - 1763328*a^5*b^12*c^6
+ 4300800*a^6*b^10*c^7 - 4423680*a^7*b^8*c^8 - 6881280*a^8*b^6*c^9 + 30146560*a
^9*b^4*c^10 - 40894464*a^10*b^2*c^11 + 20971520*a^11*c^12)*sqrt((5764801*b^8 + 4
5138800*a*b^6*c + 136380000*a^2*b^4*c^2 + 188000000*a^3*b^2*c^3 + 100000000*a^4*
c^4)/(b^30*c^2 - 60*a*b^28*c^3 + 1680*a^2*b^26*c^4 - 29120*a^3*b^24*c^5 + 349440
*a^4*b^22*c^6 - 3075072*a^5*b^20*c^7 + 20500480*a^6*b^18*c^8 - 105431040*a^7*b^1
6*c^9 + 421724160*a^8*b^14*c^10 - 1312030720*a^9*b^12*c^11 + 3148873728*a^10*b^1
0*c^12 - 5725224960*a^11*b^8*c^13 + 7633633280*a^12*b^6*c^14 - 7046430720*a^13*b
^4*c^15 + 4026531840*a^14*b^2*c^16 - 1073741824*a^15*c^17)))*sqrt(sqrt(1/2)*sqrt
(-(2401*b^9 + 86640*a*b^7*c + 413280*a^2*b^5*c^2 + 833280*a^3*b^3*c^3 + 672000*a
^4*b*c^4 + (b^20*c - 40*a*b^18*c^2 + 720*a^2*b^16*c^3 - 7680*a^3*b^14*c^4 + 5376
0*a^4*b^12*c^5 - 258048*a^5*b^10*c^6 + 860160*a^6*b^8*c^7 - 1966080*a^7*b^6*c^8
+ 2949120*a^8*b^4*c^9 - 2621440*a^9*b^2*c^10 + 1048576*a^10*c^11)*sqrt((5764801*
b^8 + 45138800*a*b^6*c + 136380000*a^2*b^4*c^2 + 188000000*a^3*b^2*c^3 + 1000000
00*a^4*c^4)/(b^30*c^2 - 60*a*b^28*c^3 + 1680*a^2*b^26*c^4 - 29120*a^3*b^24*c^5 +
349440*a^4*b^22*c^6 - 3075072*a^5*b^20*c^7 + 20500480*a^6*b^18*c^8 - 105431040*
a^7*b^16*c^9 + 421724160*a^8*b^14*c^10 - 1312030720*a^9*b^12*c^11 + 3148873728*a
^10*b^10*c^12 - 5725224960*a^11*b^8*c^13 + 7633633280*a^12*b^6*c^14 - 7046430720
*a^13*b^4*c^15 + 4026531840*a^14*b^2*c^16 - 1073741824*a^15*c^17)))/(b^20*c - 40
*a*b^18*c^2 + 720*a^2*b^16*c^3 - 7680*a^3*b^14*c^4 + 53760*a^4*b^12*c^5 - 258048
*a^5*b^10*c^6 + 860160*a^6*b^8*c^7 - 1966080*a^7*b^6*c^8 + 2949120*a^8*b^4*c^9 -
2621440*a^9*b^2*c^10 + 1048576*a^10*c^11)))) - 3*((b^4*c^2 - 8*a*b^2*c^3 + 16*a
^2*c^4)*x^8 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^6 + a^2*b^4 - 8*a^3*b^2*c
+ 16*a^4*c^2 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^4 + 2*(a*b^5 - 8*a^2*b^3*c + 16
*a^3*b*c^2)*x^2)*sqrt(sqrt(1/2)*sqrt(-(2401*b^9 + 86640*a*b^7*c + 413280*a^2*b^5
*c^2 + 833280*a^3*b^3*c^3 + 672000*a^4*b*c^4 - (b^20*c - 40*a*b^18*c^2 + 720*a^2
*b^16*c^3 - 7680*a^3*b^14*c^4 + 53760*a^4*b^12*c^5 - 258048*a^5*b^10*c^6 + 86016
0*a^6*b^8*c^7 - 1966080*a^7*b^6*c^8 + 2949120*a^8*b^4*c^9 - 2621440*a^9*b^2*c^10
+ 1048576*a^10*c^11)*sqrt((5764801*b^8 + 45138800*a*b^6*c + 136380000*a^2*b^4*c
^2 + 188000000*a^3*b^2*c^3 + 100000000*a^4*c^4)/(b^30*c^2 - 60*a*b^28*c^3 + 1680
*a^2*b^26*c^4 - 29120*a^3*b^24*c^5 + 349440*a^4*b^22*c^6 - 3075072*a^5*b^20*c^7
+ 20500480*a^6*b^18*c^8 - 105431040*a^7*b^16*c^9 + 421724160*a^8*b^14*c^10 - 131
2030720*a^9*b^12*c^11 + 3148873728*a^10*b^10*c^12 - 5725224960*a^11*b^8*c^13 + 7
633633280*a^12*b^6*c^14 - 7046430720*a^13*b^4*c^15 + 4026531840*a^14*b^2*c^16 -
1073741824*a^15*c^17)))/(b^20*c - 40*a*b^18*c^2 + 720*a^2*b^16*c^3 - 7680*a^3*b^
14*c^4 + 53760*a^4*b^12*c^5 - 258048*a^5*b^10*c^6 + 860160*a^6*b^8*c^7 - 1966080
*a^7*b^6*c^8 + 2949120*a^8*b^4*c^9 - 2621440*a^9*b^2*c^10 + 1048576*a^10*c^11)))
*log(243*(252105*b^8 + 2197104*a*b^6*c + 6748000*a^2*b^4*c^2 + 8800000*a^3*b^2*c
^3 + 4000000*a^4*c^4)*sqrt(x) + 243*(9604*b^11 - 77648*a*b^9*c + 49792*a^2*b^7*c
^2 + 710144*a^3*b^5*c^3 - 486400*a^4*b^3*c^4 - 2560000*a^5*b*c^5 - (11*b^22*c -
420*a*b^20*c^2 + 7120*a^2*b^18*c^3 - 70080*a^3*b^16*c^4 + 437760*a^4*b^14*c^5 -
1763328*a^5*b^12*c^6 + 4300800*a^6*b^10*c^7 - 4423680*a^7*b^8*c^8 - 6881280*a^8*
b^6*c^9 + 30146560*a^9*b^4*c^10 - 40894464*a^10*b^2*c^11 + 20971520*a^11*c^12)*s
qrt((5764801*b^8 + 45138800*a*b^6*c + 136380000*a^2*b^4*c^2 + 188000000*a^3*b^2*
c^3 + 100000000*a^4*c^4)/(b^30*c^2 - 60*a*b^28*c^3 + 1680*a^2*b^26*c^4 - 29120*a
^3*b^24*c^5 + 349440*a^4*b^22*c^6 - 3075072*a^5*b^20*c^7 + 20500480*a^6*b^18*c^8
- 105431040*a^7*b^16*c^9 + 421724160*a^8*b^14*c^10 - 1312030720*a^9*b^12*c^11 +
3148873728*a^10*b^10*c^12 - 5725224960*a^11*b^8*c^13 + 7633633280*a^12*b^6*c^14
- 7046430720*a^13*b^4*c^15 + 4026531840*a^14*b^2*c^16 - 1073741824*a^15*c^17)))
*sqrt(sqrt(1/2)*sqrt(-(2401*b^9 + 86640*a*b^7*c + 413280*a^2*b^5*c^2 + 833280*a^
3*b^3*c^3 + 672000*a^4*b*c^4 - (b^20*c - 40*a*b^18*c^2 + 720*a^2*b^16*c^3 - 7680
*a^3*b^14*c^4 + 53760*a^4*b^12*c^5 - 258048*a^5*b^10*c^6 + 860160*a^6*b^8*c^7 -
1966080*a^7*b^6*c^8 + 2949120*a^8*b^4*c^9 - 2621440*a^9*b^2*c^10 + 1048576*a^10*
c^11)*sqrt((5764801*b^8 + 45138800*a*b^6*c + 136380000*a^2*b^4*c^2 + 188000000*a
^3*b^2*c^3 + 100000000*a^4*c^4)/(b^30*c^2 - 60*a*b^28*c^3 + 1680*a^2*b^26*c^4 -
29120*a^3*b^24*c^5 + 349440*a^4*b^22*c^6 - 3075072*a^5*b^20*c^7 + 20500480*a^6*b
^18*c^8 - 105431040*a^7*b^16*c^9 + 421724160*a^8*b^14*c^10 - 1312030720*a^9*b^12
*c^11 + 3148873728*a^10*b^10*c^12 - 5725224960*a^11*b^8*c^13 + 7633633280*a^12*b
^6*c^14 - 7046430720*a^13*b^4*c^15 + 4026531840*a^14*b^2*c^16 - 1073741824*a^15*
c^17)))/(b^20*c - 40*a*b^18*c^2 + 720*a^2*b^16*c^3 - 7680*a^3*b^14*c^4 + 53760*a
^4*b^12*c^5 - 258048*a^5*b^10*c^6 + 860160*a^6*b^8*c^7 - 1966080*a^7*b^6*c^8 + 2
949120*a^8*b^4*c^9 - 2621440*a^9*b^2*c^10 + 1048576*a^10*c^11)))) + 3*((b^4*c^2
- 8*a*b^2*c^3 + 16*a^2*c^4)*x^8 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^6 + a
^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^4 + 2*(a*b^
5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x^2)*sqrt(sqrt(1/2)*sqrt(-(2401*b^9 + 86640*a*b^
7*c + 413280*a^2*b^5*c^2 + 833280*a^3*b^3*c^3 + 672000*a^4*b*c^4 - (b^20*c - 40*
a*b^18*c^2 + 720*a^2*b^16*c^3 - 7680*a^3*b^14*c^4 + 53760*a^4*b^12*c^5 - 258048*
a^5*b^10*c^6 + 860160*a^6*b^8*c^7 - 1966080*a^7*b^6*c^8 + 2949120*a^8*b^4*c^9 -
2621440*a^9*b^2*c^10 + 1048576*a^10*c^11)*sqrt((5764801*b^8 + 45138800*a*b^6*c +
136380000*a^2*b^4*c^2 + 188000000*a^3*b^2*c^3 + 100000000*a^4*c^4)/(b^30*c^2 -
60*a*b^28*c^3 + 1680*a^2*b^26*c^4 - 29120*a^3*b^24*c^5 + 349440*a^4*b^22*c^6 - 3
075072*a^5*b^20*c^7 + 20500480*a^6*b^18*c^8 - 105431040*a^7*b^16*c^9 + 421724160
*a^8*b^14*c^10 - 1312030720*a^9*b^12*c^11 + 3148873728*a^10*b^10*c^12 - 57252249
60*a^11*b^8*c^13 + 7633633280*a^12*b^6*c^14 - 7046430720*a^13*b^4*c^15 + 4026531
840*a^14*b^2*c^16 - 1073741824*a^15*c^17)))/(b^20*c - 40*a*b^18*c^2 + 720*a^2*b^
16*c^3 - 7680*a^3*b^14*c^4 + 53760*a^4*b^12*c^5 - 258048*a^5*b^10*c^6 + 860160*a
^6*b^8*c^7 - 1966080*a^7*b^6*c^8 + 2949120*a^8*b^4*c^9 - 2621440*a^9*b^2*c^10 +
1048576*a^10*c^11)))*log(243*(252105*b^8 + 2197104*a*b^6*c + 6748000*a^2*b^4*c^2
+ 8800000*a^3*b^2*c^3 + 4000000*a^4*c^4)*sqrt(x) - 243*(9604*b^11 - 77648*a*b^9
*c + 49792*a^2*b^7*c^2 + 710144*a^3*b^5*c^3 - 486400*a^4*b^3*c^4 - 2560000*a^5*b
*c^5 - (11*b^22*c - 420*a*b^20*c^2 + 7120*a^2*b^18*c^3 - 70080*a^3*b^16*c^4 + 43
7760*a^4*b^14*c^5 - 1763328*a^5*b^12*c^6 + 4300800*a^6*b^10*c^7 - 4423680*a^7*b^
8*c^8 - 6881280*a^8*b^6*c^9 + 30146560*a^9*b^4*c^10 - 40894464*a^10*b^2*c^11 + 2
0971520*a^11*c^12)*sqrt((5764801*b^8 + 45138800*a*b^6*c + 136380000*a^2*b^4*c^2
+ 188000000*a^3*b^2*c^3 + 100000000*a^4*c^4)/(b^30*c^2 - 60*a*b^28*c^3 + 1680*a^
2*b^26*c^4 - 29120*a^3*b^24*c^5 + 349440*a^4*b^22*c^6 - 3075072*a^5*b^20*c^7 + 2
0500480*a^6*b^18*c^8 - 105431040*a^7*b^16*c^9 + 421724160*a^8*b^14*c^10 - 131203
0720*a^9*b^12*c^11 + 3148873728*a^10*b^10*c^12 - 5725224960*a^11*b^8*c^13 + 7633
633280*a^12*b^6*c^14 - 7046430720*a^13*b^4*c^15 + 4026531840*a^14*b^2*c^16 - 107
3741824*a^15*c^17)))*sqrt(sqrt(1/2)*sqrt(-(2401*b^9 + 86640*a*b^7*c + 413280*a^2
*b^5*c^2 + 833280*a^3*b^3*c^3 + 672000*a^4*b*c^4 - (b^20*c - 40*a*b^18*c^2 + 720
*a^2*b^16*c^3 - 7680*a^3*b^14*c^4 + 53760*a^4*b^12*c^5 - 258048*a^5*b^10*c^6 + 8
60160*a^6*b^8*c^7 - 1966080*a^7*b^6*c^8 + 2949120*a^8*b^4*c^9 - 2621440*a^9*b^2*
c^10 + 1048576*a^10*c^11)*sqrt((5764801*b^8 + 45138800*a*b^6*c + 136380000*a^2*b
^4*c^2 + 188000000*a^3*b^2*c^3 + 100000000*a^4*c^4)/(b^30*c^2 - 60*a*b^28*c^3 +
1680*a^2*b^26*c^4 - 29120*a^3*b^24*c^5 + 349440*a^4*b^22*c^6 - 3075072*a^5*b^20*
c^7 + 20500480*a^6*b^18*c^8 - 105431040*a^7*b^16*c^9 + 421724160*a^8*b^14*c^10 -
1312030720*a^9*b^12*c^11 + 3148873728*a^10*b^10*c^12 - 5725224960*a^11*b^8*c^13
+ 7633633280*a^12*b^6*c^14 - 7046430720*a^13*b^4*c^15 + 4026531840*a^14*b^2*c^1
6 - 1073741824*a^15*c^17)))/(b^20*c - 40*a*b^18*c^2 + 720*a^2*b^16*c^3 - 7680*a^
3*b^14*c^4 + 53760*a^4*b^12*c^5 - 258048*a^5*b^10*c^6 + 860160*a^6*b^8*c^7 - 196
6080*a^7*b^6*c^8 + 2949120*a^8*b^4*c^9 - 2621440*a^9*b^2*c^10 + 1048576*a^10*c^1
1)))) - 4*((7*b^2*c + 20*a*c^2)*x^6 + (11*b^3 + 28*a*b*c)*x^4 + 24*a^2*b + 3*(13
*a*b^2 - 4*a^2*c)*x^2)*sqrt(x))/((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^8 + 2*(b
^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^6 + a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b
^6 - 6*a*b^4*c + 32*a^3*c^3)*x^4 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(11/2)/(c*x**4+b*x**2+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{11}{2}}}{{\left (c x^{4} + b x^{2} + a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(11/2)/(c*x^4 + b*x^2 + a)^3,x, algorithm="giac")

[Out]

integrate(x^(11/2)/(c*x^4 + b*x^2 + a)^3, x)