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

Optimal. Leaf size=422 \[ \frac{3 x \left (b^2-4 a c\right )^2 \left (16 a^2 c^2-72 a b^2 c+33 b^4\right ) \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{32768 c^{13/2} \sqrt{a x^2+b x^3+c x^4}}-\frac{b x \left (2416 a^2 c^2-1560 a b^2 c+231 b^4\right ) \sqrt{a x^2+b x^3+c x^4}}{71680 c^4}+\frac{x^2 \left (560 a^2 c^2-568 a b^2 c+99 b^4\right ) \sqrt{a x^2+b x^3+c x^4}}{35840 c^3}-\frac{b \left (-58816 a^3 c^3+81648 a^2 b^2 c^2-30660 a b^4 c+3465 b^6\right ) \sqrt{a x^2+b x^3+c x^4}}{573440 c^6 x}+\frac{\left (-6720 a^3 c^3+18896 a^2 b^2 c^2-8988 a b^4 c+1155 b^6\right ) \sqrt{a x^2+b x^3+c x^4}}{286720 c^5}-\frac{x^3 \left (10 c x \left (11 b^2-28 a c\right )+b \left (68 a c+11 b^2\right )\right ) \sqrt{a x^2+b x^3+c x^4}}{4480 c^2}+\frac{x (3 b+14 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{112 c} \]

[Out]

((1155*b^6 - 8988*a*b^4*c + 18896*a^2*b^2*c^2 - 6720*a^3*c^3)*Sqrt[a*x^2 + b*x^3
+ c*x^4])/(286720*c^5) - (b*(3465*b^6 - 30660*a*b^4*c + 81648*a^2*b^2*c^2 - 588
16*a^3*c^3)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(573440*c^6*x) - (b*(231*b^4 - 1560*a*b
^2*c + 2416*a^2*c^2)*x*Sqrt[a*x^2 + b*x^3 + c*x^4])/(71680*c^4) + ((99*b^4 - 568
*a*b^2*c + 560*a^2*c^2)*x^2*Sqrt[a*x^2 + b*x^3 + c*x^4])/(35840*c^3) - (x^3*(b*(
11*b^2 + 68*a*c) + 10*c*(11*b^2 - 28*a*c)*x)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(4480*
c^2) + (x*(3*b + 14*c*x)*(a*x^2 + b*x^3 + c*x^4)^(3/2))/(112*c) + (3*(b^2 - 4*a*
c)^2*(33*b^4 - 72*a*b^2*c + 16*a^2*c^2)*x*Sqrt[a + b*x + c*x^2]*ArcTanh[(b + 2*c
*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(32768*c^(13/2)*Sqrt[a*x^2 + b*x^3 + c*x
^4])

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Rubi [A]  time = 1.87124, antiderivative size = 422, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ \frac{3 x \left (b^2-4 a c\right )^2 \left (16 a^2 c^2-72 a b^2 c+33 b^4\right ) \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{32768 c^{13/2} \sqrt{a x^2+b x^3+c x^4}}-\frac{b x \left (2416 a^2 c^2-1560 a b^2 c+231 b^4\right ) \sqrt{a x^2+b x^3+c x^4}}{71680 c^4}+\frac{x^2 \left (560 a^2 c^2-568 a b^2 c+99 b^4\right ) \sqrt{a x^2+b x^3+c x^4}}{35840 c^3}-\frac{b \left (-58816 a^3 c^3+81648 a^2 b^2 c^2-30660 a b^4 c+3465 b^6\right ) \sqrt{a x^2+b x^3+c x^4}}{573440 c^6 x}+\frac{\left (-6720 a^3 c^3+18896 a^2 b^2 c^2-8988 a b^4 c+1155 b^6\right ) \sqrt{a x^2+b x^3+c x^4}}{286720 c^5}-\frac{x^3 \left (10 c x \left (11 b^2-28 a c\right )+b \left (68 a c+11 b^2\right )\right ) \sqrt{a x^2+b x^3+c x^4}}{4480 c^2}+\frac{x (3 b+14 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{112 c} \]

Antiderivative was successfully verified.

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

[Out]

((1155*b^6 - 8988*a*b^4*c + 18896*a^2*b^2*c^2 - 6720*a^3*c^3)*Sqrt[a*x^2 + b*x^3
+ c*x^4])/(286720*c^5) - (b*(3465*b^6 - 30660*a*b^4*c + 81648*a^2*b^2*c^2 - 588
16*a^3*c^3)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(573440*c^6*x) - (b*(231*b^4 - 1560*a*b
^2*c + 2416*a^2*c^2)*x*Sqrt[a*x^2 + b*x^3 + c*x^4])/(71680*c^4) + ((99*b^4 - 568
*a*b^2*c + 560*a^2*c^2)*x^2*Sqrt[a*x^2 + b*x^3 + c*x^4])/(35840*c^3) - (x^3*(b*(
11*b^2 + 68*a*c) + 10*c*(11*b^2 - 28*a*c)*x)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(4480*
c^2) + (x*(3*b + 14*c*x)*(a*x^2 + b*x^3 + c*x^4)^(3/2))/(112*c) + (3*(b^2 - 4*a*
c)^2*(33*b^4 - 72*a*b^2*c + 16*a^2*c^2)*x*Sqrt[a + b*x + c*x^2]*ArcTanh[(b + 2*c
*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(32768*c^(13/2)*Sqrt[a*x^2 + b*x^3 + c*x
^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*(c*x**4+b*x**3+a*x**2)**(3/2),x)

[Out]

Timed out

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Mathematica [A]  time = 0.647349, size = 304, normalized size = 0.72 \[ \frac{x \sqrt{a+x (b+c x)} \left (105 \left (16 a^2 c^2-72 a b^2 c+33 b^4\right ) \left (b^2-4 a c\right )^2 \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )+2 \sqrt{c} \sqrt{a+x (b+c x)} \left (-16 b^3 c^2 \left (5103 a^2-780 a c x^2+88 c^2 x^4\right )+32 b^2 c^3 x \left (1181 a^2-284 a c x^2+40 c^2 x^4\right )+64 b c^3 \left (919 a^3-302 a^2 c x^2+104 a c^2 x^4+1360 c^3 x^6\right )+4480 c^4 x \left (-3 a^3+2 a^2 c x^2+24 a c^2 x^4+16 c^3 x^6\right )+84 b^5 c \left (365 a-22 c x^2\right )+24 b^4 c^2 x \left (66 c x^2-749 a\right )-3465 b^7+2310 b^6 c x\right )\right )}{1146880 c^{13/2} \sqrt{x^2 (a+x (b+c x))}} \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[a + x*(b + c*x)]*(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-3465*b^7 + 2310*b^6*
c*x + 84*b^5*c*(365*a - 22*c*x^2) + 24*b^4*c^2*x*(-749*a + 66*c*x^2) + 32*b^2*c^
3*x*(1181*a^2 - 284*a*c*x^2 + 40*c^2*x^4) - 16*b^3*c^2*(5103*a^2 - 780*a*c*x^2 +
88*c^2*x^4) + 4480*c^4*x*(-3*a^3 + 2*a^2*c*x^2 + 24*a*c^2*x^4 + 16*c^3*x^6) + 6
4*b*c^3*(919*a^3 - 302*a^2*c*x^2 + 104*a*c^2*x^4 + 1360*c^3*x^6)) + 105*(b^2 - 4
*a*c)^2*(33*b^4 - 72*a*b^2*c + 16*a^2*c^2)*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*
(b + c*x)]]))/(1146880*c^(13/2)*Sqrt[x^2*(a + x*(b + c*x))])

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Maple [A]  time = 0.014, size = 649, normalized size = 1.5 \[{\frac{1}{1146880\,{x}^{3}} \left ( c{x}^{4}+b{x}^{3}+a{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 26880\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ){a}^{4}{c}^{5}+3465\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ){b}^{8}c+143360\,{x}^{3} \left ( c{x}^{2}+bx+a \right ) ^{5/2}{c}^{13/2}-59136\, \left ( c{x}^{2}+bx+a \right ) ^{5/2}{c}^{7/2}{b}^{3}+18480\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{c}^{5/2}{b}^{5}+117600\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ){a}^{2}{b}^{4}{c}^{3}-35280\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ) a{b}^{6}{c}^{2}+17920\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{c}^{11/2}x{a}^{2}+95232\, \left ( c{x}^{2}+bx+a \right ) ^{5/2}{c}^{9/2}ab+13440\,\sqrt{c{x}^{2}+bx+a}{c}^{9/2}{a}^{3}b-63840\,\sqrt{c{x}^{2}+bx+a}{c}^{7/2}{a}^{2}{b}^{3}+42840\,\sqrt{c{x}^{2}+bx+a}{c}^{5/2}a{b}^{5}-112640\, \left ( c{x}^{2}+bx+a \right ) ^{5/2}{c}^{11/2}{x}^{2}b-71680\, \left ( c{x}^{2}+bx+a \right ) ^{5/2}{c}^{11/2}xa+84480\, \left ( c{x}^{2}+bx+a \right ) ^{5/2}{c}^{9/2}x{b}^{2}+26880\,\sqrt{c{x}^{2}+bx+a}{c}^{11/2}x{a}^{3}-13860\,\sqrt{c{x}^{2}+bx+a}{c}^{5/2}x{b}^{6}-40320\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{c}^{7/2}a{b}^{3}+36960\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{c}^{7/2}x{b}^{4}+8960\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{c}^{9/2}{a}^{2}b-134400\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ){a}^{3}{b}^{2}{c}^{4}-127680\,\sqrt{c{x}^{2}+bx+a}{c}^{9/2}x{a}^{2}{b}^{2}+85680\,\sqrt{c{x}^{2}+bx+a}{c}^{7/2}xa{b}^{4}-80640\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{c}^{9/2}xa{b}^{2}-6930\,\sqrt{c{x}^{2}+bx+a}{c}^{3/2}{b}^{7} \right ) \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}{c}^{-{\frac{15}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1146880*(c*x^4+b*x^3+a*x^2)^(3/2)*(26880*ln(1/2*(2*(c*x^2+b*x+a)^(1/2)*c^(1/2)
+2*c*x+b)/c^(1/2))*a^4*c^5+3465*ln(1/2*(2*(c*x^2+b*x+a)^(1/2)*c^(1/2)+2*c*x+b)/c
^(1/2))*b^8*c+143360*x^3*(c*x^2+b*x+a)^(5/2)*c^(13/2)-59136*(c*x^2+b*x+a)^(5/2)*
c^(7/2)*b^3+18480*(c*x^2+b*x+a)^(3/2)*c^(5/2)*b^5+117600*ln(1/2*(2*(c*x^2+b*x+a)
^(1/2)*c^(1/2)+2*c*x+b)/c^(1/2))*a^2*b^4*c^3-35280*ln(1/2*(2*(c*x^2+b*x+a)^(1/2)
*c^(1/2)+2*c*x+b)/c^(1/2))*a*b^6*c^2+17920*(c*x^2+b*x+a)^(3/2)*c^(11/2)*x*a^2+95
232*(c*x^2+b*x+a)^(5/2)*c^(9/2)*a*b+13440*(c*x^2+b*x+a)^(1/2)*c^(9/2)*a^3*b-6384
0*(c*x^2+b*x+a)^(1/2)*c^(7/2)*a^2*b^3+42840*(c*x^2+b*x+a)^(1/2)*c^(5/2)*a*b^5-11
2640*(c*x^2+b*x+a)^(5/2)*c^(11/2)*x^2*b-71680*(c*x^2+b*x+a)^(5/2)*c^(11/2)*x*a+8
4480*(c*x^2+b*x+a)^(5/2)*c^(9/2)*x*b^2+26880*(c*x^2+b*x+a)^(1/2)*c^(11/2)*x*a^3-
13860*(c*x^2+b*x+a)^(1/2)*c^(5/2)*x*b^6-40320*(c*x^2+b*x+a)^(3/2)*c^(7/2)*a*b^3+
36960*(c*x^2+b*x+a)^(3/2)*c^(7/2)*x*b^4+8960*(c*x^2+b*x+a)^(3/2)*c^(9/2)*a^2*b-1
34400*ln(1/2*(2*(c*x^2+b*x+a)^(1/2)*c^(1/2)+2*c*x+b)/c^(1/2))*a^3*b^2*c^4-127680
*(c*x^2+b*x+a)^(1/2)*c^(9/2)*x*a^2*b^2+85680*(c*x^2+b*x+a)^(1/2)*c^(7/2)*x*a*b^4
-80640*(c*x^2+b*x+a)^(3/2)*c^(9/2)*x*a*b^2-6930*(c*x^2+b*x+a)^(1/2)*c^(3/2)*b^7)
/x^3/(c*x^2+b*x+a)^(3/2)/c^(15/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.374168, size = 1, normalized size = 0. \[ \left [\frac{105 \,{\left (33 \, b^{8} - 336 \, a b^{6} c + 1120 \, a^{2} b^{4} c^{2} - 1280 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} \sqrt{c} x \log \left (-\frac{4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (2 \, c^{2} x + b c\right )} +{\left (8 \, c^{2} x^{3} + 8 \, b c x^{2} +{\left (b^{2} + 4 \, a c\right )} x\right )} \sqrt{c}}{x}\right ) + 4 \,{\left (71680 \, c^{8} x^{7} + 87040 \, b c^{7} x^{6} - 3465 \, b^{7} c + 30660 \, a b^{5} c^{2} - 81648 \, a^{2} b^{3} c^{3} + 58816 \, a^{3} b c^{4} + 1280 \,{\left (b^{2} c^{6} + 84 \, a c^{7}\right )} x^{5} - 128 \,{\left (11 \, b^{3} c^{5} - 52 \, a b c^{6}\right )} x^{4} + 16 \,{\left (99 \, b^{4} c^{4} - 568 \, a b^{2} c^{5} + 560 \, a^{2} c^{6}\right )} x^{3} - 8 \,{\left (231 \, b^{5} c^{3} - 1560 \, a b^{3} c^{4} + 2416 \, a^{2} b c^{5}\right )} x^{2} + 2 \,{\left (1155 \, b^{6} c^{2} - 8988 \, a b^{4} c^{3} + 18896 \, a^{2} b^{2} c^{4} - 6720 \, a^{3} c^{5}\right )} x\right )} \sqrt{c x^{4} + b x^{3} + a x^{2}}}{2293760 \, c^{7} x}, -\frac{105 \,{\left (33 \, b^{8} - 336 \, a b^{6} c + 1120 \, a^{2} b^{4} c^{2} - 1280 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} \sqrt{-c} x \arctan \left (\frac{{\left (2 \, c x^{2} + b x\right )} \sqrt{-c}}{2 \, \sqrt{c x^{4} + b x^{3} + a x^{2}} c}\right ) - 2 \,{\left (71680 \, c^{8} x^{7} + 87040 \, b c^{7} x^{6} - 3465 \, b^{7} c + 30660 \, a b^{5} c^{2} - 81648 \, a^{2} b^{3} c^{3} + 58816 \, a^{3} b c^{4} + 1280 \,{\left (b^{2} c^{6} + 84 \, a c^{7}\right )} x^{5} - 128 \,{\left (11 \, b^{3} c^{5} - 52 \, a b c^{6}\right )} x^{4} + 16 \,{\left (99 \, b^{4} c^{4} - 568 \, a b^{2} c^{5} + 560 \, a^{2} c^{6}\right )} x^{3} - 8 \,{\left (231 \, b^{5} c^{3} - 1560 \, a b^{3} c^{4} + 2416 \, a^{2} b c^{5}\right )} x^{2} + 2 \,{\left (1155 \, b^{6} c^{2} - 8988 \, a b^{4} c^{3} + 18896 \, a^{2} b^{2} c^{4} - 6720 \, a^{3} c^{5}\right )} x\right )} \sqrt{c x^{4} + b x^{3} + a x^{2}}}{1146880 \, c^{7} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2293760*(105*(33*b^8 - 336*a*b^6*c + 1120*a^2*b^4*c^2 - 1280*a^3*b^2*c^3 + 25
6*a^4*c^4)*sqrt(c)*x*log(-(4*sqrt(c*x^4 + b*x^3 + a*x^2)*(2*c^2*x + b*c) + (8*c^
2*x^3 + 8*b*c*x^2 + (b^2 + 4*a*c)*x)*sqrt(c))/x) + 4*(71680*c^8*x^7 + 87040*b*c^
7*x^6 - 3465*b^7*c + 30660*a*b^5*c^2 - 81648*a^2*b^3*c^3 + 58816*a^3*b*c^4 + 128
0*(b^2*c^6 + 84*a*c^7)*x^5 - 128*(11*b^3*c^5 - 52*a*b*c^6)*x^4 + 16*(99*b^4*c^4
- 568*a*b^2*c^5 + 560*a^2*c^6)*x^3 - 8*(231*b^5*c^3 - 1560*a*b^3*c^4 + 2416*a^2*
b*c^5)*x^2 + 2*(1155*b^6*c^2 - 8988*a*b^4*c^3 + 18896*a^2*b^2*c^4 - 6720*a^3*c^5
)*x)*sqrt(c*x^4 + b*x^3 + a*x^2))/(c^7*x), -1/1146880*(105*(33*b^8 - 336*a*b^6*c
+ 1120*a^2*b^4*c^2 - 1280*a^3*b^2*c^3 + 256*a^4*c^4)*sqrt(-c)*x*arctan(1/2*(2*c
*x^2 + b*x)*sqrt(-c)/(sqrt(c*x^4 + b*x^3 + a*x^2)*c)) - 2*(71680*c^8*x^7 + 87040
*b*c^7*x^6 - 3465*b^7*c + 30660*a*b^5*c^2 - 81648*a^2*b^3*c^3 + 58816*a^3*b*c^4
+ 1280*(b^2*c^6 + 84*a*c^7)*x^5 - 128*(11*b^3*c^5 - 52*a*b*c^6)*x^4 + 16*(99*b^4
*c^4 - 568*a*b^2*c^5 + 560*a^2*c^6)*x^3 - 8*(231*b^5*c^3 - 1560*a*b^3*c^4 + 2416
*a^2*b*c^5)*x^2 + 2*(1155*b^6*c^2 - 8988*a*b^4*c^3 + 18896*a^2*b^2*c^4 - 6720*a^
3*c^5)*x)*sqrt(c*x^4 + b*x^3 + a*x^2))/(c^7*x)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x \left (x^{2} \left (a + b x + c x^{2}\right )\right )^{\frac{3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.339766, size = 703, normalized size = 1.67 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/573440*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*(4*(14*c*x*sign(x) + 17*b*sign(x)
)*x + (b^2*c^6*sign(x) + 84*a*c^7*sign(x))/c^7)*x - (11*b^3*c^5*sign(x) - 52*a*b
*c^6*sign(x))/c^7)*x + (99*b^4*c^4*sign(x) - 568*a*b^2*c^5*sign(x) + 560*a^2*c^6
*sign(x))/c^7)*x - (231*b^5*c^3*sign(x) - 1560*a*b^3*c^4*sign(x) + 2416*a^2*b*c^
5*sign(x))/c^7)*x + (1155*b^6*c^2*sign(x) - 8988*a*b^4*c^3*sign(x) + 18896*a^2*b
^2*c^4*sign(x) - 6720*a^3*c^5*sign(x))/c^7)*x - (3465*b^7*c*sign(x) - 30660*a*b^
5*c^2*sign(x) + 81648*a^2*b^3*c^3*sign(x) - 58816*a^3*b*c^4*sign(x))/c^7) - 3/32
768*(33*b^8*sign(x) - 336*a*b^6*c*sign(x) + 1120*a^2*b^4*c^2*sign(x) - 1280*a^3*
b^2*c^3*sign(x) + 256*a^4*c^4*sign(x))*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))*sqrt(c) - b))/c^(13/2) + 1/1146880*(3465*b^8*ln(abs(-b + 2*sqrt(a)*sqrt(c))
) - 35280*a*b^6*c*ln(abs(-b + 2*sqrt(a)*sqrt(c))) + 117600*a^2*b^4*c^2*ln(abs(-b
+ 2*sqrt(a)*sqrt(c))) - 134400*a^3*b^2*c^3*ln(abs(-b + 2*sqrt(a)*sqrt(c))) + 26
880*a^4*c^4*ln(abs(-b + 2*sqrt(a)*sqrt(c))) + 6930*sqrt(a)*b^7*sqrt(c) - 61320*a
^(3/2)*b^5*c^(3/2) + 163296*a^(5/2)*b^3*c^(5/2) - 117632*a^(7/2)*b*c^(7/2))*sign
(x)/c^(13/2)