3.303 \(\int (d+e x)^2 \left (b x+c x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=266 \[ -\frac{5 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2} \left (9 b^2 e^2-32 b c d e+32 c^2 d^2\right )}{6144 c^4}+\frac{(b+2 c x) \left (b x+c x^2\right )^{5/2} \left (9 b^2 e^2-32 b c d e+32 c^2 d^2\right )}{384 c^3}-\frac{5 b^6 \left (9 b^2 e^2-32 b c d e+32 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{16384 c^{11/2}}+\frac{5 b^4 (b+2 c x) \sqrt{b x+c x^2} \left (9 b^2 e^2-32 b c d e+32 c^2 d^2\right )}{16384 c^5}+\frac{9 e \left (b x+c x^2\right )^{7/2} (2 c d-b e)}{112 c^2}+\frac{e \left (b x+c x^2\right )^{7/2} (d+e x)}{8 c} \]

[Out]

(5*b^4*(32*c^2*d^2 - 32*b*c*d*e + 9*b^2*e^2)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(163
84*c^5) - (5*b^2*(32*c^2*d^2 - 32*b*c*d*e + 9*b^2*e^2)*(b + 2*c*x)*(b*x + c*x^2)
^(3/2))/(6144*c^4) + ((32*c^2*d^2 - 32*b*c*d*e + 9*b^2*e^2)*(b + 2*c*x)*(b*x + c
*x^2)^(5/2))/(384*c^3) + (9*e*(2*c*d - b*e)*(b*x + c*x^2)^(7/2))/(112*c^2) + (e*
(d + e*x)*(b*x + c*x^2)^(7/2))/(8*c) - (5*b^6*(32*c^2*d^2 - 32*b*c*d*e + 9*b^2*e
^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(16384*c^(11/2))

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Rubi [A]  time = 0.512527, antiderivative size = 266, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ -\frac{5 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2} \left (9 b^2 e^2-32 b c d e+32 c^2 d^2\right )}{6144 c^4}+\frac{(b+2 c x) \left (b x+c x^2\right )^{5/2} \left (9 b^2 e^2-32 b c d e+32 c^2 d^2\right )}{384 c^3}-\frac{5 b^6 \left (9 b^2 e^2-32 b c d e+32 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{16384 c^{11/2}}+\frac{5 b^4 (b+2 c x) \sqrt{b x+c x^2} \left (9 b^2 e^2-32 b c d e+32 c^2 d^2\right )}{16384 c^5}+\frac{9 e \left (b x+c x^2\right )^{7/2} (2 c d-b e)}{112 c^2}+\frac{e \left (b x+c x^2\right )^{7/2} (d+e x)}{8 c} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^2*(b*x + c*x^2)^(5/2),x]

[Out]

(5*b^4*(32*c^2*d^2 - 32*b*c*d*e + 9*b^2*e^2)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(163
84*c^5) - (5*b^2*(32*c^2*d^2 - 32*b*c*d*e + 9*b^2*e^2)*(b + 2*c*x)*(b*x + c*x^2)
^(3/2))/(6144*c^4) + ((32*c^2*d^2 - 32*b*c*d*e + 9*b^2*e^2)*(b + 2*c*x)*(b*x + c
*x^2)^(5/2))/(384*c^3) + (9*e*(2*c*d - b*e)*(b*x + c*x^2)^(7/2))/(112*c^2) + (e*
(d + e*x)*(b*x + c*x^2)^(7/2))/(8*c) - (5*b^6*(32*c^2*d^2 - 32*b*c*d*e + 9*b^2*e
^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(16384*c^(11/2))

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Rubi in Sympy [A]  time = 43.4857, size = 265, normalized size = 1. \[ - \frac{5 b^{6} \left (9 b^{2} e^{2} - 32 b c d e + 32 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{16384 c^{\frac{11}{2}}} + \frac{5 b^{4} \left (b + 2 c x\right ) \sqrt{b x + c x^{2}} \left (9 b^{2} e^{2} - 32 b c d e + 32 c^{2} d^{2}\right )}{16384 c^{5}} - \frac{5 b^{2} \left (b + 2 c x\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}} \left (9 b^{2} e^{2} - 32 b c d e + 32 c^{2} d^{2}\right )}{6144 c^{4}} + \frac{e \left (d + e x\right ) \left (b x + c x^{2}\right )^{\frac{7}{2}}}{8 c} - \frac{9 e \left (b e - 2 c d\right ) \left (b x + c x^{2}\right )^{\frac{7}{2}}}{112 c^{2}} + \frac{\left (b + 2 c x\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}} \left (9 b^{2} e^{2} - 32 b c d e + 32 c^{2} d^{2}\right )}{384 c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**2*(c*x**2+b*x)**(5/2),x)

[Out]

-5*b**6*(9*b**2*e**2 - 32*b*c*d*e + 32*c**2*d**2)*atanh(sqrt(c)*x/sqrt(b*x + c*x
**2))/(16384*c**(11/2)) + 5*b**4*(b + 2*c*x)*sqrt(b*x + c*x**2)*(9*b**2*e**2 - 3
2*b*c*d*e + 32*c**2*d**2)/(16384*c**5) - 5*b**2*(b + 2*c*x)*(b*x + c*x**2)**(3/2
)*(9*b**2*e**2 - 32*b*c*d*e + 32*c**2*d**2)/(6144*c**4) + e*(d + e*x)*(b*x + c*x
**2)**(7/2)/(8*c) - 9*e*(b*e - 2*c*d)*(b*x + c*x**2)**(7/2)/(112*c**2) + (b + 2*
c*x)*(b*x + c*x**2)**(5/2)*(9*b**2*e**2 - 32*b*c*d*e + 32*c**2*d**2)/(384*c**3)

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Mathematica [A]  time = 0.560383, size = 287, normalized size = 1.08 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} \left (945 b^7 e^2-210 b^6 c e (16 d+3 e x)+56 b^5 c^2 \left (60 d^2+40 d e x+9 e^2 x^2\right )-16 b^4 c^3 x \left (140 d^2+112 d e x+27 e^2 x^2\right )+128 b^3 c^4 x^2 \left (14 d^2+12 d e x+3 e^2 x^2\right )+256 b^2 c^5 x^3 \left (378 d^2+592 d e x+243 e^2 x^2\right )+1024 b c^6 x^4 \left (140 d^2+232 d e x+99 e^2 x^2\right )+2048 c^7 x^5 \left (28 d^2+48 d e x+21 e^2 x^2\right )\right )-\frac{105 b^6 \left (9 b^2 e^2-32 b c d e+32 c^2 d^2\right ) \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{x} \sqrt{b+c x}}\right )}{344064 c^{11/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^2*(b*x + c*x^2)^(5/2),x]

[Out]

(Sqrt[x*(b + c*x)]*(Sqrt[c]*(945*b^7*e^2 - 210*b^6*c*e*(16*d + 3*e*x) + 128*b^3*
c^4*x^2*(14*d^2 + 12*d*e*x + 3*e^2*x^2) + 56*b^5*c^2*(60*d^2 + 40*d*e*x + 9*e^2*
x^2) + 2048*c^7*x^5*(28*d^2 + 48*d*e*x + 21*e^2*x^2) - 16*b^4*c^3*x*(140*d^2 + 1
12*d*e*x + 27*e^2*x^2) + 1024*b*c^6*x^4*(140*d^2 + 232*d*e*x + 99*e^2*x^2) + 256
*b^2*c^5*x^3*(378*d^2 + 592*d*e*x + 243*e^2*x^2)) - (105*b^6*(32*c^2*d^2 - 32*b*
c*d*e + 9*b^2*e^2)*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b + c*x]])/(Sqrt[x]*Sqrt[b + c*x
])))/(344064*c^(11/2))

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Maple [B]  time = 0.013, size = 553, normalized size = 2.1 \[{\frac{{d}^{2}x}{6} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{{d}^{2}b}{12\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{b}^{2}{d}^{2}x}{96\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{d}^{2}{b}^{3}}{192\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{d}^{2}{b}^{4}x}{256\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{5\,{d}^{2}{b}^{5}}{512\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{5\,{d}^{2}{b}^{6}}{1024}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{7}{2}}}}+{\frac{{e}^{2}x}{8\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}-{\frac{9\,{e}^{2}b}{112\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}+{\frac{3\,{b}^{2}{e}^{2}x}{64\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{3\,{b}^{3}{e}^{2}}{128\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{15\,{e}^{2}{b}^{4}x}{1024\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{15\,{e}^{2}{b}^{5}}{2048\,{c}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{45\,{e}^{2}{b}^{6}x}{8192\,{c}^{4}}\sqrt{c{x}^{2}+bx}}+{\frac{45\,{e}^{2}{b}^{7}}{16384\,{c}^{5}}\sqrt{c{x}^{2}+bx}}-{\frac{45\,{e}^{2}{b}^{8}}{32768}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{11}{2}}}}+{\frac{2\,de}{7\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}-{\frac{bdex}{6\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{{b}^{2}de}{12\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{5\,de{b}^{3}x}{96\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{5\,de{b}^{4}}{192\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{5\,de{b}^{5}x}{256\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{5\,de{b}^{6}}{512\,{c}^{4}}\sqrt{c{x}^{2}+bx}}+{\frac{5\,de{b}^{7}}{1024}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^2*(c*x^2+b*x)^(5/2),x)

[Out]

1/6*d^2*(c*x^2+b*x)^(5/2)*x+1/12*d^2/c*(c*x^2+b*x)^(5/2)*b-5/96*d^2*b^2/c*(c*x^2
+b*x)^(3/2)*x-5/192*d^2*b^3/c^2*(c*x^2+b*x)^(3/2)+5/256*d^2*b^4/c^2*(c*x^2+b*x)^
(1/2)*x+5/512*d^2*b^5/c^3*(c*x^2+b*x)^(1/2)-5/1024*d^2*b^6/c^(7/2)*ln((1/2*b+c*x
)/c^(1/2)+(c*x^2+b*x)^(1/2))+1/8*e^2*x*(c*x^2+b*x)^(7/2)/c-9/112*e^2*b/c^2*(c*x^
2+b*x)^(7/2)+3/64*e^2*b^2/c^2*(c*x^2+b*x)^(5/2)*x+3/128*e^2*b^3/c^3*(c*x^2+b*x)^
(5/2)-15/1024*e^2*b^4/c^3*(c*x^2+b*x)^(3/2)*x-15/2048*e^2*b^5/c^4*(c*x^2+b*x)^(3
/2)+45/8192*e^2*b^6/c^4*(c*x^2+b*x)^(1/2)*x+45/16384*e^2*b^7/c^5*(c*x^2+b*x)^(1/
2)-45/32768*e^2*b^8/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))+2/7*d*e*(
c*x^2+b*x)^(7/2)/c-1/6*d*e*b/c*(c*x^2+b*x)^(5/2)*x-1/12*d*e*b^2/c^2*(c*x^2+b*x)^
(5/2)+5/96*d*e*b^3/c^2*(c*x^2+b*x)^(3/2)*x+5/192*d*e*b^4/c^3*(c*x^2+b*x)^(3/2)-5
/256*d*e*b^5/c^3*(c*x^2+b*x)^(1/2)*x-5/512*d*e*b^6/c^4*(c*x^2+b*x)^(1/2)+5/1024*
d*e*b^7/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))

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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^2 + b*x)^(5/2)*(e*x + d)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.239971, size = 1, normalized size = 0. \[ \left [\frac{2 \,{\left (43008 \, c^{7} e^{2} x^{7} + 3360 \, b^{5} c^{2} d^{2} - 3360 \, b^{6} c d e + 945 \, b^{7} e^{2} + 3072 \,{\left (32 \, c^{7} d e + 33 \, b c^{6} e^{2}\right )} x^{6} + 256 \,{\left (224 \, c^{7} d^{2} + 928 \, b c^{6} d e + 243 \, b^{2} c^{5} e^{2}\right )} x^{5} + 128 \,{\left (1120 \, b c^{6} d^{2} + 1184 \, b^{2} c^{5} d e + 3 \, b^{3} c^{4} e^{2}\right )} x^{4} + 48 \,{\left (2016 \, b^{2} c^{5} d^{2} + 32 \, b^{3} c^{4} d e - 9 \, b^{4} c^{3} e^{2}\right )} x^{3} + 56 \,{\left (32 \, b^{3} c^{4} d^{2} - 32 \, b^{4} c^{3} d e + 9 \, b^{5} c^{2} e^{2}\right )} x^{2} - 70 \,{\left (32 \, b^{4} c^{3} d^{2} - 32 \, b^{5} c^{2} d e + 9 \, b^{6} c e^{2}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{c} + 105 \,{\left (32 \, b^{6} c^{2} d^{2} - 32 \, b^{7} c d e + 9 \, b^{8} e^{2}\right )} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} - 2 \, \sqrt{c x^{2} + b x} c\right )}{688128 \, c^{\frac{11}{2}}}, \frac{{\left (43008 \, c^{7} e^{2} x^{7} + 3360 \, b^{5} c^{2} d^{2} - 3360 \, b^{6} c d e + 945 \, b^{7} e^{2} + 3072 \,{\left (32 \, c^{7} d e + 33 \, b c^{6} e^{2}\right )} x^{6} + 256 \,{\left (224 \, c^{7} d^{2} + 928 \, b c^{6} d e + 243 \, b^{2} c^{5} e^{2}\right )} x^{5} + 128 \,{\left (1120 \, b c^{6} d^{2} + 1184 \, b^{2} c^{5} d e + 3 \, b^{3} c^{4} e^{2}\right )} x^{4} + 48 \,{\left (2016 \, b^{2} c^{5} d^{2} + 32 \, b^{3} c^{4} d e - 9 \, b^{4} c^{3} e^{2}\right )} x^{3} + 56 \,{\left (32 \, b^{3} c^{4} d^{2} - 32 \, b^{4} c^{3} d e + 9 \, b^{5} c^{2} e^{2}\right )} x^{2} - 70 \,{\left (32 \, b^{4} c^{3} d^{2} - 32 \, b^{5} c^{2} d e + 9 \, b^{6} c e^{2}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{-c} - 105 \,{\left (32 \, b^{6} c^{2} d^{2} - 32 \, b^{7} c d e + 9 \, b^{8} e^{2}\right )} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right )}{344064 \, \sqrt{-c} c^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(e*x + d)^2,x, algorithm="fricas")

[Out]

[1/688128*(2*(43008*c^7*e^2*x^7 + 3360*b^5*c^2*d^2 - 3360*b^6*c*d*e + 945*b^7*e^
2 + 3072*(32*c^7*d*e + 33*b*c^6*e^2)*x^6 + 256*(224*c^7*d^2 + 928*b*c^6*d*e + 24
3*b^2*c^5*e^2)*x^5 + 128*(1120*b*c^6*d^2 + 1184*b^2*c^5*d*e + 3*b^3*c^4*e^2)*x^4
+ 48*(2016*b^2*c^5*d^2 + 32*b^3*c^4*d*e - 9*b^4*c^3*e^2)*x^3 + 56*(32*b^3*c^4*d
^2 - 32*b^4*c^3*d*e + 9*b^5*c^2*e^2)*x^2 - 70*(32*b^4*c^3*d^2 - 32*b^5*c^2*d*e +
9*b^6*c*e^2)*x)*sqrt(c*x^2 + b*x)*sqrt(c) + 105*(32*b^6*c^2*d^2 - 32*b^7*c*d*e
+ 9*b^8*e^2)*log((2*c*x + b)*sqrt(c) - 2*sqrt(c*x^2 + b*x)*c))/c^(11/2), 1/34406
4*((43008*c^7*e^2*x^7 + 3360*b^5*c^2*d^2 - 3360*b^6*c*d*e + 945*b^7*e^2 + 3072*(
32*c^7*d*e + 33*b*c^6*e^2)*x^6 + 256*(224*c^7*d^2 + 928*b*c^6*d*e + 243*b^2*c^5*
e^2)*x^5 + 128*(1120*b*c^6*d^2 + 1184*b^2*c^5*d*e + 3*b^3*c^4*e^2)*x^4 + 48*(201
6*b^2*c^5*d^2 + 32*b^3*c^4*d*e - 9*b^4*c^3*e^2)*x^3 + 56*(32*b^3*c^4*d^2 - 32*b^
4*c^3*d*e + 9*b^5*c^2*e^2)*x^2 - 70*(32*b^4*c^3*d^2 - 32*b^5*c^2*d*e + 9*b^6*c*e
^2)*x)*sqrt(c*x^2 + b*x)*sqrt(-c) - 105*(32*b^6*c^2*d^2 - 32*b^7*c*d*e + 9*b^8*e
^2)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)))/(sqrt(-c)*c^5)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (x \left (b + c x\right )\right )^{\frac{5}{2}} \left (d + e x\right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**2*(c*x**2+b*x)**(5/2),x)

[Out]

Integral((x*(b + c*x))**(5/2)*(d + e*x)**2, x)

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GIAC/XCAS [A]  time = 0.237898, size = 473, normalized size = 1.78 \[ \frac{1}{344064} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (2 \,{\left (12 \,{\left (14 \, c^{2} x e^{2} + \frac{32 \, c^{9} d e + 33 \, b c^{8} e^{2}}{c^{7}}\right )} x + \frac{224 \, c^{9} d^{2} + 928 \, b c^{8} d e + 243 \, b^{2} c^{7} e^{2}}{c^{7}}\right )} x + \frac{1120 \, b c^{8} d^{2} + 1184 \, b^{2} c^{7} d e + 3 \, b^{3} c^{6} e^{2}}{c^{7}}\right )} x + \frac{3 \,{\left (2016 \, b^{2} c^{7} d^{2} + 32 \, b^{3} c^{6} d e - 9 \, b^{4} c^{5} e^{2}\right )}}{c^{7}}\right )} x + \frac{7 \,{\left (32 \, b^{3} c^{6} d^{2} - 32 \, b^{4} c^{5} d e + 9 \, b^{5} c^{4} e^{2}\right )}}{c^{7}}\right )} x - \frac{35 \,{\left (32 \, b^{4} c^{5} d^{2} - 32 \, b^{5} c^{4} d e + 9 \, b^{6} c^{3} e^{2}\right )}}{c^{7}}\right )} x + \frac{105 \,{\left (32 \, b^{5} c^{4} d^{2} - 32 \, b^{6} c^{3} d e + 9 \, b^{7} c^{2} e^{2}\right )}}{c^{7}}\right )} + \frac{5 \,{\left (32 \, b^{6} c^{2} d^{2} - 32 \, b^{7} c d e + 9 \, b^{8} e^{2}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{32768 \, c^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(e*x + d)^2,x, algorithm="giac")

[Out]

1/344064*sqrt(c*x^2 + b*x)*(2*(4*(2*(8*(2*(12*(14*c^2*x*e^2 + (32*c^9*d*e + 33*b
*c^8*e^2)/c^7)*x + (224*c^9*d^2 + 928*b*c^8*d*e + 243*b^2*c^7*e^2)/c^7)*x + (112
0*b*c^8*d^2 + 1184*b^2*c^7*d*e + 3*b^3*c^6*e^2)/c^7)*x + 3*(2016*b^2*c^7*d^2 + 3
2*b^3*c^6*d*e - 9*b^4*c^5*e^2)/c^7)*x + 7*(32*b^3*c^6*d^2 - 32*b^4*c^5*d*e + 9*b
^5*c^4*e^2)/c^7)*x - 35*(32*b^4*c^5*d^2 - 32*b^5*c^4*d*e + 9*b^6*c^3*e^2)/c^7)*x
+ 105*(32*b^5*c^4*d^2 - 32*b^6*c^3*d*e + 9*b^7*c^2*e^2)/c^7) + 5/32768*(32*b^6*
c^2*d^2 - 32*b^7*c*d*e + 9*b^8*e^2)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sq
rt(c) - b))/c^(11/2)