3.1304 \(\int \frac{\sqrt{b d+2 c d x}}{\left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=192 \[ \frac{5 c^2 \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/4}}-\frac{5 c^2 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/4}}+\frac{5 c (b d+2 c d x)^{3/2}}{2 d \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{(b d+2 c d x)^{3/2}}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]

[Out]

-(b*d + 2*c*d*x)^(3/2)/(2*(b^2 - 4*a*c)*d*(a + b*x + c*x^2)^2) + (5*c*(b*d + 2*c
*d*x)^(3/2))/(2*(b^2 - 4*a*c)^2*d*(a + b*x + c*x^2)) + (5*c^2*Sqrt[d]*ArcTan[Sqr
t[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])])/(b^2 - 4*a*c)^(9/4) - (5*c^2*Sq
rt[d]*ArcTanh[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])])/(b^2 - 4*a*c)^
(9/4)

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Rubi [A]  time = 0.368558, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{5 c^2 \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/4}}-\frac{5 c^2 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/4}}+\frac{5 c (b d+2 c d x)^{3/2}}{2 d \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{(b d+2 c d x)^{3/2}}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-(b*d + 2*c*d*x)^(3/2)/(2*(b^2 - 4*a*c)*d*(a + b*x + c*x^2)^2) + (5*c*(b*d + 2*c
*d*x)^(3/2))/(2*(b^2 - 4*a*c)^2*d*(a + b*x + c*x^2)) + (5*c^2*Sqrt[d]*ArcTan[Sqr
t[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])])/(b^2 - 4*a*c)^(9/4) - (5*c^2*Sq
rt[d]*ArcTanh[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])])/(b^2 - 4*a*c)^
(9/4)

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Rubi in Sympy [A]  time = 83.8682, size = 185, normalized size = 0.96 \[ \frac{5 c^{2} \sqrt{d} \operatorname{atan}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{9}{4}}} - \frac{5 c^{2} \sqrt{d} \operatorname{atanh}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{9}{4}}} + \frac{5 c \left (b d + 2 c d x\right )^{\frac{3}{2}}}{2 d \left (- 4 a c + b^{2}\right )^{2} \left (a + b x + c x^{2}\right )} - \frac{\left (b d + 2 c d x\right )^{\frac{3}{2}}}{2 d \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*c**2*sqrt(d)*atan(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4)))/(-4*a*
c + b**2)**(9/4) - 5*c**2*sqrt(d)*atanh(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c + b
**2)**(1/4)))/(-4*a*c + b**2)**(9/4) + 5*c*(b*d + 2*c*d*x)**(3/2)/(2*d*(-4*a*c +
b**2)**2*(a + b*x + c*x**2)) - (b*d + 2*c*d*x)**(3/2)/(2*d*(-4*a*c + b**2)*(a +
b*x + c*x**2)**2)

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Mathematica [A]  time = 0.876921, size = 170, normalized size = 0.89 \[ \sqrt{d (b+2 c x)} \left (\frac{5 c^2 \tan ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/4} \sqrt{b+2 c x}}-\frac{5 c^2 \tanh ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/4} \sqrt{b+2 c x}}+\frac{(b+2 c x) \left (c \left (9 a+5 c x^2\right )-b^2+5 b c x\right )}{2 \left (b^2-4 a c\right )^2 (a+x (b+c x))^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[d*(b + 2*c*x)]*(((b + 2*c*x)*(-b^2 + 5*b*c*x + c*(9*a + 5*c*x^2)))/(2*(b^2
- 4*a*c)^2*(a + x*(b + c*x))^2) + (5*c^2*ArcTan[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1
/4)])/((b^2 - 4*a*c)^(9/4)*Sqrt[b + 2*c*x]) - (5*c^2*ArcTanh[Sqrt[b + 2*c*x]/(b^
2 - 4*a*c)^(1/4)])/((b^2 - 4*a*c)^(9/4)*Sqrt[b + 2*c*x]))

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Maple [B]  time = 0.014, size = 419, normalized size = 2.2 \[ 8\,{\frac{{c}^{2}{d}^{5} \left ( 2\,cdx+bd \right ) ^{3/2}}{ \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) \left ( 4\,{c}^{2}{d}^{2}{x}^{2}+4\,bc{d}^{2}x+4\,ac{d}^{2} \right ) ^{2}}}+10\,{\frac{{c}^{2}{d}^{5} \left ( 2\,cdx+bd \right ) ^{3/2}}{ \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{2} \left ( 4\,{c}^{2}{d}^{2}{x}^{2}+4\,bc{d}^{2}x+4\,ac{d}^{2} \right ) }}+{\frac{5\,{c}^{2}{d}^{5}\sqrt{2}}{4}\ln \left ({1 \left ( 2\,cdx+bd-\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) \left ( 2\,cdx+bd+\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) ^{-1}} \right ) \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{-{\frac{9}{4}}}}+{\frac{5\,{c}^{2}{d}^{5}\sqrt{2}}{2}\arctan \left ({\sqrt{2}\sqrt{2\,cdx+bd}{\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+1 \right ) \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{-{\frac{9}{4}}}}-{\frac{5\,{c}^{2}{d}^{5}\sqrt{2}}{2}\arctan \left ( -{\sqrt{2}\sqrt{2\,cdx+bd}{\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+1 \right ) \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{-{\frac{9}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8*c^2*d^5*(2*c*d*x+b*d)^(3/2)/(4*a*c*d^2-b^2*d^2)/(4*c^2*d^2*x^2+4*b*c*d^2*x+4*a
*c*d^2)^2+10*c^2*d^5/(4*a*c*d^2-b^2*d^2)^2*(2*c*d*x+b*d)^(3/2)/(4*c^2*d^2*x^2+4*
b*c*d^2*x+4*a*c*d^2)+5/4*c^2*d^5/(4*a*c*d^2-b^2*d^2)^(9/4)*2^(1/2)*ln((2*c*d*x+b
*d-(4*a*c*d^2-b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)*2^(1/2)+(4*a*c*d^2-b^2*d^2)^(1/
2))/(2*c*d*x+b*d+(4*a*c*d^2-b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)*2^(1/2)+(4*a*c*d^
2-b^2*d^2)^(1/2)))+5/2*c^2*d^5/(4*a*c*d^2-b^2*d^2)^(9/4)*2^(1/2)*arctan(2^(1/2)/
(4*a*c*d^2-b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)+1)-5/2*c^2*d^5/(4*a*c*d^2-b^2*d^2)
^(9/4)*2^(1/2)*arctan(-2^(1/2)/(4*a*c*d^2-b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)+1)

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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(sqrt(2*c*d*x + b*d)/(c*x^2 + b*x + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.254797, size = 2315, normalized size = 12.06 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(20*(c^8*d^2/(b^18 - 36*a*b^16*c + 576*a^2*b^14*c^2 - 5376*a^3*b^12*c^3 + 32
256*a^4*b^10*c^4 - 129024*a^5*b^8*c^5 + 344064*a^6*b^6*c^6 - 589824*a^7*b^4*c^7
+ 589824*a^8*b^2*c^8 - 262144*a^9*c^9))^(1/4)*(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^
2 + (b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^4 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b
*c^3)*x^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3
*b*c^2)*x)*arctan(-(b^14 - 28*a*b^12*c + 336*a^2*b^10*c^2 - 2240*a^3*b^8*c^3 + 8
960*a^4*b^6*c^4 - 21504*a^5*b^4*c^5 + 28672*a^6*b^2*c^6 - 16384*a^7*c^7)*(c^8*d^
2/(b^18 - 36*a*b^16*c + 576*a^2*b^14*c^2 - 5376*a^3*b^12*c^3 + 32256*a^4*b^10*c^
4 - 129024*a^5*b^8*c^5 + 344064*a^6*b^6*c^6 - 589824*a^7*b^4*c^7 + 589824*a^8*b^
2*c^8 - 262144*a^9*c^9))^(3/4)/(sqrt(2*c*d*x + b*d)*c^6*d + sqrt(2*c^13*d^3*x +
b*c^12*d^3 + (b^10*c^8 - 20*a*b^8*c^9 + 160*a^2*b^6*c^10 - 640*a^3*b^4*c^11 + 12
80*a^4*b^2*c^12 - 1024*a^5*c^13)*sqrt(c^8*d^2/(b^18 - 36*a*b^16*c + 576*a^2*b^14
*c^2 - 5376*a^3*b^12*c^3 + 32256*a^4*b^10*c^4 - 129024*a^5*b^8*c^5 + 344064*a^6*
b^6*c^6 - 589824*a^7*b^4*c^7 + 589824*a^8*b^2*c^8 - 262144*a^9*c^9))*d^2))) - 5*
(c^8*d^2/(b^18 - 36*a*b^16*c + 576*a^2*b^14*c^2 - 5376*a^3*b^12*c^3 + 32256*a^4*
b^10*c^4 - 129024*a^5*b^8*c^5 + 344064*a^6*b^6*c^6 - 589824*a^7*b^4*c^7 + 589824
*a^8*b^2*c^8 - 262144*a^9*c^9))^(1/4)*(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^4
*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^4 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^
3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*
x)*log(125*sqrt(2*c*d*x + b*d)*c^6*d + 125*(b^14 - 28*a*b^12*c + 336*a^2*b^10*c^
2 - 2240*a^3*b^8*c^3 + 8960*a^4*b^6*c^4 - 21504*a^5*b^4*c^5 + 28672*a^6*b^2*c^6
- 16384*a^7*c^7)*(c^8*d^2/(b^18 - 36*a*b^16*c + 576*a^2*b^14*c^2 - 5376*a^3*b^12
*c^3 + 32256*a^4*b^10*c^4 - 129024*a^5*b^8*c^5 + 344064*a^6*b^6*c^6 - 589824*a^7
*b^4*c^7 + 589824*a^8*b^2*c^8 - 262144*a^9*c^9))^(3/4)) + 5*(c^8*d^2/(b^18 - 36*
a*b^16*c + 576*a^2*b^14*c^2 - 5376*a^3*b^12*c^3 + 32256*a^4*b^10*c^4 - 129024*a^
5*b^8*c^5 + 344064*a^6*b^6*c^6 - 589824*a^7*b^4*c^7 + 589824*a^8*b^2*c^8 - 26214
4*a^9*c^9))^(1/4)*(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^4*c^2 - 8*a*b^2*c^3 +
16*a^2*c^4)*x^4 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^3 + (b^6 - 6*a*b^4*c
+ 32*a^3*c^3)*x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x)*log(125*sqrt(2*c*
d*x + b*d)*c^6*d - 125*(b^14 - 28*a*b^12*c + 336*a^2*b^10*c^2 - 2240*a^3*b^8*c^3
+ 8960*a^4*b^6*c^4 - 21504*a^5*b^4*c^5 + 28672*a^6*b^2*c^6 - 16384*a^7*c^7)*(c^
8*d^2/(b^18 - 36*a*b^16*c + 576*a^2*b^14*c^2 - 5376*a^3*b^12*c^3 + 32256*a^4*b^1
0*c^4 - 129024*a^5*b^8*c^5 + 344064*a^6*b^6*c^6 - 589824*a^7*b^4*c^7 + 589824*a^
8*b^2*c^8 - 262144*a^9*c^9))^(3/4)) + (10*c^3*x^3 + 15*b*c^2*x^2 - b^3 + 9*a*b*c
+ 3*(b^2*c + 6*a*c^2)*x)*sqrt(2*c*d*x + b*d))/(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c
^2 + (b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^4 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*
b*c^3)*x^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^
3*b*c^2)*x)

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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((2*c*d*x+b*d)**(1/2)/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.251443, size = 871, normalized size = 4.54 \[ -\frac{5 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} c^{2} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} + 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right )}{\sqrt{2} b^{6} d - 12 \, \sqrt{2} a b^{4} c d + 48 \, \sqrt{2} a^{2} b^{2} c^{2} d - 64 \, \sqrt{2} a^{3} c^{3} d} - \frac{5 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} c^{2} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} - 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right )}{\sqrt{2} b^{6} d - 12 \, \sqrt{2} a b^{4} c d + 48 \, \sqrt{2} a^{2} b^{2} c^{2} d - 64 \, \sqrt{2} a^{3} c^{3} d} + \frac{5 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} c^{2}{\rm ln}\left (2 \, c d x + b d + \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{2 \,{\left (\sqrt{2} b^{6} d - 12 \, \sqrt{2} a b^{4} c d + 48 \, \sqrt{2} a^{2} b^{2} c^{2} d - 64 \, \sqrt{2} a^{3} c^{3} d\right )}} - \frac{5 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} c^{2}{\rm ln}\left (2 \, c d x + b d - \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{2 \,{\left (\sqrt{2} b^{6} d - 12 \, \sqrt{2} a b^{4} c d + 48 \, \sqrt{2} a^{2} b^{2} c^{2} d - 64 \, \sqrt{2} a^{3} c^{3} d\right )}} - \frac{2 \,{\left (9 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} b^{2} c^{2} d^{3} - 36 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} a c^{3} d^{3} - 5 \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} c^{2} d\right )}}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}{\left (b^{2} d^{2} - 4 \, a c d^{2} -{\left (2 \, c d x + b d\right )}^{2}\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*c^2*arctan(1/2*sqrt(2)*(sqrt(2)*(-b^2*d^2 + 4*a*
c*d^2)^(1/4) + 2*sqrt(2*c*d*x + b*d))/(-b^2*d^2 + 4*a*c*d^2)^(1/4))/(sqrt(2)*b^6
*d - 12*sqrt(2)*a*b^4*c*d + 48*sqrt(2)*a^2*b^2*c^2*d - 64*sqrt(2)*a^3*c^3*d) - 5
*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*c^2*arctan(-1/2*sqrt(2)*(sqrt(2)*(-b^2*d^2 + 4*a*c
*d^2)^(1/4) - 2*sqrt(2*c*d*x + b*d))/(-b^2*d^2 + 4*a*c*d^2)^(1/4))/(sqrt(2)*b^6*
d - 12*sqrt(2)*a*b^4*c*d + 48*sqrt(2)*a^2*b^2*c^2*d - 64*sqrt(2)*a^3*c^3*d) + 5/
2*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*c^2*ln(2*c*d*x + b*d + sqrt(2)*(-b^2*d^2 + 4*a*c*
d^2)^(1/4)*sqrt(2*c*d*x + b*d) + sqrt(-b^2*d^2 + 4*a*c*d^2))/(sqrt(2)*b^6*d - 12
*sqrt(2)*a*b^4*c*d + 48*sqrt(2)*a^2*b^2*c^2*d - 64*sqrt(2)*a^3*c^3*d) - 5/2*(-b^
2*d^2 + 4*a*c*d^2)^(3/4)*c^2*ln(2*c*d*x + b*d - sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(
1/4)*sqrt(2*c*d*x + b*d) + sqrt(-b^2*d^2 + 4*a*c*d^2))/(sqrt(2)*b^6*d - 12*sqrt(
2)*a*b^4*c*d + 48*sqrt(2)*a^2*b^2*c^2*d - 64*sqrt(2)*a^3*c^3*d) - 2*(9*(2*c*d*x
+ b*d)^(3/2)*b^2*c^2*d^3 - 36*(2*c*d*x + b*d)^(3/2)*a*c^3*d^3 - 5*(2*c*d*x + b*d
)^(7/2)*c^2*d)/((b^4 - 8*a*b^2*c + 16*a^2*c^2)*(b^2*d^2 - 4*a*c*d^2 - (2*c*d*x +
b*d)^2)^2)