3.1305 \(\int \frac{1}{\sqrt{b d+2 c d x} \left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=192 \[ -\frac{21 c^2 \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\sqrt{d} \left (b^2-4 a c\right )^{11/4}}-\frac{21 c^2 \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\sqrt{d} \left (b^2-4 a c\right )^{11/4}}+\frac{7 c \sqrt{b d+2 c d x}}{2 d \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{\sqrt{b d+2 c d x}}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]

[Out]

-Sqrt[b*d + 2*c*d*x]/(2*(b^2 - 4*a*c)*d*(a + b*x + c*x^2)^2) + (7*c*Sqrt[b*d + 2
*c*d*x])/(2*(b^2 - 4*a*c)^2*d*(a + b*x + c*x^2)) - (21*c^2*ArcTan[Sqrt[d*(b + 2*
c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])])/((b^2 - 4*a*c)^(11/4)*Sqrt[d]) - (21*c^2*A
rcTanh[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])])/((b^2 - 4*a*c)^(11/4)
*Sqrt[d])

_______________________________________________________________________________________

Rubi [A]  time = 0.368802, 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 \[ -\frac{21 c^2 \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\sqrt{d} \left (b^2-4 a c\right )^{11/4}}-\frac{21 c^2 \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\sqrt{d} \left (b^2-4 a c\right )^{11/4}}+\frac{7 c \sqrt{b d+2 c d x}}{2 d \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{\sqrt{b d+2 c d x}}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[b*d + 2*c*d*x]/(2*(b^2 - 4*a*c)*d*(a + b*x + c*x^2)^2) + (7*c*Sqrt[b*d + 2
*c*d*x])/(2*(b^2 - 4*a*c)^2*d*(a + b*x + c*x^2)) - (21*c^2*ArcTan[Sqrt[d*(b + 2*
c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])])/((b^2 - 4*a*c)^(11/4)*Sqrt[d]) - (21*c^2*A
rcTanh[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])])/((b^2 - 4*a*c)^(11/4)
*Sqrt[d])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 82.1898, size = 185, normalized size = 0.96 \[ - \frac{21 c^{2} \operatorname{atan}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}}{\sqrt{d} \left (- 4 a c + b^{2}\right )^{\frac{11}{4}}} - \frac{21 c^{2} \operatorname{atanh}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}}{\sqrt{d} \left (- 4 a c + b^{2}\right )^{\frac{11}{4}}} + \frac{7 c \sqrt{b d + 2 c d x}}{2 d \left (- 4 a c + b^{2}\right )^{2} \left (a + b x + c x^{2}\right )} - \frac{\sqrt{b d + 2 c d x}}{2 d \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-21*c**2*atan(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4)))/(sqrt(d)*(-4
*a*c + b**2)**(11/4)) - 21*c**2*atanh(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c + b**
2)**(1/4)))/(sqrt(d)*(-4*a*c + b**2)**(11/4)) + 7*c*sqrt(b*d + 2*c*d*x)/(2*d*(-4
*a*c + b**2)**2*(a + b*x + c*x**2)) - sqrt(b*d + 2*c*d*x)/(2*d*(-4*a*c + b**2)*(
a + b*x + c*x**2)**2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.764161, size = 169, normalized size = 0.88 \[ \frac{-\frac{21 c^2 \sqrt{b+2 c x} \tan ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{11/4}}-\frac{21 c^2 \sqrt{b+2 c x} \tanh ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{11/4}}-\frac{(b+2 c x) \left (-c \left (11 a+7 c x^2\right )+b^2-7 b c x\right )}{2 \left (b^2-4 a c\right )^2 (a+x (b+c x))^2}}{\sqrt{d (b+2 c x)}} \]

Antiderivative was successfully verified.

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

[Out]

(-((b + 2*c*x)*(b^2 - 7*b*c*x - c*(11*a + 7*c*x^2)))/(2*(b^2 - 4*a*c)^2*(a + x*(
b + c*x))^2) - (21*c^2*Sqrt[b + 2*c*x]*ArcTan[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4
)])/(b^2 - 4*a*c)^(11/4) - (21*c^2*Sqrt[b + 2*c*x]*ArcTanh[Sqrt[b + 2*c*x]/(b^2
- 4*a*c)^(1/4)])/(b^2 - 4*a*c)^(11/4))/Sqrt[d*(b + 2*c*x)]

_______________________________________________________________________________________

Maple [B]  time = 0.013, size = 419, normalized size = 2.2 \[ 8\,{\frac{{c}^{2}{d}^{5}\sqrt{2\,cdx+bd}}{ \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) \left ( 4\,{c}^{2}{d}^{2}{x}^{2}+4\,bc{d}^{2}x+4\,ac{d}^{2} \right ) ^{2}}}+14\,{\frac{{c}^{2}{d}^{5}\sqrt{2\,cdx+bd}}{ \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{2} \left ( 4\,{c}^{2}{d}^{2}{x}^{2}+4\,bc{d}^{2}x+4\,ac{d}^{2} \right ) }}+{\frac{21\,{c}^{2}{d}^{5}\sqrt{2}}{4}\ln \left ({1 \left ( 2\,cdx+bd+\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) \left ( 2\,cdx+bd-\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) ^{-1}} \right ) \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{-{\frac{11}{4}}}}+{\frac{21\,{c}^{2}{d}^{5}\sqrt{2}}{2}\arctan \left ({\sqrt{2}\sqrt{2\,cdx+bd}{\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+1 \right ) \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{-{\frac{11}{4}}}}-{\frac{21\,{c}^{2}{d}^{5}\sqrt{2}}{2}\arctan \left ( -{\sqrt{2}\sqrt{2\,cdx+bd}{\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+1 \right ) \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{-{\frac{11}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(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)^(1/2)/(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+14*c^2*d^5/(4*a*c*d^2-b^2*d^2)^2*(2*c*d*x+b*d)^(1/2)/(4*c^2*d^2*x^2+4*
b*c*d^2*x+4*a*c*d^2)+21/4*c^2*d^5/(4*a*c*d^2-b^2*d^2)^(11/4)*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)))+21/2*c^2*d^5/(4*a*c*d^2-b^2*d^2)^(11/4)*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)-21/2*c^2*d^5/(4*a*c*d^2-b^2
*d^2)^(11/4)*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)

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.253992, size = 2358, normalized size = 12.28 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(2*c*d*x + b*d)*(c*x^2 + b*x + a)^3),x, algorithm="fricas")

[Out]

-1/2*(84*((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d*x^4 + 2*(b^5*c - 8*a*b^3*c^2 +
16*a^2*b*c^3)*d*x^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*d*x^2 + 2*(a*b^5 - 8*a^2*b^
3*c + 16*a^3*b*c^2)*d*x + (a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2)*d)*(c^8/((b^22 -
44*a*b^20*c + 880*a^2*b^18*c^2 - 10560*a^3*b^16*c^3 + 84480*a^4*b^14*c^4 - 47308
8*a^5*b^12*c^5 + 1892352*a^6*b^10*c^6 - 5406720*a^7*b^8*c^7 + 10813440*a^8*b^6*c
^8 - 14417920*a^9*b^4*c^9 + 11534336*a^10*b^2*c^10 - 4194304*a^11*c^11)*d^2))^(1
/4)*arctan(-(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*(c^8/((b^22 - 44*a*
b^20*c + 880*a^2*b^18*c^2 - 10560*a^3*b^16*c^3 + 84480*a^4*b^14*c^4 - 473088*a^5
*b^12*c^5 + 1892352*a^6*b^10*c^6 - 5406720*a^7*b^8*c^7 + 10813440*a^8*b^6*c^8 -
14417920*a^9*b^4*c^9 + 11534336*a^10*b^2*c^10 - 4194304*a^11*c^11)*d^2))^(1/4)*d
/(sqrt(2*c*d*x + b*d)*c^2 + sqrt(2*c^5*d*x + b*c^4*d + (b^12 - 24*a*b^10*c + 240
*a^2*b^8*c^2 - 1280*a^3*b^6*c^3 + 3840*a^4*b^4*c^4 - 6144*a^5*b^2*c^5 + 4096*a^6
*c^6)*sqrt(c^8/((b^22 - 44*a*b^20*c + 880*a^2*b^18*c^2 - 10560*a^3*b^16*c^3 + 84
480*a^4*b^14*c^4 - 473088*a^5*b^12*c^5 + 1892352*a^6*b^10*c^6 - 5406720*a^7*b^8*
c^7 + 10813440*a^8*b^6*c^8 - 14417920*a^9*b^4*c^9 + 11534336*a^10*b^2*c^10 - 419
4304*a^11*c^11)*d^2))*d^2))) + 21*((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d*x^4 +
2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d*x^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*d*
x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*d*x + (a^2*b^4 - 8*a^3*b^2*c + 16*a
^4*c^2)*d)*(c^8/((b^22 - 44*a*b^20*c + 880*a^2*b^18*c^2 - 10560*a^3*b^16*c^3 + 8
4480*a^4*b^14*c^4 - 473088*a^5*b^12*c^5 + 1892352*a^6*b^10*c^6 - 5406720*a^7*b^8
*c^7 + 10813440*a^8*b^6*c^8 - 14417920*a^9*b^4*c^9 + 11534336*a^10*b^2*c^10 - 41
94304*a^11*c^11)*d^2))^(1/4)*log(21*sqrt(2*c*d*x + b*d)*c^2 + 21*(b^6 - 12*a*b^4
*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*(c^8/((b^22 - 44*a*b^20*c + 880*a^2*b^18*c^2 -
10560*a^3*b^16*c^3 + 84480*a^4*b^14*c^4 - 473088*a^5*b^12*c^5 + 1892352*a^6*b^1
0*c^6 - 5406720*a^7*b^8*c^7 + 10813440*a^8*b^6*c^8 - 14417920*a^9*b^4*c^9 + 1153
4336*a^10*b^2*c^10 - 4194304*a^11*c^11)*d^2))^(1/4)*d) - 21*((b^4*c^2 - 8*a*b^2*
c^3 + 16*a^2*c^4)*d*x^4 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d*x^3 + (b^6 -
6*a*b^4*c + 32*a^3*c^3)*d*x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*d*x + (a^
2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2)*d)*(c^8/((b^22 - 44*a*b^20*c + 880*a^2*b^18*c^
2 - 10560*a^3*b^16*c^3 + 84480*a^4*b^14*c^4 - 473088*a^5*b^12*c^5 + 1892352*a^6*
b^10*c^6 - 5406720*a^7*b^8*c^7 + 10813440*a^8*b^6*c^8 - 14417920*a^9*b^4*c^9 + 1
1534336*a^10*b^2*c^10 - 4194304*a^11*c^11)*d^2))^(1/4)*log(21*sqrt(2*c*d*x + b*d
)*c^2 - 21*(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*(c^8/((b^22 - 44*a*b
^20*c + 880*a^2*b^18*c^2 - 10560*a^3*b^16*c^3 + 84480*a^4*b^14*c^4 - 473088*a^5*
b^12*c^5 + 1892352*a^6*b^10*c^6 - 5406720*a^7*b^8*c^7 + 10813440*a^8*b^6*c^8 - 1
4417920*a^9*b^4*c^9 + 11534336*a^10*b^2*c^10 - 4194304*a^11*c^11)*d^2))^(1/4)*d)
- (7*c^2*x^2 + 7*b*c*x - b^2 + 11*a*c)*sqrt(2*c*d*x + b*d))/((b^4*c^2 - 8*a*b^2
*c^3 + 16*a^2*c^4)*d*x^4 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d*x^3 + (b^6 -
6*a*b^4*c + 32*a^3*c^3)*d*x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*d*x + (a
^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2)*d)

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.239927, size = 871, normalized size = 4.54 \[ -\frac{21 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c^{2} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} + 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right )}{\sqrt{2} b^{6} d - 12 \, \sqrt{2} a b^{4} c d + 48 \, \sqrt{2} a^{2} b^{2} c^{2} d - 64 \, \sqrt{2} a^{3} c^{3} d} - \frac{21 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c^{2} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} - 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right )}{\sqrt{2} b^{6} d - 12 \, \sqrt{2} a b^{4} c d + 48 \, \sqrt{2} a^{2} b^{2} c^{2} d - 64 \, \sqrt{2} a^{3} c^{3} d} - \frac{21 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c^{2}{\rm ln}\left (2 \, c d x + b d + \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{2 \,{\left (\sqrt{2} b^{6} d - 12 \, \sqrt{2} a b^{4} c d + 48 \, \sqrt{2} a^{2} b^{2} c^{2} d - 64 \, \sqrt{2} a^{3} c^{3} d\right )}} + \frac{21 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c^{2}{\rm ln}\left (2 \, c d x + b d - \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{2 \,{\left (\sqrt{2} b^{6} d - 12 \, \sqrt{2} a b^{4} c d + 48 \, \sqrt{2} a^{2} b^{2} c^{2} d - 64 \, \sqrt{2} a^{3} c^{3} d\right )}} - \frac{2 \,{\left (11 \, \sqrt{2 \, c d x + b d} b^{2} c^{2} d^{3} - 44 \, \sqrt{2 \, c d x + b d} a c^{3} d^{3} - 7 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} c^{2} d\right )}}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}{\left (b^{2} d^{2} - 4 \, a c d^{2} -{\left (2 \, c d x + b d\right )}^{2}\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(2*c*d*x + b*d)*(c*x^2 + b*x + a)^3),x, algorithm="giac")

[Out]

-21*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*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) -
21*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*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) -
21/2*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*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) + 21/2*
(-b^2*d^2 + 4*a*c*d^2)^(1/4)*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*s
qrt(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*(11*sqrt
(2*c*d*x + b*d)*b^2*c^2*d^3 - 44*sqrt(2*c*d*x + b*d)*a*c^3*d^3 - 7*(2*c*d*x + b*
d)^(5/2)*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)