3.1295 \(\int \frac{1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=203 \[ -\frac{18 c \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{d^{7/2} \left (b^2-4 a c\right )^{13/4}}+\frac{18 c \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{d^{7/2} \left (b^2-4 a c\right )^{13/4}}-\frac{36 c}{d^3 \left (b^2-4 a c\right )^3 \sqrt{b d+2 c d x}}-\frac{1}{d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) (b d+2 c d x)^{5/2}}-\frac{36 c}{5 d \left (b^2-4 a c\right )^2 (b d+2 c d x)^{5/2}} \]

[Out]

(-36*c)/(5*(b^2 - 4*a*c)^2*d*(b*d + 2*c*d*x)^(5/2)) - (36*c)/((b^2 - 4*a*c)^3*d^
3*Sqrt[b*d + 2*c*d*x]) - 1/((b^2 - 4*a*c)*d*(b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x
^2)) - (18*c*ArcTan[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])])/((b^2 -
4*a*c)^(13/4)*d^(7/2)) + (18*c*ArcTanh[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*
Sqrt[d])])/((b^2 - 4*a*c)^(13/4)*d^(7/2))

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Rubi [A]  time = 0.478855, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ -\frac{18 c \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{d^{7/2} \left (b^2-4 a c\right )^{13/4}}+\frac{18 c \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{d^{7/2} \left (b^2-4 a c\right )^{13/4}}-\frac{36 c}{d^3 \left (b^2-4 a c\right )^3 \sqrt{b d+2 c d x}}-\frac{1}{d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) (b d+2 c d x)^{5/2}}-\frac{36 c}{5 d \left (b^2-4 a c\right )^2 (b d+2 c d x)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^2),x]

[Out]

(-36*c)/(5*(b^2 - 4*a*c)^2*d*(b*d + 2*c*d*x)^(5/2)) - (36*c)/((b^2 - 4*a*c)^3*d^
3*Sqrt[b*d + 2*c*d*x]) - 1/((b^2 - 4*a*c)*d*(b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x
^2)) - (18*c*ArcTan[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])])/((b^2 -
4*a*c)^(13/4)*d^(7/2)) + (18*c*ArcTanh[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*
Sqrt[d])])/((b^2 - 4*a*c)^(13/4)*d^(7/2))

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Rubi in Sympy [A]  time = 105.936, size = 201, normalized size = 0.99 \[ - \frac{36 c}{5 d \left (- 4 a c + b^{2}\right )^{2} \left (b d + 2 c d x\right )^{\frac{5}{2}}} - \frac{36 c}{d^{3} \left (- 4 a c + b^{2}\right )^{3} \sqrt{b d + 2 c d x}} - \frac{18 c \operatorname{atan}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}}{d^{\frac{7}{2}} \left (- 4 a c + b^{2}\right )^{\frac{13}{4}}} + \frac{18 c \operatorname{atanh}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}}{d^{\frac{7}{2}} \left (- 4 a c + b^{2}\right )^{\frac{13}{4}}} - \frac{1}{d \left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{5}{2}} \left (a + b x + c x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(2*c*d*x+b*d)**(7/2)/(c*x**2+b*x+a)**2,x)

[Out]

-36*c/(5*d*(-4*a*c + b**2)**2*(b*d + 2*c*d*x)**(5/2)) - 36*c/(d**3*(-4*a*c + b**
2)**3*sqrt(b*d + 2*c*d*x)) - 18*c*atan(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c + b*
*2)**(1/4)))/(d**(7/2)*(-4*a*c + b**2)**(13/4)) + 18*c*atanh(sqrt(b*d + 2*c*d*x)
/(sqrt(d)*(-4*a*c + b**2)**(1/4)))/(d**(7/2)*(-4*a*c + b**2)**(13/4)) - 1/(d*(-4
*a*c + b**2)*(b*d + 2*c*d*x)**(5/2)*(a + b*x + c*x**2))

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Mathematica [A]  time = 0.76035, size = 185, normalized size = 0.91 \[ \frac{-\frac{(b+2 c x)^5}{\left (b^2-4 a c\right )^3 (a+x (b+c x))}-\frac{32 c (b+2 c x)^3}{\left (b^2-4 a c\right )^3}-\frac{16 c (b+2 c x)}{5 \left (b^2-4 a c\right )^2}-\frac{18 c (b+2 c x)^{7/2} \tan ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{13/4}}+\frac{18 c (b+2 c x)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{13/4}}}{(d (b+2 c x))^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^2),x]

[Out]

((-16*c*(b + 2*c*x))/(5*(b^2 - 4*a*c)^2) - (32*c*(b + 2*c*x)^3)/(b^2 - 4*a*c)^3
- (b + 2*c*x)^5/((b^2 - 4*a*c)^3*(a + x*(b + c*x))) - (18*c*(b + 2*c*x)^(7/2)*Ar
cTan[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)])/(b^2 - 4*a*c)^(13/4) + (18*c*(b + 2*c
*x)^(7/2)*ArcTanh[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)])/(b^2 - 4*a*c)^(13/4))/(d
*(b + 2*c*x))^(7/2)

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Maple [B]  time = 0.026, size = 433, normalized size = 2.1 \[ -{\frac{16\,c}{5\,d \left ( 4\,ac-{b}^{2} \right ) ^{2}} \left ( 2\,cdx+bd \right ) ^{-{\frac{5}{2}}}}+32\,{\frac{c}{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{3}\sqrt{2\,cdx+bd}}}+4\,{\frac{c \left ( 2\,cdx+bd \right ) ^{3/2}}{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{3} \left ( 4\,{c}^{2}{d}^{2}{x}^{2}+4\,bc{d}^{2}x+4\,ac{d}^{2} \right ) }}+{\frac{9\,c\sqrt{2}}{2\,{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{3}}\ln \left ({1 \left ( 2\,cdx+bd-\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) \left ( 2\,cdx+bd+\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+9\,{\frac{c\sqrt{2}}{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{3}\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}\arctan \left ({\frac{\sqrt{2}\sqrt{2\,cdx+bd}}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}+1 \right ) }-9\,{\frac{c\sqrt{2}}{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{3}\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}\arctan \left ( -{\frac{\sqrt{2}\sqrt{2\,cdx+bd}}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}+1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(2*c*d*x+b*d)^(7/2)/(c*x^2+b*x+a)^2,x)

[Out]

-16/5*c/d/(4*a*c-b^2)^2/(2*c*d*x+b*d)^(5/2)+32*c/d^3/(4*a*c-b^2)^3/(2*c*d*x+b*d)
^(1/2)+4*c/d^3/(4*a*c-b^2)^3*(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)+9/2*c/d^3/(4*a*c-b^2)^3/(4*a*c*d^2-b^2*d^2)^(1/4)*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)))+9*c/d^3/(4*a*c-b^2)^3/(4*a*c*d^2-b^2*d^2)^(1/4)*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)-9*c/d^3/(4*a*c-b^2)^3/(
4*a*c*d^2-b^2*d^2)^(1/4)*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(1/((2*c*d*x + b*d)^(7/2)*(c*x^2 + b*x + a)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.274445, size = 3451, normalized size = 17. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*c*d*x + b*d)^(7/2)*(c*x^2 + b*x + a)^2),x, algorithm="fricas")

[Out]

-1/5*(720*c^4*x^4 + 1440*b*c^3*x^3 + 5*b^4 + 176*a*b^2*c - 64*a^2*c^2 + 72*(13*b
^2*c^2 + 8*a*c^3)*x^2 - 180*(4*(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3
*c^6)*d^3*x^4 + 8*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*d^3*x
^3 + (5*b^8*c - 56*a*b^6*c^2 + 192*a^2*b^4*c^3 - 128*a^3*b^2*c^4 - 256*a^4*c^5)*
d^3*x^2 + (b^9 - 8*a*b^7*c + 128*a^3*b^3*c^3 - 256*a^4*b*c^4)*d^3*x + (a*b^8 - 1
2*a^2*b^6*c + 48*a^3*b^4*c^2 - 64*a^4*b^2*c^3)*d^3)*sqrt(2*c*d*x + b*d)*(c^4/((b
^26 - 52*a*b^24*c + 1248*a^2*b^22*c^2 - 18304*a^3*b^20*c^3 + 183040*a^4*b^18*c^4
- 1317888*a^5*b^16*c^5 + 7028736*a^6*b^14*c^6 - 28114944*a^7*b^12*c^7 + 8434483
2*a^8*b^10*c^8 - 187432960*a^9*b^8*c^9 + 299892736*a^10*b^6*c^10 - 327155712*a^1
1*b^4*c^11 + 218103808*a^12*b^2*c^12 - 67108864*a^13*c^13)*d^14))^(1/4)*arctan((
b^20 - 40*a*b^18*c + 720*a^2*b^16*c^2 - 7680*a^3*b^14*c^3 + 53760*a^4*b^12*c^4 -
258048*a^5*b^10*c^5 + 860160*a^6*b^8*c^6 - 1966080*a^7*b^6*c^7 + 2949120*a^8*b^
4*c^8 - 2621440*a^9*b^2*c^9 + 1048576*a^10*c^10)*d^11*(c^4/((b^26 - 52*a*b^24*c
+ 1248*a^2*b^22*c^2 - 18304*a^3*b^20*c^3 + 183040*a^4*b^18*c^4 - 1317888*a^5*b^1
6*c^5 + 7028736*a^6*b^14*c^6 - 28114944*a^7*b^12*c^7 + 84344832*a^8*b^10*c^8 - 1
87432960*a^9*b^8*c^9 + 299892736*a^10*b^6*c^10 - 327155712*a^11*b^4*c^11 + 21810
3808*a^12*b^2*c^12 - 67108864*a^13*c^13)*d^14))^(3/4)/(sqrt(2*c*d*x + b*d)*c^3 +
sqrt((b^14*c^4 - 28*a*b^12*c^5 + 336*a^2*b^10*c^6 - 2240*a^3*b^8*c^7 + 8960*a^4
*b^6*c^8 - 21504*a^5*b^4*c^9 + 28672*a^6*b^2*c^10 - 16384*a^7*c^11)*d^8*sqrt(c^4
/((b^26 - 52*a*b^24*c + 1248*a^2*b^22*c^2 - 18304*a^3*b^20*c^3 + 183040*a^4*b^18
*c^4 - 1317888*a^5*b^16*c^5 + 7028736*a^6*b^14*c^6 - 28114944*a^7*b^12*c^7 + 843
44832*a^8*b^10*c^8 - 187432960*a^9*b^8*c^9 + 299892736*a^10*b^6*c^10 - 327155712
*a^11*b^4*c^11 + 218103808*a^12*b^2*c^12 - 67108864*a^13*c^13)*d^14)) + 2*c^7*d*
x + b*c^6*d))) - 45*(4*(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*d^
3*x^4 + 8*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*d^3*x^3 + (5*
b^8*c - 56*a*b^6*c^2 + 192*a^2*b^4*c^3 - 128*a^3*b^2*c^4 - 256*a^4*c^5)*d^3*x^2
+ (b^9 - 8*a*b^7*c + 128*a^3*b^3*c^3 - 256*a^4*b*c^4)*d^3*x + (a*b^8 - 12*a^2*b^
6*c + 48*a^3*b^4*c^2 - 64*a^4*b^2*c^3)*d^3)*sqrt(2*c*d*x + b*d)*(c^4/((b^26 - 52
*a*b^24*c + 1248*a^2*b^22*c^2 - 18304*a^3*b^20*c^3 + 183040*a^4*b^18*c^4 - 13178
88*a^5*b^16*c^5 + 7028736*a^6*b^14*c^6 - 28114944*a^7*b^12*c^7 + 84344832*a^8*b^
10*c^8 - 187432960*a^9*b^8*c^9 + 299892736*a^10*b^6*c^10 - 327155712*a^11*b^4*c^
11 + 218103808*a^12*b^2*c^12 - 67108864*a^13*c^13)*d^14))^(1/4)*log(729*(b^20 -
40*a*b^18*c + 720*a^2*b^16*c^2 - 7680*a^3*b^14*c^3 + 53760*a^4*b^12*c^4 - 258048
*a^5*b^10*c^5 + 860160*a^6*b^8*c^6 - 1966080*a^7*b^6*c^7 + 2949120*a^8*b^4*c^8 -
2621440*a^9*b^2*c^9 + 1048576*a^10*c^10)*d^11*(c^4/((b^26 - 52*a*b^24*c + 1248*
a^2*b^22*c^2 - 18304*a^3*b^20*c^3 + 183040*a^4*b^18*c^4 - 1317888*a^5*b^16*c^5 +
7028736*a^6*b^14*c^6 - 28114944*a^7*b^12*c^7 + 84344832*a^8*b^10*c^8 - 18743296
0*a^9*b^8*c^9 + 299892736*a^10*b^6*c^10 - 327155712*a^11*b^4*c^11 + 218103808*a^
12*b^2*c^12 - 67108864*a^13*c^13)*d^14))^(3/4) + 729*sqrt(2*c*d*x + b*d)*c^3) +
45*(4*(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*d^3*x^4 + 8*(b^7*c^
2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*d^3*x^3 + (5*b^8*c - 56*a*b^6*
c^2 + 192*a^2*b^4*c^3 - 128*a^3*b^2*c^4 - 256*a^4*c^5)*d^3*x^2 + (b^9 - 8*a*b^7*
c + 128*a^3*b^3*c^3 - 256*a^4*b*c^4)*d^3*x + (a*b^8 - 12*a^2*b^6*c + 48*a^3*b^4*
c^2 - 64*a^4*b^2*c^3)*d^3)*sqrt(2*c*d*x + b*d)*(c^4/((b^26 - 52*a*b^24*c + 1248*
a^2*b^22*c^2 - 18304*a^3*b^20*c^3 + 183040*a^4*b^18*c^4 - 1317888*a^5*b^16*c^5 +
7028736*a^6*b^14*c^6 - 28114944*a^7*b^12*c^7 + 84344832*a^8*b^10*c^8 - 18743296
0*a^9*b^8*c^9 + 299892736*a^10*b^6*c^10 - 327155712*a^11*b^4*c^11 + 218103808*a^
12*b^2*c^12 - 67108864*a^13*c^13)*d^14))^(1/4)*log(-729*(b^20 - 40*a*b^18*c + 72
0*a^2*b^16*c^2 - 7680*a^3*b^14*c^3 + 53760*a^4*b^12*c^4 - 258048*a^5*b^10*c^5 +
860160*a^6*b^8*c^6 - 1966080*a^7*b^6*c^7 + 2949120*a^8*b^4*c^8 - 2621440*a^9*b^2
*c^9 + 1048576*a^10*c^10)*d^11*(c^4/((b^26 - 52*a*b^24*c + 1248*a^2*b^22*c^2 - 1
8304*a^3*b^20*c^3 + 183040*a^4*b^18*c^4 - 1317888*a^5*b^16*c^5 + 7028736*a^6*b^1
4*c^6 - 28114944*a^7*b^12*c^7 + 84344832*a^8*b^10*c^8 - 187432960*a^9*b^8*c^9 +
299892736*a^10*b^6*c^10 - 327155712*a^11*b^4*c^11 + 218103808*a^12*b^2*c^12 - 67
108864*a^13*c^13)*d^14))^(3/4) + 729*sqrt(2*c*d*x + b*d)*c^3) + 72*(3*b^3*c + 8*
a*b*c^2)*x)/((4*(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*d^3*x^4 +
8*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*d^3*x^3 + (5*b^8*c -
56*a*b^6*c^2 + 192*a^2*b^4*c^3 - 128*a^3*b^2*c^4 - 256*a^4*c^5)*d^3*x^2 + (b^9
- 8*a*b^7*c + 128*a^3*b^3*c^3 - 256*a^4*b*c^4)*d^3*x + (a*b^8 - 12*a^2*b^6*c + 4
8*a^3*b^4*c^2 - 64*a^4*b^2*c^3)*d^3)*sqrt(2*c*d*x + b*d))

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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(1/(2*c*d*x+b*d)**(7/2)/(c*x**2+b*x+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.247282, size = 1052, normalized size = 5.18 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*c*d*x + b*d)^(7/2)*(c*x^2 + b*x + a)^2),x, algorithm="giac")

[Out]

9*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*c*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))/(b^8*d^
5 - 16*a*b^6*c*d^5 + 96*a^2*b^4*c^2*d^5 - 256*a^3*b^2*c^3*d^5 + 256*a^4*c^4*d^5)
+ 9*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*c*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))/(b^
8*d^5 - 16*a*b^6*c*d^5 + 96*a^2*b^4*c^2*d^5 - 256*a^3*b^2*c^3*d^5 + 256*a^4*c^4*
d^5) - 9*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*c*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^8*d
^5 - 16*sqrt(2)*a*b^6*c*d^5 + 96*sqrt(2)*a^2*b^4*c^2*d^5 - 256*sqrt(2)*a^3*b^2*c
^3*d^5 + 256*sqrt(2)*a^4*c^4*d^5) + 9*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*c*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^8*d^5 - 16*sqrt(2)*a*b^6*c*d^5 + 96*sqrt(2)*a^2*b^4*c^
2*d^5 - 256*sqrt(2)*a^3*b^2*c^3*d^5 + 256*sqrt(2)*a^4*c^4*d^5) + 4*(2*c*d*x + b*
d)^(3/2)*c/((b^6*d^3 - 12*a*b^4*c*d^3 + 48*a^2*b^2*c^2*d^3 - 64*a^3*c^3*d^3)*(b^
2*d^2 - 4*a*c*d^2 - (2*c*d*x + b*d)^2)) - 16/5*(b^2*c*d^2 - 4*a*c^2*d^2 + 10*(2*
c*d*x + b*d)^2*c)/((b^6*d^3 - 12*a*b^4*c*d^3 + 48*a^2*b^2*c^2*d^3 - 64*a^3*c^3*d
^3)*(2*c*d*x + b*d)^(5/2))