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

Optimal. Leaf size=333 \[ -\frac{x^{3/2} \left (5 a^2 d^2-2 a b c d+b^2 c^2\right )}{2 c^2 d \left (c+d x^2\right )}-\frac{2 a^2}{c \sqrt{x} \left (c+d x^2\right )}+\frac{(b c-a d) (5 a d+3 b c) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{9/4} d^{7/4}}-\frac{(b c-a d) (5 a d+3 b c) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{9/4} d^{7/4}}-\frac{(b c-a d) (5 a d+3 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{9/4} d^{7/4}}+\frac{(b c-a d) (5 a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} c^{9/4} d^{7/4}} \]

[Out]

(-2*a^2)/(c*Sqrt[x]*(c + d*x^2)) - ((b^2*c^2 - 2*a*b*c*d + 5*a^2*d^2)*x^(3/2))/(
2*c^2*d*(c + d*x^2)) - ((b*c - a*d)*(3*b*c + 5*a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*
Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(9/4)*d^(7/4)) + ((b*c - a*d)*(3*b*c + 5*a*d)*Ar
cTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(9/4)*d^(7/4)) + ((b*c
- a*d)*(3*b*c + 5*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*
x])/(8*Sqrt[2]*c^(9/4)*d^(7/4)) - ((b*c - a*d)*(3*b*c + 5*a*d)*Log[Sqrt[c] + Sqr
t[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(9/4)*d^(7/4))

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Rubi [A]  time = 0.713546, antiderivative size = 333, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ -\frac{x^{3/2} \left (5 a^2 d^2-2 a b c d+b^2 c^2\right )}{2 c^2 d \left (c+d x^2\right )}-\frac{2 a^2}{c \sqrt{x} \left (c+d x^2\right )}+\frac{(b c-a d) (5 a d+3 b c) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{9/4} d^{7/4}}-\frac{(b c-a d) (5 a d+3 b c) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{9/4} d^{7/4}}-\frac{(b c-a d) (5 a d+3 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{9/4} d^{7/4}}+\frac{(b c-a d) (5 a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} c^{9/4} d^{7/4}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*a^2)/(c*Sqrt[x]*(c + d*x^2)) - ((b^2*c^2 - 2*a*b*c*d + 5*a^2*d^2)*x^(3/2))/(
2*c^2*d*(c + d*x^2)) - ((b*c - a*d)*(3*b*c + 5*a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*
Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(9/4)*d^(7/4)) + ((b*c - a*d)*(3*b*c + 5*a*d)*Ar
cTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(9/4)*d^(7/4)) + ((b*c
- a*d)*(3*b*c + 5*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*
x])/(8*Sqrt[2]*c^(9/4)*d^(7/4)) - ((b*c - a*d)*(3*b*c + 5*a*d)*Log[Sqrt[c] + Sqr
t[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(9/4)*d^(7/4))

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Rubi in Sympy [A]  time = 96.6386, size = 308, normalized size = 0.92 \[ - \frac{2 a^{2}}{c \sqrt{x} \left (c + d x^{2}\right )} - \frac{x^{\frac{3}{2}} \left (a d \left (5 a d - 2 b c\right ) + b^{2} c^{2}\right )}{2 c^{2} d \left (c + d x^{2}\right )} - \frac{\sqrt{2} \left (a d - b c\right ) \left (5 a d + 3 b c\right ) \log{\left (- \sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x} + \sqrt{c} + \sqrt{d} x \right )}}{16 c^{\frac{9}{4}} d^{\frac{7}{4}}} + \frac{\sqrt{2} \left (a d - b c\right ) \left (5 a d + 3 b c\right ) \log{\left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x} + \sqrt{c} + \sqrt{d} x \right )}}{16 c^{\frac{9}{4}} d^{\frac{7}{4}}} + \frac{\sqrt{2} \left (a d - b c\right ) \left (5 a d + 3 b c\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}}{8 c^{\frac{9}{4}} d^{\frac{7}{4}}} - \frac{\sqrt{2} \left (a d - b c\right ) \left (5 a d + 3 b c\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}}{8 c^{\frac{9}{4}} d^{\frac{7}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**2/(c*sqrt(x)*(c + d*x**2)) - x**(3/2)*(a*d*(5*a*d - 2*b*c) + b**2*c**2)/(2
*c**2*d*(c + d*x**2)) - sqrt(2)*(a*d - b*c)*(5*a*d + 3*b*c)*log(-sqrt(2)*c**(1/4
)*d**(1/4)*sqrt(x) + sqrt(c) + sqrt(d)*x)/(16*c**(9/4)*d**(7/4)) + sqrt(2)*(a*d
- b*c)*(5*a*d + 3*b*c)*log(sqrt(2)*c**(1/4)*d**(1/4)*sqrt(x) + sqrt(c) + sqrt(d)
*x)/(16*c**(9/4)*d**(7/4)) + sqrt(2)*(a*d - b*c)*(5*a*d + 3*b*c)*atan(1 - sqrt(2
)*d**(1/4)*sqrt(x)/c**(1/4))/(8*c**(9/4)*d**(7/4)) - sqrt(2)*(a*d - b*c)*(5*a*d
+ 3*b*c)*atan(1 + sqrt(2)*d**(1/4)*sqrt(x)/c**(1/4))/(8*c**(9/4)*d**(7/4))

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Mathematica [A]  time = 0.303842, size = 317, normalized size = 0.95 \[ \frac{\frac{\sqrt{2} \left (-5 a^2 d^2+2 a b c d+3 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{d^{7/4}}+\frac{\sqrt{2} \left (5 a^2 d^2-2 a b c d-3 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{d^{7/4}}+\frac{2 \sqrt{2} \left (5 a^2 d^2-2 a b c d-3 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{d^{7/4}}+\frac{2 \sqrt{2} \left (-5 a^2 d^2+2 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{d^{7/4}}-\frac{32 a^2 \sqrt [4]{c}}{\sqrt{x}}-\frac{8 \sqrt [4]{c} x^{3/2} (b c-a d)^2}{d \left (c+d x^2\right )}}{16 c^{9/4}} \]

Antiderivative was successfully verified.

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

[Out]

((-32*a^2*c^(1/4))/Sqrt[x] - (8*c^(1/4)*(b*c - a*d)^2*x^(3/2))/(d*(c + d*x^2)) +
(2*Sqrt[2]*(-3*b^2*c^2 - 2*a*b*c*d + 5*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqr
t[x])/c^(1/4)])/d^(7/4) + (2*Sqrt[2]*(3*b^2*c^2 + 2*a*b*c*d - 5*a^2*d^2)*ArcTan[
1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/d^(7/4) + (Sqrt[2]*(3*b^2*c^2 + 2*a*b*c*
d - 5*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/d^(7/
4) + (Sqrt[2]*(-3*b^2*c^2 - 2*a*b*c*d + 5*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)
*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/d^(7/4))/(16*c^(9/4))

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Maple [A]  time = 0.026, size = 495, normalized size = 1.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/c^2*d*x^(3/2)/(d*x^2+c)*a^2+1/c*x^(3/2)/(d*x^2+c)*a*b-1/2/d*x^(3/2)/(d*x^2+
c)*b^2-5/8/c^2/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a^2+1/4
/c/d/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a*b+3/8/d^2/(c/d)
^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*b^2-5/16/c^2/(c/d)^(1/4)*2^
(1/2)*ln((x-(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2))/(x+(c/d)^(1/4)*x^(1/2)*2^(1
/2)+(c/d)^(1/2)))*a^2+1/8/c/d/(c/d)^(1/4)*2^(1/2)*ln((x-(c/d)^(1/4)*x^(1/2)*2^(1
/2)+(c/d)^(1/2))/(x+(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2)))*a*b+3/16/d^2/(c/d)
^(1/4)*2^(1/2)*ln((x-(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2))/(x+(c/d)^(1/4)*x^(
1/2)*2^(1/2)+(c/d)^(1/2)))*b^2-5/8/c^2/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^
(1/4)*x^(1/2)+1)*a^2+1/4/c/d/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1
/2)+1)*a*b+3/8/d^2/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*b^2
-2*a^2/c^2/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((b*x^2 + a)^2/((d*x^2 + c)^2*x^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.268178, size = 2039, normalized size = 6.12 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(16*a^2*c*d + 4*(b^2*c^2 - 2*a*b*c*d + 5*a^2*d^2)*x^2 + 4*(c^2*d^2*x^2 + c^
3*d)*sqrt(x)*(-(81*b^8*c^8 + 216*a*b^7*c^7*d - 324*a^2*b^6*c^6*d^2 - 984*a^3*b^5
*c^5*d^3 + 646*a^4*b^4*c^4*d^4 + 1640*a^5*b^3*c^3*d^5 - 900*a^6*b^2*c^2*d^6 - 10
00*a^7*b*c*d^7 + 625*a^8*d^8)/(c^9*d^7))^(1/4)*arctan(-c^7*d^5*(-(81*b^8*c^8 + 2
16*a*b^7*c^7*d - 324*a^2*b^6*c^6*d^2 - 984*a^3*b^5*c^5*d^3 + 646*a^4*b^4*c^4*d^4
+ 1640*a^5*b^3*c^3*d^5 - 900*a^6*b^2*c^2*d^6 - 1000*a^7*b*c*d^7 + 625*a^8*d^8)/
(c^9*d^7))^(3/4)/((27*b^6*c^6 + 54*a*b^5*c^5*d - 99*a^2*b^4*c^4*d^2 - 172*a^3*b^
3*c^3*d^3 + 165*a^4*b^2*c^2*d^4 + 150*a^5*b*c*d^5 - 125*a^6*d^6)*sqrt(x) - sqrt(
(729*b^12*c^12 + 2916*a*b^11*c^11*d - 2430*a^2*b^10*c^10*d^2 - 19980*a^3*b^9*c^9
*d^3 + 135*a^4*b^8*c^8*d^4 + 59976*a^5*b^7*c^7*d^5 + 6364*a^6*b^6*c^6*d^6 - 9996
0*a^7*b^5*c^5*d^7 + 375*a^8*b^4*c^4*d^8 + 92500*a^9*b^3*c^3*d^9 - 18750*a^10*b^2
*c^2*d^10 - 37500*a^11*b*c*d^11 + 15625*a^12*d^12)*x - (81*b^8*c^13*d^3 + 216*a*
b^7*c^12*d^4 - 324*a^2*b^6*c^11*d^5 - 984*a^3*b^5*c^10*d^6 + 646*a^4*b^4*c^9*d^7
+ 1640*a^5*b^3*c^8*d^8 - 900*a^6*b^2*c^7*d^9 - 1000*a^7*b*c^6*d^10 + 625*a^8*c^
5*d^11)*sqrt(-(81*b^8*c^8 + 216*a*b^7*c^7*d - 324*a^2*b^6*c^6*d^2 - 984*a^3*b^5*
c^5*d^3 + 646*a^4*b^4*c^4*d^4 + 1640*a^5*b^3*c^3*d^5 - 900*a^6*b^2*c^2*d^6 - 100
0*a^7*b*c*d^7 + 625*a^8*d^8)/(c^9*d^7))))) + (c^2*d^2*x^2 + c^3*d)*sqrt(x)*(-(81
*b^8*c^8 + 216*a*b^7*c^7*d - 324*a^2*b^6*c^6*d^2 - 984*a^3*b^5*c^5*d^3 + 646*a^4
*b^4*c^4*d^4 + 1640*a^5*b^3*c^3*d^5 - 900*a^6*b^2*c^2*d^6 - 1000*a^7*b*c*d^7 + 6
25*a^8*d^8)/(c^9*d^7))^(1/4)*log(c^7*d^5*(-(81*b^8*c^8 + 216*a*b^7*c^7*d - 324*a
^2*b^6*c^6*d^2 - 984*a^3*b^5*c^5*d^3 + 646*a^4*b^4*c^4*d^4 + 1640*a^5*b^3*c^3*d^
5 - 900*a^6*b^2*c^2*d^6 - 1000*a^7*b*c*d^7 + 625*a^8*d^8)/(c^9*d^7))^(3/4) - (27
*b^6*c^6 + 54*a*b^5*c^5*d - 99*a^2*b^4*c^4*d^2 - 172*a^3*b^3*c^3*d^3 + 165*a^4*b
^2*c^2*d^4 + 150*a^5*b*c*d^5 - 125*a^6*d^6)*sqrt(x)) - (c^2*d^2*x^2 + c^3*d)*sqr
t(x)*(-(81*b^8*c^8 + 216*a*b^7*c^7*d - 324*a^2*b^6*c^6*d^2 - 984*a^3*b^5*c^5*d^3
+ 646*a^4*b^4*c^4*d^4 + 1640*a^5*b^3*c^3*d^5 - 900*a^6*b^2*c^2*d^6 - 1000*a^7*b
*c*d^7 + 625*a^8*d^8)/(c^9*d^7))^(1/4)*log(-c^7*d^5*(-(81*b^8*c^8 + 216*a*b^7*c^
7*d - 324*a^2*b^6*c^6*d^2 - 984*a^3*b^5*c^5*d^3 + 646*a^4*b^4*c^4*d^4 + 1640*a^5
*b^3*c^3*d^5 - 900*a^6*b^2*c^2*d^6 - 1000*a^7*b*c*d^7 + 625*a^8*d^8)/(c^9*d^7))^
(3/4) - (27*b^6*c^6 + 54*a*b^5*c^5*d - 99*a^2*b^4*c^4*d^2 - 172*a^3*b^3*c^3*d^3
+ 165*a^4*b^2*c^2*d^4 + 150*a^5*b*c*d^5 - 125*a^6*d^6)*sqrt(x)))/((c^2*d^2*x^2 +
c^3*d)*sqrt(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((b*x**2+a)**2/x**(3/2)/(d*x**2+c)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.26034, size = 525, normalized size = 1.58 \[ -\frac{b^{2} c^{2} x^{2} - 2 \, a b c d x^{2} + 5 \, a^{2} d^{2} x^{2} + 4 \, a^{2} c d}{2 \,{\left (d x^{\frac{5}{2}} + c \sqrt{x}\right )} c^{2} d} + \frac{\sqrt{2}{\left (3 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} + 2 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d - 5 \, \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{8 \, c^{3} d^{4}} + \frac{\sqrt{2}{\left (3 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} + 2 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d - 5 \, \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{8 \, c^{3} d^{4}} - \frac{\sqrt{2}{\left (3 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} + 2 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d - 5 \, \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{16 \, c^{3} d^{4}} + \frac{\sqrt{2}{\left (3 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} + 2 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d - 5 \, \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{16 \, c^{3} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(b^2*c^2*x^2 - 2*a*b*c*d*x^2 + 5*a^2*d^2*x^2 + 4*a^2*c*d)/((d*x^(5/2) + c*s
qrt(x))*c^2*d) + 1/8*sqrt(2)*(3*(c*d^3)^(3/4)*b^2*c^2 + 2*(c*d^3)^(3/4)*a*b*c*d
- 5*(c*d^3)^(3/4)*a^2*d^2)*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/
(c/d)^(1/4))/(c^3*d^4) + 1/8*sqrt(2)*(3*(c*d^3)^(3/4)*b^2*c^2 + 2*(c*d^3)^(3/4)*
a*b*c*d - 5*(c*d^3)^(3/4)*a^2*d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*
sqrt(x))/(c/d)^(1/4))/(c^3*d^4) - 1/16*sqrt(2)*(3*(c*d^3)^(3/4)*b^2*c^2 + 2*(c*d
^3)^(3/4)*a*b*c*d - 5*(c*d^3)^(3/4)*a^2*d^2)*ln(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x
+ sqrt(c/d))/(c^3*d^4) + 1/16*sqrt(2)*(3*(c*d^3)^(3/4)*b^2*c^2 + 2*(c*d^3)^(3/4)
*a*b*c*d - 5*(c*d^3)^(3/4)*a^2*d^2)*ln(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c
/d))/(c^3*d^4)