3.53 \(\int \frac{a+b x^4}{\left (c+d x^4\right )^3} \, dx\)

Optimal. Leaf size=273 \[ -\frac{3 (7 a d+b c) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{128 \sqrt{2} c^{11/4} d^{5/4}}+\frac{3 (7 a d+b c) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{128 \sqrt{2} c^{11/4} d^{5/4}}-\frac{3 (7 a d+b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{64 \sqrt{2} c^{11/4} d^{5/4}}+\frac{3 (7 a d+b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{64 \sqrt{2} c^{11/4} d^{5/4}}+\frac{x (7 a d+b c)}{32 c^2 d \left (c+d x^4\right )}-\frac{x (b c-a d)}{8 c d \left (c+d x^4\right )^2} \]

[Out]

-((b*c - a*d)*x)/(8*c*d*(c + d*x^4)^2) + ((b*c + 7*a*d)*x)/(32*c^2*d*(c + d*x^4)
) - (3*(b*c + 7*a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/(64*Sqrt[2]*c^(11/
4)*d^(5/4)) + (3*(b*c + 7*a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/(64*Sqrt
[2]*c^(11/4)*d^(5/4)) - (3*(b*c + 7*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*x
+ Sqrt[d]*x^2])/(128*Sqrt[2]*c^(11/4)*d^(5/4)) + (3*(b*c + 7*a*d)*Log[Sqrt[c] +
Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(128*Sqrt[2]*c^(11/4)*d^(5/4))

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Rubi [A]  time = 0.356832, antiderivative size = 273, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.471 \[ -\frac{3 (7 a d+b c) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{128 \sqrt{2} c^{11/4} d^{5/4}}+\frac{3 (7 a d+b c) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{128 \sqrt{2} c^{11/4} d^{5/4}}-\frac{3 (7 a d+b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{64 \sqrt{2} c^{11/4} d^{5/4}}+\frac{3 (7 a d+b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{64 \sqrt{2} c^{11/4} d^{5/4}}+\frac{x (7 a d+b c)}{32 c^2 d \left (c+d x^4\right )}-\frac{x (b c-a d)}{8 c d \left (c+d x^4\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)/(c + d*x^4)^3,x]

[Out]

-((b*c - a*d)*x)/(8*c*d*(c + d*x^4)^2) + ((b*c + 7*a*d)*x)/(32*c^2*d*(c + d*x^4)
) - (3*(b*c + 7*a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/(64*Sqrt[2]*c^(11/
4)*d^(5/4)) + (3*(b*c + 7*a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/(64*Sqrt
[2]*c^(11/4)*d^(5/4)) - (3*(b*c + 7*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*x
+ Sqrt[d]*x^2])/(128*Sqrt[2]*c^(11/4)*d^(5/4)) + (3*(b*c + 7*a*d)*Log[Sqrt[c] +
Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(128*Sqrt[2]*c^(11/4)*d^(5/4))

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Rubi in Sympy [A]  time = 65.6386, size = 258, normalized size = 0.95 \[ \frac{x \left (a d - b c\right )}{8 c d \left (c + d x^{4}\right )^{2}} + \frac{x \left (7 a d + b c\right )}{32 c^{2} d \left (c + d x^{4}\right )} - \frac{3 \sqrt{2} \left (7 a d + b c\right ) \log{\left (- \sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x + \sqrt{c} + \sqrt{d} x^{2} \right )}}{256 c^{\frac{11}{4}} d^{\frac{5}{4}}} + \frac{3 \sqrt{2} \left (7 a d + b c\right ) \log{\left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x + \sqrt{c} + \sqrt{d} x^{2} \right )}}{256 c^{\frac{11}{4}} d^{\frac{5}{4}}} - \frac{3 \sqrt{2} \left (7 a d + b c\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}} \right )}}{128 c^{\frac{11}{4}} d^{\frac{5}{4}}} + \frac{3 \sqrt{2} \left (7 a d + b c\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}} \right )}}{128 c^{\frac{11}{4}} d^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+a)/(d*x**4+c)**3,x)

[Out]

x*(a*d - b*c)/(8*c*d*(c + d*x**4)**2) + x*(7*a*d + b*c)/(32*c**2*d*(c + d*x**4))
- 3*sqrt(2)*(7*a*d + b*c)*log(-sqrt(2)*c**(1/4)*d**(1/4)*x + sqrt(c) + sqrt(d)*
x**2)/(256*c**(11/4)*d**(5/4)) + 3*sqrt(2)*(7*a*d + b*c)*log(sqrt(2)*c**(1/4)*d*
*(1/4)*x + sqrt(c) + sqrt(d)*x**2)/(256*c**(11/4)*d**(5/4)) - 3*sqrt(2)*(7*a*d +
b*c)*atan(1 - sqrt(2)*d**(1/4)*x/c**(1/4))/(128*c**(11/4)*d**(5/4)) + 3*sqrt(2)
*(7*a*d + b*c)*atan(1 + sqrt(2)*d**(1/4)*x/c**(1/4))/(128*c**(11/4)*d**(5/4))

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Mathematica [A]  time = 0.371676, size = 243, normalized size = 0.89 \[ \frac{-\frac{32 c^{7/4} \sqrt [4]{d} x (b c-a d)}{\left (c+d x^4\right )^2}+\frac{8 c^{3/4} \sqrt [4]{d} x (7 a d+b c)}{c+d x^4}-3 \sqrt{2} (7 a d+b c) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )+3 \sqrt{2} (7 a d+b c) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )-6 \sqrt{2} (7 a d+b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )+6 \sqrt{2} (7 a d+b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{256 c^{11/4} d^{5/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)/(c + d*x^4)^3,x]

[Out]

((-32*c^(7/4)*d^(1/4)*(b*c - a*d)*x)/(c + d*x^4)^2 + (8*c^(3/4)*d^(1/4)*(b*c + 7
*a*d)*x)/(c + d*x^4) - 6*Sqrt[2]*(b*c + 7*a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^
(1/4)] + 6*Sqrt[2]*(b*c + 7*a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)] - 3*Sqr
t[2]*(b*c + 7*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2] + 3*Sq
rt[2]*(b*c + 7*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(256
*c^(11/4)*d^(5/4))

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Maple [A]  time = 0.016, size = 314, normalized size = 1.2 \[{\frac{1}{ \left ( d{x}^{4}+c \right ) ^{2}} \left ({\frac{ \left ( 7\,ad+bc \right ){x}^{5}}{32\,{c}^{2}}}+{\frac{ \left ( 11\,ad-3\,bc \right ) x}{32\,cd}} \right ) }+{\frac{21\,\sqrt{2}a}{128\,{c}^{3}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) }+{\frac{3\,\sqrt{2}b}{128\,{c}^{2}d}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) }+{\frac{21\,\sqrt{2}a}{128\,{c}^{3}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }+{\frac{3\,\sqrt{2}b}{128\,{c}^{2}d}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }+{\frac{21\,\sqrt{2}a}{256\,{c}^{3}}\sqrt [4]{{\frac{c}{d}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ) }+{\frac{3\,\sqrt{2}b}{256\,{c}^{2}d}\sqrt [4]{{\frac{c}{d}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)/(d*x^4+c)^3,x)

[Out]

(1/32*(7*a*d+b*c)/c^2*x^5+1/32*(11*a*d-3*b*c)/c/d*x)/(d*x^4+c)^2+21/128/c^3*(c/d
)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x+1)*a+3/128/c^2/d*(c/d)^(1/4)*2^(1/2
)*arctan(2^(1/2)/(c/d)^(1/4)*x+1)*b+21/128/c^3*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2
)/(c/d)^(1/4)*x-1)*a+3/128/c^2/d*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*
x-1)*b+21/256/c^3*(c/d)^(1/4)*2^(1/2)*ln((x^2+(c/d)^(1/4)*x*2^(1/2)+(c/d)^(1/2))
/(x^2-(c/d)^(1/4)*x*2^(1/2)+(c/d)^(1/2)))*a+3/256/c^2/d*(c/d)^(1/4)*2^(1/2)*ln((
x^2+(c/d)^(1/4)*x*2^(1/2)+(c/d)^(1/2))/(x^2-(c/d)^(1/4)*x*2^(1/2)+(c/d)^(1/2)))*
b

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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^4 + a)/(d*x^4 + c)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.239178, size = 941, normalized size = 3.45 \[ \frac{4 \,{\left (b c d + 7 \, a d^{2}\right )} x^{5} - 12 \,{\left (c^{2} d^{3} x^{8} + 2 \, c^{3} d^{2} x^{4} + c^{4} d\right )} \left (-\frac{b^{4} c^{4} + 28 \, a b^{3} c^{3} d + 294 \, a^{2} b^{2} c^{2} d^{2} + 1372 \, a^{3} b c d^{3} + 2401 \, a^{4} d^{4}}{c^{11} d^{5}}\right )^{\frac{1}{4}} \arctan \left (\frac{c^{3} d \left (-\frac{b^{4} c^{4} + 28 \, a b^{3} c^{3} d + 294 \, a^{2} b^{2} c^{2} d^{2} + 1372 \, a^{3} b c d^{3} + 2401 \, a^{4} d^{4}}{c^{11} d^{5}}\right )^{\frac{1}{4}}}{{\left (b c + 7 \, a d\right )} x +{\left (b c + 7 \, a d\right )} \sqrt{\frac{c^{6} d^{2} \sqrt{-\frac{b^{4} c^{4} + 28 \, a b^{3} c^{3} d + 294 \, a^{2} b^{2} c^{2} d^{2} + 1372 \, a^{3} b c d^{3} + 2401 \, a^{4} d^{4}}{c^{11} d^{5}}} +{\left (b^{2} c^{2} + 14 \, a b c d + 49 \, a^{2} d^{2}\right )} x^{2}}{b^{2} c^{2} + 14 \, a b c d + 49 \, a^{2} d^{2}}}}\right ) + 3 \,{\left (c^{2} d^{3} x^{8} + 2 \, c^{3} d^{2} x^{4} + c^{4} d\right )} \left (-\frac{b^{4} c^{4} + 28 \, a b^{3} c^{3} d + 294 \, a^{2} b^{2} c^{2} d^{2} + 1372 \, a^{3} b c d^{3} + 2401 \, a^{4} d^{4}}{c^{11} d^{5}}\right )^{\frac{1}{4}} \log \left (3 \, c^{3} d \left (-\frac{b^{4} c^{4} + 28 \, a b^{3} c^{3} d + 294 \, a^{2} b^{2} c^{2} d^{2} + 1372 \, a^{3} b c d^{3} + 2401 \, a^{4} d^{4}}{c^{11} d^{5}}\right )^{\frac{1}{4}} + 3 \,{\left (b c + 7 \, a d\right )} x\right ) - 3 \,{\left (c^{2} d^{3} x^{8} + 2 \, c^{3} d^{2} x^{4} + c^{4} d\right )} \left (-\frac{b^{4} c^{4} + 28 \, a b^{3} c^{3} d + 294 \, a^{2} b^{2} c^{2} d^{2} + 1372 \, a^{3} b c d^{3} + 2401 \, a^{4} d^{4}}{c^{11} d^{5}}\right )^{\frac{1}{4}} \log \left (-3 \, c^{3} d \left (-\frac{b^{4} c^{4} + 28 \, a b^{3} c^{3} d + 294 \, a^{2} b^{2} c^{2} d^{2} + 1372 \, a^{3} b c d^{3} + 2401 \, a^{4} d^{4}}{c^{11} d^{5}}\right )^{\frac{1}{4}} + 3 \,{\left (b c + 7 \, a d\right )} x\right ) - 4 \,{\left (3 \, b c^{2} - 11 \, a c d\right )} x}{128 \,{\left (c^{2} d^{3} x^{8} + 2 \, c^{3} d^{2} x^{4} + c^{4} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)/(d*x^4 + c)^3,x, algorithm="fricas")

[Out]

1/128*(4*(b*c*d + 7*a*d^2)*x^5 - 12*(c^2*d^3*x^8 + 2*c^3*d^2*x^4 + c^4*d)*(-(b^4
*c^4 + 28*a*b^3*c^3*d + 294*a^2*b^2*c^2*d^2 + 1372*a^3*b*c*d^3 + 2401*a^4*d^4)/(
c^11*d^5))^(1/4)*arctan(c^3*d*(-(b^4*c^4 + 28*a*b^3*c^3*d + 294*a^2*b^2*c^2*d^2
+ 1372*a^3*b*c*d^3 + 2401*a^4*d^4)/(c^11*d^5))^(1/4)/((b*c + 7*a*d)*x + (b*c + 7
*a*d)*sqrt((c^6*d^2*sqrt(-(b^4*c^4 + 28*a*b^3*c^3*d + 294*a^2*b^2*c^2*d^2 + 1372
*a^3*b*c*d^3 + 2401*a^4*d^4)/(c^11*d^5)) + (b^2*c^2 + 14*a*b*c*d + 49*a^2*d^2)*x
^2)/(b^2*c^2 + 14*a*b*c*d + 49*a^2*d^2)))) + 3*(c^2*d^3*x^8 + 2*c^3*d^2*x^4 + c^
4*d)*(-(b^4*c^4 + 28*a*b^3*c^3*d + 294*a^2*b^2*c^2*d^2 + 1372*a^3*b*c*d^3 + 2401
*a^4*d^4)/(c^11*d^5))^(1/4)*log(3*c^3*d*(-(b^4*c^4 + 28*a*b^3*c^3*d + 294*a^2*b^
2*c^2*d^2 + 1372*a^3*b*c*d^3 + 2401*a^4*d^4)/(c^11*d^5))^(1/4) + 3*(b*c + 7*a*d)
*x) - 3*(c^2*d^3*x^8 + 2*c^3*d^2*x^4 + c^4*d)*(-(b^4*c^4 + 28*a*b^3*c^3*d + 294*
a^2*b^2*c^2*d^2 + 1372*a^3*b*c*d^3 + 2401*a^4*d^4)/(c^11*d^5))^(1/4)*log(-3*c^3*
d*(-(b^4*c^4 + 28*a*b^3*c^3*d + 294*a^2*b^2*c^2*d^2 + 1372*a^3*b*c*d^3 + 2401*a^
4*d^4)/(c^11*d^5))^(1/4) + 3*(b*c + 7*a*d)*x) - 4*(3*b*c^2 - 11*a*c*d)*x)/(c^2*d
^3*x^8 + 2*c^3*d^2*x^4 + c^4*d)

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Sympy [A]  time = 6.69677, size = 151, normalized size = 0.55 \[ \frac{x^{5} \left (7 a d^{2} + b c d\right ) + x \left (11 a c d - 3 b c^{2}\right )}{32 c^{4} d + 64 c^{3} d^{2} x^{4} + 32 c^{2} d^{3} x^{8}} + \operatorname{RootSum}{\left (268435456 t^{4} c^{11} d^{5} + 194481 a^{4} d^{4} + 111132 a^{3} b c d^{3} + 23814 a^{2} b^{2} c^{2} d^{2} + 2268 a b^{3} c^{3} d + 81 b^{4} c^{4}, \left ( t \mapsto t \log{\left (\frac{128 t c^{3} d}{21 a d + 3 b c} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**4+a)/(d*x**4+c)**3,x)

[Out]

(x**5*(7*a*d**2 + b*c*d) + x*(11*a*c*d - 3*b*c**2))/(32*c**4*d + 64*c**3*d**2*x*
*4 + 32*c**2*d**3*x**8) + RootSum(268435456*_t**4*c**11*d**5 + 194481*a**4*d**4
+ 111132*a**3*b*c*d**3 + 23814*a**2*b**2*c**2*d**2 + 2268*a*b**3*c**3*d + 81*b**
4*c**4, Lambda(_t, _t*log(128*_t*c**3*d/(21*a*d + 3*b*c) + x)))

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GIAC/XCAS [A]  time = 0.221907, size = 386, normalized size = 1.41 \[ \frac{3 \, \sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b c + 7 \, \left (c d^{3}\right )^{\frac{1}{4}} a d\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{128 \, c^{3} d^{2}} + \frac{3 \, \sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b c + 7 \, \left (c d^{3}\right )^{\frac{1}{4}} a d\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{128 \, c^{3} d^{2}} + \frac{3 \, \sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b c + 7 \, \left (c d^{3}\right )^{\frac{1}{4}} a d\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{c}{d}\right )^{\frac{1}{4}} + \sqrt{\frac{c}{d}}\right )}{256 \, c^{3} d^{2}} - \frac{3 \, \sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b c + 7 \, \left (c d^{3}\right )^{\frac{1}{4}} a d\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{c}{d}\right )^{\frac{1}{4}} + \sqrt{\frac{c}{d}}\right )}{256 \, c^{3} d^{2}} + \frac{b c d x^{5} + 7 \, a d^{2} x^{5} - 3 \, b c^{2} x + 11 \, a c d x}{32 \,{\left (d x^{4} + c\right )}^{2} c^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)/(d*x^4 + c)^3,x, algorithm="giac")

[Out]

3/128*sqrt(2)*((c*d^3)^(1/4)*b*c + 7*(c*d^3)^(1/4)*a*d)*arctan(1/2*sqrt(2)*(2*x
+ sqrt(2)*(c/d)^(1/4))/(c/d)^(1/4))/(c^3*d^2) + 3/128*sqrt(2)*((c*d^3)^(1/4)*b*c
+ 7*(c*d^3)^(1/4)*a*d)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(c/d)^(1/4))/(c/d)^(1/
4))/(c^3*d^2) + 3/256*sqrt(2)*((c*d^3)^(1/4)*b*c + 7*(c*d^3)^(1/4)*a*d)*ln(x^2 +
sqrt(2)*x*(c/d)^(1/4) + sqrt(c/d))/(c^3*d^2) - 3/256*sqrt(2)*((c*d^3)^(1/4)*b*c
+ 7*(c*d^3)^(1/4)*a*d)*ln(x^2 - sqrt(2)*x*(c/d)^(1/4) + sqrt(c/d))/(c^3*d^2) +
1/32*(b*c*d*x^5 + 7*a*d^2*x^5 - 3*b*c^2*x + 11*a*c*d*x)/((d*x^4 + c)^2*c^2*d)