3.409 \(\int \frac{(d+e x)^2}{\left (a+c x^4\right )^3} \, dx\)

Optimal. Leaf size=360 \[ -\frac{\left (21 \sqrt{c} d^2-5 \sqrt{a} e^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{11/4} c^{3/4}}+\frac{\left (21 \sqrt{c} d^2-5 \sqrt{a} e^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{11/4} c^{3/4}}-\frac{\left (5 \sqrt{a} e^2+21 \sqrt{c} d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{11/4} c^{3/4}}+\frac{\left (5 \sqrt{a} e^2+21 \sqrt{c} d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} a^{11/4} c^{3/4}}+\frac{3 d e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{c}}+\frac{x \left (7 d^2+12 d e x+5 e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}+\frac{x (d+e x)^2}{8 a \left (a+c x^4\right )^2} \]

[Out]

(x*(d + e*x)^2)/(8*a*(a + c*x^4)^2) + (x*(7*d^2 + 12*d*e*x + 5*e^2*x^2))/(32*a^2
*(a + c*x^4)) + (3*d*e*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(8*a^(5/2)*Sqrt[c]) - ((21
*Sqrt[c]*d^2 + 5*Sqrt[a]*e^2)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(64*Sqrt[
2]*a^(11/4)*c^(3/4)) + ((21*Sqrt[c]*d^2 + 5*Sqrt[a]*e^2)*ArcTan[1 + (Sqrt[2]*c^(
1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(11/4)*c^(3/4)) - ((21*Sqrt[c]*d^2 - 5*Sqrt[a]*e
^2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^(11/4
)*c^(3/4)) + ((21*Sqrt[c]*d^2 - 5*Sqrt[a]*e^2)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(
1/4)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^(11/4)*c^(3/4))

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Rubi [A]  time = 0.697319, antiderivative size = 360, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.588 \[ -\frac{\left (21 \sqrt{c} d^2-5 \sqrt{a} e^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{11/4} c^{3/4}}+\frac{\left (21 \sqrt{c} d^2-5 \sqrt{a} e^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{11/4} c^{3/4}}-\frac{\left (5 \sqrt{a} e^2+21 \sqrt{c} d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{11/4} c^{3/4}}+\frac{\left (5 \sqrt{a} e^2+21 \sqrt{c} d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} a^{11/4} c^{3/4}}+\frac{3 d e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{c}}+\frac{x \left (7 d^2+12 d e x+5 e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}+\frac{x (d+e x)^2}{8 a \left (a+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^2/(a + c*x^4)^3,x]

[Out]

(x*(d + e*x)^2)/(8*a*(a + c*x^4)^2) + (x*(7*d^2 + 12*d*e*x + 5*e^2*x^2))/(32*a^2
*(a + c*x^4)) + (3*d*e*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(8*a^(5/2)*Sqrt[c]) - ((21
*Sqrt[c]*d^2 + 5*Sqrt[a]*e^2)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(64*Sqrt[
2]*a^(11/4)*c^(3/4)) + ((21*Sqrt[c]*d^2 + 5*Sqrt[a]*e^2)*ArcTan[1 + (Sqrt[2]*c^(
1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(11/4)*c^(3/4)) - ((21*Sqrt[c]*d^2 - 5*Sqrt[a]*e
^2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^(11/4
)*c^(3/4)) + ((21*Sqrt[c]*d^2 - 5*Sqrt[a]*e^2)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(
1/4)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^(11/4)*c^(3/4))

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Rubi in Sympy [A]  time = 119.628, size = 343, normalized size = 0.95 \[ \frac{x \left (d + e x\right )^{2}}{8 a \left (a + c x^{4}\right )^{2}} + \frac{x \left (7 d^{2} + 12 d e x + 5 e^{2} x^{2}\right )}{32 a^{2} \left (a + c x^{4}\right )} + \frac{3 d e \operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{8 a^{\frac{5}{2}} \sqrt{c}} + \frac{\sqrt{2} \left (5 \sqrt{a} e^{2} - 21 \sqrt{c} d^{2}\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} x + \sqrt{a} \sqrt{c} + c x^{2} \right )}}{256 a^{\frac{11}{4}} c^{\frac{3}{4}}} - \frac{\sqrt{2} \left (5 \sqrt{a} e^{2} - 21 \sqrt{c} d^{2}\right ) \log{\left (\sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} x + \sqrt{a} \sqrt{c} + c x^{2} \right )}}{256 a^{\frac{11}{4}} c^{\frac{3}{4}}} - \frac{\sqrt{2} \left (5 \sqrt{a} e^{2} + 21 \sqrt{c} d^{2}\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{128 a^{\frac{11}{4}} c^{\frac{3}{4}}} + \frac{\sqrt{2} \left (5 \sqrt{a} e^{2} + 21 \sqrt{c} d^{2}\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{128 a^{\frac{11}{4}} c^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**2/(c*x**4+a)**3,x)

[Out]

x*(d + e*x)**2/(8*a*(a + c*x**4)**2) + x*(7*d**2 + 12*d*e*x + 5*e**2*x**2)/(32*a
**2*(a + c*x**4)) + 3*d*e*atan(sqrt(c)*x**2/sqrt(a))/(8*a**(5/2)*sqrt(c)) + sqrt
(2)*(5*sqrt(a)*e**2 - 21*sqrt(c)*d**2)*log(-sqrt(2)*a**(1/4)*c**(3/4)*x + sqrt(a
)*sqrt(c) + c*x**2)/(256*a**(11/4)*c**(3/4)) - sqrt(2)*(5*sqrt(a)*e**2 - 21*sqrt
(c)*d**2)*log(sqrt(2)*a**(1/4)*c**(3/4)*x + sqrt(a)*sqrt(c) + c*x**2)/(256*a**(1
1/4)*c**(3/4)) - sqrt(2)*(5*sqrt(a)*e**2 + 21*sqrt(c)*d**2)*atan(1 - sqrt(2)*c**
(1/4)*x/a**(1/4))/(128*a**(11/4)*c**(3/4)) + sqrt(2)*(5*sqrt(a)*e**2 + 21*sqrt(c
)*d**2)*atan(1 + sqrt(2)*c**(1/4)*x/a**(1/4))/(128*a**(11/4)*c**(3/4))

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Mathematica [A]  time = 0.601113, size = 358, normalized size = 0.99 \[ \frac{\frac{\sqrt{2} \left (5 a^{3/4} e^2-21 \sqrt [4]{a} \sqrt{c} d^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{c^{3/4}}+\frac{\sqrt{2} \left (21 \sqrt [4]{a} \sqrt{c} d^2-5 a^{3/4} e^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{c^{3/4}}+\frac{32 a^2 x (d+e x)^2}{\left (a+c x^4\right )^2}-\frac{2 \sqrt [4]{a} \left (48 \sqrt [4]{a} \sqrt [4]{c} d e+5 \sqrt{2} \sqrt{a} e^2+21 \sqrt{2} \sqrt{c} d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{c^{3/4}}+\frac{2 \sqrt [4]{a} \left (-48 \sqrt [4]{a} \sqrt [4]{c} d e+5 \sqrt{2} \sqrt{a} e^2+21 \sqrt{2} \sqrt{c} d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{c^{3/4}}+\frac{8 a x \left (7 d^2+12 d e x+5 e^2 x^2\right )}{a+c x^4}}{256 a^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^2/(a + c*x^4)^3,x]

[Out]

((32*a^2*x*(d + e*x)^2)/(a + c*x^4)^2 + (8*a*x*(7*d^2 + 12*d*e*x + 5*e^2*x^2))/(
a + c*x^4) - (2*a^(1/4)*(21*Sqrt[2]*Sqrt[c]*d^2 + 48*a^(1/4)*c^(1/4)*d*e + 5*Sqr
t[2]*Sqrt[a]*e^2)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/c^(3/4) + (2*a^(1/4)*
(21*Sqrt[2]*Sqrt[c]*d^2 - 48*a^(1/4)*c^(1/4)*d*e + 5*Sqrt[2]*Sqrt[a]*e^2)*ArcTan
[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/c^(3/4) + (Sqrt[2]*(-21*a^(1/4)*Sqrt[c]*d^2 +
5*a^(3/4)*e^2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/c^(3/4)
+ (Sqrt[2]*(21*a^(1/4)*Sqrt[c]*d^2 - 5*a^(3/4)*e^2)*Log[Sqrt[a] + Sqrt[2]*a^(1/4
)*c^(1/4)*x + Sqrt[c]*x^2])/c^(3/4))/(256*a^3)

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Maple [A]  time = 0.007, size = 419, normalized size = 1.2 \[{\frac{{d}^{2}x}{8\,a \left ( c{x}^{4}+a \right ) ^{2}}}+{\frac{7\,{d}^{2}x}{32\,{a}^{2} \left ( c{x}^{4}+a \right ) }}+{\frac{21\,{d}^{2}\sqrt{2}}{256\,{a}^{3}}\sqrt [4]{{\frac{a}{c}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }+{\frac{21\,{d}^{2}\sqrt{2}}{128\,{a}^{3}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }+{\frac{21\,{d}^{2}\sqrt{2}}{128\,{a}^{3}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }+{\frac{de{x}^{2}}{4\,a \left ( c{x}^{4}+a \right ) ^{2}}}+{\frac{3\,de{x}^{2}}{8\,{a}^{2} \left ( c{x}^{4}+a \right ) }}+{\frac{3\,de}{8\,{a}^{2}}\arctan \left ({x}^{2}\sqrt{{\frac{c}{a}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{{e}^{2}{x}^{3}}{8\,a \left ( c{x}^{4}+a \right ) ^{2}}}+{\frac{5\,{e}^{2}{x}^{3}}{32\,{a}^{2} \left ( c{x}^{4}+a \right ) }}+{\frac{5\,{e}^{2}\sqrt{2}}{256\,{a}^{2}c}\ln \left ({1 \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{5\,{e}^{2}\sqrt{2}}{128\,{a}^{2}c}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{5\,{e}^{2}\sqrt{2}}{128\,{a}^{2}c}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^2/(c*x^4+a)^3,x)

[Out]

1/8*d^2*x/a/(c*x^4+a)^2+7/32*d^2/a^2*x/(c*x^4+a)+21/256*d^2/a^3*(1/c*a)^(1/4)*2^
(1/2)*ln((x^2+(1/c*a)^(1/4)*x*2^(1/2)+(1/c*a)^(1/2))/(x^2-(1/c*a)^(1/4)*x*2^(1/2
)+(1/c*a)^(1/2)))+21/128*d^2/a^3*(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1
/4)*x+1)+21/128*d^2/a^3*(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x-1)+
1/4*d*e*x^2/a/(c*x^4+a)^2+3/8*d*e/a^2*x^2/(c*x^4+a)+3/8*d*e/a^2/(a*c)^(1/2)*arct
an(x^2*(c/a)^(1/2))+1/8*e^2*x^3/a/(c*x^4+a)^2+5/32*e^2/a^2*x^3/(c*x^4+a)+5/256*e
^2/a^2/c/(1/c*a)^(1/4)*2^(1/2)*ln((x^2-(1/c*a)^(1/4)*x*2^(1/2)+(1/c*a)^(1/2))/(x
^2+(1/c*a)^(1/4)*x*2^(1/2)+(1/c*a)^(1/2)))+5/128*e^2/a^2/c/(1/c*a)^(1/4)*2^(1/2)
*arctan(2^(1/2)/(1/c*a)^(1/4)*x+1)+5/128*e^2/a^2/c/(1/c*a)^(1/4)*2^(1/2)*arctan(
2^(1/2)/(1/c*a)^(1/4)*x-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((e*x + d)^2/(c*x^4 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^2/(c*x^4 + a)^3,x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 16.8138, size = 374, normalized size = 1.04 \[ \operatorname{RootSum}{\left (268435456 t^{4} a^{11} c^{3} + 25755648 t^{2} a^{6} c^{2} d^{2} e^{2} + t \left (307200 a^{4} c d e^{5} - 5419008 a^{3} c^{2} d^{5} e\right ) + 625 a^{2} e^{8} + 111906 a c d^{4} e^{4} + 194481 c^{2} d^{8}, \left ( t \mapsto t \log{\left (x + \frac{262144000 t^{3} a^{10} c^{2} e^{6} + 46110081024 t^{3} a^{9} c^{3} d^{4} e^{2} - 1645608960 t^{2} a^{7} c^{2} d^{3} e^{5} + 3641573376 t^{2} a^{6} c^{3} d^{7} e + 32688000 t a^{5} c d^{2} e^{8} + 3128219136 t a^{4} c^{2} d^{6} e^{4} + 522764928 t a^{3} c^{3} d^{10} + 225000 a^{3} d e^{11} - 43338240 a^{2} c d^{5} e^{7} - 523431720 a c^{2} d^{9} e^{3}}{15625 a^{3} e^{12} - 21357225 a^{2} c d^{4} e^{8} - 376741449 a c^{2} d^{8} e^{4} + 85766121 c^{3} d^{12}} \right )} \right )\right )} + \frac{11 a d^{2} x + 20 a d e x^{2} + 9 a e^{2} x^{3} + 7 c d^{2} x^{5} + 12 c d e x^{6} + 5 c e^{2} x^{7}}{32 a^{4} + 64 a^{3} c x^{4} + 32 a^{2} c^{2} x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**2/(c*x**4+a)**3,x)

[Out]

RootSum(268435456*_t**4*a**11*c**3 + 25755648*_t**2*a**6*c**2*d**2*e**2 + _t*(30
7200*a**4*c*d*e**5 - 5419008*a**3*c**2*d**5*e) + 625*a**2*e**8 + 111906*a*c*d**4
*e**4 + 194481*c**2*d**8, Lambda(_t, _t*log(x + (262144000*_t**3*a**10*c**2*e**6
+ 46110081024*_t**3*a**9*c**3*d**4*e**2 - 1645608960*_t**2*a**7*c**2*d**3*e**5
+ 3641573376*_t**2*a**6*c**3*d**7*e + 32688000*_t*a**5*c*d**2*e**8 + 3128219136*
_t*a**4*c**2*d**6*e**4 + 522764928*_t*a**3*c**3*d**10 + 225000*a**3*d*e**11 - 43
338240*a**2*c*d**5*e**7 - 523431720*a*c**2*d**9*e**3)/(15625*a**3*e**12 - 213572
25*a**2*c*d**4*e**8 - 376741449*a*c**2*d**8*e**4 + 85766121*c**3*d**12)))) + (11
*a*d**2*x + 20*a*d*e*x**2 + 9*a*e**2*x**3 + 7*c*d**2*x**5 + 12*c*d*e*x**6 + 5*c*
e**2*x**7)/(32*a**4 + 64*a**3*c*x**4 + 32*a**2*c**2*x**8)

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GIAC/XCAS [A]  time = 0.274006, size = 481, normalized size = 1.34 \[ \frac{5 \, c x^{7} e^{2} + 12 \, c d x^{6} e + 7 \, c d^{2} x^{5} + 9 \, a x^{3} e^{2} + 20 \, a d x^{2} e + 11 \, a d^{2} x}{32 \,{\left (c x^{4} + a\right )}^{2} a^{2}} + \frac{\sqrt{2}{\left (24 \, \sqrt{2} \sqrt{a c} c^{2} d e + 21 \, \left (a c^{3}\right )^{\frac{1}{4}} c^{2} d^{2} + 5 \, \left (a c^{3}\right )^{\frac{3}{4}} e^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{128 \, a^{3} c^{3}} + \frac{\sqrt{2}{\left (24 \, \sqrt{2} \sqrt{a c} c^{2} d e + 21 \, \left (a c^{3}\right )^{\frac{1}{4}} c^{2} d^{2} + 5 \, \left (a c^{3}\right )^{\frac{3}{4}} e^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{128 \, a^{3} c^{3}} + \frac{\sqrt{2}{\left (21 \, \left (a c^{3}\right )^{\frac{1}{4}} c^{2} d^{2} - 5 \, \left (a c^{3}\right )^{\frac{3}{4}} e^{2}\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{256 \, a^{3} c^{3}} - \frac{\sqrt{2}{\left (21 \, \left (a c^{3}\right )^{\frac{1}{4}} c^{2} d^{2} - 5 \, \left (a c^{3}\right )^{\frac{3}{4}} e^{2}\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{256 \, a^{3} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^2/(c*x^4 + a)^3,x, algorithm="giac")

[Out]

1/32*(5*c*x^7*e^2 + 12*c*d*x^6*e + 7*c*d^2*x^5 + 9*a*x^3*e^2 + 20*a*d*x^2*e + 11
*a*d^2*x)/((c*x^4 + a)^2*a^2) + 1/128*sqrt(2)*(24*sqrt(2)*sqrt(a*c)*c^2*d*e + 21
*(a*c^3)^(1/4)*c^2*d^2 + 5*(a*c^3)^(3/4)*e^2)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*
(a/c)^(1/4))/(a/c)^(1/4))/(a^3*c^3) + 1/128*sqrt(2)*(24*sqrt(2)*sqrt(a*c)*c^2*d*
e + 21*(a*c^3)^(1/4)*c^2*d^2 + 5*(a*c^3)^(3/4)*e^2)*arctan(1/2*sqrt(2)*(2*x - sq
rt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a^3*c^3) + 1/256*sqrt(2)*(21*(a*c^3)^(1/4)*c^2*
d^2 - 5*(a*c^3)^(3/4)*e^2)*ln(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a^3*c^3)
- 1/256*sqrt(2)*(21*(a*c^3)^(1/4)*c^2*d^2 - 5*(a*c^3)^(3/4)*e^2)*ln(x^2 - sqrt(
2)*x*(a/c)^(1/4) + sqrt(a/c))/(a^3*c^3)