3.727 \(\int \frac{1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx\)

Optimal. Leaf size=422 \[ \frac{69615 b^{5/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{29/4} d^{7/2}}-\frac{69615 b^{5/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{29/4} d^{7/2}}-\frac{69615 b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{29/4} d^{7/2}}+\frac{69615 b^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{8192 \sqrt{2} a^{29/4} d^{7/2}}+\frac{69615 b}{4096 a^7 d^3 \sqrt{d x}}-\frac{13923}{4096 a^6 d (d x)^{5/2}}+\frac{7735}{4096 a^5 d (d x)^{5/2} \left (a+b x^2\right )}+\frac{595}{1024 a^4 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac{35}{128 a^3 d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac{5}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^4}+\frac{1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5} \]

[Out]

-13923/(4096*a^6*d*(d*x)^(5/2)) + (69615*b)/(4096*a^7*d^3*Sqrt[d*x]) + 1/(10*a*d
*(d*x)^(5/2)*(a + b*x^2)^5) + 5/(32*a^2*d*(d*x)^(5/2)*(a + b*x^2)^4) + 35/(128*a
^3*d*(d*x)^(5/2)*(a + b*x^2)^3) + 595/(1024*a^4*d*(d*x)^(5/2)*(a + b*x^2)^2) + 7
735/(4096*a^5*d*(d*x)^(5/2)*(a + b*x^2)) - (69615*b^(5/4)*ArcTan[1 - (Sqrt[2]*b^
(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(29/4)*d^(7/2)) + (69615*b^
(5/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a
^(29/4)*d^(7/2)) + (69615*b^(5/4)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x - Sqrt
[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(16384*Sqrt[2]*a^(29/4)*d^(7/2)) - (69615*b^(5/4
)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/
(16384*Sqrt[2]*a^(29/4)*d^(7/2))

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Rubi [A]  time = 1.1871, antiderivative size = 422, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 10, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357 \[ \frac{69615 b^{5/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{29/4} d^{7/2}}-\frac{69615 b^{5/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{29/4} d^{7/2}}-\frac{69615 b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{29/4} d^{7/2}}+\frac{69615 b^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{8192 \sqrt{2} a^{29/4} d^{7/2}}+\frac{69615 b}{4096 a^7 d^3 \sqrt{d x}}-\frac{13923}{4096 a^6 d (d x)^{5/2}}+\frac{7735}{4096 a^5 d (d x)^{5/2} \left (a+b x^2\right )}+\frac{595}{1024 a^4 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac{35}{128 a^3 d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac{5}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^4}+\frac{1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

[In]  Int[1/((d*x)^(7/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^3),x]

[Out]

-13923/(4096*a^6*d*(d*x)^(5/2)) + (69615*b)/(4096*a^7*d^3*Sqrt[d*x]) + 1/(10*a*d
*(d*x)^(5/2)*(a + b*x^2)^5) + 5/(32*a^2*d*(d*x)^(5/2)*(a + b*x^2)^4) + 35/(128*a
^3*d*(d*x)^(5/2)*(a + b*x^2)^3) + 595/(1024*a^4*d*(d*x)^(5/2)*(a + b*x^2)^2) + 7
735/(4096*a^5*d*(d*x)^(5/2)*(a + b*x^2)) - (69615*b^(5/4)*ArcTan[1 - (Sqrt[2]*b^
(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(29/4)*d^(7/2)) + (69615*b^
(5/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a
^(29/4)*d^(7/2)) + (69615*b^(5/4)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x - Sqrt
[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(16384*Sqrt[2]*a^(29/4)*d^(7/2)) - (69615*b^(5/4
)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/
(16384*Sqrt[2]*a^(29/4)*d^(7/2))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(d*x)**(7/2)/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 0.419401, size = 339, normalized size = 0.8 \[ \frac{\sqrt{d x} \left (\frac{16384 a^{17/4} b^2 x^4}{\left (a+b x^2\right )^5}+\frac{58368 a^{13/4} b^2 x^4}{\left (a+b x^2\right )^4}+\frac{145152 a^{9/4} b^2 x^4}{\left (a+b x^2\right )^3}+\frac{327136 a^{5/4} b^2 x^4}{\left (a+b x^2\right )^2}-65536 a^{5/4}+348075 \sqrt{2} b^{5/4} x^{5/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-348075 \sqrt{2} b^{5/4} x^{5/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-696150 \sqrt{2} b^{5/4} x^{5/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )+696150 \sqrt{2} b^{5/4} x^{5/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )+\frac{818520 \sqrt [4]{a} b^2 x^4}{a+b x^2}+1966080 \sqrt [4]{a} b x^2\right )}{163840 a^{29/4} d^4 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d*x)^(7/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^3),x]

[Out]

(Sqrt[d*x]*(-65536*a^(5/4) + 1966080*a^(1/4)*b*x^2 + (16384*a^(17/4)*b^2*x^4)/(a
+ b*x^2)^5 + (58368*a^(13/4)*b^2*x^4)/(a + b*x^2)^4 + (145152*a^(9/4)*b^2*x^4)/
(a + b*x^2)^3 + (327136*a^(5/4)*b^2*x^4)/(a + b*x^2)^2 + (818520*a^(1/4)*b^2*x^4
)/(a + b*x^2) - 696150*Sqrt[2]*b^(5/4)*x^(5/2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[
x])/a^(1/4)] + 696150*Sqrt[2]*b^(5/4)*x^(5/2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x
])/a^(1/4)] + 348075*Sqrt[2]*b^(5/4)*x^(5/2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/
4)*Sqrt[x] + Sqrt[b]*x] - 348075*Sqrt[2]*b^(5/4)*x^(5/2)*Log[Sqrt[a] + Sqrt[2]*a
^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]))/(163840*a^(29/4)*d^4*x^3)

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Maple [A]  time = 0.043, size = 368, normalized size = 0.9 \[ -{\frac{2}{5\,{a}^{6}d} \left ( dx \right ) ^{-{\frac{5}{2}}}}+12\,{\frac{b}{{a}^{7}{d}^{3}\sqrt{dx}}}+{\frac{34139\,{d}^{5}{b}^{2}}{4096\,{a}^{3} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}} \left ( dx \right ) ^{{\frac{3}{2}}}}+{\frac{3597\,{d}^{3}{b}^{3}}{128\,{a}^{4} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}} \left ( dx \right ) ^{{\frac{7}{2}}}}+{\frac{75471\,{b}^{4}d}{2048\,{a}^{5} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}} \left ( dx \right ) ^{{\frac{11}{2}}}}+{\frac{56269\,{b}^{5}}{2560\,{a}^{6}d \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}} \left ( dx \right ) ^{{\frac{15}{2}}}}+{\frac{20463\,{b}^{6}}{4096\,{a}^{7}{d}^{3} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}} \left ( dx \right ) ^{{\frac{19}{2}}}}+{\frac{69615\,b\sqrt{2}}{32768\,{a}^{7}{d}^{3}}\ln \left ({1 \left ( dx-\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) \left ( dx+\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+{\frac{69615\,b\sqrt{2}}{16384\,{a}^{7}{d}^{3}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+{\frac{69615\,b\sqrt{2}}{16384\,{a}^{7}{d}^{3}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(d*x)^(7/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

-2/5/a^6/d/(d*x)^(5/2)+12*b/a^7/d^3/(d*x)^(1/2)+34139/4096*d^5*b^2/a^3/(b*d^2*x^
2+a*d^2)^5*(d*x)^(3/2)+3597/128*d^3*b^3/a^4/(b*d^2*x^2+a*d^2)^5*(d*x)^(7/2)+7547
1/2048*d*b^4/a^5/(b*d^2*x^2+a*d^2)^5*(d*x)^(11/2)+56269/2560/d*b^5/a^6/(b*d^2*x^
2+a*d^2)^5*(d*x)^(15/2)+20463/4096/d^3*b^6/a^7/(b*d^2*x^2+a*d^2)^5*(d*x)^(19/2)+
69615/32768/d^3*b/a^7/(a*d^2/b)^(1/4)*2^(1/2)*ln((d*x-(a*d^2/b)^(1/4)*(d*x)^(1/2
)*2^(1/2)+(a*d^2/b)^(1/2))/(d*x+(a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)+(a*d^2/b)^(1
/2)))+69615/16384/d^3*b/a^7/(a*d^2/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a*d^2/b)^(1/
4)*(d*x)^(1/2)+1)+69615/16384/d^3*b/a^7/(a*d^2/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(
a*d^2/b)^(1/4)*(d*x)^(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/((b^2*x^4 + 2*a*b*x^2 + a^2)^3*(d*x)^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.342662, size = 784, normalized size = 1.86 \[ \frac{1392300 \, b^{6} x^{12} + 6683040 \, a b^{5} x^{10} + 12685400 \, a^{2} b^{4} x^{8} + 11804800 \, a^{3} b^{3} x^{6} + 5270300 \, a^{4} b^{2} x^{4} + 819200 \, a^{5} b x^{2} - 32768 \, a^{6} + 1392300 \,{\left (a^{7} b^{5} d^{3} x^{12} + 5 \, a^{8} b^{4} d^{3} x^{10} + 10 \, a^{9} b^{3} d^{3} x^{8} + 10 \, a^{10} b^{2} d^{3} x^{6} + 5 \, a^{11} b d^{3} x^{4} + a^{12} d^{3} x^{2}\right )} \sqrt{d x} \left (-\frac{b^{5}}{a^{29} d^{14}}\right )^{\frac{1}{4}} \arctan \left (\frac{337371570183375 \, a^{22} d^{11} \left (-\frac{b^{5}}{a^{29} d^{14}}\right )^{\frac{3}{4}}}{337371570183375 \, \sqrt{d x} b^{4} + \sqrt{-113819576367995923331126390625 \, a^{15} b^{5} d^{8} \sqrt{-\frac{b^{5}}{a^{29} d^{14}}} + 113819576367995923331126390625 \, b^{8} d x}}\right ) + 348075 \,{\left (a^{7} b^{5} d^{3} x^{12} + 5 \, a^{8} b^{4} d^{3} x^{10} + 10 \, a^{9} b^{3} d^{3} x^{8} + 10 \, a^{10} b^{2} d^{3} x^{6} + 5 \, a^{11} b d^{3} x^{4} + a^{12} d^{3} x^{2}\right )} \sqrt{d x} \left (-\frac{b^{5}}{a^{29} d^{14}}\right )^{\frac{1}{4}} \log \left (337371570183375 \, a^{22} d^{11} \left (-\frac{b^{5}}{a^{29} d^{14}}\right )^{\frac{3}{4}} + 337371570183375 \, \sqrt{d x} b^{4}\right ) - 348075 \,{\left (a^{7} b^{5} d^{3} x^{12} + 5 \, a^{8} b^{4} d^{3} x^{10} + 10 \, a^{9} b^{3} d^{3} x^{8} + 10 \, a^{10} b^{2} d^{3} x^{6} + 5 \, a^{11} b d^{3} x^{4} + a^{12} d^{3} x^{2}\right )} \sqrt{d x} \left (-\frac{b^{5}}{a^{29} d^{14}}\right )^{\frac{1}{4}} \log \left (-337371570183375 \, a^{22} d^{11} \left (-\frac{b^{5}}{a^{29} d^{14}}\right )^{\frac{3}{4}} + 337371570183375 \, \sqrt{d x} b^{4}\right )}{81920 \,{\left (a^{7} b^{5} d^{3} x^{12} + 5 \, a^{8} b^{4} d^{3} x^{10} + 10 \, a^{9} b^{3} d^{3} x^{8} + 10 \, a^{10} b^{2} d^{3} x^{6} + 5 \, a^{11} b d^{3} x^{4} + a^{12} d^{3} x^{2}\right )} \sqrt{d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^4 + 2*a*b*x^2 + a^2)^3*(d*x)^(7/2)),x, algorithm="fricas")

[Out]

1/81920*(1392300*b^6*x^12 + 6683040*a*b^5*x^10 + 12685400*a^2*b^4*x^8 + 11804800
*a^3*b^3*x^6 + 5270300*a^4*b^2*x^4 + 819200*a^5*b*x^2 - 32768*a^6 + 1392300*(a^7
*b^5*d^3*x^12 + 5*a^8*b^4*d^3*x^10 + 10*a^9*b^3*d^3*x^8 + 10*a^10*b^2*d^3*x^6 +
5*a^11*b*d^3*x^4 + a^12*d^3*x^2)*sqrt(d*x)*(-b^5/(a^29*d^14))^(1/4)*arctan(33737
1570183375*a^22*d^11*(-b^5/(a^29*d^14))^(3/4)/(337371570183375*sqrt(d*x)*b^4 + s
qrt(-113819576367995923331126390625*a^15*b^5*d^8*sqrt(-b^5/(a^29*d^14)) + 113819
576367995923331126390625*b^8*d*x))) + 348075*(a^7*b^5*d^3*x^12 + 5*a^8*b^4*d^3*x
^10 + 10*a^9*b^3*d^3*x^8 + 10*a^10*b^2*d^3*x^6 + 5*a^11*b*d^3*x^4 + a^12*d^3*x^2
)*sqrt(d*x)*(-b^5/(a^29*d^14))^(1/4)*log(337371570183375*a^22*d^11*(-b^5/(a^29*d
^14))^(3/4) + 337371570183375*sqrt(d*x)*b^4) - 348075*(a^7*b^5*d^3*x^12 + 5*a^8*
b^4*d^3*x^10 + 10*a^9*b^3*d^3*x^8 + 10*a^10*b^2*d^3*x^6 + 5*a^11*b*d^3*x^4 + a^1
2*d^3*x^2)*sqrt(d*x)*(-b^5/(a^29*d^14))^(1/4)*log(-337371570183375*a^22*d^11*(-b
^5/(a^29*d^14))^(3/4) + 337371570183375*sqrt(d*x)*b^4))/((a^7*b^5*d^3*x^12 + 5*a
^8*b^4*d^3*x^10 + 10*a^9*b^3*d^3*x^8 + 10*a^10*b^2*d^3*x^6 + 5*a^11*b*d^3*x^4 +
a^12*d^3*x^2)*sqrt(d*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(1/(d*x)**(7/2)/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.279109, size = 489, normalized size = 1.16 \[ \frac{69615 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{16384 \, a^{8} b d^{5}} + \frac{69615 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{16384 \, a^{8} b d^{5}} - \frac{69615 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}}{\rm ln}\left (d x + \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{32768 \, a^{8} b d^{5}} + \frac{69615 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}}{\rm ln}\left (d x - \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{32768 \, a^{8} b d^{5}} + \frac{348075 \, b^{6} d^{12} x^{12} + 1670760 \, a b^{5} d^{12} x^{10} + 3171350 \, a^{2} b^{4} d^{12} x^{8} + 2951200 \, a^{3} b^{3} d^{12} x^{6} + 1317575 \, a^{4} b^{2} d^{12} x^{4} + 204800 \, a^{5} b d^{12} x^{2} - 8192 \, a^{6} d^{12}}{20480 \,{\left (\sqrt{d x} b d^{2} x^{2} + \sqrt{d x} a d^{2}\right )}^{5} a^{7} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^4 + 2*a*b*x^2 + a^2)^3*(d*x)^(7/2)),x, algorithm="giac")

[Out]

69615/16384*sqrt(2)*(a*b^3*d^2)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4
) + 2*sqrt(d*x))/(a*d^2/b)^(1/4))/(a^8*b*d^5) + 69615/16384*sqrt(2)*(a*b^3*d^2)^
(3/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) - 2*sqrt(d*x))/(a*d^2/b)^(1/4
))/(a^8*b*d^5) - 69615/32768*sqrt(2)*(a*b^3*d^2)^(3/4)*ln(d*x + sqrt(2)*(a*d^2/b
)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(a^8*b*d^5) + 69615/32768*sqrt(2)*(a*b^3*d^2)
^(3/4)*ln(d*x - sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(a^8*b*d^5) +
1/20480*(348075*b^6*d^12*x^12 + 1670760*a*b^5*d^12*x^10 + 3171350*a^2*b^4*d^12*
x^8 + 2951200*a^3*b^3*d^12*x^6 + 1317575*a^4*b^2*d^12*x^4 + 204800*a^5*b*d^12*x^
2 - 8192*a^6*d^12)/((sqrt(d*x)*b*d^2*x^2 + sqrt(d*x)*a*d^2)^5*a^7*d^3)