3.705 \(\int \frac{\sqrt{d x}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx\)

Optimal. Leaf size=335 \[ \frac{15 \sqrt{d} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} a^{13/4} b^{3/4}}-\frac{15 \sqrt{d} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} a^{13/4} b^{3/4}}-\frac{15 \sqrt{d} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} a^{13/4} b^{3/4}}+\frac{15 \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{128 \sqrt{2} a^{13/4} b^{3/4}}+\frac{15 (d x)^{3/2}}{64 a^3 d \left (a+b x^2\right )}+\frac{3 (d x)^{3/2}}{16 a^2 d \left (a+b x^2\right )^2}+\frac{(d x)^{3/2}}{6 a d \left (a+b x^2\right )^3} \]

[Out]

(d*x)^(3/2)/(6*a*d*(a + b*x^2)^3) + (3*(d*x)^(3/2))/(16*a^2*d*(a + b*x^2)^2) + (
15*(d*x)^(3/2))/(64*a^3*d*(a + b*x^2)) - (15*Sqrt[d]*ArcTan[1 - (Sqrt[2]*b^(1/4)
*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(128*Sqrt[2]*a^(13/4)*b^(3/4)) + (15*Sqrt[d]*Arc
Tan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(128*Sqrt[2]*a^(13/4)*b^
(3/4)) + (15*Sqrt[d]*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x - Sqrt[2]*a^(1/4)*b
^(1/4)*Sqrt[d*x]])/(256*Sqrt[2]*a^(13/4)*b^(3/4)) - (15*Sqrt[d]*Log[Sqrt[a]*Sqrt
[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(256*Sqrt[2]*a^(13
/4)*b^(3/4))

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Rubi [A]  time = 0.729592, antiderivative size = 335, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.321 \[ \frac{15 \sqrt{d} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} a^{13/4} b^{3/4}}-\frac{15 \sqrt{d} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} a^{13/4} b^{3/4}}-\frac{15 \sqrt{d} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} a^{13/4} b^{3/4}}+\frac{15 \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{128 \sqrt{2} a^{13/4} b^{3/4}}+\frac{15 (d x)^{3/2}}{64 a^3 d \left (a+b x^2\right )}+\frac{3 (d x)^{3/2}}{16 a^2 d \left (a+b x^2\right )^2}+\frac{(d x)^{3/2}}{6 a d \left (a+b x^2\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

(d*x)^(3/2)/(6*a*d*(a + b*x^2)^3) + (3*(d*x)^(3/2))/(16*a^2*d*(a + b*x^2)^2) + (
15*(d*x)^(3/2))/(64*a^3*d*(a + b*x^2)) - (15*Sqrt[d]*ArcTan[1 - (Sqrt[2]*b^(1/4)
*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(128*Sqrt[2]*a^(13/4)*b^(3/4)) + (15*Sqrt[d]*Arc
Tan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(128*Sqrt[2]*a^(13/4)*b^
(3/4)) + (15*Sqrt[d]*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x - Sqrt[2]*a^(1/4)*b
^(1/4)*Sqrt[d*x]])/(256*Sqrt[2]*a^(13/4)*b^(3/4)) - (15*Sqrt[d]*Log[Sqrt[a]*Sqrt
[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(256*Sqrt[2]*a^(13
/4)*b^(3/4))

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Rubi in Sympy [A]  time = 147.626, size = 309, normalized size = 0.92 \[ \frac{\left (d x\right )^{\frac{3}{2}}}{6 a d \left (a + b x^{2}\right )^{3}} + \frac{3 \left (d x\right )^{\frac{3}{2}}}{16 a^{2} d \left (a + b x^{2}\right )^{2}} + \frac{15 \left (d x\right )^{\frac{3}{2}}}{64 a^{3} d \left (a + b x^{2}\right )} + \frac{15 \sqrt{2} \sqrt{d} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d} \sqrt{d x} + \sqrt{a} d + \sqrt{b} d x \right )}}{512 a^{\frac{13}{4}} b^{\frac{3}{4}}} - \frac{15 \sqrt{2} \sqrt{d} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d} \sqrt{d x} + \sqrt{a} d + \sqrt{b} d x \right )}}{512 a^{\frac{13}{4}} b^{\frac{3}{4}}} - \frac{15 \sqrt{2} \sqrt{d} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}} \right )}}{256 a^{\frac{13}{4}} b^{\frac{3}{4}}} + \frac{15 \sqrt{2} \sqrt{d} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}} \right )}}{256 a^{\frac{13}{4}} b^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(d*x)**(3/2)/(6*a*d*(a + b*x**2)**3) + 3*(d*x)**(3/2)/(16*a**2*d*(a + b*x**2)**2
) + 15*(d*x)**(3/2)/(64*a**3*d*(a + b*x**2)) + 15*sqrt(2)*sqrt(d)*log(-sqrt(2)*a
**(1/4)*b**(1/4)*sqrt(d)*sqrt(d*x) + sqrt(a)*d + sqrt(b)*d*x)/(512*a**(13/4)*b**
(3/4)) - 15*sqrt(2)*sqrt(d)*log(sqrt(2)*a**(1/4)*b**(1/4)*sqrt(d)*sqrt(d*x) + sq
rt(a)*d + sqrt(b)*d*x)/(512*a**(13/4)*b**(3/4)) - 15*sqrt(2)*sqrt(d)*atan(1 - sq
rt(2)*b**(1/4)*sqrt(d*x)/(a**(1/4)*sqrt(d)))/(256*a**(13/4)*b**(3/4)) + 15*sqrt(
2)*sqrt(d)*atan(1 + sqrt(2)*b**(1/4)*sqrt(d*x)/(a**(1/4)*sqrt(d)))/(256*a**(13/4
)*b**(3/4))

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Mathematica [A]  time = 0.260261, size = 253, normalized size = 0.76 \[ \frac{\sqrt{d x} \left (\frac{256 a^{9/4} x^{3/2}}{\left (a+b x^2\right )^3}+\frac{288 a^{5/4} x^{3/2}}{\left (a+b x^2\right )^2}+\frac{45 \sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{b^{3/4}}-\frac{45 \sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{b^{3/4}}-\frac{90 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{b^{3/4}}+\frac{90 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{b^{3/4}}+\frac{360 \sqrt [4]{a} x^{3/2}}{a+b x^2}\right )}{1536 a^{13/4} \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d*x]*((256*a^(9/4)*x^(3/2))/(a + b*x^2)^3 + (288*a^(5/4)*x^(3/2))/(a + b*x
^2)^2 + (360*a^(1/4)*x^(3/2))/(a + b*x^2) - (90*Sqrt[2]*ArcTan[1 - (Sqrt[2]*b^(1
/4)*Sqrt[x])/a^(1/4)])/b^(3/4) + (90*Sqrt[2]*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x]
)/a^(1/4)])/b^(3/4) + (45*Sqrt[2]*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x]
+ Sqrt[b]*x])/b^(3/4) - (45*Sqrt[2]*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x
] + Sqrt[b]*x])/b^(3/4)))/(1536*a^(13/4)*Sqrt[x])

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Maple [A]  time = 0.026, size = 272, normalized size = 0.8 \[{\frac{15\,{b}^{2}d}{64\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}{a}^{3}} \left ( dx \right ) ^{{\frac{11}{2}}}}+{\frac{21\,{d}^{3}b}{32\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}{a}^{2}} \left ( dx \right ) ^{{\frac{7}{2}}}}+{\frac{113\,{d}^{5}}{192\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}a} \left ( dx \right ) ^{{\frac{3}{2}}}}+{\frac{15\,d\sqrt{2}}{512\,{a}^{3}b}\ln \left ({1 \left ( dx-\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) \left ( dx+\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+{\frac{15\,d\sqrt{2}}{256\,{a}^{3}b}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+{\frac{15\,d\sqrt{2}}{256\,{a}^{3}b}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

15/64*d/(b*d^2*x^2+a*d^2)^3/a^3*b^2*(d*x)^(11/2)+21/32*d^3/(b*d^2*x^2+a*d^2)^3/a
^2*b*(d*x)^(7/2)+113/192*d^5/(b*d^2*x^2+a*d^2)^3/a*(d*x)^(3/2)+15/512*d/a^3/b/(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)))+15/256*d/a^3/b/(a*
d^2/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a*d^2/b)^(1/4)*(d*x)^(1/2)+1)+15/256*d/a^3/
b/(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(sqrt(d*x)/(b^2*x^4 + 2*a*b*x^2 + a^2)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.288892, size = 460, normalized size = 1.37 \[ \frac{180 \,{\left (a^{3} b^{3} x^{6} + 3 \, a^{4} b^{2} x^{4} + 3 \, a^{5} b x^{2} + a^{6}\right )} \left (-\frac{d^{2}}{a^{13} b^{3}}\right )^{\frac{1}{4}} \arctan \left (\frac{3375 \, a^{10} b^{2} \left (-\frac{d^{2}}{a^{13} b^{3}}\right )^{\frac{3}{4}}}{3375 \, \sqrt{d x} d + \sqrt{-11390625 \, a^{7} b d^{2} \sqrt{-\frac{d^{2}}{a^{13} b^{3}}} + 11390625 \, d^{3} x}}\right ) + 45 \,{\left (a^{3} b^{3} x^{6} + 3 \, a^{4} b^{2} x^{4} + 3 \, a^{5} b x^{2} + a^{6}\right )} \left (-\frac{d^{2}}{a^{13} b^{3}}\right )^{\frac{1}{4}} \log \left (3375 \, a^{10} b^{2} \left (-\frac{d^{2}}{a^{13} b^{3}}\right )^{\frac{3}{4}} + 3375 \, \sqrt{d x} d\right ) - 45 \,{\left (a^{3} b^{3} x^{6} + 3 \, a^{4} b^{2} x^{4} + 3 \, a^{5} b x^{2} + a^{6}\right )} \left (-\frac{d^{2}}{a^{13} b^{3}}\right )^{\frac{1}{4}} \log \left (-3375 \, a^{10} b^{2} \left (-\frac{d^{2}}{a^{13} b^{3}}\right )^{\frac{3}{4}} + 3375 \, \sqrt{d x} d\right ) + 4 \,{\left (45 \, b^{2} x^{5} + 126 \, a b x^{3} + 113 \, a^{2} x\right )} \sqrt{d x}}{768 \,{\left (a^{3} b^{3} x^{6} + 3 \, a^{4} b^{2} x^{4} + 3 \, a^{5} b x^{2} + a^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/768*(180*(a^3*b^3*x^6 + 3*a^4*b^2*x^4 + 3*a^5*b*x^2 + a^6)*(-d^2/(a^13*b^3))^(
1/4)*arctan(3375*a^10*b^2*(-d^2/(a^13*b^3))^(3/4)/(3375*sqrt(d*x)*d + sqrt(-1139
0625*a^7*b*d^2*sqrt(-d^2/(a^13*b^3)) + 11390625*d^3*x))) + 45*(a^3*b^3*x^6 + 3*a
^4*b^2*x^4 + 3*a^5*b*x^2 + a^6)*(-d^2/(a^13*b^3))^(1/4)*log(3375*a^10*b^2*(-d^2/
(a^13*b^3))^(3/4) + 3375*sqrt(d*x)*d) - 45*(a^3*b^3*x^6 + 3*a^4*b^2*x^4 + 3*a^5*
b*x^2 + a^6)*(-d^2/(a^13*b^3))^(1/4)*log(-3375*a^10*b^2*(-d^2/(a^13*b^3))^(3/4)
+ 3375*sqrt(d*x)*d) + 4*(45*b^2*x^5 + 126*a*b*x^3 + 113*a^2*x)*sqrt(d*x))/(a^3*b
^3*x^6 + 3*a^4*b^2*x^4 + 3*a^5*b*x^2 + a^6)

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Sympy [A]  time = 136.218, size = 252, normalized size = 0.75 \[ \frac{226 a^{2} d^{11} \left (d x\right )^{\frac{3}{2}}}{384 a^{6} d^{12} + 1152 a^{5} b d^{12} x^{2} + 1152 a^{4} b^{2} d^{12} x^{4} + 384 a^{3} b^{3} d^{12} x^{6}} + \frac{252 a b d^{9} \left (d x\right )^{\frac{7}{2}}}{384 a^{6} d^{12} + 1152 a^{5} b d^{12} x^{2} + 1152 a^{4} b^{2} d^{12} x^{4} + 384 a^{3} b^{3} d^{12} x^{6}} + \frac{90 b^{2} d^{7} \left (d x\right )^{\frac{11}{2}}}{384 a^{6} d^{12} + 1152 a^{5} b d^{12} x^{2} + 1152 a^{4} b^{2} d^{12} x^{4} + 384 a^{3} b^{3} d^{12} x^{6}} + 2 d^{7} \operatorname{RootSum}{\left (68719476736 t^{4} a^{13} b^{3} d^{26} + 50625, \left ( t \mapsto t \log{\left (\frac{134217728 t^{3} a^{10} b^{2} d^{20}}{3375} + \sqrt{d x} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

226*a**2*d**11*(d*x)**(3/2)/(384*a**6*d**12 + 1152*a**5*b*d**12*x**2 + 1152*a**4
*b**2*d**12*x**4 + 384*a**3*b**3*d**12*x**6) + 252*a*b*d**9*(d*x)**(7/2)/(384*a*
*6*d**12 + 1152*a**5*b*d**12*x**2 + 1152*a**4*b**2*d**12*x**4 + 384*a**3*b**3*d*
*12*x**6) + 90*b**2*d**7*(d*x)**(11/2)/(384*a**6*d**12 + 1152*a**5*b*d**12*x**2
+ 1152*a**4*b**2*d**12*x**4 + 384*a**3*b**3*d**12*x**6) + 2*d**7*RootSum(6871947
6736*_t**4*a**13*b**3*d**26 + 50625, Lambda(_t, _t*log(134217728*_t**3*a**10*b**
2*d**20/3375 + sqrt(d*x))))

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GIAC/XCAS [A]  time = 0.278439, size = 417, normalized size = 1.24 \[ \frac{45 \, \sqrt{d x} b^{2} d^{6} x^{5} + 126 \, \sqrt{d x} a b d^{6} x^{3} + 113 \, \sqrt{d x} a^{2} d^{6} x}{192 \,{\left (b d^{2} x^{2} + a d^{2}\right )}^{3} a^{3}} + \frac{15 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{256 \, a^{4} b^{3} d} + \frac{15 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{256 \, a^{4} b^{3} d} - \frac{15 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}}{\rm ln}\left (d x + \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{512 \, a^{4} b^{3} d} + \frac{15 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}}{\rm ln}\left (d x - \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{512 \, a^{4} b^{3} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/192*(45*sqrt(d*x)*b^2*d^6*x^5 + 126*sqrt(d*x)*a*b*d^6*x^3 + 113*sqrt(d*x)*a^2*
d^6*x)/((b*d^2*x^2 + a*d^2)^3*a^3) + 15/256*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^4*b^3*d) +
15/256*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^4*b^3*d) - 15/512*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^4*b^3*d) + 15/512*sq
rt(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^4*b^3*d)