3.774 \(\int \frac{(d x)^{13/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx\)

Optimal. Leaf size=557 \[ -\frac{11 d^3 (d x)^{7/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{11/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{77 d^5 (d x)^{3/2}}{1024 a b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{77 d^5 (d x)^{3/2}}{768 b^3 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{77 d^{13/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} a^{5/4} b^{15/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{77 d^{13/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} a^{5/4} b^{15/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{77 d^{13/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{2048 \sqrt{2} a^{5/4} b^{15/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{77 d^{13/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{2048 \sqrt{2} a^{5/4} b^{15/4} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

[Out]

(77*d^5*(d*x)^(3/2))/(1024*a*b^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (d*(d*x)^(11
/2))/(8*b*(a + b*x^2)^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (11*d^3*(d*x)^(7/2))/
(96*b^2*(a + b*x^2)^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (77*d^5*(d*x)^(3/2))/(7
68*b^3*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (77*d^(13/2)*(a + b*x^2)*A
rcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2]*a^(5/4)*
b^(15/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (77*d^(13/2)*(a + b*x^2)*ArcTan[1 +
(Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2]*a^(5/4)*b^(15/4)*S
qrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (77*d^(13/2)*(a + b*x^2)*Log[Sqrt[a]*Sqrt[d] +
Sqrt[b]*Sqrt[d]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(4096*Sqrt[2]*a^(5/4)*b
^(15/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (77*d^(13/2)*(a + b*x^2)*Log[Sqrt[a]*
Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(4096*Sqrt[2]*
a^(5/4)*b^(15/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])

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Rubi [A]  time = 0.955171, antiderivative size = 557, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{11 d^3 (d x)^{7/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{11/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{77 d^5 (d x)^{3/2}}{1024 a b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{77 d^5 (d x)^{3/2}}{768 b^3 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{77 d^{13/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} a^{5/4} b^{15/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{77 d^{13/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} a^{5/4} b^{15/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{77 d^{13/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{2048 \sqrt{2} a^{5/4} b^{15/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{77 d^{13/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{2048 \sqrt{2} a^{5/4} b^{15/4} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

[In]  Int[(d*x)^(13/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

(77*d^5*(d*x)^(3/2))/(1024*a*b^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (d*(d*x)^(11
/2))/(8*b*(a + b*x^2)^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (11*d^3*(d*x)^(7/2))/
(96*b^2*(a + b*x^2)^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (77*d^5*(d*x)^(3/2))/(7
68*b^3*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (77*d^(13/2)*(a + b*x^2)*A
rcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2]*a^(5/4)*
b^(15/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (77*d^(13/2)*(a + b*x^2)*ArcTan[1 +
(Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2]*a^(5/4)*b^(15/4)*S
qrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (77*d^(13/2)*(a + b*x^2)*Log[Sqrt[a]*Sqrt[d] +
Sqrt[b]*Sqrt[d]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(4096*Sqrt[2]*a^(5/4)*b
^(15/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (77*d^(13/2)*(a + b*x^2)*Log[Sqrt[a]*
Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(4096*Sqrt[2]*
a^(5/4)*b^(15/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])

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Rubi in Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x)**(13/2)/(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.353759, size = 324, normalized size = 0.58 \[ \frac{(d x)^{13/2} \left (a+b x^2\right ) \left (-3072 a^{13/4} b^{3/4} x^{3/2}-8352 a^{5/4} b^{3/4} x^{3/2} \left (a+b x^2\right )^2+8960 a^{9/4} b^{3/4} x^{3/2} \left (a+b x^2\right )+1848 \sqrt [4]{a} b^{3/4} x^{3/2} \left (a+b x^2\right )^3+231 \sqrt{2} \left (a+b x^2\right )^4 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-231 \sqrt{2} \left (a+b x^2\right )^4 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-462 \sqrt{2} \left (a+b x^2\right )^4 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )+462 \sqrt{2} \left (a+b x^2\right )^4 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )\right )}{24576 a^{5/4} b^{15/4} x^{13/2} \left (\left (a+b x^2\right )^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d*x)^(13/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

((d*x)^(13/2)*(a + b*x^2)*(-3072*a^(13/4)*b^(3/4)*x^(3/2) + 8960*a^(9/4)*b^(3/4)
*x^(3/2)*(a + b*x^2) - 8352*a^(5/4)*b^(3/4)*x^(3/2)*(a + b*x^2)^2 + 1848*a^(1/4)
*b^(3/4)*x^(3/2)*(a + b*x^2)^3 - 462*Sqrt[2]*(a + b*x^2)^4*ArcTan[1 - (Sqrt[2]*b
^(1/4)*Sqrt[x])/a^(1/4)] + 462*Sqrt[2]*(a + b*x^2)^4*ArcTan[1 + (Sqrt[2]*b^(1/4)
*Sqrt[x])/a^(1/4)] + 231*Sqrt[2]*(a + b*x^2)^4*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(
1/4)*Sqrt[x] + Sqrt[b]*x] - 231*Sqrt[2]*(a + b*x^2)^4*Log[Sqrt[a] + Sqrt[2]*a^(1
/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]))/(24576*a^(5/4)*b^(15/4)*x^(13/2)*((a + b*x^2)
^2)^(5/2))

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Maple [B]  time = 0.031, size = 1046, normalized size = 1.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^(13/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x)

[Out]

1/24576*(231*2^(1/2)*ln(-((a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)-d*x-(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)))*x^8*b^4*d^8+462*2^
(1/2)*arctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*x^8*b^4*d^8-
462*2^(1/2)*arctan((-2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*x^8*b
^4*d^8+1848*(a*d^2/b)^(1/4)*(d*x)^(15/2)*b^4+924*2^(1/2)*ln(-((a*d^2/b)^(1/4)*(d
*x)^(1/2)*2^(1/2)-d*x-(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)))*x^6*a*b^3*d^8+1848*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/
b)^(1/4))/(a*d^2/b)^(1/4))*x^6*a*b^3*d^8-1848*2^(1/2)*arctan((-2^(1/2)*(d*x)^(1/
2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*x^6*a*b^3*d^8-2808*(a*d^2/b)^(1/4)*(d*x)^(1
1/2)*a*b^3*d^2+1386*2^(1/2)*ln(-((a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)-d*x-(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)))*x^4*a^2*b^2
*d^8+2772*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*
x^4*a^2*b^2*d^8-2772*2^(1/2)*arctan((-2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^
2/b)^(1/4))*x^4*a^2*b^2*d^8-2200*(a*d^2/b)^(1/4)*(d*x)^(7/2)*a^2*b^2*d^4+924*2^(
1/2)*ln(-((a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)-d*x-(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)))*x^2*a^3*b*d^8+1848*2^(1/2)*arctan(
(2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*x^2*a^3*b*d^8-1848*2^(1/2
)*arctan((-2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*x^2*a^3*b*d^8-6
16*(a*d^2/b)^(1/4)*(d*x)^(3/2)*a^3*b*d^6+231*2^(1/2)*ln(-((a*d^2/b)^(1/4)*(d*x)^
(1/2)*2^(1/2)-d*x-(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)))*a^4*d^8+462*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(
a*d^2/b)^(1/4))*a^4*d^8-462*2^(1/2)*arctan((-2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4)
)/(a*d^2/b)^(1/4))*a^4*d^8)/d*(b*x^2+a)/(a*d^2/b)^(1/4)/b^4/a/((b*x^2+a)^2)^(5/2
)

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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((d*x)^(13/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.304049, size = 579, normalized size = 1.04 \[ \frac{924 \,{\left (a b^{7} x^{8} + 4 \, a^{2} b^{6} x^{6} + 6 \, a^{3} b^{5} x^{4} + 4 \, a^{4} b^{4} x^{2} + a^{5} b^{3}\right )} \left (-\frac{d^{26}}{a^{5} b^{15}}\right )^{\frac{1}{4}} \arctan \left (\frac{\left (-\frac{d^{26}}{a^{5} b^{15}}\right )^{\frac{3}{4}} a^{4} b^{11}}{\sqrt{d x} d^{19} + \sqrt{d^{39} x - \sqrt{-\frac{d^{26}}{a^{5} b^{15}}} a^{3} b^{7} d^{26}}}\right ) + 231 \,{\left (a b^{7} x^{8} + 4 \, a^{2} b^{6} x^{6} + 6 \, a^{3} b^{5} x^{4} + 4 \, a^{4} b^{4} x^{2} + a^{5} b^{3}\right )} \left (-\frac{d^{26}}{a^{5} b^{15}}\right )^{\frac{1}{4}} \log \left (456533 \, \sqrt{d x} d^{19} + 456533 \, \left (-\frac{d^{26}}{a^{5} b^{15}}\right )^{\frac{3}{4}} a^{4} b^{11}\right ) - 231 \,{\left (a b^{7} x^{8} + 4 \, a^{2} b^{6} x^{6} + 6 \, a^{3} b^{5} x^{4} + 4 \, a^{4} b^{4} x^{2} + a^{5} b^{3}\right )} \left (-\frac{d^{26}}{a^{5} b^{15}}\right )^{\frac{1}{4}} \log \left (456533 \, \sqrt{d x} d^{19} - 456533 \, \left (-\frac{d^{26}}{a^{5} b^{15}}\right )^{\frac{3}{4}} a^{4} b^{11}\right ) + 4 \,{\left (231 \, b^{3} d^{6} x^{7} - 351 \, a b^{2} d^{6} x^{5} - 275 \, a^{2} b d^{6} x^{3} - 77 \, a^{3} d^{6} x\right )} \sqrt{d x}}{12288 \,{\left (a b^{7} x^{8} + 4 \, a^{2} b^{6} x^{6} + 6 \, a^{3} b^{5} x^{4} + 4 \, a^{4} b^{4} x^{2} + a^{5} b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^(13/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2),x, algorithm="fricas")

[Out]

1/12288*(924*(a*b^7*x^8 + 4*a^2*b^6*x^6 + 6*a^3*b^5*x^4 + 4*a^4*b^4*x^2 + a^5*b^
3)*(-d^26/(a^5*b^15))^(1/4)*arctan((-d^26/(a^5*b^15))^(3/4)*a^4*b^11/(sqrt(d*x)*
d^19 + sqrt(d^39*x - sqrt(-d^26/(a^5*b^15))*a^3*b^7*d^26))) + 231*(a*b^7*x^8 + 4
*a^2*b^6*x^6 + 6*a^3*b^5*x^4 + 4*a^4*b^4*x^2 + a^5*b^3)*(-d^26/(a^5*b^15))^(1/4)
*log(456533*sqrt(d*x)*d^19 + 456533*(-d^26/(a^5*b^15))^(3/4)*a^4*b^11) - 231*(a*
b^7*x^8 + 4*a^2*b^6*x^6 + 6*a^3*b^5*x^4 + 4*a^4*b^4*x^2 + a^5*b^3)*(-d^26/(a^5*b
^15))^(1/4)*log(456533*sqrt(d*x)*d^19 - 456533*(-d^26/(a^5*b^15))^(3/4)*a^4*b^11
) + 4*(231*b^3*d^6*x^7 - 351*a*b^2*d^6*x^5 - 275*a^2*b*d^6*x^3 - 77*a^3*d^6*x)*s
qrt(d*x))/(a*b^7*x^8 + 4*a^2*b^6*x^6 + 6*a^3*b^5*x^4 + 4*a^4*b^4*x^2 + a^5*b^3)

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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((d*x)**(13/2)/(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.296778, size = 552, normalized size = 0.99 \[ \frac{1}{24576} \, d^{5}{\left (\frac{462 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{a^{2} b^{6}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{462 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{a^{2} b^{6}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{231 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}}{\rm ln}\left (d x + \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{a^{2} b^{6}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{231 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}}{\rm ln}\left (d x - \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{a^{2} b^{6}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{8 \,{\left (231 \, \sqrt{d x} b^{3} d^{9} x^{7} - 351 \, \sqrt{d x} a b^{2} d^{9} x^{5} - 275 \, \sqrt{d x} a^{2} b d^{9} x^{3} - 77 \, \sqrt{d x} a^{3} d^{9} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{4} a b^{3}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^(13/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2),x, algorithm="giac")

[Out]

1/24576*d^5*(462*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^2*b^6*sign(b*d^4*x^2 + a*d^4)) + 462*s
qrt(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^2*b^6*sign(b*d^4*x^2 + a*d^4)) - 231*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^2*b^6*sig
n(b*d^4*x^2 + a*d^4)) + 231*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^2*b^6*sign(b*d^4*x^2 + a*d^4)) + 8*(231*sqr
t(d*x)*b^3*d^9*x^7 - 351*sqrt(d*x)*a*b^2*d^9*x^5 - 275*sqrt(d*x)*a^2*b*d^9*x^3 -
77*sqrt(d*x)*a^3*d^9*x)/((b*d^2*x^2 + a*d^2)^4*a*b^3*sign(b*d^4*x^2 + a*d^4)))