3.375 \(\int \frac{x^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx\)

Optimal. Leaf size=310 \[ \frac{\sqrt [4]{a} (5 A b-9 a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} b^{13/4}}-\frac{\sqrt [4]{a} (5 A b-9 a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} b^{13/4}}+\frac{\sqrt [4]{a} (5 A b-9 a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} b^{13/4}}-\frac{\sqrt [4]{a} (5 A b-9 a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} b^{13/4}}+\frac{\sqrt{x} (5 A b-9 a B)}{2 b^3}-\frac{x^{5/2} (5 A b-9 a B)}{10 a b^2}+\frac{x^{9/2} (A b-a B)}{2 a b \left (a+b x^2\right )} \]

[Out]

((5*A*b - 9*a*B)*Sqrt[x])/(2*b^3) - ((5*A*b - 9*a*B)*x^(5/2))/(10*a*b^2) + ((A*b
- a*B)*x^(9/2))/(2*a*b*(a + b*x^2)) + (a^(1/4)*(5*A*b - 9*a*B)*ArcTan[1 - (Sqrt
[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*b^(13/4)) - (a^(1/4)*(5*A*b - 9*a*B)*A
rcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*b^(13/4)) + (a^(1/4)*(5
*A*b - 9*a*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqr
t[2]*b^(13/4)) - (a^(1/4)*(5*A*b - 9*a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*
Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*b^(13/4))

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Rubi [A]  time = 0.518517, antiderivative size = 310, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409 \[ \frac{\sqrt [4]{a} (5 A b-9 a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} b^{13/4}}-\frac{\sqrt [4]{a} (5 A b-9 a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} b^{13/4}}+\frac{\sqrt [4]{a} (5 A b-9 a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} b^{13/4}}-\frac{\sqrt [4]{a} (5 A b-9 a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} b^{13/4}}+\frac{\sqrt{x} (5 A b-9 a B)}{2 b^3}-\frac{x^{5/2} (5 A b-9 a B)}{10 a b^2}+\frac{x^{9/2} (A b-a B)}{2 a b \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(x^(7/2)*(A + B*x^2))/(a + b*x^2)^2,x]

[Out]

((5*A*b - 9*a*B)*Sqrt[x])/(2*b^3) - ((5*A*b - 9*a*B)*x^(5/2))/(10*a*b^2) + ((A*b
- a*B)*x^(9/2))/(2*a*b*(a + b*x^2)) + (a^(1/4)*(5*A*b - 9*a*B)*ArcTan[1 - (Sqrt
[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*b^(13/4)) - (a^(1/4)*(5*A*b - 9*a*B)*A
rcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*b^(13/4)) + (a^(1/4)*(5
*A*b - 9*a*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqr
t[2]*b^(13/4)) - (a^(1/4)*(5*A*b - 9*a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*
Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*b^(13/4))

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Rubi in Sympy [A]  time = 87.2669, size = 289, normalized size = 0.93 \[ \frac{\sqrt{2} \sqrt [4]{a} \left (5 A b - 9 B a\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{16 b^{\frac{13}{4}}} - \frac{\sqrt{2} \sqrt [4]{a} \left (5 A b - 9 B a\right ) \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{16 b^{\frac{13}{4}}} + \frac{\sqrt{2} \sqrt [4]{a} \left (5 A b - 9 B a\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 b^{\frac{13}{4}}} - \frac{\sqrt{2} \sqrt [4]{a} \left (5 A b - 9 B a\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 b^{\frac{13}{4}}} + \frac{\sqrt{x} \left (5 A b - 9 B a\right )}{2 b^{3}} + \frac{x^{\frac{9}{2}} \left (A b - B a\right )}{2 a b \left (a + b x^{2}\right )} - \frac{x^{\frac{5}{2}} \left (5 A b - 9 B a\right )}{10 a b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(7/2)*(B*x**2+A)/(b*x**2+a)**2,x)

[Out]

sqrt(2)*a**(1/4)*(5*A*b - 9*B*a)*log(-sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqrt(a
) + sqrt(b)*x)/(16*b**(13/4)) - sqrt(2)*a**(1/4)*(5*A*b - 9*B*a)*log(sqrt(2)*a**
(1/4)*b**(1/4)*sqrt(x) + sqrt(a) + sqrt(b)*x)/(16*b**(13/4)) + sqrt(2)*a**(1/4)*
(5*A*b - 9*B*a)*atan(1 - sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(8*b**(13/4)) - sqrt
(2)*a**(1/4)*(5*A*b - 9*B*a)*atan(1 + sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(8*b**(
13/4)) + sqrt(x)*(5*A*b - 9*B*a)/(2*b**3) + x**(9/2)*(A*b - B*a)/(2*a*b*(a + b*x
**2)) - x**(5/2)*(5*A*b - 9*B*a)/(10*a*b**2)

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Mathematica [A]  time = 0.394001, size = 277, normalized size = 0.89 \[ \frac{\frac{40 a \sqrt [4]{b} \sqrt{x} (A b-a B)}{a+b x^2}+160 \sqrt [4]{b} \sqrt{x} (A b-2 a B)-5 \sqrt{2} \sqrt [4]{a} (9 a B-5 A b) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+5 \sqrt{2} \sqrt [4]{a} (9 a B-5 A b) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-10 \sqrt{2} \sqrt [4]{a} (9 a B-5 A b) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )+10 \sqrt{2} \sqrt [4]{a} (9 a B-5 A b) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )+32 b^{5/4} B x^{5/2}}{80 b^{13/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(7/2)*(A + B*x^2))/(a + b*x^2)^2,x]

[Out]

(160*b^(1/4)*(A*b - 2*a*B)*Sqrt[x] + 32*b^(5/4)*B*x^(5/2) + (40*a*b^(1/4)*(A*b -
a*B)*Sqrt[x])/(a + b*x^2) - 10*Sqrt[2]*a^(1/4)*(-5*A*b + 9*a*B)*ArcTan[1 - (Sqr
t[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 10*Sqrt[2]*a^(1/4)*(-5*A*b + 9*a*B)*ArcTan[1 +
(Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 5*Sqrt[2]*a^(1/4)*(-5*A*b + 9*a*B)*Log[Sqrt
[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + 5*Sqrt[2]*a^(1/4)*(-5*A*b +
9*a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(80*b^(13/4)
)

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Maple [A]  time = 0.021, size = 339, normalized size = 1.1 \[{\frac{2\,B}{5\,{b}^{2}}{x}^{{\frac{5}{2}}}}+2\,{\frac{A\sqrt{x}}{{b}^{2}}}-4\,{\frac{B\sqrt{x}a}{{b}^{3}}}+{\frac{Aa}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }\sqrt{x}}-{\frac{{a}^{2}B}{2\,{b}^{3} \left ( b{x}^{2}+a \right ) }\sqrt{x}}-{\frac{5\,\sqrt{2}A}{8\,{b}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }-{\frac{5\,\sqrt{2}A}{16\,{b}^{2}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }-{\frac{5\,\sqrt{2}A}{8\,{b}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{9\,a\sqrt{2}B}{8\,{b}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{9\,a\sqrt{2}B}{16\,{b}^{3}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{9\,a\sqrt{2}B}{8\,{b}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(7/2)*(B*x^2+A)/(b*x^2+a)^2,x)

[Out]

2/5/b^2*B*x^(5/2)+2/b^2*A*x^(1/2)-4/b^3*B*x^(1/2)*a+1/2*a/b^2*x^(1/2)/(b*x^2+a)*
A-1/2*a^2/b^3*x^(1/2)/(b*x^2+a)*B-5/8/b^2*(a/b)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(
a/b)^(1/4)*x^(1/2)-1)-5/16/b^2*(a/b)^(1/4)*2^(1/2)*A*ln((x+(a/b)^(1/4)*x^(1/2)*2
^(1/2)+(a/b)^(1/2))/(x-(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2)))-5/8/b^2*(a/b)^(
1/4)*2^(1/2)*A*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+9/8*a/b^3*(a/b)^(1/4)*2^(1/
2)*B*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)+9/16*a/b^3*(a/b)^(1/4)*2^(1/2)*B*ln((
x+(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2))/(x-(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^
(1/2)))+9/8*a/b^3*(a/b)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(a/b)^(1/4)*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((B*x^2 + A)*x^(7/2)/(b*x^2 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.254727, size = 851, normalized size = 2.75 \[ \frac{20 \,{\left (b^{4} x^{2} + a b^{3}\right )} \left (-\frac{6561 \, B^{4} a^{5} - 14580 \, A B^{3} a^{4} b + 12150 \, A^{2} B^{2} a^{3} b^{2} - 4500 \, A^{3} B a^{2} b^{3} + 625 \, A^{4} a b^{4}}{b^{13}}\right )^{\frac{1}{4}} \arctan \left (-\frac{b^{3} \left (-\frac{6561 \, B^{4} a^{5} - 14580 \, A B^{3} a^{4} b + 12150 \, A^{2} B^{2} a^{3} b^{2} - 4500 \, A^{3} B a^{2} b^{3} + 625 \, A^{4} a b^{4}}{b^{13}}\right )^{\frac{1}{4}}}{{\left (9 \, B a - 5 \, A b\right )} \sqrt{x} - \sqrt{b^{6} \sqrt{-\frac{6561 \, B^{4} a^{5} - 14580 \, A B^{3} a^{4} b + 12150 \, A^{2} B^{2} a^{3} b^{2} - 4500 \, A^{3} B a^{2} b^{3} + 625 \, A^{4} a b^{4}}{b^{13}}} +{\left (81 \, B^{2} a^{2} - 90 \, A B a b + 25 \, A^{2} b^{2}\right )} x}}\right ) - 5 \,{\left (b^{4} x^{2} + a b^{3}\right )} \left (-\frac{6561 \, B^{4} a^{5} - 14580 \, A B^{3} a^{4} b + 12150 \, A^{2} B^{2} a^{3} b^{2} - 4500 \, A^{3} B a^{2} b^{3} + 625 \, A^{4} a b^{4}}{b^{13}}\right )^{\frac{1}{4}} \log \left (b^{3} \left (-\frac{6561 \, B^{4} a^{5} - 14580 \, A B^{3} a^{4} b + 12150 \, A^{2} B^{2} a^{3} b^{2} - 4500 \, A^{3} B a^{2} b^{3} + 625 \, A^{4} a b^{4}}{b^{13}}\right )^{\frac{1}{4}} -{\left (9 \, B a - 5 \, A b\right )} \sqrt{x}\right ) + 5 \,{\left (b^{4} x^{2} + a b^{3}\right )} \left (-\frac{6561 \, B^{4} a^{5} - 14580 \, A B^{3} a^{4} b + 12150 \, A^{2} B^{2} a^{3} b^{2} - 4500 \, A^{3} B a^{2} b^{3} + 625 \, A^{4} a b^{4}}{b^{13}}\right )^{\frac{1}{4}} \log \left (-b^{3} \left (-\frac{6561 \, B^{4} a^{5} - 14580 \, A B^{3} a^{4} b + 12150 \, A^{2} B^{2} a^{3} b^{2} - 4500 \, A^{3} B a^{2} b^{3} + 625 \, A^{4} a b^{4}}{b^{13}}\right )^{\frac{1}{4}} -{\left (9 \, B a - 5 \, A b\right )} \sqrt{x}\right ) + 4 \,{\left (4 \, B b^{2} x^{4} - 45 \, B a^{2} + 25 \, A a b - 4 \,{\left (9 \, B a b - 5 \, A b^{2}\right )} x^{2}\right )} \sqrt{x}}{40 \,{\left (b^{4} x^{2} + a b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^(7/2)/(b*x^2 + a)^2,x, algorithm="fricas")

[Out]

1/40*(20*(b^4*x^2 + a*b^3)*(-(6561*B^4*a^5 - 14580*A*B^3*a^4*b + 12150*A^2*B^2*a
^3*b^2 - 4500*A^3*B*a^2*b^3 + 625*A^4*a*b^4)/b^13)^(1/4)*arctan(-b^3*(-(6561*B^4
*a^5 - 14580*A*B^3*a^4*b + 12150*A^2*B^2*a^3*b^2 - 4500*A^3*B*a^2*b^3 + 625*A^4*
a*b^4)/b^13)^(1/4)/((9*B*a - 5*A*b)*sqrt(x) - sqrt(b^6*sqrt(-(6561*B^4*a^5 - 145
80*A*B^3*a^4*b + 12150*A^2*B^2*a^3*b^2 - 4500*A^3*B*a^2*b^3 + 625*A^4*a*b^4)/b^1
3) + (81*B^2*a^2 - 90*A*B*a*b + 25*A^2*b^2)*x))) - 5*(b^4*x^2 + a*b^3)*(-(6561*B
^4*a^5 - 14580*A*B^3*a^4*b + 12150*A^2*B^2*a^3*b^2 - 4500*A^3*B*a^2*b^3 + 625*A^
4*a*b^4)/b^13)^(1/4)*log(b^3*(-(6561*B^4*a^5 - 14580*A*B^3*a^4*b + 12150*A^2*B^2
*a^3*b^2 - 4500*A^3*B*a^2*b^3 + 625*A^4*a*b^4)/b^13)^(1/4) - (9*B*a - 5*A*b)*sqr
t(x)) + 5*(b^4*x^2 + a*b^3)*(-(6561*B^4*a^5 - 14580*A*B^3*a^4*b + 12150*A^2*B^2*
a^3*b^2 - 4500*A^3*B*a^2*b^3 + 625*A^4*a*b^4)/b^13)^(1/4)*log(-b^3*(-(6561*B^4*a
^5 - 14580*A*B^3*a^4*b + 12150*A^2*B^2*a^3*b^2 - 4500*A^3*B*a^2*b^3 + 625*A^4*a*
b^4)/b^13)^(1/4) - (9*B*a - 5*A*b)*sqrt(x)) + 4*(4*B*b^2*x^4 - 45*B*a^2 + 25*A*a
*b - 4*(9*B*a*b - 5*A*b^2)*x^2)*sqrt(x))/(b^4*x^2 + a*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(x**(7/2)*(B*x**2+A)/(b*x**2+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.249426, size = 402, normalized size = 1.3 \[ \frac{\sqrt{2}{\left (9 \, \left (a b^{3}\right )^{\frac{1}{4}} B a - 5 \, \left (a b^{3}\right )^{\frac{1}{4}} A b\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, b^{4}} + \frac{\sqrt{2}{\left (9 \, \left (a b^{3}\right )^{\frac{1}{4}} B a - 5 \, \left (a b^{3}\right )^{\frac{1}{4}} A b\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, b^{4}} + \frac{\sqrt{2}{\left (9 \, \left (a b^{3}\right )^{\frac{1}{4}} B a - 5 \, \left (a b^{3}\right )^{\frac{1}{4}} A b\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{16 \, b^{4}} - \frac{\sqrt{2}{\left (9 \, \left (a b^{3}\right )^{\frac{1}{4}} B a - 5 \, \left (a b^{3}\right )^{\frac{1}{4}} A b\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{16 \, b^{4}} - \frac{B a^{2} \sqrt{x} - A a b \sqrt{x}}{2 \,{\left (b x^{2} + a\right )} b^{3}} + \frac{2 \,{\left (B b^{8} x^{\frac{5}{2}} - 10 \, B a b^{7} \sqrt{x} + 5 \, A b^{8} \sqrt{x}\right )}}{5 \, b^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^(7/2)/(b*x^2 + a)^2,x, algorithm="giac")

[Out]

1/8*sqrt(2)*(9*(a*b^3)^(1/4)*B*a - 5*(a*b^3)^(1/4)*A*b)*arctan(1/2*sqrt(2)*(sqrt
(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/b^4 + 1/8*sqrt(2)*(9*(a*b^3)^(1/4)*B*a
- 5*(a*b^3)^(1/4)*A*b)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a
/b)^(1/4))/b^4 + 1/16*sqrt(2)*(9*(a*b^3)^(1/4)*B*a - 5*(a*b^3)^(1/4)*A*b)*ln(sqr
t(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/b^4 - 1/16*sqrt(2)*(9*(a*b^3)^(1/4)*B*
a - 5*(a*b^3)^(1/4)*A*b)*ln(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/b^4 -
1/2*(B*a^2*sqrt(x) - A*a*b*sqrt(x))/((b*x^2 + a)*b^3) + 2/5*(B*b^8*x^(5/2) - 10*
B*a*b^7*sqrt(x) + 5*A*b^8*sqrt(x))/b^10