3.390 \(\int \frac{A+B x^2}{x^{7/2} \left (a+b x^2\right )^3} \, dx\)

Optimal. Leaf size=343 \[ \frac{9 \sqrt [4]{b} (13 A b-5 a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{17/4}}-\frac{9 \sqrt [4]{b} (13 A b-5 a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{17/4}}-\frac{9 \sqrt [4]{b} (13 A b-5 a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{17/4}}+\frac{9 \sqrt [4]{b} (13 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{17/4}}+\frac{9 (13 A b-5 a B)}{16 a^4 \sqrt{x}}-\frac{9 (13 A b-5 a B)}{80 a^3 b x^{5/2}}+\frac{13 A b-5 a B}{16 a^2 b x^{5/2} \left (a+b x^2\right )}+\frac{A b-a B}{4 a b x^{5/2} \left (a+b x^2\right )^2} \]

[Out]

(-9*(13*A*b - 5*a*B))/(80*a^3*b*x^(5/2)) + (9*(13*A*b - 5*a*B))/(16*a^4*Sqrt[x])
+ (A*b - a*B)/(4*a*b*x^(5/2)*(a + b*x^2)^2) + (13*A*b - 5*a*B)/(16*a^2*b*x^(5/2
)*(a + b*x^2)) - (9*b^(1/4)*(13*A*b - 5*a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x]
)/a^(1/4)])/(32*Sqrt[2]*a^(17/4)) + (9*b^(1/4)*(13*A*b - 5*a*B)*ArcTan[1 + (Sqrt
[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(17/4)) + (9*b^(1/4)*(13*A*b - 5*a*
B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*Sqrt[2]*a^(17
/4)) - (9*b^(1/4)*(13*A*b - 5*a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x]
+ Sqrt[b]*x])/(64*Sqrt[2]*a^(17/4))

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Rubi [A]  time = 0.579375, antiderivative size = 343, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454 \[ \frac{9 \sqrt [4]{b} (13 A b-5 a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{17/4}}-\frac{9 \sqrt [4]{b} (13 A b-5 a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{17/4}}-\frac{9 \sqrt [4]{b} (13 A b-5 a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{17/4}}+\frac{9 \sqrt [4]{b} (13 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{17/4}}+\frac{9 (13 A b-5 a B)}{16 a^4 \sqrt{x}}-\frac{9 (13 A b-5 a B)}{80 a^3 b x^{5/2}}+\frac{13 A b-5 a B}{16 a^2 b x^{5/2} \left (a+b x^2\right )}+\frac{A b-a B}{4 a b x^{5/2} \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^2)/(x^(7/2)*(a + b*x^2)^3),x]

[Out]

(-9*(13*A*b - 5*a*B))/(80*a^3*b*x^(5/2)) + (9*(13*A*b - 5*a*B))/(16*a^4*Sqrt[x])
+ (A*b - a*B)/(4*a*b*x^(5/2)*(a + b*x^2)^2) + (13*A*b - 5*a*B)/(16*a^2*b*x^(5/2
)*(a + b*x^2)) - (9*b^(1/4)*(13*A*b - 5*a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x]
)/a^(1/4)])/(32*Sqrt[2]*a^(17/4)) + (9*b^(1/4)*(13*A*b - 5*a*B)*ArcTan[1 + (Sqrt
[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(17/4)) + (9*b^(1/4)*(13*A*b - 5*a*
B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*Sqrt[2]*a^(17
/4)) - (9*b^(1/4)*(13*A*b - 5*a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x]
+ Sqrt[b]*x])/(64*Sqrt[2]*a^(17/4))

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Rubi in Sympy [A]  time = 95.6378, size = 326, normalized size = 0.95 \[ \frac{A b - B a}{4 a b x^{\frac{5}{2}} \left (a + b x^{2}\right )^{2}} + \frac{13 A b - 5 B a}{16 a^{2} b x^{\frac{5}{2}} \left (a + b x^{2}\right )} - \frac{9 \left (13 A b - 5 B a\right )}{80 a^{3} b x^{\frac{5}{2}}} + \frac{9 \left (13 A b - 5 B a\right )}{16 a^{4} \sqrt{x}} + \frac{9 \sqrt{2} \sqrt [4]{b} \left (13 A b - 5 B a\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{128 a^{\frac{17}{4}}} - \frac{9 \sqrt{2} \sqrt [4]{b} \left (13 A b - 5 B a\right ) \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{128 a^{\frac{17}{4}}} - \frac{9 \sqrt{2} \sqrt [4]{b} \left (13 A b - 5 B a\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{17}{4}}} + \frac{9 \sqrt{2} \sqrt [4]{b} \left (13 A b - 5 B a\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{17}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)/x**(7/2)/(b*x**2+a)**3,x)

[Out]

(A*b - B*a)/(4*a*b*x**(5/2)*(a + b*x**2)**2) + (13*A*b - 5*B*a)/(16*a**2*b*x**(5
/2)*(a + b*x**2)) - 9*(13*A*b - 5*B*a)/(80*a**3*b*x**(5/2)) + 9*(13*A*b - 5*B*a)
/(16*a**4*sqrt(x)) + 9*sqrt(2)*b**(1/4)*(13*A*b - 5*B*a)*log(-sqrt(2)*a**(1/4)*b
**(1/4)*sqrt(x) + sqrt(a) + sqrt(b)*x)/(128*a**(17/4)) - 9*sqrt(2)*b**(1/4)*(13*
A*b - 5*B*a)*log(sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqrt(a) + sqrt(b)*x)/(128*a
**(17/4)) - 9*sqrt(2)*b**(1/4)*(13*A*b - 5*B*a)*atan(1 - sqrt(2)*b**(1/4)*sqrt(x
)/a**(1/4))/(64*a**(17/4)) + 9*sqrt(2)*b**(1/4)*(13*A*b - 5*B*a)*atan(1 + sqrt(2
)*b**(1/4)*sqrt(x)/a**(1/4))/(64*a**(17/4))

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Mathematica [A]  time = 0.539769, size = 308, normalized size = 0.9 \[ \frac{-\frac{160 a^{5/4} b x^{3/2} (a B-A b)}{\left (a+b x^2\right )^2}-\frac{256 a^{5/4} A}{x^{5/2}}-\frac{40 \sqrt [4]{a} b x^{3/2} (13 a B-21 A b)}{a+b x^2}-\frac{1280 \sqrt [4]{a} (a B-3 A b)}{\sqrt{x}}+45 \sqrt{2} \sqrt [4]{b} (13 A b-5 a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+45 \sqrt{2} \sqrt [4]{b} (5 a B-13 A b) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-90 \sqrt{2} \sqrt [4]{b} (13 A b-5 a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )+90 \sqrt{2} \sqrt [4]{b} (13 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{640 a^{17/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x^2)/(x^(7/2)*(a + b*x^2)^3),x]

[Out]

((-256*a^(5/4)*A)/x^(5/2) - (1280*a^(1/4)*(-3*A*b + a*B))/Sqrt[x] - (160*a^(5/4)
*b*(-(A*b) + a*B)*x^(3/2))/(a + b*x^2)^2 - (40*a^(1/4)*b*(-21*A*b + 13*a*B)*x^(3
/2))/(a + b*x^2) - 90*Sqrt[2]*b^(1/4)*(13*A*b - 5*a*B)*ArcTan[1 - (Sqrt[2]*b^(1/
4)*Sqrt[x])/a^(1/4)] + 90*Sqrt[2]*b^(1/4)*(13*A*b - 5*a*B)*ArcTan[1 + (Sqrt[2]*b
^(1/4)*Sqrt[x])/a^(1/4)] + 45*Sqrt[2]*b^(1/4)*(13*A*b - 5*a*B)*Log[Sqrt[a] - Sqr
t[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + 45*Sqrt[2]*b^(1/4)*(-13*A*b + 5*a*B)
*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(640*a^(17/4))

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Maple [A]  time = 0.033, size = 381, normalized size = 1.1 \[ -{\frac{2\,A}{5\,{a}^{3}}{x}^{-{\frac{5}{2}}}}+6\,{\frac{Ab}{\sqrt{x}{a}^{4}}}-2\,{\frac{B}{\sqrt{x}{a}^{3}}}+{\frac{21\,{b}^{3}A}{16\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{2}}{x}^{{\frac{7}{2}}}}-{\frac{13\,B{b}^{2}}{16\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{2}}{x}^{{\frac{7}{2}}}}+{\frac{25\,{b}^{2}A}{16\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{2}}{x}^{{\frac{3}{2}}}}-{\frac{17\,Bb}{16\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{2}}{x}^{{\frac{3}{2}}}}+{\frac{117\,b\sqrt{2}A}{128\,{a}^{4}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{117\,b\sqrt{2}A}{64\,{a}^{4}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{117\,b\sqrt{2}A}{64\,{a}^{4}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{45\,\sqrt{2}B}{128\,{a}^{3}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{45\,\sqrt{2}B}{64\,{a}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{45\,\sqrt{2}B}{64\,{a}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)/x^(7/2)/(b*x^2+a)^3,x)

[Out]

-2/5*A/a^3/x^(5/2)+6/x^(1/2)/a^4*A*b-2/x^(1/2)/a^3*B+21/16/a^4*b^3/(b*x^2+a)^2*x
^(7/2)*A-13/16/a^3*b^2/(b*x^2+a)^2*x^(7/2)*B+25/16/a^3*b^2/(b*x^2+a)^2*A*x^(3/2)
-17/16/a^2*b/(b*x^2+a)^2*B*x^(3/2)+117/128/a^4*b/(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)
))+117/64/a^4*b/(a/b)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+117/
64/a^4*b/(a/b)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)-45/128/a^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)))-45/64/a^3/(a/b)^(1/4)*2^(1/2)*B*arctan(2^(1/2)
/(a/b)^(1/4)*x^(1/2)+1)-45/64/a^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)/((b*x^2 + a)^3*x^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.261387, size = 1257, normalized size = 3.66 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/320*(180*(5*B*a*b^2 - 13*A*b^3)*x^6 + 324*(5*B*a^2*b - 13*A*a*b^2)*x^4 + 128*
A*a^3 + 128*(5*B*a^3 - 13*A*a^2*b)*x^2 - 180*(a^4*b^2*x^6 + 2*a^5*b*x^4 + a^6*x^
2)*sqrt(x)*(-(625*B^4*a^4*b - 6500*A*B^3*a^3*b^2 + 25350*A^2*B^2*a^2*b^3 - 43940
*A^3*B*a*b^4 + 28561*A^4*b^5)/a^17)^(1/4)*arctan(-a^13*(-(625*B^4*a^4*b - 6500*A
*B^3*a^3*b^2 + 25350*A^2*B^2*a^2*b^3 - 43940*A^3*B*a*b^4 + 28561*A^4*b^5)/a^17)^
(3/4)/((125*B^3*a^3*b - 975*A*B^2*a^2*b^2 + 2535*A^2*B*a*b^3 - 2197*A^3*b^4)*sqr
t(x) - sqrt((15625*B^6*a^6*b^2 - 243750*A*B^5*a^5*b^3 + 1584375*A^2*B^4*a^4*b^4
- 5492500*A^3*B^3*a^3*b^5 + 10710375*A^4*B^2*a^2*b^6 - 11138790*A^5*B*a*b^7 + 48
26809*A^6*b^8)*x - (625*B^4*a^13*b - 6500*A*B^3*a^12*b^2 + 25350*A^2*B^2*a^11*b^
3 - 43940*A^3*B*a^10*b^4 + 28561*A^4*a^9*b^5)*sqrt(-(625*B^4*a^4*b - 6500*A*B^3*
a^3*b^2 + 25350*A^2*B^2*a^2*b^3 - 43940*A^3*B*a*b^4 + 28561*A^4*b^5)/a^17)))) -
45*(a^4*b^2*x^6 + 2*a^5*b*x^4 + a^6*x^2)*sqrt(x)*(-(625*B^4*a^4*b - 6500*A*B^3*a
^3*b^2 + 25350*A^2*B^2*a^2*b^3 - 43940*A^3*B*a*b^4 + 28561*A^4*b^5)/a^17)^(1/4)*
log(729*a^13*(-(625*B^4*a^4*b - 6500*A*B^3*a^3*b^2 + 25350*A^2*B^2*a^2*b^3 - 439
40*A^3*B*a*b^4 + 28561*A^4*b^5)/a^17)^(3/4) - 729*(125*B^3*a^3*b - 975*A*B^2*a^2
*b^2 + 2535*A^2*B*a*b^3 - 2197*A^3*b^4)*sqrt(x)) + 45*(a^4*b^2*x^6 + 2*a^5*b*x^4
+ a^6*x^2)*sqrt(x)*(-(625*B^4*a^4*b - 6500*A*B^3*a^3*b^2 + 25350*A^2*B^2*a^2*b^
3 - 43940*A^3*B*a*b^4 + 28561*A^4*b^5)/a^17)^(1/4)*log(-729*a^13*(-(625*B^4*a^4*
b - 6500*A*B^3*a^3*b^2 + 25350*A^2*B^2*a^2*b^3 - 43940*A^3*B*a*b^4 + 28561*A^4*b
^5)/a^17)^(3/4) - 729*(125*B^3*a^3*b - 975*A*B^2*a^2*b^2 + 2535*A^2*B*a*b^3 - 21
97*A^3*b^4)*sqrt(x)))/((a^4*b^2*x^6 + 2*a^5*b*x^4 + a^6*x^2)*sqrt(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((B*x**2+A)/x**(7/2)/(b*x**2+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.262049, size = 440, normalized size = 1.28 \[ -\frac{9 \, \sqrt{2}{\left (5 \, \left (a b^{3}\right )^{\frac{3}{4}} B a - 13 \, \left (a b^{3}\right )^{\frac{3}{4}} A b\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{64 \, a^{5} b^{2}} - \frac{9 \, \sqrt{2}{\left (5 \, \left (a b^{3}\right )^{\frac{3}{4}} B a - 13 \, \left (a b^{3}\right )^{\frac{3}{4}} A b\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{64 \, a^{5} b^{2}} + \frac{9 \, \sqrt{2}{\left (5 \, \left (a b^{3}\right )^{\frac{3}{4}} B a - 13 \, \left (a b^{3}\right )^{\frac{3}{4}} A b\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{128 \, a^{5} b^{2}} - \frac{9 \, \sqrt{2}{\left (5 \, \left (a b^{3}\right )^{\frac{3}{4}} B a - 13 \, \left (a b^{3}\right )^{\frac{3}{4}} A b\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{128 \, a^{5} b^{2}} - \frac{13 \, B a b^{2} x^{\frac{7}{2}} - 21 \, A b^{3} x^{\frac{7}{2}} + 17 \, B a^{2} b x^{\frac{3}{2}} - 25 \, A a b^{2} x^{\frac{3}{2}}}{16 \,{\left (b x^{2} + a\right )}^{2} a^{4}} - \frac{2 \,{\left (5 \, B a x^{2} - 15 \, A b x^{2} + A a\right )}}{5 \, a^{4} x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9/64*sqrt(2)*(5*(a*b^3)^(3/4)*B*a - 13*(a*b^3)^(3/4)*A*b)*arctan(1/2*sqrt(2)*(s
qrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(a^5*b^2) - 9/64*sqrt(2)*(5*(a*b^3)
^(3/4)*B*a - 13*(a*b^3)^(3/4)*A*b)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*
sqrt(x))/(a/b)^(1/4))/(a^5*b^2) + 9/128*sqrt(2)*(5*(a*b^3)^(3/4)*B*a - 13*(a*b^3
)^(3/4)*A*b)*ln(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^5*b^2) - 9/128*s
qrt(2)*(5*(a*b^3)^(3/4)*B*a - 13*(a*b^3)^(3/4)*A*b)*ln(-sqrt(2)*sqrt(x)*(a/b)^(1
/4) + x + sqrt(a/b))/(a^5*b^2) - 1/16*(13*B*a*b^2*x^(7/2) - 21*A*b^3*x^(7/2) + 1
7*B*a^2*b*x^(3/2) - 25*A*a*b^2*x^(3/2))/((b*x^2 + a)^2*a^4) - 2/5*(5*B*a*x^2 - 1
5*A*b*x^2 + A*a)/(a^4*x^(5/2))