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

Optimal. Leaf size=293 \[ -\frac{3 (a B+7 A b) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{11/4} b^{5/4}}+\frac{3 (a B+7 A b) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{11/4} b^{5/4}}-\frac{3 (a B+7 A b) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{11/4} b^{5/4}}+\frac{3 (a B+7 A b) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{11/4} b^{5/4}}+\frac{\sqrt{x} (a B+7 A b)}{16 a^2 b \left (a+b x^2\right )}+\frac{\sqrt{x} (A b-a B)}{4 a b \left (a+b x^2\right )^2} \]

[Out]

((A*b - a*B)*Sqrt[x])/(4*a*b*(a + b*x^2)^2) + ((7*A*b + a*B)*Sqrt[x])/(16*a^2*b*
(a + b*x^2)) - (3*(7*A*b + a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(
32*Sqrt[2]*a^(11/4)*b^(5/4)) + (3*(7*A*b + a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt
[x])/a^(1/4)])/(32*Sqrt[2]*a^(11/4)*b^(5/4)) - (3*(7*A*b + a*B)*Log[Sqrt[a] - Sq
rt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*Sqrt[2]*a^(11/4)*b^(5/4)) + (3*(
7*A*b + a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*Sqr
t[2]*a^(11/4)*b^(5/4))

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Rubi [A]  time = 0.454032, antiderivative size = 293, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409 \[ -\frac{3 (a B+7 A b) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{11/4} b^{5/4}}+\frac{3 (a B+7 A b) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{11/4} b^{5/4}}-\frac{3 (a B+7 A b) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{11/4} b^{5/4}}+\frac{3 (a B+7 A b) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{11/4} b^{5/4}}+\frac{\sqrt{x} (a B+7 A b)}{16 a^2 b \left (a+b x^2\right )}+\frac{\sqrt{x} (A b-a B)}{4 a b \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

((A*b - a*B)*Sqrt[x])/(4*a*b*(a + b*x^2)^2) + ((7*A*b + a*B)*Sqrt[x])/(16*a^2*b*
(a + b*x^2)) - (3*(7*A*b + a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(
32*Sqrt[2]*a^(11/4)*b^(5/4)) + (3*(7*A*b + a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt
[x])/a^(1/4)])/(32*Sqrt[2]*a^(11/4)*b^(5/4)) - (3*(7*A*b + a*B)*Log[Sqrt[a] - Sq
rt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*Sqrt[2]*a^(11/4)*b^(5/4)) + (3*(
7*A*b + a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*Sqr
t[2]*a^(11/4)*b^(5/4))

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Rubi in Sympy [A]  time = 78.2893, size = 275, normalized size = 0.94 \[ \frac{\sqrt{x} \left (A b - B a\right )}{4 a b \left (a + b x^{2}\right )^{2}} + \frac{\sqrt{x} \left (7 A b + B a\right )}{16 a^{2} b \left (a + b x^{2}\right )} - \frac{3 \sqrt{2} \left (7 A b + B a\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{128 a^{\frac{11}{4}} b^{\frac{5}{4}}} + \frac{3 \sqrt{2} \left (7 A b + B a\right ) \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{128 a^{\frac{11}{4}} b^{\frac{5}{4}}} - \frac{3 \sqrt{2} \left (7 A b + B a\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{11}{4}} b^{\frac{5}{4}}} + \frac{3 \sqrt{2} \left (7 A b + B a\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{11}{4}} b^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)/(b*x**2+a)**3/x**(1/2),x)

[Out]

sqrt(x)*(A*b - B*a)/(4*a*b*(a + b*x**2)**2) + sqrt(x)*(7*A*b + B*a)/(16*a**2*b*(
a + b*x**2)) - 3*sqrt(2)*(7*A*b + B*a)*log(-sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x) +
sqrt(a) + sqrt(b)*x)/(128*a**(11/4)*b**(5/4)) + 3*sqrt(2)*(7*A*b + B*a)*log(sqrt
(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqrt(a) + sqrt(b)*x)/(128*a**(11/4)*b**(5/4)) -
3*sqrt(2)*(7*A*b + B*a)*atan(1 - sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(64*a**(11/4
)*b**(5/4)) + 3*sqrt(2)*(7*A*b + B*a)*atan(1 + sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4)
)/(64*a**(11/4)*b**(5/4))

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Mathematica [A]  time = 0.36774, size = 263, normalized size = 0.9 \[ \frac{-\frac{32 a^{7/4} \sqrt [4]{b} \sqrt{x} (a B-A b)}{\left (a+b x^2\right )^2}+\frac{8 a^{3/4} \sqrt [4]{b} \sqrt{x} (a B+7 A b)}{a+b x^2}-3 \sqrt{2} (a B+7 A b) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+3 \sqrt{2} (a B+7 A b) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-6 \sqrt{2} (a B+7 A b) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )+6 \sqrt{2} (a B+7 A b) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{128 a^{11/4} b^{5/4}} \]

Antiderivative was successfully verified.

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

[Out]

((-32*a^(7/4)*b^(1/4)*(-(A*b) + a*B)*Sqrt[x])/(a + b*x^2)^2 + (8*a^(3/4)*b^(1/4)
*(7*A*b + a*B)*Sqrt[x])/(a + b*x^2) - 6*Sqrt[2]*(7*A*b + a*B)*ArcTan[1 - (Sqrt[2
]*b^(1/4)*Sqrt[x])/a^(1/4)] + 6*Sqrt[2]*(7*A*b + a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4
)*Sqrt[x])/a^(1/4)] - 3*Sqrt[2]*(7*A*b + a*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1
/4)*Sqrt[x] + Sqrt[b]*x] + 3*Sqrt[2]*(7*A*b + a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)
*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(128*a^(11/4)*b^(5/4))

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Maple [A]  time = 0.022, size = 325, normalized size = 1.1 \[ 2\,{\frac{1}{ \left ( b{x}^{2}+a \right ) ^{2}} \left ( 1/32\,{\frac{ \left ( 7\,Ab+Ba \right ){x}^{5/2}}{{a}^{2}}}+1/32\,{\frac{ \left ( 11\,Ab-3\,Ba \right ) \sqrt{x}}{ab}} \right ) }+{\frac{21\,\sqrt{2}A}{64\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{21\,\sqrt{2}A}{64\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{21\,\sqrt{2}A}{128\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{3\,\sqrt{2}B}{64\,{a}^{2}b}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{3\,\sqrt{2}B}{64\,{a}^{2}b}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{3\,\sqrt{2}B}{128\,{a}^{2}b}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)/(b*x^2+a)^3/x^(1/2),x)

[Out]

2*(1/32*(7*A*b+B*a)/a^2*x^(5/2)+1/32*(11*A*b-3*B*a)/a/b*x^(1/2))/(b*x^2+a)^2+21/
64/a^3*(a/b)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+21/64/a^3*(a/
b)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)+21/128/a^3*(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)))+3/64/a^2/b*(a/b)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(a/b)^(1/4
)*x^(1/2)+1)+3/64/a^2/b*(a/b)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)
-1)+3/128/a^2/b*(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)))

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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*sqrt(x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.25412, size = 903, normalized size = 3.08 \[ -\frac{12 \,{\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )} \left (-\frac{B^{4} a^{4} + 28 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 1372 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b^{5}}\right )^{\frac{1}{4}} \arctan \left (\frac{a^{3} b \left (-\frac{B^{4} a^{4} + 28 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 1372 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b^{5}}\right )^{\frac{1}{4}}}{{\left (B a + 7 \, A b\right )} \sqrt{x} + \sqrt{a^{6} b^{2} \sqrt{-\frac{B^{4} a^{4} + 28 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 1372 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b^{5}}} +{\left (B^{2} a^{2} + 14 \, A B a b + 49 \, A^{2} b^{2}\right )} x}}\right ) - 3 \,{\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )} \left (-\frac{B^{4} a^{4} + 28 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 1372 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b^{5}}\right )^{\frac{1}{4}} \log \left (3 \, a^{3} b \left (-\frac{B^{4} a^{4} + 28 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 1372 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b^{5}}\right )^{\frac{1}{4}} + 3 \,{\left (B a + 7 \, A b\right )} \sqrt{x}\right ) + 3 \,{\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )} \left (-\frac{B^{4} a^{4} + 28 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 1372 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b^{5}}\right )^{\frac{1}{4}} \log \left (-3 \, a^{3} b \left (-\frac{B^{4} a^{4} + 28 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 1372 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b^{5}}\right )^{\frac{1}{4}} + 3 \,{\left (B a + 7 \, A b\right )} \sqrt{x}\right ) + 4 \,{\left (3 \, B a^{2} - 11 \, A a b -{\left (B a b + 7 \, A b^{2}\right )} x^{2}\right )} \sqrt{x}}{64 \,{\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/64*(12*(a^2*b^3*x^4 + 2*a^3*b^2*x^2 + a^4*b)*(-(B^4*a^4 + 28*A*B^3*a^3*b + 29
4*A^2*B^2*a^2*b^2 + 1372*A^3*B*a*b^3 + 2401*A^4*b^4)/(a^11*b^5))^(1/4)*arctan(a^
3*b*(-(B^4*a^4 + 28*A*B^3*a^3*b + 294*A^2*B^2*a^2*b^2 + 1372*A^3*B*a*b^3 + 2401*
A^4*b^4)/(a^11*b^5))^(1/4)/((B*a + 7*A*b)*sqrt(x) + sqrt(a^6*b^2*sqrt(-(B^4*a^4
+ 28*A*B^3*a^3*b + 294*A^2*B^2*a^2*b^2 + 1372*A^3*B*a*b^3 + 2401*A^4*b^4)/(a^11*
b^5)) + (B^2*a^2 + 14*A*B*a*b + 49*A^2*b^2)*x))) - 3*(a^2*b^3*x^4 + 2*a^3*b^2*x^
2 + a^4*b)*(-(B^4*a^4 + 28*A*B^3*a^3*b + 294*A^2*B^2*a^2*b^2 + 1372*A^3*B*a*b^3
+ 2401*A^4*b^4)/(a^11*b^5))^(1/4)*log(3*a^3*b*(-(B^4*a^4 + 28*A*B^3*a^3*b + 294*
A^2*B^2*a^2*b^2 + 1372*A^3*B*a*b^3 + 2401*A^4*b^4)/(a^11*b^5))^(1/4) + 3*(B*a +
7*A*b)*sqrt(x)) + 3*(a^2*b^3*x^4 + 2*a^3*b^2*x^2 + a^4*b)*(-(B^4*a^4 + 28*A*B^3*
a^3*b + 294*A^2*B^2*a^2*b^2 + 1372*A^3*B*a*b^3 + 2401*A^4*b^4)/(a^11*b^5))^(1/4)
*log(-3*a^3*b*(-(B^4*a^4 + 28*A*B^3*a^3*b + 294*A^2*B^2*a^2*b^2 + 1372*A^3*B*a*b
^3 + 2401*A^4*b^4)/(a^11*b^5))^(1/4) + 3*(B*a + 7*A*b)*sqrt(x)) + 4*(3*B*a^2 - 1
1*A*a*b - (B*a*b + 7*A*b^2)*x^2)*sqrt(x))/(a^2*b^3*x^4 + 2*a^3*b^2*x^2 + a^4*b)

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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)/(b*x**2+a)**3/x**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.244603, size = 396, normalized size = 1.35 \[ \frac{3 \, \sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} B a + 7 \, \left (a b^{3}\right )^{\frac{1}{4}} A b\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{64 \, a^{3} b^{2}} + \frac{3 \, \sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} B a + 7 \, \left (a b^{3}\right )^{\frac{1}{4}} A b\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{64 \, a^{3} b^{2}} + \frac{3 \, \sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} B a + 7 \, \left (a b^{3}\right )^{\frac{1}{4}} A b\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{128 \, a^{3} b^{2}} - \frac{3 \, \sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} B a + 7 \, \left (a b^{3}\right )^{\frac{1}{4}} A b\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{128 \, a^{3} b^{2}} + \frac{B a b x^{\frac{5}{2}} + 7 \, A b^{2} x^{\frac{5}{2}} - 3 \, B a^{2} \sqrt{x} + 11 \, A a b \sqrt{x}}{16 \,{\left (b x^{2} + a\right )}^{2} a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/64*sqrt(2)*((a*b^3)^(1/4)*B*a + 7*(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))/(a^3*b^2) + 3/64*sqrt(2)*((a*b^3)^(1/4)
*B*a + 7*(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))/(a^3*b^2) + 3/128*sqrt(2)*((a*b^3)^(1/4)*B*a + 7*(a*b^3)^(1/4)*A*
b)*ln(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^3*b^2) - 3/128*sqrt(2)*((a
*b^3)^(1/4)*B*a + 7*(a*b^3)^(1/4)*A*b)*ln(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqr
t(a/b))/(a^3*b^2) + 1/16*(B*a*b*x^(5/2) + 7*A*b^2*x^(5/2) - 3*B*a^2*sqrt(x) + 11
*A*a*b*sqrt(x))/((b*x^2 + a)^2*a^2*b)