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

Optimal. Leaf size=333 \[ -\frac{3 \left (7 \sqrt{a} B+15 A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{64 \sqrt{2} a^{13/4} \sqrt [4]{c}}+\frac{3 \left (7 \sqrt{a} B+15 A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{64 \sqrt{2} a^{13/4} \sqrt [4]{c}}-\frac{3 \left (7 \sqrt{a} B-15 A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{13/4} \sqrt [4]{c}}+\frac{3 \left (7 \sqrt{a} B-15 A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{13/4} \sqrt [4]{c}}-\frac{45 A}{16 a^3 \sqrt{x}}+\frac{9 A+7 B x}{16 a^2 \sqrt{x} \left (a+c x^2\right )}+\frac{A+B x}{4 a \sqrt{x} \left (a+c x^2\right )^2} \]

[Out]

(-45*A)/(16*a^3*Sqrt[x]) + (A + B*x)/(4*a*Sqrt[x]*(a + c*x^2)^2) + (9*A + 7*B*x)
/(16*a^2*Sqrt[x]*(a + c*x^2)) - (3*(7*Sqrt[a]*B - 15*A*Sqrt[c])*ArcTan[1 - (Sqrt
[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(13/4)*c^(1/4)) + (3*(7*Sqrt[a]*B -
15*A*Sqrt[c])*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(13/
4)*c^(1/4)) - (3*(7*Sqrt[a]*B + 15*A*Sqrt[c])*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1
/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*a^(13/4)*c^(1/4)) + (3*(7*Sqrt[a]*B + 15*A
*Sqrt[c])*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2
]*a^(13/4)*c^(1/4))

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Rubi [A]  time = 0.754228, antiderivative size = 333, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45 \[ -\frac{3 \left (7 \sqrt{a} B+15 A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{64 \sqrt{2} a^{13/4} \sqrt [4]{c}}+\frac{3 \left (7 \sqrt{a} B+15 A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{64 \sqrt{2} a^{13/4} \sqrt [4]{c}}-\frac{3 \left (7 \sqrt{a} B-15 A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{13/4} \sqrt [4]{c}}+\frac{3 \left (7 \sqrt{a} B-15 A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{13/4} \sqrt [4]{c}}-\frac{45 A}{16 a^3 \sqrt{x}}+\frac{9 A+7 B x}{16 a^2 \sqrt{x} \left (a+c x^2\right )}+\frac{A+B x}{4 a \sqrt{x} \left (a+c x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(-45*A)/(16*a^3*Sqrt[x]) + (A + B*x)/(4*a*Sqrt[x]*(a + c*x^2)^2) + (9*A + 7*B*x)
/(16*a^2*Sqrt[x]*(a + c*x^2)) - (3*(7*Sqrt[a]*B - 15*A*Sqrt[c])*ArcTan[1 - (Sqrt
[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(13/4)*c^(1/4)) + (3*(7*Sqrt[a]*B -
15*A*Sqrt[c])*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(13/
4)*c^(1/4)) - (3*(7*Sqrt[a]*B + 15*A*Sqrt[c])*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1
/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*a^(13/4)*c^(1/4)) + (3*(7*Sqrt[a]*B + 15*A
*Sqrt[c])*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2
]*a^(13/4)*c^(1/4))

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Rubi in Sympy [A]  time = 136.895, size = 325, normalized size = 0.98 \[ - \frac{45 A}{16 a^{3} \sqrt{x}} + \frac{A + B x}{4 a \sqrt{x} \left (a + c x^{2}\right )^{2}} + \frac{\frac{9 A}{2} + \frac{7 B x}{2}}{8 a^{2} \sqrt{x} \left (a + c x^{2}\right )} + \frac{3 \sqrt{2} \left (15 A \sqrt{c} - 7 B \sqrt{a}\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{13}{4}} \sqrt [4]{c}} - \frac{3 \sqrt{2} \left (15 A \sqrt{c} - 7 B \sqrt{a}\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{13}{4}} \sqrt [4]{c}} - \frac{3 \sqrt{2} \left (15 A \sqrt{c} + 7 B \sqrt{a}\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} \sqrt{x} + \sqrt{a} \sqrt{c} + c x \right )}}{128 a^{\frac{13}{4}} \sqrt [4]{c}} + \frac{3 \sqrt{2} \left (15 A \sqrt{c} + 7 B \sqrt{a}\right ) \log{\left (\sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} \sqrt{x} + \sqrt{a} \sqrt{c} + c x \right )}}{128 a^{\frac{13}{4}} \sqrt [4]{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-45*A/(16*a**3*sqrt(x)) + (A + B*x)/(4*a*sqrt(x)*(a + c*x**2)**2) + (9*A/2 + 7*B
*x/2)/(8*a**2*sqrt(x)*(a + c*x**2)) + 3*sqrt(2)*(15*A*sqrt(c) - 7*B*sqrt(a))*ata
n(1 - sqrt(2)*c**(1/4)*sqrt(x)/a**(1/4))/(64*a**(13/4)*c**(1/4)) - 3*sqrt(2)*(15
*A*sqrt(c) - 7*B*sqrt(a))*atan(1 + sqrt(2)*c**(1/4)*sqrt(x)/a**(1/4))/(64*a**(13
/4)*c**(1/4)) - 3*sqrt(2)*(15*A*sqrt(c) + 7*B*sqrt(a))*log(-sqrt(2)*a**(1/4)*c**
(3/4)*sqrt(x) + sqrt(a)*sqrt(c) + c*x)/(128*a**(13/4)*c**(1/4)) + 3*sqrt(2)*(15*
A*sqrt(c) + 7*B*sqrt(a))*log(sqrt(2)*a**(1/4)*c**(3/4)*sqrt(x) + sqrt(a)*sqrt(c)
+ c*x)/(128*a**(13/4)*c**(1/4))

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Mathematica [A]  time = 0.786849, size = 328, normalized size = 0.98 \[ \frac{-\frac{3 \sqrt{2} \left (15 a^{3/4} A \sqrt{c}+7 a^{5/4} B\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{\sqrt [4]{c}}+\frac{3 \sqrt{2} \left (15 a^{3/4} A \sqrt{c}+7 a^{5/4} B\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{\sqrt [4]{c}}-\frac{6 \sqrt{2} a^{3/4} \left (7 \sqrt{a} B-15 A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{c}}+\frac{6 \sqrt{2} a^{3/4} \left (7 \sqrt{a} B-15 A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{c}}+\frac{32 a^2 \sqrt{x} (a B-A c x)}{\left (a+c x^2\right )^2}+\frac{8 a \sqrt{x} (7 a B-13 A c x)}{a+c x^2}-\frac{256 a A}{\sqrt{x}}}{128 a^4} \]

Antiderivative was successfully verified.

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

[Out]

((-256*a*A)/Sqrt[x] + (32*a^2*Sqrt[x]*(a*B - A*c*x))/(a + c*x^2)^2 + (8*a*Sqrt[x
]*(7*a*B - 13*A*c*x))/(a + c*x^2) - (6*Sqrt[2]*a^(3/4)*(7*Sqrt[a]*B - 15*A*Sqrt[
c])*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/c^(1/4) + (6*Sqrt[2]*a^(3/4)*
(7*Sqrt[a]*B - 15*A*Sqrt[c])*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/c^(1
/4) - (3*Sqrt[2]*(7*a^(5/4)*B + 15*a^(3/4)*A*Sqrt[c])*Log[Sqrt[a] - Sqrt[2]*a^(1
/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/c^(1/4) + (3*Sqrt[2]*(7*a^(5/4)*B + 15*a^(3/4)
*A*Sqrt[c])*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/c^(1/4))
/(128*a^4)

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Maple [A]  time = 0.029, size = 354, normalized size = 1.1 \[ -2\,{\frac{A}{{a}^{3}\sqrt{x}}}-{\frac{13\,A{c}^{2}}{16\,{a}^{3} \left ( c{x}^{2}+a \right ) ^{2}}{x}^{{\frac{7}{2}}}}+{\frac{7\,Bc}{16\,{a}^{2} \left ( c{x}^{2}+a \right ) ^{2}}{x}^{{\frac{5}{2}}}}-{\frac{17\,Ac}{16\,{a}^{2} \left ( c{x}^{2}+a \right ) ^{2}}{x}^{{\frac{3}{2}}}}+{\frac{11\,B}{16\,a \left ( c{x}^{2}+a \right ) ^{2}}\sqrt{x}}+{\frac{21\,B\sqrt{2}}{64\,{a}^{3}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }+{\frac{21\,B\sqrt{2}}{128\,{a}^{3}}\sqrt [4]{{\frac{a}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }+{\frac{21\,B\sqrt{2}}{64\,{a}^{3}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }-{\frac{45\,A\sqrt{2}}{128\,{a}^{3}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-{\frac{45\,A\sqrt{2}}{64\,{a}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-{\frac{45\,A\sqrt{2}}{64\,{a}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A/a^3/x^(1/2)-13/16/a^3/(c*x^2+a)^2*A*x^(7/2)*c^2+7/16/a^2/(c*x^2+a)^2*B*x^(5
/2)*c-17/16/a^2/(c*x^2+a)^2*A*x^(3/2)*c+11/16/a/(c*x^2+a)^2*B*x^(1/2)+21/64/a^3*
B*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)-1)+21/128/a^3*B*(a/c)^(
1/4)*2^(1/2)*ln((x+(a/c)^(1/4)*x^(1/2)*2^(1/2)+(a/c)^(1/2))/(x-(a/c)^(1/4)*x^(1/
2)*2^(1/2)+(a/c)^(1/2)))+21/64/a^3*B*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1
/4)*x^(1/2)+1)-45/128/a^3*A/(a/c)^(1/4)*2^(1/2)*ln((x-(a/c)^(1/4)*x^(1/2)*2^(1/2
)+(a/c)^(1/2))/(x+(a/c)^(1/4)*x^(1/2)*2^(1/2)+(a/c)^(1/2)))-45/64/a^3*A/(a/c)^(1
/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)+1)-45/64/a^3*A/(a/c)^(1/4)*2^(1/2
)*arctan(2^(1/2)/(a/c)^(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 + A)/((c*x^2 + a)^3*x^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.306877, size = 1292, normalized size = 3.88 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/64*(180*A*c^2*x^4 - 28*B*a*c*x^3 + 324*A*a*c*x^2 - 44*B*a^2*x + 128*A*a^2 + 3
*(a^3*c^2*x^4 + 2*a^4*c*x^2 + a^5)*sqrt(x)*sqrt((a^6*sqrt(-(2401*B^4*a^2 - 22050
*A^2*B^2*a*c + 50625*A^4*c^2)/(a^13*c)) + 210*A*B)/a^6)*log(-9*(2401*B^4*a^2 - 5
0625*A^4*c^2)*sqrt(x) + 9*(15*A*a^10*c*sqrt(-(2401*B^4*a^2 - 22050*A^2*B^2*a*c +
50625*A^4*c^2)/(a^13*c)) + 343*B^3*a^5 - 1575*A^2*B*a^4*c)*sqrt((a^6*sqrt(-(240
1*B^4*a^2 - 22050*A^2*B^2*a*c + 50625*A^4*c^2)/(a^13*c)) + 210*A*B)/a^6)) - 3*(a
^3*c^2*x^4 + 2*a^4*c*x^2 + a^5)*sqrt(x)*sqrt((a^6*sqrt(-(2401*B^4*a^2 - 22050*A^
2*B^2*a*c + 50625*A^4*c^2)/(a^13*c)) + 210*A*B)/a^6)*log(-9*(2401*B^4*a^2 - 5062
5*A^4*c^2)*sqrt(x) - 9*(15*A*a^10*c*sqrt(-(2401*B^4*a^2 - 22050*A^2*B^2*a*c + 50
625*A^4*c^2)/(a^13*c)) + 343*B^3*a^5 - 1575*A^2*B*a^4*c)*sqrt((a^6*sqrt(-(2401*B
^4*a^2 - 22050*A^2*B^2*a*c + 50625*A^4*c^2)/(a^13*c)) + 210*A*B)/a^6)) - 3*(a^3*
c^2*x^4 + 2*a^4*c*x^2 + a^5)*sqrt(x)*sqrt(-(a^6*sqrt(-(2401*B^4*a^2 - 22050*A^2*
B^2*a*c + 50625*A^4*c^2)/(a^13*c)) - 210*A*B)/a^6)*log(-9*(2401*B^4*a^2 - 50625*
A^4*c^2)*sqrt(x) + 9*(15*A*a^10*c*sqrt(-(2401*B^4*a^2 - 22050*A^2*B^2*a*c + 5062
5*A^4*c^2)/(a^13*c)) - 343*B^3*a^5 + 1575*A^2*B*a^4*c)*sqrt(-(a^6*sqrt(-(2401*B^
4*a^2 - 22050*A^2*B^2*a*c + 50625*A^4*c^2)/(a^13*c)) - 210*A*B)/a^6)) + 3*(a^3*c
^2*x^4 + 2*a^4*c*x^2 + a^5)*sqrt(x)*sqrt(-(a^6*sqrt(-(2401*B^4*a^2 - 22050*A^2*B
^2*a*c + 50625*A^4*c^2)/(a^13*c)) - 210*A*B)/a^6)*log(-9*(2401*B^4*a^2 - 50625*A
^4*c^2)*sqrt(x) - 9*(15*A*a^10*c*sqrt(-(2401*B^4*a^2 - 22050*A^2*B^2*a*c + 50625
*A^4*c^2)/(a^13*c)) - 343*B^3*a^5 + 1575*A^2*B*a^4*c)*sqrt(-(a^6*sqrt(-(2401*B^4
*a^2 - 22050*A^2*B^2*a*c + 50625*A^4*c^2)/(a^13*c)) - 210*A*B)/a^6)))/((a^3*c^2*
x^4 + 2*a^4*c*x^2 + a^5)*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+A)/x**(3/2)/(c*x**2+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.287466, size = 410, normalized size = 1.23 \[ -\frac{2 \, A}{a^{3} \sqrt{x}} + \frac{3 \, \sqrt{2}{\left (7 \, \left (a c^{3}\right )^{\frac{1}{4}} B a c - 15 \, \left (a c^{3}\right )^{\frac{3}{4}} A\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{64 \, a^{4} c^{2}} + \frac{3 \, \sqrt{2}{\left (7 \, \left (a c^{3}\right )^{\frac{1}{4}} B a c - 15 \, \left (a c^{3}\right )^{\frac{3}{4}} A\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{64 \, a^{4} c^{2}} + \frac{3 \, \sqrt{2}{\left (7 \, \left (a c^{3}\right )^{\frac{1}{4}} B a c + 15 \, \left (a c^{3}\right )^{\frac{3}{4}} A\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{128 \, a^{4} c^{2}} - \frac{3 \, \sqrt{2}{\left (7 \, \left (a c^{3}\right )^{\frac{1}{4}} B a c + 15 \, \left (a c^{3}\right )^{\frac{3}{4}} A\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{128 \, a^{4} c^{2}} - \frac{13 \, A c^{2} x^{\frac{7}{2}} - 7 \, B a c x^{\frac{5}{2}} + 17 \, A a c x^{\frac{3}{2}} - 11 \, B a^{2} \sqrt{x}}{16 \,{\left (c x^{2} + a\right )}^{2} a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A/(a^3*sqrt(x)) + 3/64*sqrt(2)*(7*(a*c^3)^(1/4)*B*a*c - 15*(a*c^3)^(3/4)*A)*a
rctan(1/2*sqrt(2)*(sqrt(2)*(a/c)^(1/4) + 2*sqrt(x))/(a/c)^(1/4))/(a^4*c^2) + 3/6
4*sqrt(2)*(7*(a*c^3)^(1/4)*B*a*c - 15*(a*c^3)^(3/4)*A)*arctan(-1/2*sqrt(2)*(sqrt
(2)*(a/c)^(1/4) - 2*sqrt(x))/(a/c)^(1/4))/(a^4*c^2) + 3/128*sqrt(2)*(7*(a*c^3)^(
1/4)*B*a*c + 15*(a*c^3)^(3/4)*A)*ln(sqrt(2)*sqrt(x)*(a/c)^(1/4) + x + sqrt(a/c))
/(a^4*c^2) - 3/128*sqrt(2)*(7*(a*c^3)^(1/4)*B*a*c + 15*(a*c^3)^(3/4)*A)*ln(-sqrt
(2)*sqrt(x)*(a/c)^(1/4) + x + sqrt(a/c))/(a^4*c^2) - 1/16*(13*A*c^2*x^(7/2) - 7*
B*a*c*x^(5/2) + 17*A*a*c*x^(3/2) - 11*B*a^2*sqrt(x))/((c*x^2 + a)^2*a^3)