3.417 \(\int \frac{A+B x}{x^{5/2} \left (a+c x^2\right )} \, dx\)

Optimal. Leaf size=278 \[ -\frac{\sqrt [4]{c} \left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{2 \sqrt{2} a^{7/4}}+\frac{\sqrt [4]{c} \left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{2 \sqrt{2} a^{7/4}}+\frac{\sqrt [4]{c} \left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{7/4}}-\frac{\sqrt [4]{c} \left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{7/4}}-\frac{2 A}{3 a x^{3/2}}-\frac{2 B}{a \sqrt{x}} \]

[Out]

(-2*A)/(3*a*x^(3/2)) - (2*B)/(a*Sqrt[x]) + ((Sqrt[a]*B + A*Sqrt[c])*c^(1/4)*ArcT
an[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(7/4)) - ((Sqrt[a]*B + A*S
qrt[c])*c^(1/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(7/4))
- ((Sqrt[a]*B - A*Sqrt[c])*c^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x
] + Sqrt[c]*x])/(2*Sqrt[2]*a^(7/4)) + ((Sqrt[a]*B - A*Sqrt[c])*c^(1/4)*Log[Sqrt[
a] + Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(2*Sqrt[2]*a^(7/4))

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Rubi [A]  time = 0.56495, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{\sqrt [4]{c} \left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{2 \sqrt{2} a^{7/4}}+\frac{\sqrt [4]{c} \left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{2 \sqrt{2} a^{7/4}}+\frac{\sqrt [4]{c} \left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{7/4}}-\frac{\sqrt [4]{c} \left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{7/4}}-\frac{2 A}{3 a x^{3/2}}-\frac{2 B}{a \sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(x^(5/2)*(a + c*x^2)),x]

[Out]

(-2*A)/(3*a*x^(3/2)) - (2*B)/(a*Sqrt[x]) + ((Sqrt[a]*B + A*Sqrt[c])*c^(1/4)*ArcT
an[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(7/4)) - ((Sqrt[a]*B + A*S
qrt[c])*c^(1/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(7/4))
- ((Sqrt[a]*B - A*Sqrt[c])*c^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x
] + Sqrt[c]*x])/(2*Sqrt[2]*a^(7/4)) + ((Sqrt[a]*B - A*Sqrt[c])*c^(1/4)*Log[Sqrt[
a] + Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(2*Sqrt[2]*a^(7/4))

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Rubi in Sympy [A]  time = 103.115, size = 262, normalized size = 0.94 \[ - \frac{2 A}{3 a x^{\frac{3}{2}}} - \frac{2 B}{a \sqrt{x}} + \frac{\sqrt{2} \sqrt [4]{c} \left (A \sqrt{c} - B \sqrt{a}\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} \sqrt{x} + \sqrt{a} \sqrt{c} + c x \right )}}{4 a^{\frac{7}{4}}} - \frac{\sqrt{2} \sqrt [4]{c} \left (A \sqrt{c} - B \sqrt{a}\right ) \log{\left (\sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} \sqrt{x} + \sqrt{a} \sqrt{c} + c x \right )}}{4 a^{\frac{7}{4}}} + \frac{\sqrt{2} \sqrt [4]{c} \left (A \sqrt{c} + B \sqrt{a}\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 a^{\frac{7}{4}}} - \frac{\sqrt{2} \sqrt [4]{c} \left (A \sqrt{c} + B \sqrt{a}\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 a^{\frac{7}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x**(5/2)/(c*x**2+a),x)

[Out]

-2*A/(3*a*x**(3/2)) - 2*B/(a*sqrt(x)) + sqrt(2)*c**(1/4)*(A*sqrt(c) - B*sqrt(a))
*log(-sqrt(2)*a**(1/4)*c**(3/4)*sqrt(x) + sqrt(a)*sqrt(c) + c*x)/(4*a**(7/4)) -
sqrt(2)*c**(1/4)*(A*sqrt(c) - B*sqrt(a))*log(sqrt(2)*a**(1/4)*c**(3/4)*sqrt(x) +
sqrt(a)*sqrt(c) + c*x)/(4*a**(7/4)) + sqrt(2)*c**(1/4)*(A*sqrt(c) + B*sqrt(a))*
atan(1 - sqrt(2)*c**(1/4)*sqrt(x)/a**(1/4))/(2*a**(7/4)) - sqrt(2)*c**(1/4)*(A*s
qrt(c) + B*sqrt(a))*atan(1 + sqrt(2)*c**(1/4)*sqrt(x)/a**(1/4))/(2*a**(7/4))

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Mathematica [A]  time = 0.776322, size = 276, normalized size = 0.99 \[ \frac{3 \sqrt{2} \sqrt [4]{c} \left (\sqrt [4]{a} A \sqrt{c}-a^{3/4} B\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )+3 \sqrt{2} \sqrt [4]{c} \left (a^{3/4} B-\sqrt [4]{a} A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )+6 \sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )-6 \sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )-\frac{8 a A}{x^{3/2}}-\frac{24 a B}{\sqrt{x}}}{12 a^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(x^(5/2)*(a + c*x^2)),x]

[Out]

((-8*a*A)/x^(3/2) - (24*a*B)/Sqrt[x] + 6*Sqrt[2]*a^(1/4)*(Sqrt[a]*B + A*Sqrt[c])
*c^(1/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)] - 6*Sqrt[2]*a^(1/4)*(Sqrt
[a]*B + A*Sqrt[c])*c^(1/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)] + 3*Sqr
t[2]*(-(a^(3/4)*B) + a^(1/4)*A*Sqrt[c])*c^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^
(1/4)*Sqrt[x] + Sqrt[c]*x] + 3*Sqrt[2]*(a^(3/4)*B - a^(1/4)*A*Sqrt[c])*c^(1/4)*L
og[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(12*a^2)

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Maple [A]  time = 0.017, size = 289, normalized size = 1. \[ -{\frac{2\,A}{3\,a}{x}^{-{\frac{3}{2}}}}-2\,{\frac{B}{\sqrt{x}a}}-{\frac{Ac\sqrt{2}}{4\,{a}^{2}}\sqrt [4]{{\frac{a}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }-{\frac{Ac\sqrt{2}}{2\,{a}^{2}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }-{\frac{Ac\sqrt{2}}{2\,{a}^{2}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }-{\frac{B\sqrt{2}}{4\,a}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-{\frac{B\sqrt{2}}{2\,a}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-{\frac{B\sqrt{2}}{2\,a}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^(5/2)/(c*x^2+a),x)

[Out]

-2/3*A/a/x^(3/2)-2*B/a/x^(1/2)-1/4/a^2*c*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)))-1/2/a
^2*c*A*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)+1)-1/2/a^2*c*A*(a/
c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)-1)-1/4/a*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)))-1/2/a*B/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)+1)
-1/2/a*B/(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)*x^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.299191, size = 1077, normalized size = 3.87 \[ -\frac{3 \, a x^{\frac{3}{2}} \sqrt{-\frac{a^{3} \sqrt{-\frac{B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} + 2 \, A B c}{a^{3}}} \log \left (-{\left (B^{4} a^{2} c - A^{4} c^{3}\right )} \sqrt{x} +{\left (B a^{6} \sqrt{-\frac{B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} - A B^{2} a^{3} c + A^{3} a^{2} c^{2}\right )} \sqrt{-\frac{a^{3} \sqrt{-\frac{B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} + 2 \, A B c}{a^{3}}}\right ) - 3 \, a x^{\frac{3}{2}} \sqrt{-\frac{a^{3} \sqrt{-\frac{B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} + 2 \, A B c}{a^{3}}} \log \left (-{\left (B^{4} a^{2} c - A^{4} c^{3}\right )} \sqrt{x} -{\left (B a^{6} \sqrt{-\frac{B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} - A B^{2} a^{3} c + A^{3} a^{2} c^{2}\right )} \sqrt{-\frac{a^{3} \sqrt{-\frac{B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} + 2 \, A B c}{a^{3}}}\right ) - 3 \, a x^{\frac{3}{2}} \sqrt{\frac{a^{3} \sqrt{-\frac{B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} - 2 \, A B c}{a^{3}}} \log \left (-{\left (B^{4} a^{2} c - A^{4} c^{3}\right )} \sqrt{x} +{\left (B a^{6} \sqrt{-\frac{B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} + A B^{2} a^{3} c - A^{3} a^{2} c^{2}\right )} \sqrt{\frac{a^{3} \sqrt{-\frac{B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} - 2 \, A B c}{a^{3}}}\right ) + 3 \, a x^{\frac{3}{2}} \sqrt{\frac{a^{3} \sqrt{-\frac{B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} - 2 \, A B c}{a^{3}}} \log \left (-{\left (B^{4} a^{2} c - A^{4} c^{3}\right )} \sqrt{x} -{\left (B a^{6} \sqrt{-\frac{B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} + A B^{2} a^{3} c - A^{3} a^{2} c^{2}\right )} \sqrt{\frac{a^{3} \sqrt{-\frac{B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} - 2 \, A B c}{a^{3}}}\right ) + 12 \, B x + 4 \, A}{6 \, a x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + a)*x^(5/2)),x, algorithm="fricas")

[Out]

-1/6*(3*a*x^(3/2)*sqrt(-(a^3*sqrt(-(B^4*a^2*c - 2*A^2*B^2*a*c^2 + A^4*c^3)/a^7)
+ 2*A*B*c)/a^3)*log(-(B^4*a^2*c - A^4*c^3)*sqrt(x) + (B*a^6*sqrt(-(B^4*a^2*c - 2
*A^2*B^2*a*c^2 + A^4*c^3)/a^7) - A*B^2*a^3*c + A^3*a^2*c^2)*sqrt(-(a^3*sqrt(-(B^
4*a^2*c - 2*A^2*B^2*a*c^2 + A^4*c^3)/a^7) + 2*A*B*c)/a^3)) - 3*a*x^(3/2)*sqrt(-(
a^3*sqrt(-(B^4*a^2*c - 2*A^2*B^2*a*c^2 + A^4*c^3)/a^7) + 2*A*B*c)/a^3)*log(-(B^4
*a^2*c - A^4*c^3)*sqrt(x) - (B*a^6*sqrt(-(B^4*a^2*c - 2*A^2*B^2*a*c^2 + A^4*c^3)
/a^7) - A*B^2*a^3*c + A^3*a^2*c^2)*sqrt(-(a^3*sqrt(-(B^4*a^2*c - 2*A^2*B^2*a*c^2
+ A^4*c^3)/a^7) + 2*A*B*c)/a^3)) - 3*a*x^(3/2)*sqrt((a^3*sqrt(-(B^4*a^2*c - 2*A
^2*B^2*a*c^2 + A^4*c^3)/a^7) - 2*A*B*c)/a^3)*log(-(B^4*a^2*c - A^4*c^3)*sqrt(x)
+ (B*a^6*sqrt(-(B^4*a^2*c - 2*A^2*B^2*a*c^2 + A^4*c^3)/a^7) + A*B^2*a^3*c - A^3*
a^2*c^2)*sqrt((a^3*sqrt(-(B^4*a^2*c - 2*A^2*B^2*a*c^2 + A^4*c^3)/a^7) - 2*A*B*c)
/a^3)) + 3*a*x^(3/2)*sqrt((a^3*sqrt(-(B^4*a^2*c - 2*A^2*B^2*a*c^2 + A^4*c^3)/a^7
) - 2*A*B*c)/a^3)*log(-(B^4*a^2*c - A^4*c^3)*sqrt(x) - (B*a^6*sqrt(-(B^4*a^2*c -
2*A^2*B^2*a*c^2 + A^4*c^3)/a^7) + A*B^2*a^3*c - A^3*a^2*c^2)*sqrt((a^3*sqrt(-(B
^4*a^2*c - 2*A^2*B^2*a*c^2 + A^4*c^3)/a^7) - 2*A*B*c)/a^3)) + 12*B*x + 4*A)/(a*x
^(3/2))

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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**(5/2)/(c*x**2+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.283472, size = 348, normalized size = 1.25 \[ -\frac{2 \,{\left (3 \, B x + A\right )}}{3 \, a x^{\frac{3}{2}}} - \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} A c^{2} + \left (a c^{3}\right )^{\frac{3}{4}} B\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{2 \, a^{2} c^{2}} - \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} A c^{2} + \left (a c^{3}\right )^{\frac{3}{4}} B\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{2 \, a^{2} c^{2}} - \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} A c^{2} - \left (a c^{3}\right )^{\frac{3}{4}} B\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{4 \, a^{2} c^{2}} + \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} A c^{2} - \left (a c^{3}\right )^{\frac{3}{4}} B\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{4 \, a^{2} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + a)*x^(5/2)),x, algorithm="giac")

[Out]

-2/3*(3*B*x + A)/(a*x^(3/2)) - 1/2*sqrt(2)*((a*c^3)^(1/4)*A*c^2 + (a*c^3)^(3/4)*
B)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/c)^(1/4) + 2*sqrt(x))/(a/c)^(1/4))/(a^2*c^2) -
1/2*sqrt(2)*((a*c^3)^(1/4)*A*c^2 + (a*c^3)^(3/4)*B)*arctan(-1/2*sqrt(2)*(sqrt(2
)*(a/c)^(1/4) - 2*sqrt(x))/(a/c)^(1/4))/(a^2*c^2) - 1/4*sqrt(2)*((a*c^3)^(1/4)*A
*c^2 - (a*c^3)^(3/4)*B)*ln(sqrt(2)*sqrt(x)*(a/c)^(1/4) + x + sqrt(a/c))/(a^2*c^2
) + 1/4*sqrt(2)*((a*c^3)^(1/4)*A*c^2 - (a*c^3)^(3/4)*B)*ln(-sqrt(2)*sqrt(x)*(a/c
)^(1/4) + x + sqrt(a/c))/(a^2*c^2)