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

Optimal. Leaf size=289 \[ -\frac{(a B+5 A b) \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{12 \sqrt{3} a^{11/6} b^{7/6}}+\frac{(a B+5 A b) \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{12 \sqrt{3} a^{11/6} b^{7/6}}-\frac{(a B+5 A b) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{18 a^{11/6} b^{7/6}}+\frac{(a B+5 A b) \tan ^{-1}\left (\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}+\sqrt{3}\right )}{18 a^{11/6} b^{7/6}}+\frac{(a B+5 A b) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{9 a^{11/6} b^{7/6}}+\frac{\sqrt{x} (A b-a B)}{3 a b \left (a+b x^3\right )} \]

[Out]

((A*b - a*B)*Sqrt[x])/(3*a*b*(a + b*x^3)) - ((5*A*b + a*B)*ArcTan[Sqrt[3] - (2*b
^(1/6)*Sqrt[x])/a^(1/6)])/(18*a^(11/6)*b^(7/6)) + ((5*A*b + a*B)*ArcTan[Sqrt[3]
+ (2*b^(1/6)*Sqrt[x])/a^(1/6)])/(18*a^(11/6)*b^(7/6)) + ((5*A*b + a*B)*ArcTan[(b
^(1/6)*Sqrt[x])/a^(1/6)])/(9*a^(11/6)*b^(7/6)) - ((5*A*b + a*B)*Log[a^(1/3) - Sq
rt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/(12*Sqrt[3]*a^(11/6)*b^(7/6)) + ((5*
A*b + a*B)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/(12*Sqrt[
3]*a^(11/6)*b^(7/6))

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Rubi [A]  time = 1.02019, antiderivative size = 289, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ -\frac{(a B+5 A b) \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{12 \sqrt{3} a^{11/6} b^{7/6}}+\frac{(a B+5 A b) \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{12 \sqrt{3} a^{11/6} b^{7/6}}-\frac{(a B+5 A b) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{18 a^{11/6} b^{7/6}}+\frac{(a B+5 A b) \tan ^{-1}\left (\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}+\sqrt{3}\right )}{18 a^{11/6} b^{7/6}}+\frac{(a B+5 A b) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{9 a^{11/6} b^{7/6}}+\frac{\sqrt{x} (A b-a B)}{3 a b \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

((A*b - a*B)*Sqrt[x])/(3*a*b*(a + b*x^3)) - ((5*A*b + a*B)*ArcTan[Sqrt[3] - (2*b
^(1/6)*Sqrt[x])/a^(1/6)])/(18*a^(11/6)*b^(7/6)) + ((5*A*b + a*B)*ArcTan[Sqrt[3]
+ (2*b^(1/6)*Sqrt[x])/a^(1/6)])/(18*a^(11/6)*b^(7/6)) + ((5*A*b + a*B)*ArcTan[(b
^(1/6)*Sqrt[x])/a^(1/6)])/(9*a^(11/6)*b^(7/6)) - ((5*A*b + a*B)*Log[a^(1/3) - Sq
rt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/(12*Sqrt[3]*a^(11/6)*b^(7/6)) + ((5*
A*b + a*B)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/(12*Sqrt[
3]*a^(11/6)*b^(7/6))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.374282, size = 244, normalized size = 0.84 \[ \frac{-\frac{12 a^{5/6} \sqrt [6]{b} \sqrt{x} (a B-A b)}{a+b x^3}-\sqrt{3} (a B+5 A b) \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )+\sqrt{3} (a B+5 A b) \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )-2 (a B+5 A b) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )+2 (a B+5 A b) \tan ^{-1}\left (\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}+\sqrt{3}\right )+4 (a B+5 A b) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{36 a^{11/6} b^{7/6}} \]

Antiderivative was successfully verified.

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

[Out]

((-12*a^(5/6)*b^(1/6)*(-(A*b) + a*B)*Sqrt[x])/(a + b*x^3) - 2*(5*A*b + a*B)*ArcT
an[Sqrt[3] - (2*b^(1/6)*Sqrt[x])/a^(1/6)] + 2*(5*A*b + a*B)*ArcTan[Sqrt[3] + (2*
b^(1/6)*Sqrt[x])/a^(1/6)] + 4*(5*A*b + a*B)*ArcTan[(b^(1/6)*Sqrt[x])/a^(1/6)] -
Sqrt[3]*(5*A*b + a*B)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x]
+ Sqrt[3]*(5*A*b + a*B)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)
*x])/(36*a^(11/6)*b^(7/6))

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Maple [A]  time = 0.058, size = 381, normalized size = 1.3 \[{\frac{Ab-Ba}{3\,ab \left ( b{x}^{3}+a \right ) }\sqrt{x}}+{\frac{5\,A}{9\,{a}^{2}}\sqrt [6]{{\frac{a}{b}}}\arctan \left ({1\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ) }+{\frac{B}{9\,ab}\sqrt [6]{{\frac{a}{b}}}\arctan \left ({1\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ) }-{\frac{5\,\sqrt{3}A}{36\,{a}^{2}}\sqrt [6]{{\frac{a}{b}}}\ln \left ( x-\sqrt{3}\sqrt [6]{{\frac{a}{b}}}\sqrt{x}+\sqrt [3]{{\frac{a}{b}}} \right ) }-{\frac{\sqrt{3}B}{36\,ab}\sqrt [6]{{\frac{a}{b}}}\ln \left ( x-\sqrt{3}\sqrt [6]{{\frac{a}{b}}}\sqrt{x}+\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{5\,A}{18\,{a}^{2}}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( -\sqrt{3}+2\,{\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ) }+{\frac{B}{18\,ab}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( -\sqrt{3}+2\,{\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ) }+{\frac{5\,\sqrt{3}A}{36\,{a}^{2}}\sqrt [6]{{\frac{a}{b}}}\ln \left ( x+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}\sqrt{x}+\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{\sqrt{3}B}{36\,ab}\sqrt [6]{{\frac{a}{b}}}\ln \left ( x+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}\sqrt{x}+\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{5\,A}{18\,{a}^{2}}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( 2\,{\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+\sqrt{3} \right ) }+{\frac{B}{18\,ab}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( 2\,{\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+\sqrt{3} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(A*b-B*a)*x^(1/2)/a/b/(b*x^3+a)+5/9/a^2*(a/b)^(1/6)*arctan(x^(1/2)/(a/b)^(1/
6))*A+1/9/b/a*(a/b)^(1/6)*arctan(x^(1/2)/(a/b)^(1/6))*B-5/36/a^2*3^(1/2)*(a/b)^(
1/6)*ln(x-3^(1/2)*(a/b)^(1/6)*x^(1/2)+(a/b)^(1/3))*A-1/36/b/a*3^(1/2)*(a/b)^(1/6
)*ln(x-3^(1/2)*(a/b)^(1/6)*x^(1/2)+(a/b)^(1/3))*B+5/18/a^2*(a/b)^(1/6)*arctan(-3
^(1/2)+2*x^(1/2)/(a/b)^(1/6))*A+1/18/b/a*(a/b)^(1/6)*arctan(-3^(1/2)+2*x^(1/2)/(
a/b)^(1/6))*B+5/36/a^2*3^(1/2)*(a/b)^(1/6)*ln(x+3^(1/2)*(a/b)^(1/6)*x^(1/2)+(a/b
)^(1/3))*A+1/36/b/a*3^(1/2)*(a/b)^(1/6)*ln(x+3^(1/2)*(a/b)^(1/6)*x^(1/2)+(a/b)^(
1/3))*B+5/18/a^2*(a/b)^(1/6)*arctan(2*x^(1/2)/(a/b)^(1/6)+3^(1/2))*A+1/18/b/a*(a
/b)^(1/6)*arctan(2*x^(1/2)/(a/b)^(1/6)+3^(1/2))*B

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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^3 + A)/((b*x^3 + a)^2*sqrt(x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.276739, size = 2988, normalized size = 10.34 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/36*(4*sqrt(3)*(a*b^2*x^3 + a^2*b)*(-(B^6*a^6 + 30*A*B^5*a^5*b + 375*A^2*B^4*a
^4*b^2 + 2500*A^3*B^3*a^3*b^3 + 9375*A^4*B^2*a^2*b^4 + 18750*A^5*B*a*b^5 + 15625
*A^6*b^6)/(a^11*b^7))^(1/6)*arctan(sqrt(3)*a^2*b*(-(B^6*a^6 + 30*A*B^5*a^5*b + 3
75*A^2*B^4*a^4*b^2 + 2500*A^3*B^3*a^3*b^3 + 9375*A^4*B^2*a^2*b^4 + 18750*A^5*B*a
*b^5 + 15625*A^6*b^6)/(a^11*b^7))^(1/6)/(a^2*b*(-(B^6*a^6 + 30*A*B^5*a^5*b + 375
*A^2*B^4*a^4*b^2 + 2500*A^3*B^3*a^3*b^3 + 9375*A^4*B^2*a^2*b^4 + 18750*A^5*B*a*b
^5 + 15625*A^6*b^6)/(a^11*b^7))^(1/6) + 2*(B*a + 5*A*b)*sqrt(x) + 2*sqrt(a^4*b^2
*(-(B^6*a^6 + 30*A*B^5*a^5*b + 375*A^2*B^4*a^4*b^2 + 2500*A^3*B^3*a^3*b^3 + 9375
*A^4*B^2*a^2*b^4 + 18750*A^5*B*a*b^5 + 15625*A^6*b^6)/(a^11*b^7))^(1/3) + (B^2*a
^2 + 10*A*B*a*b + 25*A^2*b^2)*x + (B*a^3*b + 5*A*a^2*b^2)*sqrt(x)*(-(B^6*a^6 + 3
0*A*B^5*a^5*b + 375*A^2*B^4*a^4*b^2 + 2500*A^3*B^3*a^3*b^3 + 9375*A^4*B^2*a^2*b^
4 + 18750*A^5*B*a*b^5 + 15625*A^6*b^6)/(a^11*b^7))^(1/6)))) + 4*sqrt(3)*(a*b^2*x
^3 + a^2*b)*(-(B^6*a^6 + 30*A*B^5*a^5*b + 375*A^2*B^4*a^4*b^2 + 2500*A^3*B^3*a^3
*b^3 + 9375*A^4*B^2*a^2*b^4 + 18750*A^5*B*a*b^5 + 15625*A^6*b^6)/(a^11*b^7))^(1/
6)*arctan(-sqrt(3)*a^2*b*(-(B^6*a^6 + 30*A*B^5*a^5*b + 375*A^2*B^4*a^4*b^2 + 250
0*A^3*B^3*a^3*b^3 + 9375*A^4*B^2*a^2*b^4 + 18750*A^5*B*a*b^5 + 15625*A^6*b^6)/(a
^11*b^7))^(1/6)/(a^2*b*(-(B^6*a^6 + 30*A*B^5*a^5*b + 375*A^2*B^4*a^4*b^2 + 2500*
A^3*B^3*a^3*b^3 + 9375*A^4*B^2*a^2*b^4 + 18750*A^5*B*a*b^5 + 15625*A^6*b^6)/(a^1
1*b^7))^(1/6) - 2*(B*a + 5*A*b)*sqrt(x) - 2*sqrt(a^4*b^2*(-(B^6*a^6 + 30*A*B^5*a
^5*b + 375*A^2*B^4*a^4*b^2 + 2500*A^3*B^3*a^3*b^3 + 9375*A^4*B^2*a^2*b^4 + 18750
*A^5*B*a*b^5 + 15625*A^6*b^6)/(a^11*b^7))^(1/3) + (B^2*a^2 + 10*A*B*a*b + 25*A^2
*b^2)*x - (B*a^3*b + 5*A*a^2*b^2)*sqrt(x)*(-(B^6*a^6 + 30*A*B^5*a^5*b + 375*A^2*
B^4*a^4*b^2 + 2500*A^3*B^3*a^3*b^3 + 9375*A^4*B^2*a^2*b^4 + 18750*A^5*B*a*b^5 +
15625*A^6*b^6)/(a^11*b^7))^(1/6)))) - (a*b^2*x^3 + a^2*b)*(-(B^6*a^6 + 30*A*B^5*
a^5*b + 375*A^2*B^4*a^4*b^2 + 2500*A^3*B^3*a^3*b^3 + 9375*A^4*B^2*a^2*b^4 + 1875
0*A^5*B*a*b^5 + 15625*A^6*b^6)/(a^11*b^7))^(1/6)*log(a^4*b^2*(-(B^6*a^6 + 30*A*B
^5*a^5*b + 375*A^2*B^4*a^4*b^2 + 2500*A^3*B^3*a^3*b^3 + 9375*A^4*B^2*a^2*b^4 + 1
8750*A^5*B*a*b^5 + 15625*A^6*b^6)/(a^11*b^7))^(1/3) + (B^2*a^2 + 10*A*B*a*b + 25
*A^2*b^2)*x + (B*a^3*b + 5*A*a^2*b^2)*sqrt(x)*(-(B^6*a^6 + 30*A*B^5*a^5*b + 375*
A^2*B^4*a^4*b^2 + 2500*A^3*B^3*a^3*b^3 + 9375*A^4*B^2*a^2*b^4 + 18750*A^5*B*a*b^
5 + 15625*A^6*b^6)/(a^11*b^7))^(1/6)) + (a*b^2*x^3 + a^2*b)*(-(B^6*a^6 + 30*A*B^
5*a^5*b + 375*A^2*B^4*a^4*b^2 + 2500*A^3*B^3*a^3*b^3 + 9375*A^4*B^2*a^2*b^4 + 18
750*A^5*B*a*b^5 + 15625*A^6*b^6)/(a^11*b^7))^(1/6)*log(a^4*b^2*(-(B^6*a^6 + 30*A
*B^5*a^5*b + 375*A^2*B^4*a^4*b^2 + 2500*A^3*B^3*a^3*b^3 + 9375*A^4*B^2*a^2*b^4 +
18750*A^5*B*a*b^5 + 15625*A^6*b^6)/(a^11*b^7))^(1/3) + (B^2*a^2 + 10*A*B*a*b +
25*A^2*b^2)*x - (B*a^3*b + 5*A*a^2*b^2)*sqrt(x)*(-(B^6*a^6 + 30*A*B^5*a^5*b + 37
5*A^2*B^4*a^4*b^2 + 2500*A^3*B^3*a^3*b^3 + 9375*A^4*B^2*a^2*b^4 + 18750*A^5*B*a*
b^5 + 15625*A^6*b^6)/(a^11*b^7))^(1/6)) - 2*(a*b^2*x^3 + a^2*b)*(-(B^6*a^6 + 30*
A*B^5*a^5*b + 375*A^2*B^4*a^4*b^2 + 2500*A^3*B^3*a^3*b^3 + 9375*A^4*B^2*a^2*b^4
+ 18750*A^5*B*a*b^5 + 15625*A^6*b^6)/(a^11*b^7))^(1/6)*log(a^2*b*(-(B^6*a^6 + 30
*A*B^5*a^5*b + 375*A^2*B^4*a^4*b^2 + 2500*A^3*B^3*a^3*b^3 + 9375*A^4*B^2*a^2*b^4
+ 18750*A^5*B*a*b^5 + 15625*A^6*b^6)/(a^11*b^7))^(1/6) + (B*a + 5*A*b)*sqrt(x))
+ 2*(a*b^2*x^3 + a^2*b)*(-(B^6*a^6 + 30*A*B^5*a^5*b + 375*A^2*B^4*a^4*b^2 + 250
0*A^3*B^3*a^3*b^3 + 9375*A^4*B^2*a^2*b^4 + 18750*A^5*B*a*b^5 + 15625*A^6*b^6)/(a
^11*b^7))^(1/6)*log(-a^2*b*(-(B^6*a^6 + 30*A*B^5*a^5*b + 375*A^2*B^4*a^4*b^2 + 2
500*A^3*B^3*a^3*b^3 + 9375*A^4*B^2*a^2*b^4 + 18750*A^5*B*a*b^5 + 15625*A^6*b^6)/
(a^11*b^7))^(1/6) + (B*a + 5*A*b)*sqrt(x)) + 12*(B*a - A*b)*sqrt(x))/(a*b^2*x^3
+ a^2*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**3+A)/(b*x**3+a)**2/x**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.236299, size = 408, normalized size = 1.41 \[ \frac{\sqrt{3}{\left (\left (a b^{5}\right )^{\frac{1}{6}} B a + 5 \, \left (a b^{5}\right )^{\frac{1}{6}} A b\right )}{\rm ln}\left (\sqrt{3} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{6}} + x + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right )}{36 \, a^{2} b^{2}} - \frac{\sqrt{3}{\left (\left (a b^{5}\right )^{\frac{1}{6}} B a + 5 \, \left (a b^{5}\right )^{\frac{1}{6}} A b\right )}{\rm ln}\left (-\sqrt{3} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{6}} + x + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right )}{36 \, a^{2} b^{2}} - \frac{B a \sqrt{x} - A b \sqrt{x}}{3 \,{\left (b x^{3} + a\right )} a b} + \frac{{\left (\left (a b^{5}\right )^{\frac{1}{6}} B a + 5 \, \left (a b^{5}\right )^{\frac{1}{6}} A b\right )} \arctan \left (\frac{\sqrt{3} \left (\frac{a}{b}\right )^{\frac{1}{6}} + 2 \, \sqrt{x}}{\left (\frac{a}{b}\right )^{\frac{1}{6}}}\right )}{18 \, a^{2} b^{2}} + \frac{{\left (\left (a b^{5}\right )^{\frac{1}{6}} B a + 5 \, \left (a b^{5}\right )^{\frac{1}{6}} A b\right )} \arctan \left (-\frac{\sqrt{3} \left (\frac{a}{b}\right )^{\frac{1}{6}} - 2 \, \sqrt{x}}{\left (\frac{a}{b}\right )^{\frac{1}{6}}}\right )}{18 \, a^{2} b^{2}} + \frac{{\left (\left (a b^{5}\right )^{\frac{1}{6}} B a + 5 \, \left (a b^{5}\right )^{\frac{1}{6}} A b\right )} \arctan \left (\frac{\sqrt{x}}{\left (\frac{a}{b}\right )^{\frac{1}{6}}}\right )}{9 \, a^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/36*sqrt(3)*((a*b^5)^(1/6)*B*a + 5*(a*b^5)^(1/6)*A*b)*ln(sqrt(3)*sqrt(x)*(a/b)^
(1/6) + x + (a/b)^(1/3))/(a^2*b^2) - 1/36*sqrt(3)*((a*b^5)^(1/6)*B*a + 5*(a*b^5)
^(1/6)*A*b)*ln(-sqrt(3)*sqrt(x)*(a/b)^(1/6) + x + (a/b)^(1/3))/(a^2*b^2) - 1/3*(
B*a*sqrt(x) - A*b*sqrt(x))/((b*x^3 + a)*a*b) + 1/18*((a*b^5)^(1/6)*B*a + 5*(a*b^
5)^(1/6)*A*b)*arctan((sqrt(3)*(a/b)^(1/6) + 2*sqrt(x))/(a/b)^(1/6))/(a^2*b^2) +
1/18*((a*b^5)^(1/6)*B*a + 5*(a*b^5)^(1/6)*A*b)*arctan(-(sqrt(3)*(a/b)^(1/6) - 2*
sqrt(x))/(a/b)^(1/6))/(a^2*b^2) + 1/9*((a*b^5)^(1/6)*B*a + 5*(a*b^5)^(1/6)*A*b)*
arctan(sqrt(x)/(a/b)^(1/6))/(a^2*b^2)