3.21 \(\int \frac{\left (c+d x^3\right )^4}{\left (a+b x^3\right )^2} \, dx\)

Optimal. Leaf size=267 \[ -\frac{(b c-a d)^3 (5 a d+b c) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 a^{5/3} b^{13/3}}+\frac{2 (b c-a d)^3 (5 a d+b c) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{13/3}}-\frac{2 (b c-a d)^3 (5 a d+b c) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} b^{13/3}}+\frac{d^2 x \left (3 a^2 d^2-8 a b c d+6 b^2 c^2\right )}{b^4}+\frac{x (b c-a d)^4}{3 a b^4 \left (a+b x^3\right )}+\frac{d^3 x^4 (2 b c-a d)}{2 b^3}+\frac{d^4 x^7}{7 b^2} \]

[Out]

(d^2*(6*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*x)/b^4 + (d^3*(2*b*c - a*d)*x^4)/(2*b^3
) + (d^4*x^7)/(7*b^2) + ((b*c - a*d)^4*x)/(3*a*b^4*(a + b*x^3)) - (2*(b*c - a*d)
^3*(b*c + 5*a*d)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a
^(5/3)*b^(13/3)) + (2*(b*c - a*d)^3*(b*c + 5*a*d)*Log[a^(1/3) + b^(1/3)*x])/(9*a
^(5/3)*b^(13/3)) - ((b*c - a*d)^3*(b*c + 5*a*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x
+ b^(2/3)*x^2])/(9*a^(5/3)*b^(13/3))

_______________________________________________________________________________________

Rubi [A]  time = 0.497773, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421 \[ -\frac{(b c-a d)^3 (5 a d+b c) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 a^{5/3} b^{13/3}}+\frac{2 (b c-a d)^3 (5 a d+b c) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{13/3}}-\frac{2 (b c-a d)^3 (5 a d+b c) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} b^{13/3}}+\frac{d^2 x \left (3 a^2 d^2-8 a b c d+6 b^2 c^2\right )}{b^4}+\frac{x (b c-a d)^4}{3 a b^4 \left (a+b x^3\right )}+\frac{d^3 x^4 (2 b c-a d)}{2 b^3}+\frac{d^4 x^7}{7 b^2} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^3)^4/(a + b*x^3)^2,x]

[Out]

(d^2*(6*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*x)/b^4 + (d^3*(2*b*c - a*d)*x^4)/(2*b^3
) + (d^4*x^7)/(7*b^2) + ((b*c - a*d)^4*x)/(3*a*b^4*(a + b*x^3)) - (2*(b*c - a*d)
^3*(b*c + 5*a*d)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a
^(5/3)*b^(13/3)) + (2*(b*c - a*d)^3*(b*c + 5*a*d)*Log[a^(1/3) + b^(1/3)*x])/(9*a
^(5/3)*b^(13/3)) - ((b*c - a*d)^3*(b*c + 5*a*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x
+ b^(2/3)*x^2])/(9*a^(5/3)*b^(13/3))

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ d^{2} \left (3 a^{2} d^{2} - 8 a b c d + 6 b^{2} c^{2}\right ) \int \frac{1}{b^{4}}\, dx + \frac{d^{4} x^{7}}{7 b^{2}} - \frac{d^{3} x^{4} \left (a d - 2 b c\right )}{2 b^{3}} + \frac{x \left (a d - b c\right )^{4}}{3 a b^{4} \left (a + b x^{3}\right )} - \frac{2 \left (a d - b c\right )^{3} \left (5 a d + b c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{9 a^{\frac{5}{3}} b^{\frac{13}{3}}} + \frac{\left (a d - b c\right )^{3} \left (5 a d + b c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{9 a^{\frac{5}{3}} b^{\frac{13}{3}}} + \frac{2 \sqrt{3} \left (a d - b c\right )^{3} \left (5 a d + b c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{5}{3}} b^{\frac{13}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**3+c)**4/(b*x**3+a)**2,x)

[Out]

d**2*(3*a**2*d**2 - 8*a*b*c*d + 6*b**2*c**2)*Integral(b**(-4), x) + d**4*x**7/(7
*b**2) - d**3*x**4*(a*d - 2*b*c)/(2*b**3) + x*(a*d - b*c)**4/(3*a*b**4*(a + b*x*
*3)) - 2*(a*d - b*c)**3*(5*a*d + b*c)*log(a**(1/3) + b**(1/3)*x)/(9*a**(5/3)*b**
(13/3)) + (a*d - b*c)**3*(5*a*d + b*c)*log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(
2/3)*x**2)/(9*a**(5/3)*b**(13/3)) + 2*sqrt(3)*(a*d - b*c)**3*(5*a*d + b*c)*atan(
sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(9*a**(5/3)*b**(13/3))

_______________________________________________________________________________________

Mathematica [A]  time = 0.363644, size = 260, normalized size = 0.97 \[ \frac{\frac{14 (a d-b c)^3 (5 a d+b c) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{5/3}}+\frac{28 (b c-a d)^3 (5 a d+b c) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{5/3}}+\frac{28 \sqrt{3} (b c-a d)^3 (5 a d+b c) \tan ^{-1}\left (\frac{2 \sqrt [3]{b} x-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{5/3}}+126 \sqrt [3]{b} d^2 x \left (3 a^2 d^2-8 a b c d+6 b^2 c^2\right )+63 b^{4/3} d^3 x^4 (2 b c-a d)+\frac{42 \sqrt [3]{b} x (b c-a d)^4}{a \left (a+b x^3\right )}+18 b^{7/3} d^4 x^7}{126 b^{13/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^3)^4/(a + b*x^3)^2,x]

[Out]

(126*b^(1/3)*d^2*(6*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*x + 63*b^(4/3)*d^3*(2*b*c -
a*d)*x^4 + 18*b^(7/3)*d^4*x^7 + (42*b^(1/3)*(b*c - a*d)^4*x)/(a*(a + b*x^3)) +
(28*Sqrt[3]*(b*c - a*d)^3*(b*c + 5*a*d)*ArcTan[(-a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]
*a^(1/3))])/a^(5/3) + (28*(b*c - a*d)^3*(b*c + 5*a*d)*Log[a^(1/3) + b^(1/3)*x])/
a^(5/3) + (14*(-(b*c) + a*d)^3*(b*c + 5*a*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b
^(2/3)*x^2])/a^(5/3))/(126*b^(13/3))

_______________________________________________________________________________________

Maple [B]  time = 0.016, size = 708, normalized size = 2.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^3+c)^4/(b*x^3+a)^2,x)

[Out]

-8*d^3/b^3*a*c*x+5/9/b^5*a^3/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*d^4-2
/9/b^2/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*c^3*d+28/9/b^4*a^2/(a/b)^(2
/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*c*d^3-8/3/b^3*a/(a/b)^(2/3)*
3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*c^2*d^2-1/2*d^4/b^3*x^4*a+d^3/b^
2*x^4*c+3*d^4/b^4*a^2*x+6*d^2/b^2*c^2*x+1/3*x/a/(b*x^3+a)*c^4-1/9/b/a/(a/b)^(2/3
)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*c^4-10/9/b^5*a^3/(a/b)^(2/3)*ln(x+(a/b)^(1/3
))*d^4+4/9/b^2/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*c^3*d+2/9/b/a/(a/b)^(2/3)*ln(x+(a/b
)^(1/3))*c^4+1/3/b^4*x*a^3/(b*x^3+a)*d^4-4/3/b*x/(b*x^3+a)*c^3*d+1/7*d^4*x^7/b^2
+28/9/b^4*a^2/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*c*d^3-8/3/b^3*a/(a/b)^(2/3)*ln(x+(a/
b)^(1/3))*c^2*d^2-14/9/b^4*a^2/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*c*d
^3+4/3/b^3*a/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*c^2*d^2-10/9/b^5*a^3/
(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*d^4+2/9/b/a/(a/b)^(2
/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*c^4+2/b^2*x*a/(b*x^3+a)*c^2*
d^2+4/9/b^2/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*c^3*d-4/
3/b^3*x*a^2/(b*x^3+a)*c*d^3

_______________________________________________________________________________________

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((d*x^3 + c)^4/(b*x^3 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.22305, size = 794, normalized size = 2.97 \[ \frac{\sqrt{3}{\left (14 \, \sqrt{3}{\left (a b^{4} c^{4} + 2 \, a^{2} b^{3} c^{3} d - 12 \, a^{3} b^{2} c^{2} d^{2} + 14 \, a^{4} b c d^{3} - 5 \, a^{5} d^{4} +{\left (b^{5} c^{4} + 2 \, a b^{4} c^{3} d - 12 \, a^{2} b^{3} c^{2} d^{2} + 14 \, a^{3} b^{2} c d^{3} - 5 \, a^{4} b d^{4}\right )} x^{3}\right )} \log \left (\left (-a^{2} b\right )^{\frac{2}{3}} x^{2} + \left (-a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 28 \, \sqrt{3}{\left (a b^{4} c^{4} + 2 \, a^{2} b^{3} c^{3} d - 12 \, a^{3} b^{2} c^{2} d^{2} + 14 \, a^{4} b c d^{3} - 5 \, a^{5} d^{4} +{\left (b^{5} c^{4} + 2 \, a b^{4} c^{3} d - 12 \, a^{2} b^{3} c^{2} d^{2} + 14 \, a^{3} b^{2} c d^{3} - 5 \, a^{4} b d^{4}\right )} x^{3}\right )} \log \left (\left (-a^{2} b\right )^{\frac{1}{3}} x - a\right ) + 84 \,{\left (a b^{4} c^{4} + 2 \, a^{2} b^{3} c^{3} d - 12 \, a^{3} b^{2} c^{2} d^{2} + 14 \, a^{4} b c d^{3} - 5 \, a^{5} d^{4} +{\left (b^{5} c^{4} + 2 \, a b^{4} c^{3} d - 12 \, a^{2} b^{3} c^{2} d^{2} + 14 \, a^{3} b^{2} c d^{3} - 5 \, a^{4} b d^{4}\right )} x^{3}\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (-a^{2} b\right )^{\frac{1}{3}} x + \sqrt{3} a}{3 \, a}\right ) + 3 \, \sqrt{3}{\left (6 \, a b^{3} d^{4} x^{10} + 3 \,{\left (14 \, a b^{3} c d^{3} - 5 \, a^{2} b^{2} d^{4}\right )} x^{7} + 21 \,{\left (12 \, a b^{3} c^{2} d^{2} - 14 \, a^{2} b^{2} c d^{3} + 5 \, a^{3} b d^{4}\right )} x^{4} + 14 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 24 \, a^{2} b^{2} c^{2} d^{2} - 28 \, a^{3} b c d^{3} + 10 \, a^{4} d^{4}\right )} x\right )} \left (-a^{2} b\right )^{\frac{1}{3}}\right )}}{378 \,{\left (a b^{5} x^{3} + a^{2} b^{4}\right )} \left (-a^{2} b\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^3 + c)^4/(b*x^3 + a)^2,x, algorithm="fricas")

[Out]

1/378*sqrt(3)*(14*sqrt(3)*(a*b^4*c^4 + 2*a^2*b^3*c^3*d - 12*a^3*b^2*c^2*d^2 + 14
*a^4*b*c*d^3 - 5*a^5*d^4 + (b^5*c^4 + 2*a*b^4*c^3*d - 12*a^2*b^3*c^2*d^2 + 14*a^
3*b^2*c*d^3 - 5*a^4*b*d^4)*x^3)*log((-a^2*b)^(2/3)*x^2 + (-a^2*b)^(1/3)*a*x + a^
2) - 28*sqrt(3)*(a*b^4*c^4 + 2*a^2*b^3*c^3*d - 12*a^3*b^2*c^2*d^2 + 14*a^4*b*c*d
^3 - 5*a^5*d^4 + (b^5*c^4 + 2*a*b^4*c^3*d - 12*a^2*b^3*c^2*d^2 + 14*a^3*b^2*c*d^
3 - 5*a^4*b*d^4)*x^3)*log((-a^2*b)^(1/3)*x - a) + 84*(a*b^4*c^4 + 2*a^2*b^3*c^3*
d - 12*a^3*b^2*c^2*d^2 + 14*a^4*b*c*d^3 - 5*a^5*d^4 + (b^5*c^4 + 2*a*b^4*c^3*d -
12*a^2*b^3*c^2*d^2 + 14*a^3*b^2*c*d^3 - 5*a^4*b*d^4)*x^3)*arctan(1/3*(2*sqrt(3)
*(-a^2*b)^(1/3)*x + sqrt(3)*a)/a) + 3*sqrt(3)*(6*a*b^3*d^4*x^10 + 3*(14*a*b^3*c*
d^3 - 5*a^2*b^2*d^4)*x^7 + 21*(12*a*b^3*c^2*d^2 - 14*a^2*b^2*c*d^3 + 5*a^3*b*d^4
)*x^4 + 14*(b^4*c^4 - 4*a*b^3*c^3*d + 24*a^2*b^2*c^2*d^2 - 28*a^3*b*c*d^3 + 10*a
^4*d^4)*x)*(-a^2*b)^(1/3))/((a*b^5*x^3 + a^2*b^4)*(-a^2*b)^(1/3))

_______________________________________________________________________________________

Sympy [A]  time = 14.071, size = 403, normalized size = 1.51 \[ \frac{x \left (a^{4} d^{4} - 4 a^{3} b c d^{3} + 6 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d + b^{4} c^{4}\right )}{3 a^{2} b^{4} + 3 a b^{5} x^{3}} + \operatorname{RootSum}{\left (729 t^{3} a^{5} b^{13} + 1000 a^{12} d^{12} - 8400 a^{11} b c d^{11} + 30720 a^{10} b^{2} c^{2} d^{10} - 63472 a^{9} b^{3} c^{3} d^{9} + 79848 a^{8} b^{4} c^{4} d^{8} - 60192 a^{7} b^{5} c^{5} d^{7} + 22848 a^{6} b^{6} c^{6} d^{6} + 288 a^{5} b^{7} c^{7} d^{5} - 3528 a^{4} b^{8} c^{8} d^{4} + 752 a^{3} b^{9} c^{9} d^{3} + 192 a^{2} b^{10} c^{10} d^{2} - 48 a b^{11} c^{11} d - 8 b^{12} c^{12}, \left ( t \mapsto t \log{\left (- \frac{9 t a^{2} b^{4}}{10 a^{4} d^{4} - 28 a^{3} b c d^{3} + 24 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d - 2 b^{4} c^{4}} + x \right )} \right )\right )} + \frac{d^{4} x^{7}}{7 b^{2}} - \frac{x^{4} \left (a d^{4} - 2 b c d^{3}\right )}{2 b^{3}} + \frac{x \left (3 a^{2} d^{4} - 8 a b c d^{3} + 6 b^{2} c^{2} d^{2}\right )}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x**3+c)**4/(b*x**3+a)**2,x)

[Out]

x*(a**4*d**4 - 4*a**3*b*c*d**3 + 6*a**2*b**2*c**2*d**2 - 4*a*b**3*c**3*d + b**4*
c**4)/(3*a**2*b**4 + 3*a*b**5*x**3) + RootSum(729*_t**3*a**5*b**13 + 1000*a**12*
d**12 - 8400*a**11*b*c*d**11 + 30720*a**10*b**2*c**2*d**10 - 63472*a**9*b**3*c**
3*d**9 + 79848*a**8*b**4*c**4*d**8 - 60192*a**7*b**5*c**5*d**7 + 22848*a**6*b**6
*c**6*d**6 + 288*a**5*b**7*c**7*d**5 - 3528*a**4*b**8*c**8*d**4 + 752*a**3*b**9*
c**9*d**3 + 192*a**2*b**10*c**10*d**2 - 48*a*b**11*c**11*d - 8*b**12*c**12, Lamb
da(_t, _t*log(-9*_t*a**2*b**4/(10*a**4*d**4 - 28*a**3*b*c*d**3 + 24*a**2*b**2*c*
*2*d**2 - 4*a*b**3*c**3*d - 2*b**4*c**4) + x))) + d**4*x**7/(7*b**2) - x**4*(a*d
**4 - 2*b*c*d**3)/(2*b**3) + x*(3*a**2*d**4 - 8*a*b*c*d**3 + 6*b**2*c**2*d**2)/b
**4

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.220824, size = 643, normalized size = 2.41 \[ -\frac{2 \,{\left (b^{4} c^{4} + 2 \, a b^{3} c^{3} d - 12 \, a^{2} b^{2} c^{2} d^{2} + 14 \, a^{3} b c d^{3} - 5 \, a^{4} d^{4}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{2} b^{4}} + \frac{2 \, \sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{4} c^{4} + 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} c^{3} d - 12 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b^{2} c^{2} d^{2} + 14 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} b c d^{3} - 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{4} d^{4}\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{2} b^{5}} + \frac{b^{4} c^{4} x - 4 \, a b^{3} c^{3} d x + 6 \, a^{2} b^{2} c^{2} d^{2} x - 4 \, a^{3} b c d^{3} x + a^{4} d^{4} x}{3 \,{\left (b x^{3} + a\right )} a b^{4}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{4} c^{4} + 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} c^{3} d - 12 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b^{2} c^{2} d^{2} + 14 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} b c d^{3} - 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{4} d^{4}\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{9 \, a^{2} b^{5}} + \frac{2 \, b^{12} d^{4} x^{7} + 14 \, b^{12} c d^{3} x^{4} - 7 \, a b^{11} d^{4} x^{4} + 84 \, b^{12} c^{2} d^{2} x - 112 \, a b^{11} c d^{3} x + 42 \, a^{2} b^{10} d^{4} x}{14 \, b^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^3 + c)^4/(b*x^3 + a)^2,x, algorithm="giac")

[Out]

-2/9*(b^4*c^4 + 2*a*b^3*c^3*d - 12*a^2*b^2*c^2*d^2 + 14*a^3*b*c*d^3 - 5*a^4*d^4)
*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a^2*b^4) + 2/9*sqrt(3)*((-a*b^2)^(1/3)*
b^4*c^4 + 2*(-a*b^2)^(1/3)*a*b^3*c^3*d - 12*(-a*b^2)^(1/3)*a^2*b^2*c^2*d^2 + 14*
(-a*b^2)^(1/3)*a^3*b*c*d^3 - 5*(-a*b^2)^(1/3)*a^4*d^4)*arctan(1/3*sqrt(3)*(2*x +
(-a/b)^(1/3))/(-a/b)^(1/3))/(a^2*b^5) + 1/3*(b^4*c^4*x - 4*a*b^3*c^3*d*x + 6*a^
2*b^2*c^2*d^2*x - 4*a^3*b*c*d^3*x + a^4*d^4*x)/((b*x^3 + a)*a*b^4) + 1/9*((-a*b^
2)^(1/3)*b^4*c^4 + 2*(-a*b^2)^(1/3)*a*b^3*c^3*d - 12*(-a*b^2)^(1/3)*a^2*b^2*c^2*
d^2 + 14*(-a*b^2)^(1/3)*a^3*b*c*d^3 - 5*(-a*b^2)^(1/3)*a^4*d^4)*ln(x^2 + x*(-a/b
)^(1/3) + (-a/b)^(2/3))/(a^2*b^5) + 1/14*(2*b^12*d^4*x^7 + 14*b^12*c*d^3*x^4 - 7
*a*b^11*d^4*x^4 + 84*b^12*c^2*d^2*x - 112*a*b^11*c*d^3*x + 42*a^2*b^10*d^4*x)/b^
14