3.233 \(\int \frac{x^9 \left (c+d x^3+e x^6+f x^9\right )}{a+b x^3} \, dx\)

Optimal. Leaf size=348 \[ \frac{x^{10} \left (a^2 f-a b e+b^2 d\right )}{10 b^3}+\frac{a^2 x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{b^6}-\frac{a x^4 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{4 b^5}+\frac{x^7 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{7 b^4}+\frac{a^{7/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^{19/3}}-\frac{a^{7/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^{19/3}}+\frac{a^{7/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt{3} b^{19/3}}+\frac{x^{13} (b e-a f)}{13 b^2}+\frac{f x^{16}}{16 b} \]

[Out]

(a^2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/b^6 - (a*(b^3*c - a*b^2*d + a^2*b*e
- a^3*f)*x^4)/(4*b^5) + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^7)/(7*b^4) + ((b^
2*d - a*b*e + a^2*f)*x^10)/(10*b^3) + ((b*e - a*f)*x^13)/(13*b^2) + (f*x^16)/(16
*b) + (a^(7/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x
)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*b^(19/3)) - (a^(7/3)*(b^3*c - a*b^2*d + a^2*b*e -
a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(3*b^(19/3)) + (a^(7/3)*(b^3*c - a*b^2*d + a^2
*b*e - a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*b^(19/3))

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Rubi [A]  time = 0.774286, antiderivative size = 348, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{x^{10} \left (a^2 f-a b e+b^2 d\right )}{10 b^3}+\frac{a^2 x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{b^6}-\frac{a x^4 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{4 b^5}+\frac{x^7 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{7 b^4}+\frac{a^{7/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^{19/3}}-\frac{a^{7/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^{19/3}}+\frac{a^{7/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt{3} b^{19/3}}+\frac{x^{13} (b e-a f)}{13 b^2}+\frac{f x^{16}}{16 b} \]

Antiderivative was successfully verified.

[In]  Int[(x^9*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3),x]

[Out]

(a^2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/b^6 - (a*(b^3*c - a*b^2*d + a^2*b*e
- a^3*f)*x^4)/(4*b^5) + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^7)/(7*b^4) + ((b^
2*d - a*b*e + a^2*f)*x^10)/(10*b^3) + ((b*e - a*f)*x^13)/(13*b^2) + (f*x^16)/(16
*b) + (a^(7/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x
)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*b^(19/3)) - (a^(7/3)*(b^3*c - a*b^2*d + a^2*b*e -
a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(3*b^(19/3)) + (a^(7/3)*(b^3*c - a*b^2*d + a^2
*b*e - a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*b^(19/3))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{\frac{7}{3}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 b^{\frac{19}{3}}} - \frac{a^{\frac{7}{3}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 b^{\frac{19}{3}}} - \frac{\sqrt{3} a^{\frac{7}{3}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 b^{\frac{19}{3}}} + \frac{a x^{4} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{4 b^{5}} + \frac{f x^{16}}{16 b} - \frac{x^{13} \left (a f - b e\right )}{13 b^{2}} + \frac{x^{10} \left (a^{2} f - a b e + b^{2} d\right )}{10 b^{3}} - \frac{x^{7} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{7 b^{4}} - \frac{\left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \int a^{2}\, dx}{b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**9*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a),x)

[Out]

a**(7/3)*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)*log(a**(1/3) + b**(1/3)*x)/(3*b
**(19/3)) - a**(7/3)*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)*log(a**(2/3) - a**(
1/3)*b**(1/3)*x + b**(2/3)*x**2)/(6*b**(19/3)) - sqrt(3)*a**(7/3)*(a**3*f - a**2
*b*e + a*b**2*d - b**3*c)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(
3*b**(19/3)) + a*x**4*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)/(4*b**5) + f*x**16
/(16*b) - x**13*(a*f - b*e)/(13*b**2) + x**10*(a**2*f - a*b*e + b**2*d)/(10*b**3
) - x**7*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)/(7*b**4) - (a**3*f - a**2*b*e +
a*b**2*d - b**3*c)*Integral(a**2, x)/b**6

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Mathematica [A]  time = 0.144861, size = 351, normalized size = 1.01 \[ \frac{x^{10} \left (a^2 f-a b e+b^2 d\right )}{10 b^3}-\frac{a^2 x \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{b^6}+\frac{a x^4 \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{4 b^5}+\frac{x^7 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{7 b^4}-\frac{a^{7/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{6 b^{19/3}}+\frac{a^{7/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{3 b^{19/3}}+\frac{a^{7/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} x-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{\sqrt{3} b^{19/3}}+\frac{x^{13} (b e-a f)}{13 b^2}+\frac{f x^{16}}{16 b} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^9*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3),x]

[Out]

-((a^2*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/b^6) + (a*(-(b^3*c) + a*b^2*d -
a^2*b*e + a^3*f)*x^4)/(4*b^5) + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^7)/(7*b^
4) + ((b^2*d - a*b*e + a^2*f)*x^10)/(10*b^3) + ((b*e - a*f)*x^13)/(13*b^2) + (f*
x^16)/(16*b) + (a^(7/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*ArcTan[(-a^(1/3)
+ 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*b^(19/3)) + (a^(7/3)*(-(b^3*c) + a*b
^2*d - a^2*b*e + a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(3*b^(19/3)) - (a^(7/3)*(-(b^3
*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])
/(6*b^(19/3))

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Maple [A]  time = 0.008, size = 592, normalized size = 1.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^9*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a),x)

[Out]

1/13/b*x^13*e+1/10/b*x^10*d+1/7/b*x^7*c-1/13/b^2*x^13*a*f+1/10/b^3*x^10*a^2*f-1/
10/b^2*x^10*a*e-1/7/b^4*x^7*a^3*f+1/7/b^3*x^7*a^2*e-1/7/b^2*x^7*a*d-1/3*a^3/b^4/
(a/b)^(2/3)*ln(x+(a/b)^(1/3))*c-1/6*a^6/b^7/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/
b)^(2/3))*f+1/16*f*x^16/b+1/3*a^4/b^5/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/
(a/b)^(1/3)*x-1))*d-1/3*a^3/b^4/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^
(1/3)*x-1))*c+1/3*a^6/b^7/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*
x-1))*f-1/3*a^5/b^6/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*
e+1/4/b^5*x^4*a^4*f-1/4/b^4*x^4*a^3*e+1/4/b^3*x^4*a^2*d-1/4/b^2*x^4*a*c-1/b^6*a^
5*f*x+1/b^5*a^4*e*x-1/b^4*a^3*d*x+1/b^3*a^2*c*x+1/6*a^5/b^6/(a/b)^(2/3)*ln(x^2-x
*(a/b)^(1/3)+(a/b)^(2/3))*e+1/3*a^6/b^7/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*f-1/3*a^5/
b^6/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*e+1/3*a^4/b^5/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*d-
1/6*a^4/b^5/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*d+1/6*a^3/b^4/(a/b)^(2
/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*c

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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((f*x^9 + e*x^6 + d*x^3 + c)*x^9/(b*x^3 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.223335, size = 481, normalized size = 1.38 \[ \frac{\sqrt{3}{\left (3640 \, \sqrt{3}{\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} \left (\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x^{2} - x \left (\frac{a}{b}\right )^{\frac{1}{3}} + \left (\frac{a}{b}\right )^{\frac{2}{3}}\right ) - 7280 \, \sqrt{3}{\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} \left (\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right ) + 21840 \,{\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} \left (\frac{a}{b}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} x - \sqrt{3} \left (\frac{a}{b}\right )^{\frac{1}{3}}}{3 \, \left (\frac{a}{b}\right )^{\frac{1}{3}}}\right ) + 3 \, \sqrt{3}{\left (455 \, b^{5} f x^{16} + 560 \,{\left (b^{5} e - a b^{4} f\right )} x^{13} + 728 \,{\left (b^{5} d - a b^{4} e + a^{2} b^{3} f\right )} x^{10} + 1040 \,{\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{7} - 1820 \,{\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x^{4} + 7280 \,{\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} x\right )}\right )}}{65520 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^9/(b*x^3 + a),x, algorithm="fricas")

[Out]

1/65520*sqrt(3)*(3640*sqrt(3)*(a^2*b^3*c - a^3*b^2*d + a^4*b*e - a^5*f)*(a/b)^(1
/3)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3)) - 7280*sqrt(3)*(a^2*b^3*c - a^3*b^2*d
+ a^4*b*e - a^5*f)*(a/b)^(1/3)*log(x + (a/b)^(1/3)) + 21840*(a^2*b^3*c - a^3*b^
2*d + a^4*b*e - a^5*f)*(a/b)^(1/3)*arctan(-1/3*(2*sqrt(3)*x - sqrt(3)*(a/b)^(1/3
))/(a/b)^(1/3)) + 3*sqrt(3)*(455*b^5*f*x^16 + 560*(b^5*e - a*b^4*f)*x^13 + 728*(
b^5*d - a*b^4*e + a^2*b^3*f)*x^10 + 1040*(b^5*c - a*b^4*d + a^2*b^3*e - a^3*b^2*
f)*x^7 - 1820*(a*b^4*c - a^2*b^3*d + a^3*b^2*e - a^4*b*f)*x^4 + 7280*(a^2*b^3*c
- a^3*b^2*d + a^4*b*e - a^5*f)*x))/b^6

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Sympy [A]  time = 3.69756, size = 450, normalized size = 1.29 \[ \operatorname{RootSum}{\left (27 t^{3} b^{19} - a^{16} f^{3} + 3 a^{15} b e f^{2} - 3 a^{14} b^{2} d f^{2} - 3 a^{14} b^{2} e^{2} f + 3 a^{13} b^{3} c f^{2} + 6 a^{13} b^{3} d e f + a^{13} b^{3} e^{3} - 6 a^{12} b^{4} c e f - 3 a^{12} b^{4} d^{2} f - 3 a^{12} b^{4} d e^{2} + 6 a^{11} b^{5} c d f + 3 a^{11} b^{5} c e^{2} + 3 a^{11} b^{5} d^{2} e - 3 a^{10} b^{6} c^{2} f - 6 a^{10} b^{6} c d e - a^{10} b^{6} d^{3} + 3 a^{9} b^{7} c^{2} e + 3 a^{9} b^{7} c d^{2} - 3 a^{8} b^{8} c^{2} d + a^{7} b^{9} c^{3}, \left ( t \mapsto t \log{\left (\frac{3 t b^{6}}{a^{5} f - a^{4} b e + a^{3} b^{2} d - a^{2} b^{3} c} + x \right )} \right )\right )} + \frac{f x^{16}}{16 b} - \frac{x^{13} \left (a f - b e\right )}{13 b^{2}} + \frac{x^{10} \left (a^{2} f - a b e + b^{2} d\right )}{10 b^{3}} - \frac{x^{7} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{7 b^{4}} + \frac{x^{4} \left (a^{4} f - a^{3} b e + a^{2} b^{2} d - a b^{3} c\right )}{4 b^{5}} - \frac{x \left (a^{5} f - a^{4} b e + a^{3} b^{2} d - a^{2} b^{3} c\right )}{b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**9*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a),x)

[Out]

RootSum(27*_t**3*b**19 - a**16*f**3 + 3*a**15*b*e*f**2 - 3*a**14*b**2*d*f**2 - 3
*a**14*b**2*e**2*f + 3*a**13*b**3*c*f**2 + 6*a**13*b**3*d*e*f + a**13*b**3*e**3
- 6*a**12*b**4*c*e*f - 3*a**12*b**4*d**2*f - 3*a**12*b**4*d*e**2 + 6*a**11*b**5*
c*d*f + 3*a**11*b**5*c*e**2 + 3*a**11*b**5*d**2*e - 3*a**10*b**6*c**2*f - 6*a**1
0*b**6*c*d*e - a**10*b**6*d**3 + 3*a**9*b**7*c**2*e + 3*a**9*b**7*c*d**2 - 3*a**
8*b**8*c**2*d + a**7*b**9*c**3, Lambda(_t, _t*log(3*_t*b**6/(a**5*f - a**4*b*e +
a**3*b**2*d - a**2*b**3*c) + x))) + f*x**16/(16*b) - x**13*(a*f - b*e)/(13*b**2
) + x**10*(a**2*f - a*b*e + b**2*d)/(10*b**3) - x**7*(a**3*f - a**2*b*e + a*b**2
*d - b**3*c)/(7*b**4) + x**4*(a**4*f - a**3*b*e + a**2*b**2*d - a*b**3*c)/(4*b**
5) - x*(a**5*f - a**4*b*e + a**3*b**2*d - a**2*b**3*c)/b**6

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GIAC/XCAS [A]  time = 0.217692, size = 613, normalized size = 1.76 \[ -\frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b^{3} c - \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} b^{2} d - \left (-a b^{2}\right )^{\frac{1}{3}} a^{5} f + \left (-a b^{2}\right )^{\frac{1}{3}} a^{4} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, b^{7}} - \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b^{3} c - \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} b^{2} d - \left (-a b^{2}\right )^{\frac{1}{3}} a^{5} f + \left (-a b^{2}\right )^{\frac{1}{3}} a^{4} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, b^{7}} + \frac{{\left (a^{3} b^{13} c - a^{4} b^{12} d - a^{6} b^{10} f + a^{5} b^{11} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b^{16}} + \frac{455 \, b^{15} f x^{16} - 560 \, a b^{14} f x^{13} + 560 \, b^{15} x^{13} e + 728 \, b^{15} d x^{10} + 728 \, a^{2} b^{13} f x^{10} - 728 \, a b^{14} x^{10} e + 1040 \, b^{15} c x^{7} - 1040 \, a b^{14} d x^{7} - 1040 \, a^{3} b^{12} f x^{7} + 1040 \, a^{2} b^{13} x^{7} e - 1820 \, a b^{14} c x^{4} + 1820 \, a^{2} b^{13} d x^{4} + 1820 \, a^{4} b^{11} f x^{4} - 1820 \, a^{3} b^{12} x^{4} e + 7280 \, a^{2} b^{13} c x - 7280 \, a^{3} b^{12} d x - 7280 \, a^{5} b^{10} f x + 7280 \, a^{4} b^{11} x e}{7280 \, b^{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^9/(b*x^3 + a),x, algorithm="giac")

[Out]

-1/3*sqrt(3)*((-a*b^2)^(1/3)*a^2*b^3*c - (-a*b^2)^(1/3)*a^3*b^2*d - (-a*b^2)^(1/
3)*a^5*f + (-a*b^2)^(1/3)*a^4*b*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b
)^(1/3))/b^7 - 1/6*((-a*b^2)^(1/3)*a^2*b^3*c - (-a*b^2)^(1/3)*a^3*b^2*d - (-a*b^
2)^(1/3)*a^5*f + (-a*b^2)^(1/3)*a^4*b*e)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))
/b^7 + 1/3*(a^3*b^13*c - a^4*b^12*d - a^6*b^10*f + a^5*b^11*e)*(-a/b)^(1/3)*ln(a
bs(x - (-a/b)^(1/3)))/(a*b^16) + 1/7280*(455*b^15*f*x^16 - 560*a*b^14*f*x^13 + 5
60*b^15*x^13*e + 728*b^15*d*x^10 + 728*a^2*b^13*f*x^10 - 728*a*b^14*x^10*e + 104
0*b^15*c*x^7 - 1040*a*b^14*d*x^7 - 1040*a^3*b^12*f*x^7 + 1040*a^2*b^13*x^7*e - 1
820*a*b^14*c*x^4 + 1820*a^2*b^13*d*x^4 + 1820*a^4*b^11*f*x^4 - 1820*a^3*b^12*x^4
*e + 7280*a^2*b^13*c*x - 7280*a^3*b^12*d*x - 7280*a^5*b^10*f*x + 7280*a^4*b^11*x
*e)/b^16