3.260 \(\int \frac{x^9 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx\)

Optimal. Leaf size=369 \[ \frac{x^7 \left (3 a^2 f-2 a b e+b^2 d\right )}{7 b^4}-\frac{a^2 x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^6 \left (a+b x^3\right )}-\frac{a x \left (-5 a^3 f+4 a^2 b e-3 a b^2 d+2 b^3 c\right )}{b^6}+\frac{x^4 \left (-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right )}{4 b^5}-\frac{a^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-16 a^3 f+13 a^2 b e-10 a b^2 d+7 b^3 c\right )}{18 b^{19/3}}+\frac{a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-16 a^3 f+13 a^2 b e-10 a b^2 d+7 b^3 c\right )}{9 b^{19/3}}-\frac{a^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-16 a^3 f+13 a^2 b e-10 a b^2 d+7 b^3 c\right )}{3 \sqrt{3} b^{19/3}}+\frac{x^{10} (b e-2 a f)}{10 b^3}+\frac{f x^{13}}{13 b^2} \]

[Out]

-((a*(2*b^3*c - 3*a*b^2*d + 4*a^2*b*e - 5*a^3*f)*x)/b^6) + ((b^3*c - 2*a*b^2*d +
3*a^2*b*e - 4*a^3*f)*x^4)/(4*b^5) + ((b^2*d - 2*a*b*e + 3*a^2*f)*x^7)/(7*b^4) +
((b*e - 2*a*f)*x^10)/(10*b^3) + (f*x^13)/(13*b^2) - (a^2*(b^3*c - a*b^2*d + a^2
*b*e - a^3*f)*x)/(3*b^6*(a + b*x^3)) - (a^(4/3)*(7*b^3*c - 10*a*b^2*d + 13*a^2*b
*e - 16*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*b^(
19/3)) + (a^(4/3)*(7*b^3*c - 10*a*b^2*d + 13*a^2*b*e - 16*a^3*f)*Log[a^(1/3) + b
^(1/3)*x])/(9*b^(19/3)) - (a^(4/3)*(7*b^3*c - 10*a*b^2*d + 13*a^2*b*e - 16*a^3*f
)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*b^(19/3))

_______________________________________________________________________________________

Rubi [A]  time = 0.974218, antiderivative size = 369, 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 \[ \frac{x^7 \left (3 a^2 f-2 a b e+b^2 d\right )}{7 b^4}-\frac{a^2 x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^6 \left (a+b x^3\right )}-\frac{a x \left (-5 a^3 f+4 a^2 b e-3 a b^2 d+2 b^3 c\right )}{b^6}+\frac{x^4 \left (-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right )}{4 b^5}-\frac{a^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-16 a^3 f+13 a^2 b e-10 a b^2 d+7 b^3 c\right )}{18 b^{19/3}}+\frac{a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-16 a^3 f+13 a^2 b e-10 a b^2 d+7 b^3 c\right )}{9 b^{19/3}}-\frac{a^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-16 a^3 f+13 a^2 b e-10 a b^2 d+7 b^3 c\right )}{3 \sqrt{3} b^{19/3}}+\frac{x^{10} (b e-2 a f)}{10 b^3}+\frac{f x^{13}}{13 b^2} \]

Antiderivative was successfully verified.

[In]  Int[(x^9*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^2,x]

[Out]

-((a*(2*b^3*c - 3*a*b^2*d + 4*a^2*b*e - 5*a^3*f)*x)/b^6) + ((b^3*c - 2*a*b^2*d +
3*a^2*b*e - 4*a^3*f)*x^4)/(4*b^5) + ((b^2*d - 2*a*b*e + 3*a^2*f)*x^7)/(7*b^4) +
((b*e - 2*a*f)*x^10)/(10*b^3) + (f*x^13)/(13*b^2) - (a^2*(b^3*c - a*b^2*d + a^2
*b*e - a^3*f)*x)/(3*b^6*(a + b*x^3)) - (a^(4/3)*(7*b^3*c - 10*a*b^2*d + 13*a^2*b
*e - 16*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*b^(
19/3)) + (a^(4/3)*(7*b^3*c - 10*a*b^2*d + 13*a^2*b*e - 16*a^3*f)*Log[a^(1/3) + b
^(1/3)*x])/(9*b^(19/3)) - (a^(4/3)*(7*b^3*c - 10*a*b^2*d + 13*a^2*b*e - 16*a^3*f
)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*b^(19/3))

_______________________________________________________________________________________

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(x**9*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 0.879941, size = 364, normalized size = 0.99 \[ \frac{x^7 \left (3 a^2 f-2 a b e+b^2 d\right )}{7 b^4}+\frac{a^2 x \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{3 b^6 \left (a+b x^3\right )}+\frac{a x \left (5 a^3 f-4 a^2 b e+3 a b^2 d-2 b^3 c\right )}{b^6}+\frac{x^4 \left (-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right )}{4 b^5}+\frac{a^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (16 a^3 f-13 a^2 b e+10 a b^2 d-7 b^3 c\right )}{18 b^{19/3}}-\frac{a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (16 a^3 f-13 a^2 b e+10 a b^2 d-7 b^3 c\right )}{9 b^{19/3}}+\frac{a^{4/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (16 a^3 f-13 a^2 b e+10 a b^2 d-7 b^3 c\right )}{3 \sqrt{3} b^{19/3}}+\frac{x^{10} (b e-2 a f)}{10 b^3}+\frac{f x^{13}}{13 b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^9*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^2,x]

[Out]

(a*(-2*b^3*c + 3*a*b^2*d - 4*a^2*b*e + 5*a^3*f)*x)/b^6 + ((b^3*c - 2*a*b^2*d + 3
*a^2*b*e - 4*a^3*f)*x^4)/(4*b^5) + ((b^2*d - 2*a*b*e + 3*a^2*f)*x^7)/(7*b^4) + (
(b*e - 2*a*f)*x^10)/(10*b^3) + (f*x^13)/(13*b^2) + (a^2*(-(b^3*c) + a*b^2*d - a^
2*b*e + a^3*f)*x)/(3*b^6*(a + b*x^3)) + (a^(4/3)*(-7*b^3*c + 10*a*b^2*d - 13*a^2
*b*e + 16*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/(3*Sqrt[3]*b^(19/3
)) - (a^(4/3)*(-7*b^3*c + 10*a*b^2*d - 13*a^2*b*e + 16*a^3*f)*Log[a^(1/3) + b^(1
/3)*x])/(9*b^(19/3)) + (a^(4/3)*(-7*b^3*c + 10*a*b^2*d - 13*a^2*b*e + 16*a^3*f)*
Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*b^(19/3))

_______________________________________________________________________________________

Maple [A]  time = 0.015, size = 622, 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)^2,x)

[Out]

1/10/b^2*x^10*e+1/7/b^2*x^7*d+1/4/b^2*x^4*c-16/9*a^5/b^7*f/(a/b)^(2/3)*3^(1/2)*a
rctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+13/9*a^4/b^6*e/(a/b)^(2/3)*3^(1/2)*arctan
(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-10/9*a^3/b^5*d/(a/b)^(2/3)*3^(1/2)*arctan(1/3*
3^(1/2)*(2/(a/b)^(1/3)*x-1))+7/9*a^2/b^4*c/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2
)*(2/(a/b)^(1/3)*x-1))+1/13*f*x^13/b^2-13/18*a^4/b^6*e/(a/b)^(2/3)*ln(x^2-x*(a/b
)^(1/3)+(a/b)^(2/3))-10/9*a^3/b^5*d/(a/b)^(2/3)*ln(x+(a/b)^(1/3))+5/9*a^3/b^5*d/
(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+7/9*a^2/b^4*c/(a/b)^(2/3)*ln(x+(a/
b)^(1/3))-7/18*a^2/b^4*c/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+1/3*a^5/b
^6*x/(b*x^3+a)*f-1/3*a^4/b^5*x/(b*x^3+a)*e-1/5/b^3*x^10*a*f+8/9*a^5/b^7*f/(a/b)^
(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+13/9*a^4/b^6*e/(a/b)^(2/3)*ln(x+(a/b)^(1
/3))+1/3*a^3/b^4*x/(b*x^3+a)*d-1/3*a^2/b^3*x/(b*x^3+a)*c-16/9*a^5/b^7*f/(a/b)^(2
/3)*ln(x+(a/b)^(1/3))+3/4/b^4*x^4*a^2*e-1/2/b^3*x^4*a*d+5/b^6*a^4*f*x-4/b^5*a^3*
e*x+3/b^4*a^2*d*x-2/b^3*a*c*x+3/7/b^4*x^7*a^2*f-2/7/b^3*x^7*a*e-1/b^5*x^4*a^3*f

_______________________________________________________________________________________

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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.225067, size = 678, normalized size = 1.84 \[ \frac{\sqrt{3}{\left (910 \, \sqrt{3}{\left (7 \, a^{2} b^{3} c - 10 \, a^{3} b^{2} d + 13 \, a^{4} b e - 16 \, a^{5} f +{\left (7 \, a b^{4} c - 10 \, a^{2} b^{3} d + 13 \, a^{3} b^{2} e - 16 \, a^{4} b f\right )} x^{3}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right ) - 1820 \, \sqrt{3}{\left (7 \, a^{2} b^{3} c - 10 \, a^{3} b^{2} d + 13 \, a^{4} b e - 16 \, a^{5} f +{\left (7 \, a b^{4} c - 10 \, a^{2} b^{3} d + 13 \, a^{3} b^{2} e - 16 \, a^{4} b f\right )} x^{3}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x - \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right ) + 5460 \,{\left (7 \, a^{2} b^{3} c - 10 \, a^{3} b^{2} d + 13 \, a^{4} b e - 16 \, a^{5} f +{\left (7 \, a b^{4} c - 10 \, a^{2} b^{3} d + 13 \, a^{3} b^{2} e - 16 \, a^{4} b f\right )} x^{3}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} x + \sqrt{3} \left (-\frac{a}{b}\right )^{\frac{1}{3}}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right ) + 3 \, \sqrt{3}{\left (420 \, b^{5} f x^{16} + 42 \,{\left (13 \, b^{5} e - 16 \, a b^{4} f\right )} x^{13} + 78 \,{\left (10 \, b^{5} d - 13 \, a b^{4} e + 16 \, a^{2} b^{3} f\right )} x^{10} + 195 \,{\left (7 \, b^{5} c - 10 \, a b^{4} d + 13 \, a^{2} b^{3} e - 16 \, a^{3} b^{2} f\right )} x^{7} - 1365 \,{\left (7 \, a b^{4} c - 10 \, a^{2} b^{3} d + 13 \, a^{3} b^{2} e - 16 \, a^{4} b f\right )} x^{4} - 1820 \,{\left (7 \, a^{2} b^{3} c - 10 \, a^{3} b^{2} d + 13 \, a^{4} b e - 16 \, a^{5} f\right )} x\right )}\right )}}{49140 \,{\left (b^{7} x^{3} + a b^{6}\right )}} \]

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)^2,x, algorithm="fricas")

[Out]

1/49140*sqrt(3)*(910*sqrt(3)*(7*a^2*b^3*c - 10*a^3*b^2*d + 13*a^4*b*e - 16*a^5*f
+ (7*a*b^4*c - 10*a^2*b^3*d + 13*a^3*b^2*e - 16*a^4*b*f)*x^3)*(-a/b)^(1/3)*log(
x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3)) - 1820*sqrt(3)*(7*a^2*b^3*c - 10*a^3*b^2*d
+ 13*a^4*b*e - 16*a^5*f + (7*a*b^4*c - 10*a^2*b^3*d + 13*a^3*b^2*e - 16*a^4*b*f)
*x^3)*(-a/b)^(1/3)*log(x - (-a/b)^(1/3)) + 5460*(7*a^2*b^3*c - 10*a^3*b^2*d + 13
*a^4*b*e - 16*a^5*f + (7*a*b^4*c - 10*a^2*b^3*d + 13*a^3*b^2*e - 16*a^4*b*f)*x^3
)*(-a/b)^(1/3)*arctan(1/3*(2*sqrt(3)*x + sqrt(3)*(-a/b)^(1/3))/(-a/b)^(1/3)) + 3
*sqrt(3)*(420*b^5*f*x^16 + 42*(13*b^5*e - 16*a*b^4*f)*x^13 + 78*(10*b^5*d - 13*a
*b^4*e + 16*a^2*b^3*f)*x^10 + 195*(7*b^5*c - 10*a*b^4*d + 13*a^2*b^3*e - 16*a^3*
b^2*f)*x^7 - 1365*(7*a*b^4*c - 10*a^2*b^3*d + 13*a^3*b^2*e - 16*a^4*b*f)*x^4 - 1
820*(7*a^2*b^3*c - 10*a^3*b^2*d + 13*a^4*b*e - 16*a^5*f)*x))/(b^7*x^3 + a*b^6)

_______________________________________________________________________________________

Sympy [A]  time = 18.4096, size = 490, normalized size = 1.33 \[ \frac{x \left (a^{5} f - a^{4} b e + a^{3} b^{2} d - a^{2} b^{3} c\right )}{3 a b^{6} + 3 b^{7} x^{3}} + \operatorname{RootSum}{\left (729 t^{3} b^{19} + 4096 a^{13} f^{3} - 9984 a^{12} b e f^{2} + 7680 a^{11} b^{2} d f^{2} + 8112 a^{11} b^{2} e^{2} f - 5376 a^{10} b^{3} c f^{2} - 12480 a^{10} b^{3} d e f - 2197 a^{10} b^{3} e^{3} + 8736 a^{9} b^{4} c e f + 4800 a^{9} b^{4} d^{2} f + 5070 a^{9} b^{4} d e^{2} - 6720 a^{8} b^{5} c d f - 3549 a^{8} b^{5} c e^{2} - 3900 a^{8} b^{5} d^{2} e + 2352 a^{7} b^{6} c^{2} f + 5460 a^{7} b^{6} c d e + 1000 a^{7} b^{6} d^{3} - 1911 a^{6} b^{7} c^{2} e - 2100 a^{6} b^{7} c d^{2} + 1470 a^{5} b^{8} c^{2} d - 343 a^{4} b^{9} c^{3}, \left ( t \mapsto t \log{\left (- \frac{9 t b^{6}}{16 a^{4} f - 13 a^{3} b e + 10 a^{2} b^{2} d - 7 a b^{3} c} + x \right )} \right )\right )} + \frac{f x^{13}}{13 b^{2}} - \frac{x^{10} \left (2 a f - b e\right )}{10 b^{3}} + \frac{x^{7} \left (3 a^{2} f - 2 a b e + b^{2} d\right )}{7 b^{4}} - \frac{x^{4} \left (4 a^{3} f - 3 a^{2} b e + 2 a b^{2} d - b^{3} c\right )}{4 b^{5}} + \frac{x \left (5 a^{4} f - 4 a^{3} b e + 3 a^{2} b^{2} d - 2 a b^{3} c\right )}{b^{6}} \]

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)**2,x)

[Out]

x*(a**5*f - a**4*b*e + a**3*b**2*d - a**2*b**3*c)/(3*a*b**6 + 3*b**7*x**3) + Roo
tSum(729*_t**3*b**19 + 4096*a**13*f**3 - 9984*a**12*b*e*f**2 + 7680*a**11*b**2*d
*f**2 + 8112*a**11*b**2*e**2*f - 5376*a**10*b**3*c*f**2 - 12480*a**10*b**3*d*e*f
- 2197*a**10*b**3*e**3 + 8736*a**9*b**4*c*e*f + 4800*a**9*b**4*d**2*f + 5070*a*
*9*b**4*d*e**2 - 6720*a**8*b**5*c*d*f - 3549*a**8*b**5*c*e**2 - 3900*a**8*b**5*d
**2*e + 2352*a**7*b**6*c**2*f + 5460*a**7*b**6*c*d*e + 1000*a**7*b**6*d**3 - 191
1*a**6*b**7*c**2*e - 2100*a**6*b**7*c*d**2 + 1470*a**5*b**8*c**2*d - 343*a**4*b*
*9*c**3, Lambda(_t, _t*log(-9*_t*b**6/(16*a**4*f - 13*a**3*b*e + 10*a**2*b**2*d
- 7*a*b**3*c) + x))) + f*x**13/(13*b**2) - x**10*(2*a*f - b*e)/(10*b**3) + x**7*
(3*a**2*f - 2*a*b*e + b**2*d)/(7*b**4) - x**4*(4*a**3*f - 3*a**2*b*e + 2*a*b**2*
d - b**3*c)/(4*b**5) + x*(5*a**4*f - 4*a**3*b*e + 3*a**2*b**2*d - 2*a*b**3*c)/b*
*6

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.216323, size = 609, normalized size = 1.65 \[ \frac{\sqrt{3}{\left (7 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} c - 10 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b^{2} d - 16 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{4} f + 13 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, b^{7}} - \frac{{\left (7 \, a^{2} b^{3} c - 10 \, a^{3} b^{2} d - 16 \, a^{5} f + 13 \, a^{4} b e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a b^{6}} + \frac{{\left (7 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} c - 10 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b^{2} d - 16 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{4} f + 13 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, b^{7}} - \frac{a^{2} b^{3} c x - a^{3} b^{2} d x - a^{5} f x + a^{4} b x e}{3 \,{\left (b x^{3} + a\right )} b^{6}} + \frac{140 \, b^{24} f x^{13} - 364 \, a b^{23} f x^{10} + 182 \, b^{24} x^{10} e + 260 \, b^{24} d x^{7} + 780 \, a^{2} b^{22} f x^{7} - 520 \, a b^{23} x^{7} e + 455 \, b^{24} c x^{4} - 910 \, a b^{23} d x^{4} - 1820 \, a^{3} b^{21} f x^{4} + 1365 \, a^{2} b^{22} x^{4} e - 3640 \, a b^{23} c x + 5460 \, a^{2} b^{22} d x + 9100 \, a^{4} b^{20} f x - 7280 \, a^{3} b^{21} x e}{1820 \, b^{26}} \]

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)^2,x, algorithm="giac")

[Out]

1/9*sqrt(3)*(7*(-a*b^2)^(1/3)*a*b^3*c - 10*(-a*b^2)^(1/3)*a^2*b^2*d - 16*(-a*b^2
)^(1/3)*a^4*f + 13*(-a*b^2)^(1/3)*a^3*b*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3
))/(-a/b)^(1/3))/b^7 - 1/9*(7*a^2*b^3*c - 10*a^3*b^2*d - 16*a^5*f + 13*a^4*b*e)*
(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a*b^6) + 1/18*(7*(-a*b^2)^(1/3)*a*b^3*c
- 10*(-a*b^2)^(1/3)*a^2*b^2*d - 16*(-a*b^2)^(1/3)*a^4*f + 13*(-a*b^2)^(1/3)*a^3*
b*e)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/b^7 - 1/3*(a^2*b^3*c*x - a^3*b^2*d*
x - a^5*f*x + a^4*b*x*e)/((b*x^3 + a)*b^6) + 1/1820*(140*b^24*f*x^13 - 364*a*b^2
3*f*x^10 + 182*b^24*x^10*e + 260*b^24*d*x^7 + 780*a^2*b^22*f*x^7 - 520*a*b^23*x^
7*e + 455*b^24*c*x^4 - 910*a*b^23*d*x^4 - 1820*a^3*b^21*f*x^4 + 1365*a^2*b^22*x^
4*e - 3640*a*b^23*c*x + 5460*a^2*b^22*d*x + 9100*a^4*b^20*f*x - 7280*a^3*b^21*x*
e)/b^26