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

Optimal. Leaf size=265 \[ -\frac{c}{a^2 x}-\frac{x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^2 b^2 \left (a+b x^3\right )}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (5 a^3 f-2 a^2 b e-a b^2 d+4 b^3 c\right )}{18 a^{7/3} b^{8/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (5 a^3 f-2 a^2 b e-a b^2 d+4 b^3 c\right )}{9 a^{7/3} b^{8/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (5 a^3 f-2 a^2 b e-a b^2 d+4 b^3 c\right )}{3 \sqrt{3} a^{7/3} b^{8/3}}+\frac{f x^2}{2 b^2} \]

[Out]

-(c/(a^2*x)) + (f*x^2)/(2*b^2) - ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(3*a^
2*b^2*(a + b*x^3)) + ((4*b^3*c - a*b^2*d - 2*a^2*b*e + 5*a^3*f)*ArcTan[(a^(1/3)
- 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(7/3)*b^(8/3)) + ((4*b^3*c - a*b
^2*d - 2*a^2*b*e + 5*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(7/3)*b^(8/3)) - ((4*
b^3*c - a*b^2*d - 2*a^2*b*e + 5*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)
*x^2])/(18*a^(7/3)*b^(8/3))

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Rubi [A]  time = 0.594864, antiderivative size = 265, 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 \[ -\frac{c}{a^2 x}-\frac{x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^2 b^2 \left (a+b x^3\right )}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (5 a^3 f-2 a^2 b e-a b^2 d+4 b^3 c\right )}{18 a^{7/3} b^{8/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (5 a^3 f-2 a^2 b e-a b^2 d+4 b^3 c\right )}{9 a^{7/3} b^{8/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (5 a^3 f-2 a^2 b e-a b^2 d+4 b^3 c\right )}{3 \sqrt{3} a^{7/3} b^{8/3}}+\frac{f x^2}{2 b^2} \]

Antiderivative was successfully verified.

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

[Out]

-(c/(a^2*x)) + (f*x^2)/(2*b^2) - ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(3*a^
2*b^2*(a + b*x^3)) + ((4*b^3*c - a*b^2*d - 2*a^2*b*e + 5*a^3*f)*ArcTan[(a^(1/3)
- 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(7/3)*b^(8/3)) + ((4*b^3*c - a*b
^2*d - 2*a^2*b*e + 5*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(7/3)*b^(8/3)) - ((4*
b^3*c - a*b^2*d - 2*a^2*b*e + 5*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)
*x^2])/(18*a^(7/3)*b^(8/3))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

f*Integral(x, x)/b**2 - x*(a**3*f/x**2 - a**2*b*e/x**2 + a*b**2*d/x**2 - b**3*c/
x**2)/(3*a*b**3*(a + b*x**3)) - (a**2*f - a*b*e + b**2*d)/(a*b**3*x) + (3*a**2*f
- 2*a*b*e + b**2*d)*log(a**(1/3) + b**(1/3)*x)/(3*a**(4/3)*b**(8/3)) - (3*a**2*
f - 2*a*b*e + b**2*d)*log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(6*a**
(4/3)*b**(8/3)) + sqrt(3)*(3*a**2*f - 2*a*b*e + b**2*d)*atan(sqrt(3)*(a**(1/3)/3
- 2*b**(1/3)*x/3)/a**(1/3))/(3*a**(4/3)*b**(8/3))

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Mathematica [A]  time = 0.290292, size = 255, normalized size = 0.96 \[ \frac{1}{18} \left (-\frac{18 c}{a^2 x}+\frac{6 x^2 \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{a^2 b^2 \left (a+b x^3\right )}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (5 a^3 f-2 a^2 b e-a b^2 d+4 b^3 c\right )}{a^{7/3} b^{8/3}}+\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (5 a^3 f-2 a^2 b e-a b^2 d+4 b^3 c\right )}{a^{7/3} b^{8/3}}+\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (5 a^3 f-2 a^2 b e-a b^2 d+4 b^3 c\right )}{a^{7/3} b^{8/3}}+\frac{9 f x^2}{b^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-18*c)/(a^2*x) + (9*f*x^2)/b^2 + (6*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x^2
)/(a^2*b^2*(a + b*x^3)) + (2*Sqrt[3]*(4*b^3*c - a*b^2*d - 2*a^2*b*e + 5*a^3*f)*A
rcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/(a^(7/3)*b^(8/3)) + (2*(4*b^3*c - a*
b^2*d - 2*a^2*b*e + 5*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(a^(7/3)*b^(8/3)) - ((4*b
^3*c - a*b^2*d - 2*a^2*b*e + 5*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*
x^2])/(a^(7/3)*b^(8/3)))/18

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Maple [B]  time = 0.018, size = 474, normalized size = 1.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*f*x^2/b^2-1/a^2*c/x+1/3*a/b^2*x^2/(b*x^3+a)*f-1/3/b*x^2/(b*x^3+a)*e+1/3*d*x^
2/a/(b*x^3+a)-1/3/a^2*b*x^2/(b*x^3+a)*c+5/9*a/b^3*f/(a/b)^(1/3)*ln(x+(a/b)^(1/3)
)-5/18*a/b^3*f/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-5/9*a/b^3*f*3^(1/2)
/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-1/9*d/a/b/(a/b)^(1/3)*ln(x+
(a/b)^(1/3))+1/18*d/a/b/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+1/9*d/a*3^
(1/2)/b/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+4/9/a^2*c/(a/b)^(1/3
)*ln(x+(a/b)^(1/3))-2/9/a^2*c/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-4/9/
a^2*c*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-2/9/b^2*e/(a/b
)^(1/3)*ln(x+(a/b)^(1/3))+1/9/b^2*e/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3)
)+2/9/b^2*e*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-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((f*x^9 + e*x^6 + d*x^3 + c)/((b*x^3 + a)^2*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238255, size = 510, normalized size = 1.92 \[ -\frac{\sqrt{3}{\left (\sqrt{3}{\left ({\left (4 \, b^{4} c - a b^{3} d - 2 \, a^{2} b^{2} e + 5 \, a^{3} b f\right )} x^{4} +{\left (4 \, a b^{3} c - a^{2} b^{2} d - 2 \, a^{3} b e + 5 \, a^{4} f\right )} x\right )} \log \left (\left (a b^{2}\right )^{\frac{1}{3}} b x^{2} + a b - \left (a b^{2}\right )^{\frac{2}{3}} x\right ) - 2 \, \sqrt{3}{\left ({\left (4 \, b^{4} c - a b^{3} d - 2 \, a^{2} b^{2} e + 5 \, a^{3} b f\right )} x^{4} +{\left (4 \, a b^{3} c - a^{2} b^{2} d - 2 \, a^{3} b e + 5 \, a^{4} f\right )} x\right )} \log \left (a b + \left (a b^{2}\right )^{\frac{2}{3}} x\right ) + 6 \,{\left ({\left (4 \, b^{4} c - a b^{3} d - 2 \, a^{2} b^{2} e + 5 \, a^{3} b f\right )} x^{4} +{\left (4 \, a b^{3} c - a^{2} b^{2} d - 2 \, a^{3} b e + 5 \, a^{4} f\right )} x\right )} \arctan \left (-\frac{\sqrt{3} a b - 2 \, \sqrt{3} \left (a b^{2}\right )^{\frac{2}{3}} x}{3 \, a b}\right ) - 3 \, \sqrt{3}{\left (3 \, a^{2} b f x^{6} - 6 \, a b^{2} c -{\left (8 \, b^{3} c - 2 \, a b^{2} d + 2 \, a^{2} b e - 5 \, a^{3} f\right )} x^{3}\right )} \left (a b^{2}\right )^{\frac{1}{3}}\right )}}{54 \,{\left (a^{2} b^{3} x^{4} + a^{3} b^{2} x\right )} \left (a b^{2}\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)/((b*x^3 + a)^2*x^2),x, algorithm="fricas")

[Out]

-1/54*sqrt(3)*(sqrt(3)*((4*b^4*c - a*b^3*d - 2*a^2*b^2*e + 5*a^3*b*f)*x^4 + (4*a
*b^3*c - a^2*b^2*d - 2*a^3*b*e + 5*a^4*f)*x)*log((a*b^2)^(1/3)*b*x^2 + a*b - (a*
b^2)^(2/3)*x) - 2*sqrt(3)*((4*b^4*c - a*b^3*d - 2*a^2*b^2*e + 5*a^3*b*f)*x^4 + (
4*a*b^3*c - a^2*b^2*d - 2*a^3*b*e + 5*a^4*f)*x)*log(a*b + (a*b^2)^(2/3)*x) + 6*(
(4*b^4*c - a*b^3*d - 2*a^2*b^2*e + 5*a^3*b*f)*x^4 + (4*a*b^3*c - a^2*b^2*d - 2*a
^3*b*e + 5*a^4*f)*x)*arctan(-1/3*(sqrt(3)*a*b - 2*sqrt(3)*(a*b^2)^(2/3)*x)/(a*b)
) - 3*sqrt(3)*(3*a^2*b*f*x^6 - 6*a*b^2*c - (8*b^3*c - 2*a*b^2*d + 2*a^2*b*e - 5*
a^3*f)*x^3)*(a*b^2)^(1/3))/((a^2*b^3*x^4 + a^3*b^2*x)*(a*b^2)^(1/3))

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Sympy [A]  time = 49.411, size = 457, normalized size = 1.72 \[ \frac{- 3 a b^{2} c + x^{3} \left (a^{3} f - a^{2} b e + a b^{2} d - 4 b^{3} c\right )}{3 a^{3} b^{2} x + 3 a^{2} b^{3} x^{4}} + \operatorname{RootSum}{\left (729 t^{3} a^{7} b^{8} - 125 a^{9} f^{3} + 150 a^{8} b e f^{2} + 75 a^{7} b^{2} d f^{2} - 60 a^{7} b^{2} e^{2} f - 300 a^{6} b^{3} c f^{2} - 60 a^{6} b^{3} d e f + 8 a^{6} b^{3} e^{3} + 240 a^{5} b^{4} c e f - 15 a^{5} b^{4} d^{2} f + 12 a^{5} b^{4} d e^{2} + 120 a^{4} b^{5} c d f - 48 a^{4} b^{5} c e^{2} + 6 a^{4} b^{5} d^{2} e - 240 a^{3} b^{6} c^{2} f - 48 a^{3} b^{6} c d e + a^{3} b^{6} d^{3} + 96 a^{2} b^{7} c^{2} e - 12 a^{2} b^{7} c d^{2} + 48 a b^{8} c^{2} d - 64 b^{9} c^{3}, \left ( t \mapsto t \log{\left (\frac{81 t^{2} a^{5} b^{5}}{25 a^{6} f^{2} - 20 a^{5} b e f - 10 a^{4} b^{2} d f + 4 a^{4} b^{2} e^{2} + 40 a^{3} b^{3} c f + 4 a^{3} b^{3} d e - 16 a^{2} b^{4} c e + a^{2} b^{4} d^{2} - 8 a b^{5} c d + 16 b^{6} c^{2}} + x \right )} \right )\right )} + \frac{f x^{2}}{2 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-3*a*b**2*c + x**3*(a**3*f - a**2*b*e + a*b**2*d - 4*b**3*c))/(3*a**3*b**2*x +
3*a**2*b**3*x**4) + RootSum(729*_t**3*a**7*b**8 - 125*a**9*f**3 + 150*a**8*b*e*f
**2 + 75*a**7*b**2*d*f**2 - 60*a**7*b**2*e**2*f - 300*a**6*b**3*c*f**2 - 60*a**6
*b**3*d*e*f + 8*a**6*b**3*e**3 + 240*a**5*b**4*c*e*f - 15*a**5*b**4*d**2*f + 12*
a**5*b**4*d*e**2 + 120*a**4*b**5*c*d*f - 48*a**4*b**5*c*e**2 + 6*a**4*b**5*d**2*
e - 240*a**3*b**6*c**2*f - 48*a**3*b**6*c*d*e + a**3*b**6*d**3 + 96*a**2*b**7*c*
*2*e - 12*a**2*b**7*c*d**2 + 48*a*b**8*c**2*d - 64*b**9*c**3, Lambda(_t, _t*log(
81*_t**2*a**5*b**5/(25*a**6*f**2 - 20*a**5*b*e*f - 10*a**4*b**2*d*f + 4*a**4*b**
2*e**2 + 40*a**3*b**3*c*f + 4*a**3*b**3*d*e - 16*a**2*b**4*c*e + a**2*b**4*d**2
- 8*a*b**5*c*d + 16*b**6*c**2) + x))) + f*x**2/(2*b**2)

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GIAC/XCAS [A]  time = 0.219279, size = 477, normalized size = 1.8 \[ \frac{f x^{2}}{2 \, b^{2}} + \frac{{\left (4 \, b^{3} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a b^{2} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 5 \, a^{3} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 2 \, a^{2} b \left (-\frac{a}{b}\right )^{\frac{1}{3}} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{3} b^{2}} - \frac{4 \, b^{3} c x^{3} - a b^{2} d x^{3} - a^{3} f x^{3} + a^{2} b x^{3} e + 3 \, a b^{2} c}{3 \,{\left (b x^{4} + a x\right )} a^{2} b^{2}} + \frac{\sqrt{3}{\left (4 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d + 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{3} b^{4}} - \frac{{\left (4 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d + 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{3} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)/((b*x^3 + a)^2*x^2),x, algorithm="giac")

[Out]

1/2*f*x^2/b^2 + 1/9*(4*b^3*c*(-a/b)^(1/3) - a*b^2*d*(-a/b)^(1/3) + 5*a^3*f*(-a/b
)^(1/3) - 2*a^2*b*(-a/b)^(1/3)*e)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a^3*b^
2) - 1/3*(4*b^3*c*x^3 - a*b^2*d*x^3 - a^3*f*x^3 + a^2*b*x^3*e + 3*a*b^2*c)/((b*x
^4 + a*x)*a^2*b^2) + 1/9*sqrt(3)*(4*(-a*b^2)^(2/3)*b^3*c - (-a*b^2)^(2/3)*a*b^2*
d + 5*(-a*b^2)^(2/3)*a^3*f - 2*(-a*b^2)^(2/3)*a^2*b*e)*arctan(1/3*sqrt(3)*(2*x +
(-a/b)^(1/3))/(-a/b)^(1/3))/(a^3*b^4) - 1/18*(4*(-a*b^2)^(2/3)*b^3*c - (-a*b^2)
^(2/3)*a*b^2*d + 5*(-a*b^2)^(2/3)*a^3*f - 2*(-a*b^2)^(2/3)*a^2*b*e)*ln(x^2 + x*(
-a/b)^(1/3) + (-a/b)^(2/3))/(a^3*b^4)