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

Optimal. Leaf size=260 \[ -\frac{c}{2 a^2 x^2}-\frac{x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^2 b^2 \left (a+b x^3\right )}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (4 a^3 f-a^2 b e-2 a b^2 d+5 b^3 c\right )}{18 a^{8/3} b^{7/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (4 a^3 f-a^2 b e-2 a b^2 d+5 b^3 c\right )}{9 a^{8/3} b^{7/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (4 a^3 f-a^2 b e-2 a b^2 d+5 b^3 c\right )}{3 \sqrt{3} a^{8/3} b^{7/3}}+\frac{f x}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.573303, antiderivative size = 260, 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 \[ -\frac{c}{2 a^2 x^2}-\frac{x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^2 b^2 \left (a+b x^3\right )}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (4 a^3 f-a^2 b e-2 a b^2 d+5 b^3 c\right )}{18 a^{8/3} b^{7/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (4 a^3 f-a^2 b e-2 a b^2 d+5 b^3 c\right )}{9 a^{8/3} b^{7/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (4 a^3 f-a^2 b e-2 a b^2 d+5 b^3 c\right )}{3 \sqrt{3} a^{8/3} b^{7/3}}+\frac{f x}{b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 132.498, size = 250, normalized size = 0.96 \[ \frac{f x}{b^{2}} - \frac{x \left (\frac{a^{3} f}{x^{3}} - \frac{a^{2} b e}{x^{3}} + \frac{a b^{2} d}{x^{3}} - \frac{b^{3} c}{x^{3}}\right )}{3 a b^{3} \left (a + b x^{3}\right )} - \frac{a^{2} f - a b e + b^{2} d}{2 a b^{3} x^{2}} - \frac{\left (3 a^{2} f - 2 a b e + b^{2} d\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{5}{3}} b^{\frac{7}{3}}} + \frac{\left (3 a^{2} f - 2 a b e + b^{2} d\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{5}{3}} b^{\frac{7}{3}}} + \frac{\sqrt{3} \left (3 a^{2} f - 2 a b e + b^{2} d\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{5}{3}} b^{\frac{7}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

f*x/b**2 - x*(a**3*f/x**3 - a**2*b*e/x**3 + a*b**2*d/x**3 - b**3*c/x**3)/(3*a*b*
*3*(a + b*x**3)) - (a**2*f - a*b*e + b**2*d)/(2*a*b**3*x**2) - (3*a**2*f - 2*a*b
*e + b**2*d)*log(a**(1/3) + b**(1/3)*x)/(3*a**(5/3)*b**(7/3)) + (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**(5/3)*b*
*(7/3)) + 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**(5/3)*b**(7/3))

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Mathematica [A]  time = 0.282511, size = 250, normalized size = 0.96 \[ \frac{1}{18} \left (-\frac{9 c}{a^2 x^2}+\frac{6 x \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{a^2 b^2 \left (a+b x^3\right )}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (4 a^3 f-a^2 b e-2 a b^2 d+5 b^3 c\right )}{a^{8/3} b^{7/3}}-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (4 a^3 f-a^2 b e-2 a b^2 d+5 b^3 c\right )}{a^{8/3} b^{7/3}}+\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (4 a^3 f-a^2 b e-2 a b^2 d+5 b^3 c\right )}{a^{8/3} b^{7/3}}+\frac{18 f x}{b^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.016, size = 463, 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^3/(b*x^3+a)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.239142, size = 522, normalized size = 2.01 \[ -\frac{\sqrt{3}{\left (\sqrt{3}{\left ({\left (5 \, b^{4} c - 2 \, a b^{3} d - a^{2} b^{2} e + 4 \, a^{3} b f\right )} x^{5} +{\left (5 \, a b^{3} c - 2 \, a^{2} b^{2} d - a^{3} b e + 4 \, a^{4} f\right )} x^{2}\right )} \log \left (\left (-a^{2} b\right )^{\frac{2}{3}} x^{2} + \left (-a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 2 \, \sqrt{3}{\left ({\left (5 \, b^{4} c - 2 \, a b^{3} d - a^{2} b^{2} e + 4 \, a^{3} b f\right )} x^{5} +{\left (5 \, a b^{3} c - 2 \, a^{2} b^{2} d - a^{3} b e + 4 \, a^{4} f\right )} x^{2}\right )} \log \left (\left (-a^{2} b\right )^{\frac{1}{3}} x - a\right ) + 6 \,{\left ({\left (5 \, b^{4} c - 2 \, a b^{3} d - a^{2} b^{2} e + 4 \, a^{3} b f\right )} x^{5} +{\left (5 \, a b^{3} c - 2 \, a^{2} b^{2} d - a^{3} b e + 4 \, a^{4} f\right )} x^{2}\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (-a^{2} b\right )^{\frac{1}{3}} x + \sqrt{3} a}{3 \, a}\right ) - 3 \, \sqrt{3}{\left (6 \, a^{2} b f x^{6} - 3 \, a b^{2} c -{\left (5 \, b^{3} c - 2 \, a b^{2} d + 2 \, a^{2} b e - 8 \, a^{3} f\right )} x^{3}\right )} \left (-a^{2} b\right )^{\frac{1}{3}}\right )}}{54 \,{\left (a^{2} b^{3} x^{5} + a^{3} b^{2} x^{2}\right )} \left (-a^{2} b\right )^{\frac{1}{3}}} \]

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^3),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 107.283, size = 381, normalized size = 1.47 \[ \frac{- 3 a b^{2} c + x^{3} \left (2 a^{3} f - 2 a^{2} b e + 2 a b^{2} d - 5 b^{3} c\right )}{6 a^{3} b^{2} x^{2} + 6 a^{2} b^{3} x^{5}} + \operatorname{RootSum}{\left (729 t^{3} a^{8} b^{7} + 64 a^{9} f^{3} - 48 a^{8} b e f^{2} - 96 a^{7} b^{2} d f^{2} + 12 a^{7} b^{2} e^{2} f + 240 a^{6} b^{3} c f^{2} + 48 a^{6} b^{3} d e f - a^{6} b^{3} e^{3} - 120 a^{5} b^{4} c e f + 48 a^{5} b^{4} d^{2} f - 6 a^{5} b^{4} d e^{2} - 240 a^{4} b^{5} c d f + 15 a^{4} b^{5} c e^{2} - 12 a^{4} b^{5} d^{2} e + 300 a^{3} b^{6} c^{2} f + 60 a^{3} b^{6} c d e - 8 a^{3} b^{6} d^{3} - 75 a^{2} b^{7} c^{2} e + 60 a^{2} b^{7} c d^{2} - 150 a b^{8} c^{2} d + 125 b^{9} c^{3}, \left ( t \mapsto t \log{\left (- \frac{9 t a^{3} b^{2}}{4 a^{3} f - a^{2} b e - 2 a b^{2} d + 5 b^{3} c} + x \right )} \right )\right )} + \frac{f x}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.217019, size = 417, normalized size = 1.6 \[ \frac{f x}{b^{2}} + \frac{{\left (5 \, b^{3} c - 2 \, a b^{2} d + 4 \, a^{3} f - a^{2} b e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{3} b^{2}} - \frac{c}{2 \, a^{2} x^{2}} - \frac{\sqrt{3}{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d + 4 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f - \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{3} b^{3}} - \frac{b^{3} c x - a b^{2} d x - a^{3} f x + a^{2} b x e}{3 \,{\left (b x^{3} + a\right )} a^{2} b^{2}} - \frac{{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d + 4 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f - \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{3} b^{3}} \]

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^3),x, algorithm="giac")

[Out]

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