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

Optimal. Leaf size=297 \[ \frac{2 b c-a d}{5 a^3 x^5}-\frac{c}{8 a^2 x^8}-\frac{a^2 e-2 a b d+3 b^2 c}{2 a^4 x^2}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-2 a^3 f+5 a^2 b e-8 a b^2 d+11 b^3 c\right )}{9 a^{14/3} \sqrt [3]{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-2 a^3 f+5 a^2 b e-8 a b^2 d+11 b^3 c\right )}{3 \sqrt{3} a^{14/3} \sqrt [3]{b}}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-2 a^3 f+5 a^2 b e-8 a b^2 d+11 b^3 c\right )}{18 a^{14/3} \sqrt [3]{b}}-\frac{x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^4 \left (a+b x^3\right )} \]

[Out]

-c/(8*a^2*x^8) + (2*b*c - a*d)/(5*a^3*x^5) - (3*b^2*c - 2*a*b*d + a^2*e)/(2*a^4*
x^2) - ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(3*a^4*(a + b*x^3)) + ((11*b^3*c
- 8*a*b^2*d + 5*a^2*b*e - 2*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/
3))])/(3*Sqrt[3]*a^(14/3)*b^(1/3)) - ((11*b^3*c - 8*a*b^2*d + 5*a^2*b*e - 2*a^3*
f)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(14/3)*b^(1/3)) + ((11*b^3*c - 8*a*b^2*d + 5*a
^2*b*e - 2*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(14/3)*b
^(1/3))

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Rubi [A]  time = 0.758633, antiderivative size = 297, 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 \[ \frac{2 b c-a d}{5 a^3 x^5}-\frac{c}{8 a^2 x^8}-\frac{a^2 e-2 a b d+3 b^2 c}{2 a^4 x^2}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-2 a^3 f+5 a^2 b e-8 a b^2 d+11 b^3 c\right )}{9 a^{14/3} \sqrt [3]{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-2 a^3 f+5 a^2 b e-8 a b^2 d+11 b^3 c\right )}{3 \sqrt{3} a^{14/3} \sqrt [3]{b}}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-2 a^3 f+5 a^2 b e-8 a b^2 d+11 b^3 c\right )}{18 a^{14/3} \sqrt [3]{b}}-\frac{x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^4 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

-c/(8*a^2*x^8) + (2*b*c - a*d)/(5*a^3*x^5) - (3*b^2*c - 2*a*b*d + a^2*e)/(2*a^4*
x^2) - ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(3*a^4*(a + b*x^3)) + ((11*b^3*c
- 8*a*b^2*d + 5*a^2*b*e - 2*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/
3))])/(3*Sqrt[3]*a^(14/3)*b^(1/3)) - ((11*b^3*c - 8*a*b^2*d + 5*a^2*b*e - 2*a^3*
f)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(14/3)*b^(1/3)) + ((11*b^3*c - 8*a*b^2*d + 5*a
^2*b*e - 2*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(14/3)*b
^(1/3))

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Rubi in Sympy [A]  time = 153.905, size = 303, normalized size = 1.02 \[ - \frac{x \left (\frac{a^{3} f}{x^{9}} - \frac{a^{2} b e}{x^{9}} + \frac{a b^{2} d}{x^{9}} - \frac{b^{3} c}{x^{9}}\right )}{3 a b^{3} \left (a + b x^{3}\right )} - \frac{a^{2} f - a b e + b^{2} d}{8 a b^{3} x^{8}} + \frac{2 a^{2} f - 2 a b e + b^{2} d}{5 a^{2} b^{2} x^{5}} - \frac{3 a^{2} f - 2 a b e + b^{2} d}{2 a^{3} b x^{2}} - \frac{\left (3 a^{2} f - 2 a b e + b^{2} d\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{11}{3}} \sqrt [3]{b}} + \frac{\left (3 a^{2} f - 2 a b e + b^{2} d\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{11}{3}} \sqrt [3]{b}} + \frac{\sqrt{3} \left (3 a^{2} f - 2 a b e + b^{2} d\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{11}{3}} \sqrt [3]{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x*(a**3*f/x**9 - a**2*b*e/x**9 + a*b**2*d/x**9 - b**3*c/x**9)/(3*a*b**3*(a + b*
x**3)) - (a**2*f - a*b*e + b**2*d)/(8*a*b**3*x**8) + (2*a**2*f - 2*a*b*e + b**2*
d)/(5*a**2*b**2*x**5) - (3*a**2*f - 2*a*b*e + b**2*d)/(2*a**3*b*x**2) - (3*a**2*
f - 2*a*b*e + b**2*d)*log(a**(1/3) + b**(1/3)*x)/(3*a**(11/3)*b**(1/3)) + (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
**(11/3)*b**(1/3)) + 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**(11/3)*b**(1/3))

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Mathematica [A]  time = 0.335302, size = 280, normalized size = 0.94 \[ \frac{-\frac{72 a^{5/3} (a d-2 b c)}{x^5}-\frac{45 a^{8/3} c}{x^8}-\frac{180 a^{2/3} \left (a^2 e-2 a b d+3 b^2 c\right )}{x^2}+\frac{40 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (2 a^3 f-5 a^2 b e+8 a b^2 d-11 b^3 c\right )}{\sqrt [3]{b}}+\frac{40 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-2 a^3 f+5 a^2 b e-8 a b^2 d+11 b^3 c\right )}{\sqrt [3]{b}}+\frac{120 a^{2/3} x \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{a+b x^3}+\frac{20 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-2 a^3 f+5 a^2 b e-8 a b^2 d+11 b^3 c\right )}{\sqrt [3]{b}}}{360 a^{14/3}} \]

Antiderivative was successfully verified.

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

[Out]

((-45*a^(8/3)*c)/x^8 - (72*a^(5/3)*(-2*b*c + a*d))/x^5 - (180*a^(2/3)*(3*b^2*c -
2*a*b*d + a^2*e))/x^2 + (120*a^(2/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/
(a + b*x^3) + (40*Sqrt[3]*(11*b^3*c - 8*a*b^2*d + 5*a^2*b*e - 2*a^3*f)*ArcTan[(1
- (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3) + (40*(-11*b^3*c + 8*a*b^2*d - 5*a^2
*b*e + 2*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/b^(1/3) + (20*(11*b^3*c - 8*a*b^2*d +
5*a^2*b*e - 2*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(1/3))/(3
60*a^(14/3))

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Maple [B]  time = 0.02, size = 520, 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^9/(b*x^3+a)^2,x)

[Out]

-1/8*c/a^2/x^8-1/5/a^2/x^5*d+2/5/a^3/x^5*b*c-1/2/a^2/x^2*e+1/a^3/x^2*b*d-3/2/a^4
/x^2*b^2*c+1/3/a*x/(b*x^3+a)*f-1/3/a^2*x/(b*x^3+a)*b*e+1/3/a^3*x/(b*x^3+a)*b^2*d
-1/3/a^4*x/(b*x^3+a)*b^3*c-5/9/a^2*e/(a/b)^(2/3)*ln(x+(a/b)^(1/3))+5/18/a^2*e/(a
/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-5/9/a^2*e/(a/b)^(2/3)*3^(1/2)*arctan
(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+8/9/a^3*b*d/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-4/9/
a^3*b*d/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+8/9/a^3*b*d/(a/b)^(2/3)*3^
(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-11/9/a^4*b^2*c/(a/b)^(2/3)*ln(x+(a
/b)^(1/3))+11/18/a^4*b^2*c/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-11/9/a^
4*b^2*c/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+2/9/a*f/b/(a
/b)^(2/3)*ln(x+(a/b)^(1/3))-1/9/a*f/b/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/
3))+2/9/a*f/b/(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^9),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.25194, size = 555, normalized size = 1.87 \[ \frac{\sqrt{3}{\left (20 \, \sqrt{3}{\left ({\left (11 \, b^{4} c - 8 \, a b^{3} d + 5 \, a^{2} b^{2} e - 2 \, a^{3} b f\right )} x^{11} +{\left (11 \, a b^{3} c - 8 \, a^{2} b^{2} d + 5 \, a^{3} b e - 2 \, a^{4} f\right )} x^{8}\right )} \log \left (\left (a^{2} b\right )^{\frac{2}{3}} x^{2} - \left (a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 40 \, \sqrt{3}{\left ({\left (11 \, b^{4} c - 8 \, a b^{3} d + 5 \, a^{2} b^{2} e - 2 \, a^{3} b f\right )} x^{11} +{\left (11 \, a b^{3} c - 8 \, a^{2} b^{2} d + 5 \, a^{3} b e - 2 \, a^{4} f\right )} x^{8}\right )} \log \left (\left (a^{2} b\right )^{\frac{1}{3}} x + a\right ) - 120 \,{\left ({\left (11 \, b^{4} c - 8 \, a b^{3} d + 5 \, a^{2} b^{2} e - 2 \, a^{3} b f\right )} x^{11} +{\left (11 \, a b^{3} c - 8 \, a^{2} b^{2} d + 5 \, a^{3} b e - 2 \, a^{4} f\right )} x^{8}\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} x - \sqrt{3} a}{3 \, a}\right ) - 3 \, \sqrt{3}{\left (20 \,{\left (11 \, b^{3} c - 8 \, a b^{2} d + 5 \, a^{2} b e - 2 \, a^{3} f\right )} x^{9} + 12 \,{\left (11 \, a b^{2} c - 8 \, a^{2} b d + 5 \, a^{3} e\right )} x^{6} + 15 \, a^{3} c - 3 \,{\left (11 \, a^{2} b c - 8 \, a^{3} d\right )} x^{3}\right )} \left (a^{2} b\right )^{\frac{1}{3}}\right )}}{1080 \,{\left (a^{4} b x^{11} + a^{5} x^{8}\right )} \left (a^{2} b\right )^{\frac{1}{3}}} \]

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^9),x, algorithm="fricas")

[Out]

1/1080*sqrt(3)*(20*sqrt(3)*((11*b^4*c - 8*a*b^3*d + 5*a^2*b^2*e - 2*a^3*b*f)*x^1
1 + (11*a*b^3*c - 8*a^2*b^2*d + 5*a^3*b*e - 2*a^4*f)*x^8)*log((a^2*b)^(2/3)*x^2
- (a^2*b)^(1/3)*a*x + a^2) - 40*sqrt(3)*((11*b^4*c - 8*a*b^3*d + 5*a^2*b^2*e - 2
*a^3*b*f)*x^11 + (11*a*b^3*c - 8*a^2*b^2*d + 5*a^3*b*e - 2*a^4*f)*x^8)*log((a^2*
b)^(1/3)*x + a) - 120*((11*b^4*c - 8*a*b^3*d + 5*a^2*b^2*e - 2*a^3*b*f)*x^11 + (
11*a*b^3*c - 8*a^2*b^2*d + 5*a^3*b*e - 2*a^4*f)*x^8)*arctan(1/3*(2*sqrt(3)*(a^2*
b)^(1/3)*x - sqrt(3)*a)/a) - 3*sqrt(3)*(20*(11*b^3*c - 8*a*b^2*d + 5*a^2*b*e - 2
*a^3*f)*x^9 + 12*(11*a*b^2*c - 8*a^2*b*d + 5*a^3*e)*x^6 + 15*a^3*c - 3*(11*a^2*b
*c - 8*a^3*d)*x^3)*(a^2*b)^(1/3))/((a^4*b*x^11 + a^5*x^8)*(a^2*b)^(1/3))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.217116, size = 468, normalized size = 1.58 \[ \frac{{\left (11 \, b^{3} c - 8 \, a b^{2} d - 2 \, a^{3} f + 5 \, a^{2} b e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{5}} - \frac{\sqrt{3}{\left (11 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 8 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{5} b} - \frac{b^{3} c x - a b^{2} d x - a^{3} f x + a^{2} b x e}{3 \,{\left (b x^{3} + a\right )} a^{4}} - \frac{{\left (11 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 8 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{5} b} - \frac{60 \, b^{2} c x^{6} - 40 \, a b d x^{6} + 20 \, a^{2} x^{6} e - 16 \, a b c x^{3} + 8 \, a^{2} d x^{3} + 5 \, a^{2} c}{40 \, a^{4} x^{8}} \]

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^9),x, algorithm="giac")

[Out]

1/9*(11*b^3*c - 8*a*b^2*d - 2*a^3*f + 5*a^2*b*e)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^
(1/3)))/a^5 - 1/9*sqrt(3)*(11*(-a*b^2)^(1/3)*b^3*c - 8*(-a*b^2)^(1/3)*a*b^2*d -
2*(-a*b^2)^(1/3)*a^3*f + 5*(-a*b^2)^(1/3)*a^2*b*e)*arctan(1/3*sqrt(3)*(2*x + (-a
/b)^(1/3))/(-a/b)^(1/3))/(a^5*b) - 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^4) - 1/18*(11*(-a*b^2)^(1/3)*b^3*c - 8*(-a*b^2)^(1/3)*a*b^2*d
- 2*(-a*b^2)^(1/3)*a^3*f + 5*(-a*b^2)^(1/3)*a^2*b*e)*ln(x^2 + x*(-a/b)^(1/3) + (
-a/b)^(2/3))/(a^5*b) - 1/40*(60*b^2*c*x^6 - 40*a*b*d*x^6 + 20*a^2*x^6*e - 16*a*b
*c*x^3 + 8*a^2*d*x^3 + 5*a^2*c)/(a^4*x^8)