3.274 \(\int \frac{c+d x^3+e x^6+f x^9}{x^{12} \left (a+b x^3\right )^2} \, dx\)

Optimal. Leaf size=335 \[ \frac{2 b c-a d}{8 a^3 x^8}-\frac{c}{11 a^2 x^{11}}-\frac{a^2 e-2 a b d+3 b^2 c}{5 a^4 x^5}-\frac{b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-5 a^3 f+8 a^2 b e-11 a b^2 d+14 b^3 c\right )}{18 a^{17/3}}+\frac{b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-5 a^3 f+8 a^2 b e-11 a b^2 d+14 b^3 c\right )}{9 a^{17/3}}-\frac{b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-5 a^3 f+8 a^2 b e-11 a b^2 d+14 b^3 c\right )}{3 \sqrt{3} a^{17/3}}+\frac{b x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^5 \left (a+b x^3\right )}+\frac{a^3 (-f)+2 a^2 b e-3 a b^2 d+4 b^3 c}{2 a^5 x^2} \]

[Out]

-c/(11*a^2*x^11) + (2*b*c - a*d)/(8*a^3*x^8) - (3*b^2*c - 2*a*b*d + a^2*e)/(5*a^
4*x^5) + (4*b^3*c - 3*a*b^2*d + 2*a^2*b*e - a^3*f)/(2*a^5*x^2) + (b*(b^3*c - a*b
^2*d + a^2*b*e - a^3*f)*x)/(3*a^5*(a + b*x^3)) - (b^(2/3)*(14*b^3*c - 11*a*b^2*d
+ 8*a^2*b*e - 5*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sq
rt[3]*a^(17/3)) + (b^(2/3)*(14*b^3*c - 11*a*b^2*d + 8*a^2*b*e - 5*a^3*f)*Log[a^(
1/3) + b^(1/3)*x])/(9*a^(17/3)) - (b^(2/3)*(14*b^3*c - 11*a*b^2*d + 8*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^(17/3))

_______________________________________________________________________________________

Rubi [A]  time = 0.965721, antiderivative size = 335, 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 \[ \frac{2 b c-a d}{8 a^3 x^8}-\frac{c}{11 a^2 x^{11}}-\frac{a^2 e-2 a b d+3 b^2 c}{5 a^4 x^5}-\frac{b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-5 a^3 f+8 a^2 b e-11 a b^2 d+14 b^3 c\right )}{18 a^{17/3}}+\frac{b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-5 a^3 f+8 a^2 b e-11 a b^2 d+14 b^3 c\right )}{9 a^{17/3}}-\frac{b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-5 a^3 f+8 a^2 b e-11 a b^2 d+14 b^3 c\right )}{3 \sqrt{3} a^{17/3}}+\frac{b x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^5 \left (a+b x^3\right )}+\frac{a^3 (-f)+2 a^2 b e-3 a b^2 d+4 b^3 c}{2 a^5 x^2} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^12*(a + b*x^3)^2),x]

[Out]

-c/(11*a^2*x^11) + (2*b*c - a*d)/(8*a^3*x^8) - (3*b^2*c - 2*a*b*d + a^2*e)/(5*a^
4*x^5) + (4*b^3*c - 3*a*b^2*d + 2*a^2*b*e - a^3*f)/(2*a^5*x^2) + (b*(b^3*c - a*b
^2*d + a^2*b*e - a^3*f)*x)/(3*a^5*(a + b*x^3)) - (b^(2/3)*(14*b^3*c - 11*a*b^2*d
+ 8*a^2*b*e - 5*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sq
rt[3]*a^(17/3)) + (b^(2/3)*(14*b^3*c - 11*a*b^2*d + 8*a^2*b*e - 5*a^3*f)*Log[a^(
1/3) + b^(1/3)*x])/(9*a^(17/3)) - (b^(2/3)*(14*b^3*c - 11*a*b^2*d + 8*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^(17/3))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 172.126, size = 330, normalized size = 0.99 \[ - \frac{x \left (\frac{a^{3} f}{x^{12}} - \frac{a^{2} b e}{x^{12}} + \frac{a b^{2} d}{x^{12}} - \frac{b^{3} c}{x^{12}}\right )}{3 a b^{3} \left (a + b x^{3}\right )} - \frac{a^{2} f - a b e + b^{2} d}{11 a b^{3} x^{11}} + \frac{2 a^{2} f - 2 a b e + b^{2} d}{8 a^{2} b^{2} x^{8}} - \frac{3 a^{2} f - 2 a b e + b^{2} d}{5 a^{3} b x^{5}} + \frac{3 a^{2} f - 2 a b e + b^{2} d}{2 a^{4} x^{2}} + \frac{b^{\frac{2}{3}} \left (3 a^{2} f - 2 a b e + b^{2} d\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{14}{3}}} - \frac{b^{\frac{2}{3}} \left (3 a^{2} f - 2 a b e + b^{2} d\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{14}{3}}} - \frac{\sqrt{3} b^{\frac{2}{3}} \left (3 a^{2} f - 2 a b e + b^{2} d\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{14}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((f*x**9+e*x**6+d*x**3+c)/x**12/(b*x**3+a)**2,x)

[Out]

-x*(a**3*f/x**12 - a**2*b*e/x**12 + a*b**2*d/x**12 - b**3*c/x**12)/(3*a*b**3*(a
+ b*x**3)) - (a**2*f - a*b*e + b**2*d)/(11*a*b**3*x**11) + (2*a**2*f - 2*a*b*e +
b**2*d)/(8*a**2*b**2*x**8) - (3*a**2*f - 2*a*b*e + b**2*d)/(5*a**3*b*x**5) + (3
*a**2*f - 2*a*b*e + b**2*d)/(2*a**4*x**2) + b**(2/3)*(3*a**2*f - 2*a*b*e + b**2*
d)*log(a**(1/3) + b**(1/3)*x)/(3*a**(14/3)) - b**(2/3)*(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**(14/3)) - sqrt(3)
*b**(2/3)*(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**(14/3))

_______________________________________________________________________________________

Mathematica [A]  time = 0.345357, size = 317, normalized size = 0.95 \[ \frac{-\frac{495 a^{8/3} (a d-2 b c)}{x^8}-\frac{360 a^{11/3} c}{x^{11}}-\frac{792 a^{5/3} \left (a^2 e-2 a b d+3 b^2 c\right )}{x^5}+440 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-5 a^3 f+8 a^2 b e-11 a b^2 d+14 b^3 c\right )-440 \sqrt{3} b^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-5 a^3 f+8 a^2 b e-11 a b^2 d+14 b^3 c\right )-\frac{1320 a^{2/3} b x \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{a+b x^3}-\frac{1980 a^{2/3} \left (a^3 f-2 a^2 b e+3 a b^2 d-4 b^3 c\right )}{x^2}+220 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (5 a^3 f-8 a^2 b e+11 a b^2 d-14 b^3 c\right )}{3960 a^{17/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^12*(a + b*x^3)^2),x]

[Out]

((-360*a^(11/3)*c)/x^11 - (495*a^(8/3)*(-2*b*c + a*d))/x^8 - (792*a^(5/3)*(3*b^2
*c - 2*a*b*d + a^2*e))/x^5 - (1980*a^(2/3)*(-4*b^3*c + 3*a*b^2*d - 2*a^2*b*e + a
^3*f))/x^2 - (1320*a^(2/3)*b*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/(a + b*x^
3) - 440*Sqrt[3]*b^(2/3)*(14*b^3*c - 11*a*b^2*d + 8*a^2*b*e - 5*a^3*f)*ArcTan[(1
- (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 440*b^(2/3)*(14*b^3*c - 11*a*b^2*d + 8*a^2*
b*e - 5*a^3*f)*Log[a^(1/3) + b^(1/3)*x] + 220*b^(2/3)*(-14*b^3*c + 11*a*b^2*d -
8*a^2*b*e + 5*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(3960*a^(17
/3))

_______________________________________________________________________________________

Maple [A]  time = 0.023, size = 566, normalized size = 1.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x^9+e*x^6+d*x^3+c)/x^12/(b*x^3+a)^2,x)

[Out]

1/3*b^2/a^3*x/(b*x^3+a)*e-1/3*b^3/a^4*x/(b*x^3+a)*d-1/8/a^2/x^8*d-1/5/a^2/x^5*e-
1/2/a^2/x^2*f-11/9*b^2/a^4*d/(a/b)^(2/3)*ln(x+(a/b)^(1/3))+11/18*b^2/a^4*d/(a/b)
^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+14/9*b^3/a^5*c/(a/b)^(2/3)*ln(x+(a/b)^(
1/3))-7/9*b^3/a^5*c/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-1/3*b/a^2*x/(b
*x^3+a)*f+8/9*b/a^3*e/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-4/9*b/a^3*e/(a/b)^(2/3)*ln(x
^2-x*(a/b)^(1/3)+(a/b)^(2/3))+1/3*b^4/a^5*x/(b*x^3+a)*c-5/9/a^2*f/(a/b)^(2/3)*3^
(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+1/4/a^3/x^8*b*c+2/5/a^3/x^5*b*d-3/
5/a^4/x^5*b^2*c+1/a^3/x^2*b*e-3/2/a^4/x^2*b^2*d+2/a^5/x^2*b^3*c-5/9/a^2*f/(a/b)^
(2/3)*ln(x+(a/b)^(1/3))+5/18/a^2*f/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))
+8/9*b/a^3*e/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-11/9*b^
2/a^4*d/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+14/9*b^3/a^5
*c/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-1/11*c/a^2/x^11

_______________________________________________________________________________________

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^12),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.23277, size = 670, normalized size = 2. \[ \frac{\sqrt{3}{\left (220 \, \sqrt{3}{\left ({\left (14 \, b^{4} c - 11 \, a b^{3} d + 8 \, a^{2} b^{2} e - 5 \, a^{3} b f\right )} x^{14} +{\left (14 \, a b^{3} c - 11 \, a^{2} b^{2} d + 8 \, a^{3} b e - 5 \, a^{4} f\right )} x^{11}\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b^{2} x^{2} + a b x \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} + a^{2} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}\right ) - 440 \, \sqrt{3}{\left ({\left (14 \, b^{4} c - 11 \, a b^{3} d + 8 \, a^{2} b^{2} e - 5 \, a^{3} b f\right )} x^{14} +{\left (14 \, a b^{3} c - 11 \, a^{2} b^{2} d + 8 \, a^{3} b e - 5 \, a^{4} f\right )} x^{11}\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b x - a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}\right ) + 1320 \,{\left ({\left (14 \, b^{4} c - 11 \, a b^{3} d + 8 \, a^{2} b^{2} e - 5 \, a^{3} b f\right )} x^{14} +{\left (14 \, a b^{3} c - 11 \, a^{2} b^{2} d + 8 \, a^{3} b e - 5 \, a^{4} f\right )} x^{11}\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} b x + \sqrt{3} a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}{3 \, a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}\right ) + 3 \, \sqrt{3}{\left (220 \,{\left (14 \, b^{4} c - 11 \, a b^{3} d + 8 \, a^{2} b^{2} e - 5 \, a^{3} b f\right )} x^{12} + 132 \,{\left (14 \, a b^{3} c - 11 \, a^{2} b^{2} d + 8 \, a^{3} b e - 5 \, a^{4} f\right )} x^{9} - 33 \,{\left (14 \, a^{2} b^{2} c - 11 \, a^{3} b d + 8 \, a^{4} e\right )} x^{6} - 120 \, a^{4} c + 15 \,{\left (14 \, a^{3} b c - 11 \, a^{4} d\right )} x^{3}\right )}\right )}}{11880 \,{\left (a^{5} b x^{14} + a^{6} x^{11}\right )}} \]

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^12),x, algorithm="fricas")

[Out]

1/11880*sqrt(3)*(220*sqrt(3)*((14*b^4*c - 11*a*b^3*d + 8*a^2*b^2*e - 5*a^3*b*f)*
x^14 + (14*a*b^3*c - 11*a^2*b^2*d + 8*a^3*b*e - 5*a^4*f)*x^11)*(-b^2/a^2)^(1/3)*
log(b^2*x^2 + a*b*x*(-b^2/a^2)^(1/3) + a^2*(-b^2/a^2)^(2/3)) - 440*sqrt(3)*((14*
b^4*c - 11*a*b^3*d + 8*a^2*b^2*e - 5*a^3*b*f)*x^14 + (14*a*b^3*c - 11*a^2*b^2*d
+ 8*a^3*b*e - 5*a^4*f)*x^11)*(-b^2/a^2)^(1/3)*log(b*x - a*(-b^2/a^2)^(1/3)) + 13
20*((14*b^4*c - 11*a*b^3*d + 8*a^2*b^2*e - 5*a^3*b*f)*x^14 + (14*a*b^3*c - 11*a^
2*b^2*d + 8*a^3*b*e - 5*a^4*f)*x^11)*(-b^2/a^2)^(1/3)*arctan(1/3*(2*sqrt(3)*b*x
+ sqrt(3)*a*(-b^2/a^2)^(1/3))/(a*(-b^2/a^2)^(1/3))) + 3*sqrt(3)*(220*(14*b^4*c -
11*a*b^3*d + 8*a^2*b^2*e - 5*a^3*b*f)*x^12 + 132*(14*a*b^3*c - 11*a^2*b^2*d + 8
*a^3*b*e - 5*a^4*f)*x^9 - 33*(14*a^2*b^2*c - 11*a^3*b*d + 8*a^4*e)*x^6 - 120*a^4
*c + 15*(14*a^3*b*c - 11*a^4*d)*x^3))/(a^5*b*x^14 + a^6*x^11)

_______________________________________________________________________________________

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**12/(b*x**3+a)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.217836, size = 528, normalized size = 1.58 \[ \frac{\sqrt{3}{\left (14 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 11 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 8 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{6}} - \frac{{\left (14 \, b^{4} c - 11 \, a b^{3} d - 5 \, a^{3} b f + 8 \, a^{2} b^{2} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{6}} + \frac{{\left (14 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 11 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 8 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{6}} + \frac{b^{4} c x - a b^{3} d x - a^{3} b f x + a^{2} b^{2} x e}{3 \,{\left (b x^{3} + a\right )} a^{5}} + \frac{880 \, b^{3} c x^{9} - 660 \, a b^{2} d x^{9} - 220 \, a^{3} f x^{9} + 440 \, a^{2} b x^{9} e - 264 \, a b^{2} c x^{6} + 176 \, a^{2} b d x^{6} - 88 \, a^{3} x^{6} e + 110 \, a^{2} b c x^{3} - 55 \, a^{3} d x^{3} - 40 \, a^{3} c}{440 \, a^{5} x^{11}} \]

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^12),x, algorithm="giac")

[Out]

1/9*sqrt(3)*(14*(-a*b^2)^(1/3)*b^3*c - 11*(-a*b^2)^(1/3)*a*b^2*d - 5*(-a*b^2)^(1
/3)*a^3*f + 8*(-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^6 - 1/9*(14*b^4*c - 11*a*b^3*d - 5*a^3*b*f + 8*a^2*b^2*e)*(-a/b)^(
1/3)*ln(abs(x - (-a/b)^(1/3)))/a^6 + 1/18*(14*(-a*b^2)^(1/3)*b^3*c - 11*(-a*b^2)
^(1/3)*a*b^2*d - 5*(-a*b^2)^(1/3)*a^3*f + 8*(-a*b^2)^(1/3)*a^2*b*e)*ln(x^2 + x*(
-a/b)^(1/3) + (-a/b)^(2/3))/a^6 + 1/3*(b^4*c*x - a*b^3*d*x - a^3*b*f*x + a^2*b^2
*x*e)/((b*x^3 + a)*a^5) + 1/440*(880*b^3*c*x^9 - 660*a*b^2*d*x^9 - 220*a^3*f*x^9
+ 440*a^2*b*x^9*e - 264*a*b^2*c*x^6 + 176*a^2*b*d*x^6 - 88*a^3*x^6*e + 110*a^2*
b*c*x^3 - 55*a^3*d*x^3 - 40*a^3*c)/(a^5*x^11)