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

Optimal. Leaf size=341 \[ \frac{3 b c-a d}{5 a^4 x^5}-\frac{c}{8 a^3 x^8}-\frac{a^2 e-3 a b d+6 b^2 c}{2 a^5 x^2}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-5 a^3 f+20 a^2 b e-44 a b^2 d+77 b^3 c\right )}{27 a^{17/3} \sqrt [3]{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-5 a^3 f+20 a^2 b e-44 a b^2 d+77 b^3 c\right )}{9 \sqrt{3} a^{17/3} \sqrt [3]{b}}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-5 a^3 f+20 a^2 b e-44 a b^2 d+77 b^3 c\right )}{54 a^{17/3} \sqrt [3]{b}}-\frac{x \left (-5 a^3 f+11 a^2 b e-17 a b^2 d+23 b^3 c\right )}{18 a^5 \left (a+b x^3\right )}-\frac{x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^4 \left (a+b x^3\right )^2} \]

[Out]

-c/(8*a^3*x^8) + (3*b*c - a*d)/(5*a^4*x^5) - (6*b^2*c - 3*a*b*d + a^2*e)/(2*a^5*
x^2) - ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(6*a^4*(a + b*x^3)^2) - ((23*b^3*
c - 17*a*b^2*d + 11*a^2*b*e - 5*a^3*f)*x)/(18*a^5*(a + b*x^3)) + ((77*b^3*c - 44
*a*b^2*d + 20*a^2*b*e - 5*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3)
)])/(9*Sqrt[3]*a^(17/3)*b^(1/3)) - ((77*b^3*c - 44*a*b^2*d + 20*a^2*b*e - 5*a^3*
f)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(17/3)*b^(1/3)) + ((77*b^3*c - 44*a*b^2*d + 2
0*a^2*b*e - 5*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(17/3
)*b^(1/3))

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Rubi [A]  time = 1.12027, antiderivative size = 341, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{3 b c-a d}{5 a^4 x^5}-\frac{c}{8 a^3 x^8}-\frac{a^2 e-3 a b d+6 b^2 c}{2 a^5 x^2}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-5 a^3 f+20 a^2 b e-44 a b^2 d+77 b^3 c\right )}{27 a^{17/3} \sqrt [3]{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-5 a^3 f+20 a^2 b e-44 a b^2 d+77 b^3 c\right )}{9 \sqrt{3} a^{17/3} \sqrt [3]{b}}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-5 a^3 f+20 a^2 b e-44 a b^2 d+77 b^3 c\right )}{54 a^{17/3} \sqrt [3]{b}}-\frac{x \left (-5 a^3 f+11 a^2 b e-17 a b^2 d+23 b^3 c\right )}{18 a^5 \left (a+b x^3\right )}-\frac{x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^4 \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-c/(8*a^3*x^8) + (3*b*c - a*d)/(5*a^4*x^5) - (6*b^2*c - 3*a*b*d + a^2*e)/(2*a^5*
x^2) - ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(6*a^4*(a + b*x^3)^2) - ((23*b^3*
c - 17*a*b^2*d + 11*a^2*b*e - 5*a^3*f)*x)/(18*a^5*(a + b*x^3)) + ((77*b^3*c - 44
*a*b^2*d + 20*a^2*b*e - 5*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3)
)])/(9*Sqrt[3]*a^(17/3)*b^(1/3)) - ((77*b^3*c - 44*a*b^2*d + 20*a^2*b*e - 5*a^3*
f)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(17/3)*b^(1/3)) + ((77*b^3*c - 44*a*b^2*d + 2
0*a^2*b*e - 5*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(17/3
)*b^(1/3))

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

[Out]

Timed out

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Mathematica [A]  time = 0.481443, size = 324, normalized size = 0.95 \[ \frac{-\frac{216 a^{5/3} (a d-3 b c)}{x^5}-\frac{135 a^{8/3} c}{x^8}-\frac{540 a^{2/3} \left (a^2 e-3 a b d+6 b^2 c\right )}{x^2}+\frac{40 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (5 a^3 f-20 a^2 b e+44 a b^2 d-77 b^3 c\right )}{\sqrt [3]{b}}+\frac{40 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-5 a^3 f+20 a^2 b e-44 a b^2 d+77 b^3 c\right )}{\sqrt [3]{b}}+\frac{180 a^{5/3} x \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{\left (a+b x^3\right )^2}+\frac{60 a^{2/3} x \left (5 a^3 f-11 a^2 b e+17 a b^2 d-23 b^3 c\right )}{a+b x^3}+\frac{20 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-5 a^3 f+20 a^2 b e-44 a b^2 d+77 b^3 c\right )}{\sqrt [3]{b}}}{1080 a^{17/3}} \]

Antiderivative was successfully verified.

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

[Out]

((-135*a^(8/3)*c)/x^8 - (216*a^(5/3)*(-3*b*c + a*d))/x^5 - (540*a^(2/3)*(6*b^2*c
- 3*a*b*d + a^2*e))/x^2 + (180*a^(5/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x
)/(a + b*x^3)^2 + (60*a^(2/3)*(-23*b^3*c + 17*a*b^2*d - 11*a^2*b*e + 5*a^3*f)*x)
/(a + b*x^3) + (40*Sqrt[3]*(77*b^3*c - 44*a*b^2*d + 20*a^2*b*e - 5*a^3*f)*ArcTan
[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3) + (40*(-77*b^3*c + 44*a*b^2*d - 2
0*a^2*b*e + 5*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/b^(1/3) + (20*(77*b^3*c - 44*a*b^
2*d + 20*a^2*b*e - 5*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(1
/3))/(1080*a^(17/3))

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Maple [B]  time = 0.025, size = 603, 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)^3,x)

[Out]

5/27/a^2*f/b/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+44/27/a
^4*b*d/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-77/27/a^5*b^2
*c/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+4/9/a/(b*x^3+a)^2
*f*x-20/27/a^3*e/(a/b)^(2/3)*ln(x+(a/b)^(1/3))+10/27/a^3*e/(a/b)^(2/3)*ln(x^2-x*
(a/b)^(1/3)+(a/b)^(2/3))+3/5/a^4/x^5*b*c+3/2/a^4/x^2*b*d-3/a^5/x^2*b^2*c+10/9/a^
3/(b*x^3+a)^2*b^2*d*x-13/9/a^4/(b*x^3+a)^2*b^3*c*x+5/27/a^2*f/b/(a/b)^(2/3)*ln(x
+(a/b)^(1/3))-5/54/a^2*f/b/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-20/27/a
^3*e/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+44/27/a^4*b*d/(
a/b)^(2/3)*ln(x+(a/b)^(1/3))-1/8*c/a^3/x^8-1/5/a^3/x^5*d-1/2/a^3/x^2*e-11/18/a^3
/(b*x^3+a)^2*x^4*b^2*e-22/27/a^4*b*d/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3
))-77/27/a^5*b^2*c/(a/b)^(2/3)*ln(x+(a/b)^(1/3))+77/54/a^5*b^2*c/(a/b)^(2/3)*ln(
x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-23/18/a^5/(b*x^3+a)^2*x^4*b^4*c-7/9/a^2/(b*x^3+a)
^2*b*e*x+17/18/a^4/(b*x^3+a)^2*x^4*b^3*d+5/18/a^2/(b*x^3+a)^2*x^4*b*f

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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)^3*x^9),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224288, size = 794, normalized size = 2.33 \[ \frac{\sqrt{3}{\left (20 \, \sqrt{3}{\left ({\left (77 \, b^{5} c - 44 \, a b^{4} d + 20 \, a^{2} b^{3} e - 5 \, a^{3} b^{2} f\right )} x^{14} + 2 \,{\left (77 \, a b^{4} c - 44 \, a^{2} b^{3} d + 20 \, a^{3} b^{2} e - 5 \, a^{4} b f\right )} x^{11} +{\left (77 \, a^{2} b^{3} c - 44 \, a^{3} b^{2} d + 20 \, a^{4} b e - 5 \, a^{5} f\right )} x^{8}\right )} \log \left (\left (a^{2} b\right )^{\frac{2}{3}} x^{2} - \left (a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 40 \, \sqrt{3}{\left ({\left (77 \, b^{5} c - 44 \, a b^{4} d + 20 \, a^{2} b^{3} e - 5 \, a^{3} b^{2} f\right )} x^{14} + 2 \,{\left (77 \, a b^{4} c - 44 \, a^{2} b^{3} d + 20 \, a^{3} b^{2} e - 5 \, a^{4} b f\right )} x^{11} +{\left (77 \, a^{2} b^{3} c - 44 \, a^{3} b^{2} d + 20 \, a^{4} b e - 5 \, a^{5} f\right )} x^{8}\right )} \log \left (\left (a^{2} b\right )^{\frac{1}{3}} x + a\right ) - 120 \,{\left ({\left (77 \, b^{5} c - 44 \, a b^{4} d + 20 \, a^{2} b^{3} e - 5 \, a^{3} b^{2} f\right )} x^{14} + 2 \,{\left (77 \, a b^{4} c - 44 \, a^{2} b^{3} d + 20 \, a^{3} b^{2} e - 5 \, a^{4} b f\right )} x^{11} +{\left (77 \, a^{2} b^{3} c - 44 \, a^{3} b^{2} d + 20 \, a^{4} b e - 5 \, a^{5} f\right )} x^{8}\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} x - \sqrt{3} a}{3 \, a}\right ) - 3 \, \sqrt{3}{\left (20 \,{\left (77 \, b^{4} c - 44 \, a b^{3} d + 20 \, a^{2} b^{2} e - 5 \, a^{3} b f\right )} x^{12} + 32 \,{\left (77 \, a b^{3} c - 44 \, a^{2} b^{2} d + 20 \, a^{3} b e - 5 \, a^{4} f\right )} x^{9} + 9 \,{\left (77 \, a^{2} b^{2} c - 44 \, a^{3} b d + 20 \, a^{4} e\right )} x^{6} + 45 \, a^{4} c - 18 \,{\left (7 \, a^{3} b c - 4 \, a^{4} d\right )} x^{3}\right )} \left (a^{2} b\right )^{\frac{1}{3}}\right )}}{3240 \,{\left (a^{5} b^{2} x^{14} + 2 \, a^{6} b x^{11} + a^{7} x^{8}\right )} \left (a^{2} b\right )^{\frac{1}{3}}} \]

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)^3*x^9),x, algorithm="fricas")

[Out]

1/3240*sqrt(3)*(20*sqrt(3)*((77*b^5*c - 44*a*b^4*d + 20*a^2*b^3*e - 5*a^3*b^2*f)
*x^14 + 2*(77*a*b^4*c - 44*a^2*b^3*d + 20*a^3*b^2*e - 5*a^4*b*f)*x^11 + (77*a^2*
b^3*c - 44*a^3*b^2*d + 20*a^4*b*e - 5*a^5*f)*x^8)*log((a^2*b)^(2/3)*x^2 - (a^2*b
)^(1/3)*a*x + a^2) - 40*sqrt(3)*((77*b^5*c - 44*a*b^4*d + 20*a^2*b^3*e - 5*a^3*b
^2*f)*x^14 + 2*(77*a*b^4*c - 44*a^2*b^3*d + 20*a^3*b^2*e - 5*a^4*b*f)*x^11 + (77
*a^2*b^3*c - 44*a^3*b^2*d + 20*a^4*b*e - 5*a^5*f)*x^8)*log((a^2*b)^(1/3)*x + a)
- 120*((77*b^5*c - 44*a*b^4*d + 20*a^2*b^3*e - 5*a^3*b^2*f)*x^14 + 2*(77*a*b^4*c
- 44*a^2*b^3*d + 20*a^3*b^2*e - 5*a^4*b*f)*x^11 + (77*a^2*b^3*c - 44*a^3*b^2*d
+ 20*a^4*b*e - 5*a^5*f)*x^8)*arctan(1/3*(2*sqrt(3)*(a^2*b)^(1/3)*x - sqrt(3)*a)/
a) - 3*sqrt(3)*(20*(77*b^4*c - 44*a*b^3*d + 20*a^2*b^2*e - 5*a^3*b*f)*x^12 + 32*
(77*a*b^3*c - 44*a^2*b^2*d + 20*a^3*b*e - 5*a^4*f)*x^9 + 9*(77*a^2*b^2*c - 44*a^
3*b*d + 20*a^4*e)*x^6 + 45*a^4*c - 18*(7*a^3*b*c - 4*a^4*d)*x^3)*(a^2*b)^(1/3))/
((a^5*b^2*x^14 + 2*a^6*b*x^11 + a^7*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)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.219569, size = 532, normalized size = 1.56 \[ \frac{{\left (77 \, b^{3} c - 44 \, a b^{2} d - 5 \, a^{3} f + 20 \, a^{2} b e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{6}} - \frac{\sqrt{3}{\left (77 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 44 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 20 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{6} b} - \frac{{\left (77 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 44 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 20 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{6} b} - \frac{23 \, b^{4} c x^{4} - 17 \, a b^{3} d x^{4} - 5 \, a^{3} b f x^{4} + 11 \, a^{2} b^{2} x^{4} e + 26 \, a b^{3} c x - 20 \, a^{2} b^{2} d x - 8 \, a^{4} f x + 14 \, a^{3} b x e}{18 \,{\left (b x^{3} + a\right )}^{2} a^{5}} - \frac{120 \, b^{2} c x^{6} - 60 \, a b d x^{6} + 20 \, a^{2} x^{6} e - 24 \, a b c x^{3} + 8 \, a^{2} d x^{3} + 5 \, a^{2} c}{40 \, a^{5} x^{8}} \]

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)^3*x^9),x, algorithm="giac")

[Out]

1/27*(77*b^3*c - 44*a*b^2*d - 5*a^3*f + 20*a^2*b*e)*(-a/b)^(1/3)*ln(abs(x - (-a/
b)^(1/3)))/a^6 - 1/27*sqrt(3)*(77*(-a*b^2)^(1/3)*b^3*c - 44*(-a*b^2)^(1/3)*a*b^2
*d - 5*(-a*b^2)^(1/3)*a^3*f + 20*(-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*b) - 1/54*(77*(-a*b^2)^(1/3)*b^3*c - 44*(-a*
b^2)^(1/3)*a*b^2*d - 5*(-a*b^2)^(1/3)*a^3*f + 20*(-a*b^2)^(1/3)*a^2*b*e)*ln(x^2
+ x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^6*b) - 1/18*(23*b^4*c*x^4 - 17*a*b^3*d*x^4 -
5*a^3*b*f*x^4 + 11*a^2*b^2*x^4*e + 26*a*b^3*c*x - 20*a^2*b^2*d*x - 8*a^4*f*x +
14*a^3*b*x*e)/((b*x^3 + a)^2*a^5) - 1/40*(120*b^2*c*x^6 - 60*a*b*d*x^6 + 20*a^2*
x^6*e - 24*a*b*c*x^3 + 8*a^2*d*x^3 + 5*a^2*c)/(a^5*x^8)