3.299 \(\int \frac{c+d x^3+e x^6+f x^9}{x^8 \left (a+b x^3\right )^3} \, dx\)

Optimal. Leaf size=343 \[ \frac{3 b c-a d}{4 a^4 x^4}-\frac{c}{7 a^3 x^7}-\frac{a^2 e-3 a b d+6 b^2 c}{a^5 x}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-2 a^3 f+14 a^2 b e-35 a b^2 d+65 b^3 c\right )}{54 a^{16/3} b^{2/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-2 a^3 f+14 a^2 b e-35 a b^2 d+65 b^3 c\right )}{27 a^{16/3} b^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-2 a^3 f+14 a^2 b e-35 a b^2 d+65 b^3 c\right )}{9 \sqrt{3} a^{16/3} b^{2/3}}-\frac{x^2 \left (-2 a^3 f+5 a^2 b e-8 a b^2 d+11 b^3 c\right )}{9 a^5 \left (a+b x^3\right )}-\frac{x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^4 \left (a+b x^3\right )^2} \]

[Out]

-c/(7*a^3*x^7) + (3*b*c - a*d)/(4*a^4*x^4) - (6*b^2*c - 3*a*b*d + a^2*e)/(a^5*x)
- ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(6*a^4*(a + b*x^3)^2) - ((11*b^3*c
- 8*a*b^2*d + 5*a^2*b*e - 2*a^3*f)*x^2)/(9*a^5*(a + b*x^3)) + ((65*b^3*c - 35*a*
b^2*d + 14*a^2*b*e - 2*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])
/(9*Sqrt[3]*a^(16/3)*b^(2/3)) + ((65*b^3*c - 35*a*b^2*d + 14*a^2*b*e - 2*a^3*f)*
Log[a^(1/3) + b^(1/3)*x])/(27*a^(16/3)*b^(2/3)) - ((65*b^3*c - 35*a*b^2*d + 14*a
^2*b*e - 2*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(16/3)*b
^(2/3))

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Rubi [A]  time = 1.14773, antiderivative size = 343, 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 \[ \frac{3 b c-a d}{4 a^4 x^4}-\frac{c}{7 a^3 x^7}-\frac{a^2 e-3 a b d+6 b^2 c}{a^5 x}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-2 a^3 f+14 a^2 b e-35 a b^2 d+65 b^3 c\right )}{54 a^{16/3} b^{2/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-2 a^3 f+14 a^2 b e-35 a b^2 d+65 b^3 c\right )}{27 a^{16/3} b^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-2 a^3 f+14 a^2 b e-35 a b^2 d+65 b^3 c\right )}{9 \sqrt{3} a^{16/3} b^{2/3}}-\frac{x^2 \left (-2 a^3 f+5 a^2 b e-8 a b^2 d+11 b^3 c\right )}{9 a^5 \left (a+b x^3\right )}-\frac{x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^4 \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^8*(a + b*x^3)^3),x]

[Out]

-c/(7*a^3*x^7) + (3*b*c - a*d)/(4*a^4*x^4) - (6*b^2*c - 3*a*b*d + a^2*e)/(a^5*x)
- ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(6*a^4*(a + b*x^3)^2) - ((11*b^3*c
- 8*a*b^2*d + 5*a^2*b*e - 2*a^3*f)*x^2)/(9*a^5*(a + b*x^3)) + ((65*b^3*c - 35*a*
b^2*d + 14*a^2*b*e - 2*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])
/(9*Sqrt[3]*a^(16/3)*b^(2/3)) + ((65*b^3*c - 35*a*b^2*d + 14*a^2*b*e - 2*a^3*f)*
Log[a^(1/3) + b^(1/3)*x])/(27*a^(16/3)*b^(2/3)) - ((65*b^3*c - 35*a*b^2*d + 14*a
^2*b*e - 2*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(16/3)*b
^(2/3))

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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**8/(b*x**3+a)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 0.461329, size = 328, normalized size = 0.96 \[ \frac{-\frac{189 a^{4/3} (a d-3 b c)}{x^4}-\frac{108 a^{7/3} c}{x^7}-\frac{756 \sqrt [3]{a} \left (a^2 e-3 a b d+6 b^2 c\right )}{x}+\frac{84 \sqrt [3]{a} x^2 \left (2 a^3 f-5 a^2 b e+8 a b^2 d-11 b^3 c\right )}{a+b x^3}+\frac{28 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-2 a^3 f+14 a^2 b e-35 a b^2 d+65 b^3 c\right )}{b^{2/3}}+\frac{28 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-2 a^3 f+14 a^2 b e-35 a b^2 d+65 b^3 c\right )}{b^{2/3}}+\frac{126 a^{4/3} x^2 \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{\left (a+b x^3\right )^2}+\frac{14 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (2 a^3 f-14 a^2 b e+35 a b^2 d-65 b^3 c\right )}{b^{2/3}}}{756 a^{16/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^8*(a + b*x^3)^3),x]

[Out]

((-108*a^(7/3)*c)/x^7 - (189*a^(4/3)*(-3*b*c + a*d))/x^4 - (756*a^(1/3)*(6*b^2*c
- 3*a*b*d + a^2*e))/x + (126*a^(4/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x^2
)/(a + b*x^3)^2 + (84*a^(1/3)*(-11*b^3*c + 8*a*b^2*d - 5*a^2*b*e + 2*a^3*f)*x^2)
/(a + b*x^3) + (28*Sqrt[3]*(65*b^3*c - 35*a*b^2*d + 14*a^2*b*e - 2*a^3*f)*ArcTan
[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3) + (28*(65*b^3*c - 35*a*b^2*d + 14
*a^2*b*e - 2*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/b^(2/3) + (14*(-65*b^3*c + 35*a*b^
2*d - 14*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^(2
/3))/(756*a^(16/3))

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Maple [B]  time = 0.024, size = 611, 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^8/(b*x^3+a)^3,x)

[Out]

7/18/a/(b*x^3+a)^2*x^2*f+14/27/a^3*e/(a/b)^(1/3)*ln(x+(a/b)^(1/3))-7/27/a^3*e/(a
/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+3/4/a^4/x^4*b*c+3/a^4/x*b*d-6/a^5/x*
b^2*c-13/18/a^2/(b*x^3+a)^2*x^2*b*e+19/18/a^3/(b*x^3+a)^2*x^2*b^2*d-25/18/a^4/(b
*x^3+a)^2*x^2*b^3*c+1/27/a^2*f/b/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-1
4/27/a^3*e*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-35/27/a^4
*b*d/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+35/54/a^4*b*d/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3
)+(a/b)^(2/3))+65/27/a^5*b^2*c/(a/b)^(1/3)*ln(x+(a/b)^(1/3))-65/54/a^5*b^2*c/(a/
b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-1/7*c/a^3/x^7+2/27/a^2*f*3^(1/2)/b/(a
/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+35/27/a^4*b*d*3^(1/2)/(a/b)^(1
/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-65/27/a^5*b^2*c*3^(1/2)/(a/b)^(1/3)*
arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-1/4/a^3/x^4*d-e/a^3/x-5/9/a^3/(b*x^3+a)^
2*x^5*e*b^2+8/9/a^4/(b*x^3+a)^2*x^5*d*b^3-2/27/a^2*f/b/(a/b)^(1/3)*ln(x+(a/b)^(1
/3))+2/9/a^2/(b*x^3+a)^2*x^5*f*b-11/9/a^5/(b*x^3+a)^2*x^5*b^4*c

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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^8),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.222887, size = 809, normalized size = 2.36 \[ \frac{\sqrt{3}{\left (14 \, \sqrt{3}{\left ({\left (65 \, b^{5} c - 35 \, a b^{4} d + 14 \, a^{2} b^{3} e - 2 \, a^{3} b^{2} f\right )} x^{13} + 2 \,{\left (65 \, a b^{4} c - 35 \, a^{2} b^{3} d + 14 \, a^{3} b^{2} e - 2 \, a^{4} b f\right )} x^{10} +{\left (65 \, a^{2} b^{3} c - 35 \, a^{3} b^{2} d + 14 \, a^{4} b e - 2 \, a^{5} f\right )} x^{7}\right )} \log \left (\left (-a b^{2}\right )^{\frac{1}{3}} b x^{2} - a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) - 28 \, \sqrt{3}{\left ({\left (65 \, b^{5} c - 35 \, a b^{4} d + 14 \, a^{2} b^{3} e - 2 \, a^{3} b^{2} f\right )} x^{13} + 2 \,{\left (65 \, a b^{4} c - 35 \, a^{2} b^{3} d + 14 \, a^{3} b^{2} e - 2 \, a^{4} b f\right )} x^{10} +{\left (65 \, a^{2} b^{3} c - 35 \, a^{3} b^{2} d + 14 \, a^{4} b e - 2 \, a^{5} f\right )} x^{7}\right )} \log \left (a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) + 84 \,{\left ({\left (65 \, b^{5} c - 35 \, a b^{4} d + 14 \, a^{2} b^{3} e - 2 \, a^{3} b^{2} f\right )} x^{13} + 2 \,{\left (65 \, a b^{4} c - 35 \, a^{2} b^{3} d + 14 \, a^{3} b^{2} e - 2 \, a^{4} b f\right )} x^{10} +{\left (65 \, a^{2} b^{3} c - 35 \, a^{3} b^{2} d + 14 \, a^{4} b e - 2 \, a^{5} f\right )} x^{7}\right )} \arctan \left (-\frac{\sqrt{3} a b - 2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} x}{3 \, a b}\right ) - 3 \, \sqrt{3}{\left (28 \,{\left (65 \, b^{4} c - 35 \, a b^{3} d + 14 \, a^{2} b^{2} e - 2 \, a^{3} b f\right )} x^{12} + 49 \,{\left (65 \, a b^{3} c - 35 \, a^{2} b^{2} d + 14 \, a^{3} b e - 2 \, a^{4} f\right )} x^{9} + 18 \,{\left (65 \, a^{2} b^{2} c - 35 \, a^{3} b d + 14 \, a^{4} e\right )} x^{6} + 36 \, a^{4} c - 9 \,{\left (13 \, a^{3} b c - 7 \, a^{4} d\right )} x^{3}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}\right )}}{2268 \,{\left (a^{5} b^{2} x^{13} + 2 \, a^{6} b x^{10} + a^{7} x^{7}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}} \]

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)^3*x^8),x, algorithm="fricas")

[Out]

1/2268*sqrt(3)*(14*sqrt(3)*((65*b^5*c - 35*a*b^4*d + 14*a^2*b^3*e - 2*a^3*b^2*f)
*x^13 + 2*(65*a*b^4*c - 35*a^2*b^3*d + 14*a^3*b^2*e - 2*a^4*b*f)*x^10 + (65*a^2*
b^3*c - 35*a^3*b^2*d + 14*a^4*b*e - 2*a^5*f)*x^7)*log((-a*b^2)^(1/3)*b*x^2 - a*b
+ (-a*b^2)^(2/3)*x) - 28*sqrt(3)*((65*b^5*c - 35*a*b^4*d + 14*a^2*b^3*e - 2*a^3
*b^2*f)*x^13 + 2*(65*a*b^4*c - 35*a^2*b^3*d + 14*a^3*b^2*e - 2*a^4*b*f)*x^10 + (
65*a^2*b^3*c - 35*a^3*b^2*d + 14*a^4*b*e - 2*a^5*f)*x^7)*log(a*b + (-a*b^2)^(2/3
)*x) + 84*((65*b^5*c - 35*a*b^4*d + 14*a^2*b^3*e - 2*a^3*b^2*f)*x^13 + 2*(65*a*b
^4*c - 35*a^2*b^3*d + 14*a^3*b^2*e - 2*a^4*b*f)*x^10 + (65*a^2*b^3*c - 35*a^3*b^
2*d + 14*a^4*b*e - 2*a^5*f)*x^7)*arctan(-1/3*(sqrt(3)*a*b - 2*sqrt(3)*(-a*b^2)^(
2/3)*x)/(a*b)) - 3*sqrt(3)*(28*(65*b^4*c - 35*a*b^3*d + 14*a^2*b^2*e - 2*a^3*b*f
)*x^12 + 49*(65*a*b^3*c - 35*a^2*b^2*d + 14*a^3*b*e - 2*a^4*f)*x^9 + 18*(65*a^2*
b^2*c - 35*a^3*b*d + 14*a^4*e)*x^6 + 36*a^4*c - 9*(13*a^3*b*c - 7*a^4*d)*x^3)*(-
a*b^2)^(1/3))/((a^5*b^2*x^13 + 2*a^6*b*x^10 + a^7*x^7)*(-a*b^2)^(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**8/(b*x**3+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.221184, size = 586, normalized size = 1.71 \[ \frac{{\left (65 \, b^{3} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 35 \, a b^{2} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 2 \, a^{3} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 14 \, a^{2} b \left (-\frac{a}{b}\right )^{\frac{1}{3}} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{6}} + \frac{\sqrt{3}{\left (65 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 35 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 14 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{6} b^{2}} - \frac{22 \, b^{4} c x^{5} - 16 \, a b^{3} d x^{5} - 4 \, a^{3} b f x^{5} + 10 \, a^{2} b^{2} x^{5} e + 25 \, a b^{3} c x^{2} - 19 \, a^{2} b^{2} d x^{2} - 7 \, a^{4} f x^{2} + 13 \, a^{3} b x^{2} e}{18 \,{\left (b x^{3} + a\right )}^{2} a^{5}} - \frac{{\left (65 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 35 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 14 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{6} b^{2}} - \frac{168 \, b^{2} c x^{6} - 84 \, a b d x^{6} + 28 \, a^{2} x^{6} e - 21 \, a b c x^{3} + 7 \, a^{2} d x^{3} + 4 \, a^{2} c}{28 \, a^{5} x^{7}} \]

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^8),x, algorithm="giac")

[Out]

1/27*(65*b^3*c*(-a/b)^(1/3) - 35*a*b^2*d*(-a/b)^(1/3) - 2*a^3*f*(-a/b)^(1/3) + 1
4*a^2*b*(-a/b)^(1/3)*e)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/a^6 + 1/27*sqrt(3
)*(65*(-a*b^2)^(2/3)*b^3*c - 35*(-a*b^2)^(2/3)*a*b^2*d - 2*(-a*b^2)^(2/3)*a^3*f
+ 14*(-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^6*b^2) - 1/18*(22*b^4*c*x^5 - 16*a*b^3*d*x^5 - 4*a^3*b*f*x^5 + 10*a^2*b^2*
x^5*e + 25*a*b^3*c*x^2 - 19*a^2*b^2*d*x^2 - 7*a^4*f*x^2 + 13*a^3*b*x^2*e)/((b*x^
3 + a)^2*a^5) - 1/54*(65*(-a*b^2)^(2/3)*b^3*c - 35*(-a*b^2)^(2/3)*a*b^2*d - 2*(-
a*b^2)^(2/3)*a^3*f + 14*(-a*b^2)^(2/3)*a^2*b*e)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)
^(2/3))/(a^6*b^2) - 1/28*(168*b^2*c*x^6 - 84*a*b*d*x^6 + 28*a^2*x^6*e - 21*a*b*c
*x^3 + 7*a^2*d*x^3 + 4*a^2*c)/(a^5*x^7)