3.271 \(\int \frac{c+d x^3+e x^6+f x^9}{x^8 \left (a+b x^3\right )^2} \, dx\)

Optimal. Leaf size=297 \[ \frac{2 b c-a d}{4 a^3 x^4}-\frac{c}{7 a^2 x^7}-\frac{a^2 e-2 a b d+3 b^2 c}{a^4 x}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+4 a^2 b e-7 a b^2 d+10 b^3 c\right )}{18 a^{13/3} b^{2/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+4 a^2 b e-7 a b^2 d+10 b^3 c\right )}{9 a^{13/3} b^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+4 a^2 b e-7 a b^2 d+10 b^3 c\right )}{3 \sqrt{3} a^{13/3} b^{2/3}}-\frac{x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^4 \left (a+b x^3\right )} \]

[Out]

-c/(7*a^2*x^7) + (2*b*c - a*d)/(4*a^3*x^4) - (3*b^2*c - 2*a*b*d + a^2*e)/(a^4*x)
- ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(3*a^4*(a + b*x^3)) + ((10*b^3*c -
7*a*b^2*d + 4*a^2*b*e - a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))]
)/(3*Sqrt[3]*a^(13/3)*b^(2/3)) + ((10*b^3*c - 7*a*b^2*d + 4*a^2*b*e - a^3*f)*Log
[a^(1/3) + b^(1/3)*x])/(9*a^(13/3)*b^(2/3)) - ((10*b^3*c - 7*a*b^2*d + 4*a^2*b*e
- a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(13/3)*b^(2/3))

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Rubi [A]  time = 0.784527, 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 \[ \frac{2 b c-a d}{4 a^3 x^4}-\frac{c}{7 a^2 x^7}-\frac{a^2 e-2 a b d+3 b^2 c}{a^4 x}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+4 a^2 b e-7 a b^2 d+10 b^3 c\right )}{18 a^{13/3} b^{2/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+4 a^2 b e-7 a b^2 d+10 b^3 c\right )}{9 a^{13/3} b^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+4 a^2 b e-7 a b^2 d+10 b^3 c\right )}{3 \sqrt{3} a^{13/3} b^{2/3}}-\frac{x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^4 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^8*(a + b*x^3)^2),x]

[Out]

-c/(7*a^2*x^7) + (2*b*c - a*d)/(4*a^3*x^4) - (3*b^2*c - 2*a*b*d + a^2*e)/(a^4*x)
- ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(3*a^4*(a + b*x^3)) + ((10*b^3*c -
7*a*b^2*d + 4*a^2*b*e - a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))]
)/(3*Sqrt[3]*a^(13/3)*b^(2/3)) + ((10*b^3*c - 7*a*b^2*d + 4*a^2*b*e - a^3*f)*Log
[a^(1/3) + b^(1/3)*x])/(9*a^(13/3)*b^(2/3)) - ((10*b^3*c - 7*a*b^2*d + 4*a^2*b*e
- a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(13/3)*b^(2/3))

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Rubi in Sympy [A]  time = 147.304, size = 299, normalized size = 1.01 \[ - \frac{x \left (\frac{a^{3} f}{x^{8}} - \frac{a^{2} b e}{x^{8}} + \frac{a b^{2} d}{x^{8}} - \frac{b^{3} c}{x^{8}}\right )}{3 a b^{3} \left (a + b x^{3}\right )} - \frac{a^{2} f - a b e + b^{2} d}{7 a b^{3} x^{7}} + \frac{2 a^{2} f - 2 a b e + b^{2} d}{4 a^{2} b^{2} x^{4}} - \frac{3 a^{2} f - 2 a b e + b^{2} d}{a^{3} b x} + \frac{\left (3 a^{2} f - 2 a b e + b^{2} d\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{10}{3}} b^{\frac{2}{3}}} - \frac{\left (3 a^{2} f - 2 a b e + b^{2} d\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{10}{3}} b^{\frac{2}{3}}} + \frac{\sqrt{3} \left (3 a^{2} f - 2 a b e + b^{2} d\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{10}{3}} b^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((f*x**9+e*x**6+d*x**3+c)/x**8/(b*x**3+a)**2,x)

[Out]

-x*(a**3*f/x**8 - a**2*b*e/x**8 + a*b**2*d/x**8 - b**3*c/x**8)/(3*a*b**3*(a + b*
x**3)) - (a**2*f - a*b*e + b**2*d)/(7*a*b**3*x**7) + (2*a**2*f - 2*a*b*e + b**2*
d)/(4*a**2*b**2*x**4) - (3*a**2*f - 2*a*b*e + b**2*d)/(a**3*b*x) + (3*a**2*f - 2
*a*b*e + b**2*d)*log(a**(1/3) + b**(1/3)*x)/(3*a**(10/3)*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**(10
/3)*b**(2/3)) + 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**(10/3)*b**(2/3))

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Mathematica [A]  time = 0.340983, size = 281, normalized size = 0.95 \[ \frac{-\frac{63 a^{4/3} (a d-2 b c)}{x^4}-\frac{36 a^{7/3} c}{x^7}-\frac{252 \sqrt [3]{a} \left (a^2 e-2 a b d+3 b^2 c\right )}{x}+\frac{84 \sqrt [3]{a} x^2 \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{a+b x^3}+\frac{28 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+4 a^2 b e-7 a b^2 d+10 b^3 c\right )}{b^{2/3}}+\frac{28 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^3 (-f)+4 a^2 b e-7 a b^2 d+10 b^3 c\right )}{b^{2/3}}+\frac{14 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 f-4 a^2 b e+7 a b^2 d-10 b^3 c\right )}{b^{2/3}}}{252 a^{13/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^8*(a + b*x^3)^2),x]

[Out]

((-36*a^(7/3)*c)/x^7 - (63*a^(4/3)*(-2*b*c + a*d))/x^4 - (252*a^(1/3)*(3*b^2*c -
2*a*b*d + a^2*e))/x + (84*a^(1/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x^2)/(
a + b*x^3) + (28*Sqrt[3]*(10*b^3*c - 7*a*b^2*d + 4*a^2*b*e - a^3*f)*ArcTan[(1 -
(2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3) + (28*(10*b^3*c - 7*a*b^2*d + 4*a^2*b*e
- a^3*f)*Log[a^(1/3) + b^(1/3)*x])/b^(2/3) + (14*(-10*b^3*c + 7*a*b^2*d - 4*a^2
*b*e + a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(2/3))/(252*a^(1
3/3))

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Maple [B]  time = 0.021, size = 529, 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)^2,x)

[Out]

-1/7*c/a^2/x^7-1/4/a^2/x^4*d+1/2/a^3/x^4*b*c-e/a^2/x+2/a^3/x*b*d-3/a^4/x*b^2*c+1
/3/a*x^2/(b*x^3+a)*f-1/3/a^2*x^2/(b*x^3+a)*b*e+1/3/a^3*x^2/(b*x^3+a)*b^2*d-1/3/a
^4*x^2/(b*x^3+a)*b^3*c+4/9/a^2*e/(a/b)^(1/3)*ln(x+(a/b)^(1/3))-2/9/a^2*e/(a/b)^(
1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-4/9/a^2*e*3^(1/2)/(a/b)^(1/3)*arctan(1/3*
3^(1/2)*(2/(a/b)^(1/3)*x-1))-7/9/a^3*b*d/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+7/18/a^3*
b*d/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+7/9/a^3*b*d*3^(1/2)/(a/b)^(1/3
)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+10/9/a^4*b^2*c/(a/b)^(1/3)*ln(x+(a/b)^
(1/3))-5/9/a^4*b^2*c/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-10/9/a^4*b^2*
c*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-1/9/a*f/b/(a/b)^(1
/3)*ln(x+(a/b)^(1/3))+1/18/a*f/b/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+1
/9/a*f*3^(1/2)/b/(a/b)^(1/3)*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^8),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.248018, size = 570, normalized size = 1.92 \[ \frac{\sqrt{3}{\left (14 \, \sqrt{3}{\left ({\left (10 \, b^{4} c - 7 \, a b^{3} d + 4 \, a^{2} b^{2} e - a^{3} b f\right )} x^{10} +{\left (10 \, a b^{3} c - 7 \, a^{2} b^{2} d + 4 \, a^{3} b e - a^{4} f\right )} x^{7}\right )} \log \left (\left (-a b^{2}\right )^{\frac{1}{3}} b x^{2} - a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) - 28 \, \sqrt{3}{\left ({\left (10 \, b^{4} c - 7 \, a b^{3} d + 4 \, a^{2} b^{2} e - a^{3} b f\right )} x^{10} +{\left (10 \, a b^{3} c - 7 \, a^{2} b^{2} d + 4 \, a^{3} b e - a^{4} f\right )} x^{7}\right )} \log \left (a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) + 84 \,{\left ({\left (10 \, b^{4} c - 7 \, a b^{3} d + 4 \, a^{2} b^{2} e - a^{3} b f\right )} x^{10} +{\left (10 \, a b^{3} c - 7 \, a^{2} b^{2} d + 4 \, a^{3} b e - a^{4} f\right )} x^{7}\right )} \arctan \left (-\frac{\sqrt{3} a b - 2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} x}{3 \, a b}\right ) - 3 \, \sqrt{3}{\left (28 \,{\left (10 \, b^{3} c - 7 \, a b^{2} d + 4 \, a^{2} b e - a^{3} f\right )} x^{9} + 21 \,{\left (10 \, a b^{2} c - 7 \, a^{2} b d + 4 \, a^{3} e\right )} x^{6} + 12 \, a^{3} c - 3 \,{\left (10 \, a^{2} b c - 7 \, a^{3} d\right )} x^{3}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}\right )}}{756 \,{\left (a^{4} b x^{10} + a^{5} x^{7}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}} \]

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)^2*x^8),x, algorithm="fricas")

[Out]

1/756*sqrt(3)*(14*sqrt(3)*((10*b^4*c - 7*a*b^3*d + 4*a^2*b^2*e - a^3*b*f)*x^10 +
(10*a*b^3*c - 7*a^2*b^2*d + 4*a^3*b*e - a^4*f)*x^7)*log((-a*b^2)^(1/3)*b*x^2 -
a*b + (-a*b^2)^(2/3)*x) - 28*sqrt(3)*((10*b^4*c - 7*a*b^3*d + 4*a^2*b^2*e - a^3*
b*f)*x^10 + (10*a*b^3*c - 7*a^2*b^2*d + 4*a^3*b*e - a^4*f)*x^7)*log(a*b + (-a*b^
2)^(2/3)*x) + 84*((10*b^4*c - 7*a*b^3*d + 4*a^2*b^2*e - a^3*b*f)*x^10 + (10*a*b^
3*c - 7*a^2*b^2*d + 4*a^3*b*e - a^4*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*(10*b^3*c - 7*a*b^2*d + 4*a^2*b*e - a^
3*f)*x^9 + 21*(10*a*b^2*c - 7*a^2*b*d + 4*a^3*e)*x^6 + 12*a^3*c - 3*(10*a^2*b*c
- 7*a^3*d)*x^3)*(-a*b^2)^(1/3))/((a^4*b*x^10 + a^5*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)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.218856, size = 522, normalized size = 1.76 \[ \frac{{\left (10 \, b^{3} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 7 \, a b^{2} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a^{3} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 4 \, a^{2} b \left (-\frac{a}{b}\right )^{\frac{1}{3}} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{5}} - \frac{b^{3} c x^{2} - a b^{2} d x^{2} - a^{3} f x^{2} + a^{2} b x^{2} e}{3 \,{\left (b x^{3} + a\right )} a^{4}} + \frac{\sqrt{3}{\left (10 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 7 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 4 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{5} b^{2}} - \frac{{\left (10 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 7 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 4 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{5} b^{2}} - \frac{84 \, b^{2} c x^{6} - 56 \, a b d x^{6} + 28 \, a^{2} x^{6} e - 14 \, a b c x^{3} + 7 \, a^{2} d x^{3} + 4 \, a^{2} c}{28 \, a^{4} x^{7}} \]

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)^2*x^8),x, algorithm="giac")

[Out]

1/9*(10*b^3*c*(-a/b)^(1/3) - 7*a*b^2*d*(-a/b)^(1/3) - a^3*f*(-a/b)^(1/3) + 4*a^2
*b*(-a/b)^(1/3)*e)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/a^5 - 1/3*(b^3*c*x^2 -
a*b^2*d*x^2 - a^3*f*x^2 + a^2*b*x^2*e)/((b*x^3 + a)*a^4) + 1/9*sqrt(3)*(10*(-a*
b^2)^(2/3)*b^3*c - 7*(-a*b^2)^(2/3)*a*b^2*d - (-a*b^2)^(2/3)*a^3*f + 4*(-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^5*b^2) -
1/18*(10*(-a*b^2)^(2/3)*b^3*c - 7*(-a*b^2)^(2/3)*a*b^2*d - (-a*b^2)^(2/3)*a^3*f
+ 4*(-a*b^2)^(2/3)*a^2*b*e)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^5*b^2) -
1/28*(84*b^2*c*x^6 - 56*a*b*d*x^6 + 28*a^2*x^6*e - 14*a*b*c*x^3 + 7*a^2*d*x^3 +
4*a^2*c)/(a^4*x^7)