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

Optimal. Leaf size=303 \[ -\frac{c}{a^3 x}-\frac{x^2 \left (4 a^3 f-a^2 b e-2 a b^2 d+5 b^3 c\right )}{9 a^3 b^2 \left (a+b x^3\right )}-\frac{x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^2 b^2 \left (a+b x^3\right )^2}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-5 a^3 f-a^2 b e-2 a b^2 d+14 b^3 c\right )}{54 a^{10/3} b^{8/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-5 a^3 f-a^2 b e-2 a b^2 d+14 b^3 c\right )}{27 a^{10/3} b^{8/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-5 a^3 f-a^2 b e-2 a b^2 d+14 b^3 c\right )}{9 \sqrt{3} a^{10/3} b^{8/3}} \]

[Out]

-(c/(a^3*x)) - ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(6*a^2*b^2*(a + b*x^3)^
2) - ((5*b^3*c - 2*a*b^2*d - a^2*b*e + 4*a^3*f)*x^2)/(9*a^3*b^2*(a + b*x^3)) + (
(14*b^3*c - 2*a*b^2*d - 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^(10/3)*b^(8/3)) + ((14*b^3*c - 2*a*b^2*d - a^2*b*e -
5*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(10/3)*b^(8/3)) - ((14*b^3*c - 2*a*b^2*
d - 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^(10
/3)*b^(8/3))

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Rubi [A]  time = 0.805318, antiderivative size = 303, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ -\frac{c}{a^3 x}-\frac{x^2 \left (4 a^3 f-a^2 b e-2 a b^2 d+5 b^3 c\right )}{9 a^3 b^2 \left (a+b x^3\right )}-\frac{x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^2 b^2 \left (a+b x^3\right )^2}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-5 a^3 f-a^2 b e-2 a b^2 d+14 b^3 c\right )}{54 a^{10/3} b^{8/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-5 a^3 f-a^2 b e-2 a b^2 d+14 b^3 c\right )}{27 a^{10/3} b^{8/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-5 a^3 f-a^2 b e-2 a b^2 d+14 b^3 c\right )}{9 \sqrt{3} a^{10/3} b^{8/3}} \]

Antiderivative was successfully verified.

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

[Out]

-(c/(a^3*x)) - ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(6*a^2*b^2*(a + b*x^3)^
2) - ((5*b^3*c - 2*a*b^2*d - a^2*b*e + 4*a^3*f)*x^2)/(9*a^3*b^2*(a + b*x^3)) + (
(14*b^3*c - 2*a*b^2*d - 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^(10/3)*b^(8/3)) + ((14*b^3*c - 2*a*b^2*d - a^2*b*e -
5*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(10/3)*b^(8/3)) - ((14*b^3*c - 2*a*b^2*
d - 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^(10
/3)*b^(8/3))

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Rubi in Sympy [A]  time = 145.367, size = 286, normalized size = 0.94 \[ - \frac{x \left (\frac{a^{3} f}{x^{2}} - \frac{a^{2} b e}{x^{2}} + \frac{a b^{2} d}{x^{2}} - \frac{b^{3} c}{x^{2}}\right )}{6 a b^{3} \left (a + b x^{3}\right )^{2}} - \frac{x^{2} \left (3 a^{2} f - 2 a b e + b^{2} d\right )}{3 a^{2} b^{2} \left (a + b x^{3}\right )} - \frac{a^{2} f - a b e + b^{2} d}{a^{2} b^{3} x} + \frac{\left (3 a^{2} f - 5 a b e + 4 b^{2} d\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{9 a^{\frac{7}{3}} b^{\frac{8}{3}}} - \frac{\left (3 a^{2} f - 5 a b e + 4 b^{2} d\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{18 a^{\frac{7}{3}} b^{\frac{8}{3}}} + \frac{\sqrt{3} \left (3 a^{2} f - 5 a b e + 4 b^{2} d\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{7}{3}} b^{\frac{8}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x*(a**3*f/x**2 - a**2*b*e/x**2 + a*b**2*d/x**2 - b**3*c/x**2)/(6*a*b**3*(a + b*
x**3)**2) - x**2*(3*a**2*f - 2*a*b*e + b**2*d)/(3*a**2*b**2*(a + b*x**3)) - (a**
2*f - a*b*e + b**2*d)/(a**2*b**3*x) + (3*a**2*f - 5*a*b*e + 4*b**2*d)*log(a**(1/
3) + b**(1/3)*x)/(9*a**(7/3)*b**(8/3)) - (3*a**2*f - 5*a*b*e + 4*b**2*d)*log(a**
(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(18*a**(7/3)*b**(8/3)) + sqrt(3)*(3
*a**2*f - 5*a*b*e + 4*b**2*d)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3
))/(9*a**(7/3)*b**(8/3))

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Mathematica [A]  time = 0.41739, size = 286, normalized size = 0.94 \[ \frac{-\frac{6 \sqrt [3]{a} x^2 \left (4 a^3 f-a^2 b e-2 a b^2 d+5 b^3 c\right )}{b^2 \left (a+b x^3\right )}-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (5 a^3 f+a^2 b e+2 a b^2 d-14 b^3 c\right )}{b^{8/3}}+\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-5 a^3 f-a^2 b e-2 a b^2 d+14 b^3 c\right )}{b^{8/3}}+\frac{9 a^{4/3} x^2 \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{b^2 \left (a+b x^3\right )^2}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (5 a^3 f+a^2 b e+2 a b^2 d-14 b^3 c\right )}{b^{8/3}}-\frac{54 \sqrt [3]{a} c}{x}}{54 a^{10/3}} \]

Antiderivative was successfully verified.

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

[Out]

((-54*a^(1/3)*c)/x + (9*a^(4/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x^2)/(b^2
*(a + b*x^3)^2) - (6*a^(1/3)*(5*b^3*c - 2*a*b^2*d - a^2*b*e + 4*a^3*f)*x^2)/(b^2
*(a + b*x^3)) + (2*Sqrt[3]*(14*b^3*c - 2*a*b^2*d - a^2*b*e - 5*a^3*f)*ArcTan[(1
- (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(8/3) - (2*(-14*b^3*c + 2*a*b^2*d + a^2*b*e
+ 5*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/b^(8/3) + ((-14*b^3*c + 2*a*b^2*d + 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^(8/3))/(54*a^(10/
3))

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Maple [B]  time = 0.02, size = 547, 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^2/(b*x^3+a)^3,x)

[Out]

-c/a^3/x-4/9/(b*x^3+a)^2/b*x^5*f+1/9/a/(b*x^3+a)^2*x^5*e+2/9/a^2/(b*x^3+a)^2*b*x
^5*d-5/9/a^3/(b*x^3+a)^2*b^2*x^5*c-5/18*a/(b*x^3+a)^2/b^2*x^2*f-1/18/(b*x^3+a)^2
/b*x^2*e+7/18*d/a*x^2/(b*x^3+a)^2-13/18/a^2/(b*x^3+a)^2*x^2*b*c-5/27/b^3/(a/b)^(
1/3)*ln(x+(a/b)^(1/3))*f-1/27/a/b^2/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*e-2/27*d/a^2/b
/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+14/27/a^3/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*c+5/54/b^
3/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*f+1/54/a/b^2/(a/b)^(1/3)*ln(x^2-
x*(a/b)^(1/3)+(a/b)^(2/3))*e+1/27*d/a^2/b/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)
^(2/3))-7/27/a^3/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*c+5/27/b^3*3^(1/2
)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*f+1/27/a/b^2*3^(1/2)/(a/b)
^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*e+2/27*d/a^2*3^(1/2)/b/(a/b)^(1/3
)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-14/27/a^3*3^(1/2)/(a/b)^(1/3)*arctan(1
/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*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^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.222033, size = 741, normalized size = 2.45 \[ \frac{\sqrt{3}{\left (\sqrt{3}{\left ({\left (14 \, b^{5} c - 2 \, a b^{4} d - a^{2} b^{3} e - 5 \, a^{3} b^{2} f\right )} x^{7} + 2 \,{\left (14 \, a b^{4} c - 2 \, a^{2} b^{3} d - a^{3} b^{2} e - 5 \, a^{4} b f\right )} x^{4} +{\left (14 \, a^{2} b^{3} c - 2 \, a^{3} b^{2} d - a^{4} b e - 5 \, a^{5} f\right )} x\right )} \log \left (\left (-a b^{2}\right )^{\frac{1}{3}} b x^{2} - a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) - 2 \, \sqrt{3}{\left ({\left (14 \, b^{5} c - 2 \, a b^{4} d - a^{2} b^{3} e - 5 \, a^{3} b^{2} f\right )} x^{7} + 2 \,{\left (14 \, a b^{4} c - 2 \, a^{2} b^{3} d - a^{3} b^{2} e - 5 \, a^{4} b f\right )} x^{4} +{\left (14 \, a^{2} b^{3} c - 2 \, a^{3} b^{2} d - a^{4} b e - 5 \, a^{5} f\right )} x\right )} \log \left (a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) + 6 \,{\left ({\left (14 \, b^{5} c - 2 \, a b^{4} d - a^{2} b^{3} e - 5 \, a^{3} b^{2} f\right )} x^{7} + 2 \,{\left (14 \, a b^{4} c - 2 \, a^{2} b^{3} d - a^{3} b^{2} e - 5 \, a^{4} b f\right )} x^{4} +{\left (14 \, a^{2} b^{3} c - 2 \, a^{3} b^{2} d - a^{4} b e - 5 \, a^{5} f\right )} x\right )} \arctan \left (-\frac{\sqrt{3} a b - 2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} x}{3 \, a b}\right ) - 3 \, \sqrt{3}{\left (2 \,{\left (14 \, b^{4} c - 2 \, a b^{3} d - a^{2} b^{2} e + 4 \, a^{3} b f\right )} x^{6} + 18 \, a^{2} b^{2} c +{\left (49 \, a b^{3} c - 7 \, a^{2} b^{2} d + a^{3} b e + 5 \, a^{4} f\right )} x^{3}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}\right )}}{162 \,{\left (a^{3} b^{4} x^{7} + 2 \, a^{4} b^{3} x^{4} + a^{5} b^{2} x\right )} \left (-a b^{2}\right )^{\frac{1}{3}}} \]

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^2),x, algorithm="fricas")

[Out]

1/162*sqrt(3)*(sqrt(3)*((14*b^5*c - 2*a*b^4*d - a^2*b^3*e - 5*a^3*b^2*f)*x^7 + 2
*(14*a*b^4*c - 2*a^2*b^3*d - a^3*b^2*e - 5*a^4*b*f)*x^4 + (14*a^2*b^3*c - 2*a^3*
b^2*d - a^4*b*e - 5*a^5*f)*x)*log((-a*b^2)^(1/3)*b*x^2 - a*b + (-a*b^2)^(2/3)*x)
- 2*sqrt(3)*((14*b^5*c - 2*a*b^4*d - a^2*b^3*e - 5*a^3*b^2*f)*x^7 + 2*(14*a*b^4
*c - 2*a^2*b^3*d - a^3*b^2*e - 5*a^4*b*f)*x^4 + (14*a^2*b^3*c - 2*a^3*b^2*d - a^
4*b*e - 5*a^5*f)*x)*log(a*b + (-a*b^2)^(2/3)*x) + 6*((14*b^5*c - 2*a*b^4*d - a^2
*b^3*e - 5*a^3*b^2*f)*x^7 + 2*(14*a*b^4*c - 2*a^2*b^3*d - a^3*b^2*e - 5*a^4*b*f)
*x^4 + (14*a^2*b^3*c - 2*a^3*b^2*d - a^4*b*e - 5*a^5*f)*x)*arctan(-1/3*(sqrt(3)*
a*b - 2*sqrt(3)*(-a*b^2)^(2/3)*x)/(a*b)) - 3*sqrt(3)*(2*(14*b^4*c - 2*a*b^3*d -
a^2*b^2*e + 4*a^3*b*f)*x^6 + 18*a^2*b^2*c + (49*a*b^3*c - 7*a^2*b^2*d + a^3*b*e
+ 5*a^4*f)*x^3)*(-a*b^2)^(1/3))/((a^3*b^4*x^7 + 2*a^4*b^3*x^4 + a^5*b^2*x)*(-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**2/(b*x**3+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.222157, size = 525, normalized size = 1.73 \[ -\frac{c}{a^{3} x} + \frac{{\left (14 \, b^{3} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 2 \, a b^{2} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 5 \, a^{3} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a^{2} b \left (-\frac{a}{b}\right )^{\frac{1}{3}} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{4} b^{2}} + \frac{\sqrt{3}{\left (14 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f - \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{4} b^{4}} - \frac{10 \, b^{4} c x^{5} - 4 \, a b^{3} d x^{5} + 8 \, a^{3} b f x^{5} - 2 \, a^{2} b^{2} x^{5} e + 13 \, a b^{3} c x^{2} - 7 \, a^{2} b^{2} d x^{2} + 5 \, a^{4} f x^{2} + a^{3} b x^{2} e}{18 \,{\left (b x^{3} + a\right )}^{2} a^{3} b^{2}} - \frac{{\left (14 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f - \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{4} b^{4}} \]

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^2),x, algorithm="giac")

[Out]

-c/(a^3*x) + 1/27*(14*b^3*c*(-a/b)^(1/3) - 2*a*b^2*d*(-a/b)^(1/3) - 5*a^3*f*(-a/
b)^(1/3) - a^2*b*(-a/b)^(1/3)*e)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a^4*b^2
) + 1/27*sqrt(3)*(14*(-a*b^2)^(2/3)*b^3*c - 2*(-a*b^2)^(2/3)*a*b^2*d - 5*(-a*b^2
)^(2/3)*a^3*f - (-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^4*b^4) - 1/18*(10*b^4*c*x^5 - 4*a*b^3*d*x^5 + 8*a^3*b*f*x^5 - 2
*a^2*b^2*x^5*e + 13*a*b^3*c*x^2 - 7*a^2*b^2*d*x^2 + 5*a^4*f*x^2 + a^3*b*x^2*e)/(
(b*x^3 + a)^2*a^3*b^2) - 1/54*(14*(-a*b^2)^(2/3)*b^3*c - 2*(-a*b^2)^(2/3)*a*b^2*
d - 5*(-a*b^2)^(2/3)*a^3*f - (-a*b^2)^(2/3)*a^2*b*e)*ln(x^2 + x*(-a/b)^(1/3) + (
-a/b)^(2/3))/(a^4*b^4)