3.273 \(\int \frac{c+d x^3+e x^6+f x^9}{x^{11} \left (a+b x^3\right )^2} \, dx\)

Optimal. Leaf size=334 \[ \frac{2 b c-a d}{7 a^3 x^7}-\frac{c}{10 a^2 x^{10}}-\frac{a^2 e-2 a b d+3 b^2 c}{4 a^4 x^4}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-4 a^3 f+7 a^2 b e-10 a b^2 d+13 b^3 c\right )}{9 a^{16/3}}-\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-4 a^3 f+7 a^2 b e-10 a b^2 d+13 b^3 c\right )}{3 \sqrt{3} a^{16/3}}+\frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-4 a^3 f+7 a^2 b e-10 a b^2 d+13 b^3 c\right )}{18 a^{16/3}}+\frac{b x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^5 \left (a+b x^3\right )}+\frac{a^3 (-f)+2 a^2 b e-3 a b^2 d+4 b^3 c}{a^5 x} \]

[Out]

-c/(10*a^2*x^10) + (2*b*c - a*d)/(7*a^3*x^7) - (3*b^2*c - 2*a*b*d + a^2*e)/(4*a^
4*x^4) + (4*b^3*c - 3*a*b^2*d + 2*a^2*b*e - a^3*f)/(a^5*x) + (b*(b^3*c - a*b^2*d
+ a^2*b*e - a^3*f)*x^2)/(3*a^5*(a + b*x^3)) - (b^(1/3)*(13*b^3*c - 10*a*b^2*d +
7*a^2*b*e - 4*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt
[3]*a^(16/3)) - (b^(1/3)*(13*b^3*c - 10*a*b^2*d + 7*a^2*b*e - 4*a^3*f)*Log[a^(1/
3) + b^(1/3)*x])/(9*a^(16/3)) + (b^(1/3)*(13*b^3*c - 10*a*b^2*d + 7*a^2*b*e - 4*
a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(16/3))

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Rubi [A]  time = 0.948896, antiderivative size = 334, 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 \[ \frac{2 b c-a d}{7 a^3 x^7}-\frac{c}{10 a^2 x^{10}}-\frac{a^2 e-2 a b d+3 b^2 c}{4 a^4 x^4}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-4 a^3 f+7 a^2 b e-10 a b^2 d+13 b^3 c\right )}{9 a^{16/3}}-\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-4 a^3 f+7 a^2 b e-10 a b^2 d+13 b^3 c\right )}{3 \sqrt{3} a^{16/3}}+\frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-4 a^3 f+7 a^2 b e-10 a b^2 d+13 b^3 c\right )}{18 a^{16/3}}+\frac{b x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^5 \left (a+b x^3\right )}+\frac{a^3 (-f)+2 a^2 b e-3 a b^2 d+4 b^3 c}{a^5 x} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^11*(a + b*x^3)^2),x]

[Out]

-c/(10*a^2*x^10) + (2*b*c - a*d)/(7*a^3*x^7) - (3*b^2*c - 2*a*b*d + a^2*e)/(4*a^
4*x^4) + (4*b^3*c - 3*a*b^2*d + 2*a^2*b*e - a^3*f)/(a^5*x) + (b*(b^3*c - a*b^2*d
+ a^2*b*e - a^3*f)*x^2)/(3*a^5*(a + b*x^3)) - (b^(1/3)*(13*b^3*c - 10*a*b^2*d +
7*a^2*b*e - 4*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt
[3]*a^(16/3)) - (b^(1/3)*(13*b^3*c - 10*a*b^2*d + 7*a^2*b*e - 4*a^3*f)*Log[a^(1/
3) + b^(1/3)*x])/(9*a^(16/3)) + (b^(1/3)*(13*b^3*c - 10*a*b^2*d + 7*a^2*b*e - 4*
a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(16/3))

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Rubi in Sympy [A]  time = 162.534, size = 326, normalized size = 0.98 \[ - \frac{x \left (\frac{a^{3} f}{x^{11}} - \frac{a^{2} b e}{x^{11}} + \frac{a b^{2} d}{x^{11}} - \frac{b^{3} c}{x^{11}}\right )}{3 a b^{3} \left (a + b x^{3}\right )} - \frac{a^{2} f - a b e + b^{2} d}{10 a b^{3} x^{10}} + \frac{2 a^{2} f - 2 a b e + b^{2} d}{7 a^{2} b^{2} x^{7}} - \frac{3 a^{2} f - 2 a b e + b^{2} d}{4 a^{3} b x^{4}} + \frac{3 a^{2} f - 2 a b e + b^{2} d}{a^{4} x} - \frac{\sqrt [3]{b} \left (3 a^{2} f - 2 a b e + b^{2} d\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{13}{3}}} + \frac{\sqrt [3]{b} \left (3 a^{2} f - 2 a b e + b^{2} d\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{13}{3}}} - \frac{\sqrt{3} \sqrt [3]{b} \left (3 a^{2} f - 2 a b e + b^{2} d\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{13}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((f*x**9+e*x**6+d*x**3+c)/x**11/(b*x**3+a)**2,x)

[Out]

-x*(a**3*f/x**11 - a**2*b*e/x**11 + a*b**2*d/x**11 - b**3*c/x**11)/(3*a*b**3*(a
+ b*x**3)) - (a**2*f - a*b*e + b**2*d)/(10*a*b**3*x**10) + (2*a**2*f - 2*a*b*e +
b**2*d)/(7*a**2*b**2*x**7) - (3*a**2*f - 2*a*b*e + b**2*d)/(4*a**3*b*x**4) + (3
*a**2*f - 2*a*b*e + b**2*d)/(a**4*x) - b**(1/3)*(3*a**2*f - 2*a*b*e + b**2*d)*lo
g(a**(1/3) + b**(1/3)*x)/(3*a**(13/3)) + b**(1/3)*(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**(13/3)) - sqrt(3)*b**(
1/3)*(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**(13/3))

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Mathematica [A]  time = 0.343361, size = 319, normalized size = 0.96 \[ \frac{-\frac{180 a^{7/3} (a d-2 b c)}{x^7}-\frac{126 a^{10/3} c}{x^{10}}-\frac{315 a^{4/3} \left (a^2 e-2 a b d+3 b^2 c\right )}{x^4}-\frac{420 \sqrt [3]{a} b x^2 \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{a+b x^3}-\frac{1260 \sqrt [3]{a} \left (a^3 f-2 a^2 b e+3 a b^2 d-4 b^3 c\right )}{x}+140 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (4 a^3 f-7 a^2 b e+10 a b^2 d-13 b^3 c\right )-140 \sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-4 a^3 f+7 a^2 b e-10 a b^2 d+13 b^3 c\right )+70 \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-4 a^3 f+7 a^2 b e-10 a b^2 d+13 b^3 c\right )}{1260 a^{16/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^11*(a + b*x^3)^2),x]

[Out]

((-126*a^(10/3)*c)/x^10 - (180*a^(7/3)*(-2*b*c + a*d))/x^7 - (315*a^(4/3)*(3*b^2
*c - 2*a*b*d + a^2*e))/x^4 - (1260*a^(1/3)*(-4*b^3*c + 3*a*b^2*d - 2*a^2*b*e + a
^3*f))/x - (420*a^(1/3)*b*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x^2)/(a + b*x^3
) - 140*Sqrt[3]*b^(1/3)*(13*b^3*c - 10*a*b^2*d + 7*a^2*b*e - 4*a^3*f)*ArcTan[(1
- (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 140*b^(1/3)*(-13*b^3*c + 10*a*b^2*d - 7*a^2*
b*e + 4*a^3*f)*Log[a^(1/3) + b^(1/3)*x] + 70*b^(1/3)*(13*b^3*c - 10*a*b^2*d + 7*
a^2*b*e - 4*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(1260*a^(16/3
))

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Maple [A]  time = 0.025, size = 575, 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^11/(b*x^3+a)^2,x)

[Out]

-1/3*b/a^2*x^2/(b*x^3+a)*f+1/3*b^2/a^3*x^2/(b*x^3+a)*e+13/18*b^3/a^5*c/(a/b)^(1/
3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+1/3*b^4/a^5*x^2/(b*x^3+a)*c-7/9*b/a^3*e/(a/
b)^(1/3)*ln(x+(a/b)^(1/3))+7/18*b/a^3*e/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(
2/3))-4/9/a^2*f*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-1/7/
a^2/x^7*d-1/4/a^2/x^4*e-1/a^2/x*f-1/3*b^3/a^4*x^2/(b*x^3+a)*d+10/9*b^2/a^4*d/(a/
b)^(1/3)*ln(x+(a/b)^(1/3))-5/9*b^2/a^4*d/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^
(2/3))-13/9*b^3/a^5*c/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+2/7/a^3/x^7*b*c+1/2/a^3/x^4*
b*d-3/4/a^4/x^4*b^2*c+2/a^3/x*b*e-3/a^4/x*b^2*d+4/a^5/x*b^3*c+4/9/a^2*f/(a/b)^(1
/3)*ln(x+(a/b)^(1/3))-2/9/a^2*f/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+7/
9*b/a^3*e*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-10/9*b^2/a
^4*d*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+13/9*b^3/a^5*c*
3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-1/10*c/a^2/x^10

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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^11),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.247194, size = 632, normalized size = 1.89 \[ \frac{\sqrt{3}{\left (70 \, \sqrt{3}{\left ({\left (13 \, b^{4} c - 10 \, a b^{3} d + 7 \, a^{2} b^{2} e - 4 \, a^{3} b f\right )} x^{13} +{\left (13 \, a b^{3} c - 10 \, a^{2} b^{2} d + 7 \, a^{3} b e - 4 \, a^{4} f\right )} x^{10}\right )} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x^{2} - a x \left (\frac{b}{a}\right )^{\frac{2}{3}} + a \left (\frac{b}{a}\right )^{\frac{1}{3}}\right ) - 140 \, \sqrt{3}{\left ({\left (13 \, b^{4} c - 10 \, a b^{3} d + 7 \, a^{2} b^{2} e - 4 \, a^{3} b f\right )} x^{13} +{\left (13 \, a b^{3} c - 10 \, a^{2} b^{2} d + 7 \, a^{3} b e - 4 \, a^{4} f\right )} x^{10}\right )} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x + a \left (\frac{b}{a}\right )^{\frac{2}{3}}\right ) - 420 \,{\left ({\left (13 \, b^{4} c - 10 \, a b^{3} d + 7 \, a^{2} b^{2} e - 4 \, a^{3} b f\right )} x^{13} +{\left (13 \, a b^{3} c - 10 \, a^{2} b^{2} d + 7 \, a^{3} b e - 4 \, a^{4} f\right )} x^{10}\right )} \left (\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} b x - \sqrt{3} a \left (\frac{b}{a}\right )^{\frac{2}{3}}}{3 \, a \left (\frac{b}{a}\right )^{\frac{2}{3}}}\right ) + 3 \, \sqrt{3}{\left (140 \,{\left (13 \, b^{4} c - 10 \, a b^{3} d + 7 \, a^{2} b^{2} e - 4 \, a^{3} b f\right )} x^{12} + 105 \,{\left (13 \, a b^{3} c - 10 \, a^{2} b^{2} d + 7 \, a^{3} b e - 4 \, a^{4} f\right )} x^{9} - 15 \,{\left (13 \, a^{2} b^{2} c - 10 \, a^{3} b d + 7 \, a^{4} e\right )} x^{6} - 42 \, a^{4} c + 6 \,{\left (13 \, a^{3} b c - 10 \, a^{4} d\right )} x^{3}\right )}\right )}}{3780 \,{\left (a^{5} b x^{13} + a^{6} x^{10}\right )}} \]

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^11),x, algorithm="fricas")

[Out]

1/3780*sqrt(3)*(70*sqrt(3)*((13*b^4*c - 10*a*b^3*d + 7*a^2*b^2*e - 4*a^3*b*f)*x^
13 + (13*a*b^3*c - 10*a^2*b^2*d + 7*a^3*b*e - 4*a^4*f)*x^10)*(b/a)^(1/3)*log(b*x
^2 - a*x*(b/a)^(2/3) + a*(b/a)^(1/3)) - 140*sqrt(3)*((13*b^4*c - 10*a*b^3*d + 7*
a^2*b^2*e - 4*a^3*b*f)*x^13 + (13*a*b^3*c - 10*a^2*b^2*d + 7*a^3*b*e - 4*a^4*f)*
x^10)*(b/a)^(1/3)*log(b*x + a*(b/a)^(2/3)) - 420*((13*b^4*c - 10*a*b^3*d + 7*a^2
*b^2*e - 4*a^3*b*f)*x^13 + (13*a*b^3*c - 10*a^2*b^2*d + 7*a^3*b*e - 4*a^4*f)*x^1
0)*(b/a)^(1/3)*arctan(-1/3*(2*sqrt(3)*b*x - sqrt(3)*a*(b/a)^(2/3))/(a*(b/a)^(2/3
))) + 3*sqrt(3)*(140*(13*b^4*c - 10*a*b^3*d + 7*a^2*b^2*e - 4*a^3*b*f)*x^12 + 10
5*(13*a*b^3*c - 10*a^2*b^2*d + 7*a^3*b*e - 4*a^4*f)*x^9 - 15*(13*a^2*b^2*c - 10*
a^3*b*d + 7*a^4*e)*x^6 - 42*a^4*c + 6*(13*a^3*b*c - 10*a^4*d)*x^3))/(a^5*b*x^13
+ a^6*x^10)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.217663, size = 590, normalized size = 1.77 \[ -\frac{{\left (13 \, b^{4} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 10 \, a b^{3} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 4 \, a^{3} b f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 7 \, a^{2} b^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{6}} - \frac{\sqrt{3}{\left (13 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 10 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 4 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 7 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{6} b} + \frac{b^{4} c x^{2} - a b^{3} d x^{2} - a^{3} b f x^{2} + a^{2} b^{2} x^{2} e}{3 \,{\left (b x^{3} + a\right )} a^{5}} + \frac{{\left (13 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 10 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 4 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 7 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{6} b} + \frac{560 \, b^{3} c x^{9} - 420 \, a b^{2} d x^{9} - 140 \, a^{3} f x^{9} + 280 \, a^{2} b x^{9} e - 105 \, a b^{2} c x^{6} + 70 \, a^{2} b d x^{6} - 35 \, a^{3} x^{6} e + 40 \, a^{2} b c x^{3} - 20 \, a^{3} d x^{3} - 14 \, a^{3} c}{140 \, a^{5} x^{10}} \]

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^11),x, algorithm="giac")

[Out]

-1/9*(13*b^4*c*(-a/b)^(1/3) - 10*a*b^3*d*(-a/b)^(1/3) - 4*a^3*b*f*(-a/b)^(1/3) +
7*a^2*b^2*(-a/b)^(1/3)*e)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/a^6 - 1/9*sqrt
(3)*(13*(-a*b^2)^(2/3)*b^3*c - 10*(-a*b^2)^(2/3)*a*b^2*d - 4*(-a*b^2)^(2/3)*a^3*
f + 7*(-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) + 1/3*(b^4*c*x^2 - a*b^3*d*x^2 - a^3*b*f*x^2 + a^2*b^2*x^2*e)/((b*x^
3 + a)*a^5) + 1/18*(13*(-a*b^2)^(2/3)*b^3*c - 10*(-a*b^2)^(2/3)*a*b^2*d - 4*(-a*
b^2)^(2/3)*a^3*f + 7*(-a*b^2)^(2/3)*a^2*b*e)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2
/3))/(a^6*b) + 1/140*(560*b^3*c*x^9 - 420*a*b^2*d*x^9 - 140*a^3*f*x^9 + 280*a^2*
b*x^9*e - 105*a*b^2*c*x^6 + 70*a^2*b*d*x^6 - 35*a^3*x^6*e + 40*a^2*b*c*x^3 - 20*
a^3*d*x^3 - 14*a^3*c)/(a^5*x^10)