3.247 \(\int \frac{c+d x^3+e x^6+f x^9}{x^{12} \left (a+b x^3\right )} \, dx\)

Optimal. Leaf size=280 \[ \frac{b c-a d}{8 a^2 x^8}-\frac{a^2 e-a b d+b^2 c}{5 a^3 x^5}-\frac{b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^{14/3}}+\frac{b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^{14/3}}-\frac{b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt{3} a^{14/3}}+\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{2 a^4 x^2}-\frac{c}{11 a x^{11}} \]

[Out]

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

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Rubi [A]  time = 0.472228, antiderivative size = 280, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.233 \[ \frac{b c-a d}{8 a^2 x^8}-\frac{a^2 e-a b d+b^2 c}{5 a^3 x^5}-\frac{b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^{14/3}}+\frac{b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^{14/3}}-\frac{b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt{3} a^{14/3}}+\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{2 a^4 x^2}-\frac{c}{11 a x^{11}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^12*(a + b*x^3)),x]

[Out]

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

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Rubi in Sympy [A]  time = 84.0703, size = 260, normalized size = 0.93 \[ - \frac{c}{11 a x^{11}} - \frac{a d - b c}{8 a^{2} x^{8}} - \frac{a^{2} e - a b d + b^{2} c}{5 a^{3} x^{5}} - \frac{a^{3} f - a^{2} b e + a b^{2} d - b^{3} c}{2 a^{4} x^{2}} - \frac{b^{\frac{2}{3}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{14}{3}}} + \frac{b^{\frac{2}{3}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{14}{3}}} + \frac{\sqrt{3} b^{\frac{2}{3}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{14}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((f*x**9+e*x**6+d*x**3+c)/x**12/(b*x**3+a),x)

[Out]

-c/(11*a*x**11) - (a*d - b*c)/(8*a**2*x**8) - (a**2*e - a*b*d + b**2*c)/(5*a**3*
x**5) - (a**3*f - a**2*b*e + a*b**2*d - b**3*c)/(2*a**4*x**2) - b**(2/3)*(a**3*f
- a**2*b*e + a*b**2*d - b**3*c)*log(a**(1/3) + b**(1/3)*x)/(3*a**(14/3)) + b**(
2/3)*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)*log(a**(2/3) - a**(1/3)*b**(1/3)*x
+ b**(2/3)*x**2)/(6*a**(14/3)) + sqrt(3)*b**(2/3)*(a**3*f - a**2*b*e + a*b**2*d
- b**3*c)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(3*a**(14/3))

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Mathematica [A]  time = 0.215224, size = 266, normalized size = 0.95 \[ \frac{\frac{165 a^{8/3} (b c-a d)}{x^8}-\frac{120 a^{11/3} c}{x^{11}}-\frac{264 a^{5/3} \left (a^2 e-a b d+b^2 c\right )}{x^5}+440 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )-440 \sqrt{3} b^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )+\frac{660 a^{2/3} \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{x^2}+220 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{1320 a^{14/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^12*(a + b*x^3)),x]

[Out]

((-120*a^(11/3)*c)/x^11 + (165*a^(8/3)*(b*c - a*d))/x^8 - (264*a^(5/3)*(b^2*c -
a*b*d + a^2*e))/x^5 + (660*a^(2/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f))/x^2 - 44
0*Sqrt[3]*b^(2/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/
a^(1/3))/Sqrt[3]] + 440*b^(2/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(1/3)
+ b^(1/3)*x] + 220*b^(2/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a^(2/3) -
a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(1320*a^(14/3))

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Maple [B]  time = 0.011, size = 493, 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^12/(b*x^3+a),x)

[Out]

-1/11*c/a/x^11-1/8/a/x^8*d+1/8/a^2/x^8*b*c-1/5/a/x^5*e+1/5/a^2/x^5*b*d-1/5/a^3/x
^5*b^2*c-1/2/a/x^2*f+1/2/a^2/x^2*b*e-1/2/a^3/x^2*b^2*d+1/2/a^4/x^2*b^3*c-1/3/a/(
a/b)^(2/3)*ln(x+(a/b)^(1/3))*f+1/3*b/a^2/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*e-1/3*b^2
/a^3/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*d+1/3*b^3/a^4/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*c
+1/6/a/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*f-1/6*b/a^2/(a/b)^(2/3)*ln(
x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*e+1/6*b^2/a^3/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a
/b)^(2/3))*d-1/6*b^3/a^4/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*c-1/3/a/(
a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*f+1/3*b/a^2/(a/b)^(2/
3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*e-1/3*b^2/a^3/(a/b)^(2/3)*3^(
1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*d+1/3*b^3/a^4/(a/b)^(2/3)*3^(1/2)*a
rctan(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)*x^12),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.223798, size = 427, normalized size = 1.52 \[ \frac{\sqrt{3}{\left (220 \, \sqrt{3}{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{11} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b^{2} x^{2} + a b x \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} + a^{2} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}\right ) - 440 \, \sqrt{3}{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{11} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b x - a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}\right ) + 1320 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{11} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} b x + \sqrt{3} a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}{3 \, a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}\right ) + 3 \, \sqrt{3}{\left (220 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{9} - 88 \,{\left (a b^{2} c - a^{2} b d + a^{3} e\right )} x^{6} - 40 \, a^{3} c + 55 \,{\left (a^{2} b c - a^{3} d\right )} x^{3}\right )}\right )}}{3960 \, a^{4} x^{11}} \]

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)*x^12),x, algorithm="fricas")

[Out]

1/3960*sqrt(3)*(220*sqrt(3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^11*(-b^2/a^2)^
(1/3)*log(b^2*x^2 + a*b*x*(-b^2/a^2)^(1/3) + a^2*(-b^2/a^2)^(2/3)) - 440*sqrt(3)
*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^11*(-b^2/a^2)^(1/3)*log(b*x - a*(-b^2/a^2
)^(1/3)) + 1320*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^11*(-b^2/a^2)^(1/3)*arctan
(1/3*(2*sqrt(3)*b*x + sqrt(3)*a*(-b^2/a^2)^(1/3))/(a*(-b^2/a^2)^(1/3))) + 3*sqrt
(3)*(220*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^9 - 88*(a*b^2*c - a^2*b*d + a^3*e
)*x^6 - 40*a^3*c + 55*(a^2*b*c - a^3*d)*x^3))/(a^4*x^11)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.217835, size = 456, normalized size = 1.63 \[ \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a^{5}} - \frac{{\left (b^{4} c - a b^{3} d - a^{3} b f + a^{2} b^{2} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a^{5}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a^{5}} + \frac{220 \, b^{3} c x^{9} - 220 \, a b^{2} d x^{9} - 220 \, a^{3} f x^{9} + 220 \, a^{2} b x^{9} e - 88 \, a b^{2} c x^{6} + 88 \, a^{2} b d x^{6} - 88 \, a^{3} x^{6} e + 55 \, a^{2} b c x^{3} - 55 \, a^{3} d x^{3} - 40 \, a^{3} c}{440 \, a^{4} x^{11}} \]

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)*x^12),x, algorithm="giac")

[Out]

1/3*sqrt(3)*((-a*b^2)^(1/3)*b^3*c - (-a*b^2)^(1/3)*a*b^2*d - (-a*b^2)^(1/3)*a^3*
f + (-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^5 - 1/3*(b^4*c - a*b^3*d - a^3*b*f + a^2*b^2*e)*(-a/b)^(1/3)*ln(abs(x - (-a/
b)^(1/3)))/a^5 + 1/6*((-a*b^2)^(1/3)*b^3*c - (-a*b^2)^(1/3)*a*b^2*d - (-a*b^2)^(
1/3)*a^3*f + (-a*b^2)^(1/3)*a^2*b*e)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/a^5
+ 1/440*(220*b^3*c*x^9 - 220*a*b^2*d*x^9 - 220*a^3*f*x^9 + 220*a^2*b*x^9*e - 88
*a*b^2*c*x^6 + 88*a^2*b*d*x^6 - 88*a^3*x^6*e + 55*a^2*b*c*x^3 - 55*a^3*d*x^3 - 4
0*a^3*c)/(a^4*x^11)