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

Optimal. Leaf size=225 \[ \frac{b c-a d}{2 a^2 x^2}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^{8/3} b^{4/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^{8/3} b^{4/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt{3} a^{8/3} b^{4/3}}-\frac{c}{5 a x^5}+\frac{f x}{b} \]

[Out]

-c/(5*a*x^5) + (b*c - a*d)/(2*a^2*x^2) + (f*x)/b - ((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^(8/3)*b^(4
/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^(8/3)
*b^(4/3)) - ((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^(8/3)*b^(4/3))

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Rubi [A]  time = 0.381287, antiderivative size = 225, 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 \[ \frac{b c-a d}{2 a^2 x^2}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^{8/3} b^{4/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^{8/3} b^{4/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt{3} a^{8/3} b^{4/3}}-\frac{c}{5 a x^5}+\frac{f x}{b} \]

Antiderivative was successfully verified.

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

[Out]

-c/(5*a*x^5) + (b*c - a*d)/(2*a^2*x^2) + (f*x)/b - ((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^(8/3)*b^(4
/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^(8/3)
*b^(4/3)) - ((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^(8/3)*b^(4/3))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\int f\, dx}{b} - \frac{c}{5 a x^{5}} - \frac{a d - b c}{2 a^{2} x^{2}} - \frac{\left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{8}{3}} b^{\frac{4}{3}}} + \frac{\left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{8}{3}} b^{\frac{4}{3}}} + \frac{\sqrt{3} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{8}{3}} b^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(f, x)/b - c/(5*a*x**5) - (a*d - b*c)/(2*a**2*x**2) - (a**3*f - a**2*b*e
+ a*b**2*d - b**3*c)*log(a**(1/3) + b**(1/3)*x)/(3*a**(8/3)*b**(4/3)) + (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**(8/3)*b**(4/3)) + sqrt(3)*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)*ata
n(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(3*a**(8/3)*b**(4/3))

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Mathematica [A]  time = 0.153976, size = 220, normalized size = 0.98 \[ \frac{b c-a d}{2 a^2 x^2}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{6 a^{8/3} b^{4/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^{8/3} b^{4/3}}+\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{\sqrt{3} a^{8/3} b^{4/3}}-\frac{c}{5 a x^5}+\frac{f x}{b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.012, size = 410, 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^6/(b*x^3+a),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23851, size = 313, normalized size = 1.39 \[ \frac{\sqrt{3}{\left (5 \, \sqrt{3}{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{5} \log \left (\left (-a^{2} b\right )^{\frac{2}{3}} x^{2} + \left (-a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 10 \, \sqrt{3}{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{5} \log \left (\left (-a^{2} b\right )^{\frac{1}{3}} x - a\right ) + 30 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{5} \arctan \left (\frac{2 \, \sqrt{3} \left (-a^{2} b\right )^{\frac{1}{3}} x + \sqrt{3} a}{3 \, a}\right ) + 3 \, \sqrt{3}{\left (10 \, a^{2} f x^{6} + 5 \,{\left (b^{2} c - a b d\right )} x^{3} - 2 \, a b c\right )} \left (-a^{2} b\right )^{\frac{1}{3}}\right )}}{90 \, \left (-a^{2} b\right )^{\frac{1}{3}} a^{2} b x^{5}} \]

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^6),x, algorithm="fricas")

[Out]

1/90*sqrt(3)*(5*sqrt(3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^5*log((-a^2*b)^(2/
3)*x^2 + (-a^2*b)^(1/3)*a*x + a^2) - 10*sqrt(3)*(b^3*c - a*b^2*d + a^2*b*e - a^3
*f)*x^5*log((-a^2*b)^(1/3)*x - a) + 30*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^5*a
rctan(1/3*(2*sqrt(3)*(-a^2*b)^(1/3)*x + sqrt(3)*a)/a) + 3*sqrt(3)*(10*a^2*f*x^6
+ 5*(b^2*c - a*b*d)*x^3 - 2*a*b*c)*(-a^2*b)^(1/3))/((-a^2*b)^(1/3)*a^2*b*x^5)

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Sympy [A]  time = 20.5395, size = 328, normalized size = 1.46 \[ \operatorname{RootSum}{\left (27 t^{3} a^{8} b^{4} + a^{9} f^{3} - 3 a^{8} b e f^{2} + 3 a^{7} b^{2} d f^{2} + 3 a^{7} b^{2} e^{2} f - 3 a^{6} b^{3} c f^{2} - 6 a^{6} b^{3} d e f - a^{6} b^{3} e^{3} + 6 a^{5} b^{4} c e f + 3 a^{5} b^{4} d^{2} f + 3 a^{5} b^{4} d e^{2} - 6 a^{4} b^{5} c d f - 3 a^{4} b^{5} c e^{2} - 3 a^{4} b^{5} d^{2} e + 3 a^{3} b^{6} c^{2} f + 6 a^{3} b^{6} c d e + a^{3} b^{6} d^{3} - 3 a^{2} b^{7} c^{2} e - 3 a^{2} b^{7} c d^{2} + 3 a b^{8} c^{2} d - b^{9} c^{3}, \left ( t \mapsto t \log{\left (- \frac{3 t a^{3} b}{a^{3} f - a^{2} b e + a b^{2} d - b^{3} c} + x \right )} \right )\right )} + \frac{f x}{b} - \frac{2 a c + x^{3} \left (5 a d - 5 b c\right )}{10 a^{2} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(27*_t**3*a**8*b**4 + a**9*f**3 - 3*a**8*b*e*f**2 + 3*a**7*b**2*d*f**2 +
3*a**7*b**2*e**2*f - 3*a**6*b**3*c*f**2 - 6*a**6*b**3*d*e*f - a**6*b**3*e**3 + 6
*a**5*b**4*c*e*f + 3*a**5*b**4*d**2*f + 3*a**5*b**4*d*e**2 - 6*a**4*b**5*c*d*f -
3*a**4*b**5*c*e**2 - 3*a**4*b**5*d**2*e + 3*a**3*b**6*c**2*f + 6*a**3*b**6*c*d*
e + a**3*b**6*d**3 - 3*a**2*b**7*c**2*e - 3*a**2*b**7*c*d**2 + 3*a*b**8*c**2*d -
b**9*c**3, Lambda(_t, _t*log(-3*_t*a**3*b/(a**3*f - a**2*b*e + a*b**2*d - b**3*
c) + x))) + f*x/b - (2*a*c + x**3*(5*a*d - 5*b*c))/(10*a**2*x**5)

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GIAC/XCAS [A]  time = 0.217755, size = 370, normalized size = 1.64 \[ \frac{f x}{b} - \frac{{\left (b^{3} c - a b^{2} d - a^{3} f + a^{2} b e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a^{3} b} + \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a^{3} b^{2}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a^{3} b^{2}} + \frac{5 \, b c x^{3} - 5 \, a d x^{3} - 2 \, a c}{10 \, a^{2} x^{5}} \]

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^6),x, algorithm="giac")

[Out]

f*x/b - 1/3*(b^3*c - a*b^2*d - a^3*f + a^2*b*e)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(
1/3)))/(a^3*b) + 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^3*b^2) + 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^3*b^2) + 1/10*(5*b*c*x^3 - 5*a*d*x^3 - 2*a*c)/(a^2*x^5)