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

Optimal. Leaf size=244 \[ \frac{b c-a d}{5 a^2 x^5}-\frac{a^2 e-a b d+b^2 c}{2 a^3 x^2}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^{11/3} \sqrt [3]{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt{3} a^{11/3} \sqrt [3]{b}}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^{11/3} \sqrt [3]{b}}-\frac{c}{8 a x^8} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.410485, antiderivative size = 244, 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 \[ \frac{b c-a d}{5 a^2 x^5}-\frac{a^2 e-a b d+b^2 c}{2 a^3 x^2}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^{11/3} \sqrt [3]{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt{3} a^{11/3} \sqrt [3]{b}}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^{11/3} \sqrt [3]{b}}-\frac{c}{8 a x^8} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 65.8415, size = 228, normalized size = 0.93 \[ - \frac{c}{8 a x^{8}} - \frac{a d - b c}{5 a^{2} x^{5}} - \frac{a^{2} e - a b d + b^{2} c}{2 a^{3} x^{2}} + \frac{\left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{11}{3}} \sqrt [3]{b}} - \frac{\left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{11}{3}} \sqrt [3]{b}} - \frac{\sqrt{3} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{11}{3}} \sqrt [3]{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-c/(8*a*x**8) - (a*d - b*c)/(5*a**2*x**5) - (a**2*e - a*b*d + b**2*c)/(2*a**3*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**(
11/3)*b**(1/3)) - (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**(11/3)*b**(1/3)) - sqrt(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
**(11/3)*b**(1/3))

_______________________________________________________________________________________

Mathematica [A]  time = 0.207986, size = 231, normalized size = 0.95 \[ \frac{\frac{24 a^{5/3} (b c-a d)}{x^5}-\frac{15 a^{8/3} c}{x^8}-\frac{60 a^{2/3} \left (a^2 e-a b d+b^2 c\right )}{x^2}+\frac{40 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{\sqrt [3]{b}}+\frac{40 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt [3]{b}}+\frac{20 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt [3]{b}}}{120 a^{11/3}} \]

Antiderivative was successfully verified.

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

[Out]

((-15*a^(8/3)*c)/x^8 + (24*a^(5/3)*(b*c - a*d))/x^5 - (60*a^(2/3)*(b^2*c - a*b*d
+ a^2*e))/x^2 + (40*Sqrt[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]])/b^(1/3) + (40*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*
f)*Log[a^(1/3) + b^(1/3)*x])/b^(1/3) + (20*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*L
og[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(1/3))/(120*a^(11/3))

_______________________________________________________________________________________

Maple [B]  time = 0.01, size = 441, 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^9/(b*x^3+a),x)

[Out]

-1/8*c/a/x^8-1/5/a/x^5*d+1/5/a^2/x^5*b*c-1/2/a/x^2*e+1/2/a^2/x^2*b*d-1/2/a^3/x^2
*b^2*c+1/3/b/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*f-1/3/a/(a/b)^(2/3)*ln(x+(a/b)^(1/3))
*e+1/3/a^2*b/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*d-1/3/a^3*b^2/(a/b)^(2/3)*ln(x+(a/b)^
(1/3))*c-1/6/b/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*f+1/6/a/(a/b)^(2/3)
*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*e-1/6/a^2*b/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+
(a/b)^(2/3))*d+1/6/a^3*b^2/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*c+1/3/b
/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*f-1/3/a/(a/b)^(2/3)
*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*e+1/3/a^2*b/(a/b)^(2/3)*3^(1/2)
*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*d-1/3/a^3*b^2/(a/b)^(2/3)*3^(1/2)*arcta
n(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*c

_______________________________________________________________________________________

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^9),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.221883, size = 319, normalized size = 1.31 \[ \frac{\sqrt{3}{\left (20 \, \sqrt{3}{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{8} \log \left (\left (a^{2} b\right )^{\frac{2}{3}} x^{2} - \left (a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 40 \, \sqrt{3}{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{8} \log \left (\left (a^{2} b\right )^{\frac{1}{3}} x + a\right ) - 120 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{8} \arctan \left (\frac{2 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} x - \sqrt{3} a}{3 \, a}\right ) - 3 \, \sqrt{3}{\left (20 \,{\left (b^{2} c - a b d + a^{2} e\right )} x^{6} - 8 \,{\left (a b c - a^{2} d\right )} x^{3} + 5 \, a^{2} c\right )} \left (a^{2} b\right )^{\frac{1}{3}}\right )}}{360 \, \left (a^{2} b\right )^{\frac{1}{3}} a^{3} x^{8}} \]

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^9),x, algorithm="fricas")

[Out]

1/360*sqrt(3)*(20*sqrt(3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^8*log((a^2*b)^(2
/3)*x^2 - (a^2*b)^(1/3)*a*x + a^2) - 40*sqrt(3)*(b^3*c - a*b^2*d + a^2*b*e - a^3
*f)*x^8*log((a^2*b)^(1/3)*x + a) - 120*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^8*a
rctan(1/3*(2*sqrt(3)*(a^2*b)^(1/3)*x - sqrt(3)*a)/a) - 3*sqrt(3)*(20*(b^2*c - a*
b*d + a^2*e)*x^6 - 8*(a*b*c - a^2*d)*x^3 + 5*a^2*c)*(a^2*b)^(1/3))/((a^2*b)^(1/3
)*a^3*x^8)

_______________________________________________________________________________________

Sympy [A]  time = 106.427, size = 348, normalized size = 1.43 \[ \operatorname{RootSum}{\left (27 t^{3} a^{11} b - a^{9} f^{3} + 3 a^{8} b e f^{2} - 3 a^{7} b^{2} d f^{2} - 3 a^{7} b^{2} e^{2} f + 3 a^{6} b^{3} c f^{2} + 6 a^{6} b^{3} d e f + a^{6} b^{3} e^{3} - 6 a^{5} b^{4} c e f - 3 a^{5} b^{4} d^{2} f - 3 a^{5} b^{4} d e^{2} + 6 a^{4} b^{5} c d f + 3 a^{4} b^{5} c e^{2} + 3 a^{4} b^{5} d^{2} e - 3 a^{3} b^{6} c^{2} f - 6 a^{3} b^{6} c d e - a^{3} b^{6} d^{3} + 3 a^{2} b^{7} c^{2} e + 3 a^{2} b^{7} c d^{2} - 3 a b^{8} c^{2} d + b^{9} c^{3}, \left ( t \mapsto t \log{\left (\frac{3 t a^{4}}{a^{3} f - a^{2} b e + a b^{2} d - b^{3} c} + x \right )} \right )\right )} - \frac{5 a^{2} c + x^{6} \left (20 a^{2} e - 20 a b d + 20 b^{2} c\right ) + x^{3} \left (8 a^{2} d - 8 a b c\right )}{40 a^{3} x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(27*_t**3*a**11*b - 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**4/(a**3*f - a**2*b*e + a*b**2*d - b**3*c) +
x))) - (5*a**2*c + x**6*(20*a**2*e - 20*a*b*d + 20*b**2*c) + x**3*(8*a**2*d - 8*
a*b*c))/(40*a**3*x**8)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.217408, size = 401, normalized size = 1.64 \[ \frac{{\left (b^{3} c - a b^{2} d - a^{3} f + a^{2} b e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a^{4}} - \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a^{4} b} - \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a^{4} b} - \frac{20 \, b^{2} c x^{6} - 20 \, a b d x^{6} + 20 \, a^{2} x^{6} e - 8 \, a b c x^{3} + 8 \, a^{2} d x^{3} + 5 \, a^{2} c}{40 \, a^{3} x^{8}} \]

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^9),x, algorithm="giac")

[Out]

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
^4 - 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^4*b) - 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^
4*b) - 1/40*(20*b^2*c*x^6 - 20*a*b*d*x^6 + 20*a^2*x^6*e - 8*a*b*c*x^3 + 8*a^2*d*
x^3 + 5*a^2*c)/(a^3*x^8)