3.230 \(\int \frac{c+d x^3+e x^6+f x^9}{x^{10} \left (a+b x^3\right )} \, dx\)

Optimal. Leaf size=128 \[ \frac{b c-a d}{6 a^2 x^6}-\frac{a^2 e-a b d+b^2 c}{3 a^3 x^3}+\frac{\log \left (a+b x^3\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^4}-\frac{\log (x) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a^4}-\frac{c}{9 a x^9} \]

[Out]

-c/(9*a*x^9) + (b*c - a*d)/(6*a^2*x^6) - (b^2*c - a*b*d + a^2*e)/(3*a^3*x^3) - (
(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[x])/a^4 + ((b^3*c - a*b^2*d + a^2*b*e -
a^3*f)*Log[a + b*x^3])/(3*a^4)

_______________________________________________________________________________________

Rubi [A]  time = 0.307128, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{b c-a d}{6 a^2 x^6}-\frac{a^2 e-a b d+b^2 c}{3 a^3 x^3}+\frac{\log \left (a+b x^3\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^4}-\frac{\log (x) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a^4}-\frac{c}{9 a x^9} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^10*(a + b*x^3)),x]

[Out]

-c/(9*a*x^9) + (b*c - a*d)/(6*a^2*x^6) - (b^2*c - a*b*d + a^2*e)/(3*a^3*x^3) - (
(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[x])/a^4 + ((b^3*c - a*b^2*d + a^2*b*e -
a^3*f)*Log[a + b*x^3])/(3*a^4)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 50.1633, size = 117, normalized size = 0.91 \[ - \frac{c}{9 a x^{9}} - \frac{a d - b c}{6 a^{2} x^{6}} - \frac{a^{2} e - a b d + b^{2} c}{3 a^{3} x^{3}} + \frac{\left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (x^{3} \right )}}{3 a^{4}} - \frac{\left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (a + b x^{3} \right )}}{3 a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((f*x**9+e*x**6+d*x**3+c)/x**10/(b*x**3+a),x)

[Out]

-c/(9*a*x**9) - (a*d - b*c)/(6*a**2*x**6) - (a**2*e - a*b*d + b**2*c)/(3*a**3*x*
*3) + (a**3*f - a**2*b*e + a*b**2*d - b**3*c)*log(x**3)/(3*a**4) - (a**3*f - a**
2*b*e + a*b**2*d - b**3*c)*log(a + b*x**3)/(3*a**4)

_______________________________________________________________________________________

Mathematica [A]  time = 0.129909, size = 128, normalized size = 1. \[ \frac{b c-a d}{6 a^2 x^6}+\frac{a^2 (-e)+a b d-b^2 c}{3 a^3 x^3}+\frac{\log \left (a+b x^3\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^4}+\frac{\log (x) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{a^4}-\frac{c}{9 a x^9} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^10*(a + b*x^3)),x]

[Out]

-c/(9*a*x^9) + (b*c - a*d)/(6*a^2*x^6) + (-(b^2*c) + a*b*d - a^2*e)/(3*a^3*x^3)
+ ((-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[x])/a^4 + ((b^3*c - a*b^2*d + a^2*
b*e - a^3*f)*Log[a + b*x^3])/(3*a^4)

_______________________________________________________________________________________

Maple [A]  time = 0.012, size = 161, normalized size = 1.3 \[ -{\frac{c}{9\,a{x}^{9}}}-{\frac{d}{6\,a{x}^{6}}}+{\frac{bc}{6\,{a}^{2}{x}^{6}}}-{\frac{e}{3\,a{x}^{3}}}+{\frac{bd}{3\,{a}^{2}{x}^{3}}}-{\frac{{b}^{2}c}{3\,{a}^{3}{x}^{3}}}+{\frac{\ln \left ( x \right ) f}{a}}-{\frac{\ln \left ( x \right ) be}{{a}^{2}}}+{\frac{\ln \left ( x \right ){b}^{2}d}{{a}^{3}}}-{\frac{\ln \left ( x \right ){b}^{3}c}{{a}^{4}}}-{\frac{\ln \left ( b{x}^{3}+a \right ) f}{3\,a}}+{\frac{\ln \left ( b{x}^{3}+a \right ) be}{3\,{a}^{2}}}-{\frac{\ln \left ( b{x}^{3}+a \right ){b}^{2}d}{3\,{a}^{3}}}+{\frac{\ln \left ( b{x}^{3}+a \right ){b}^{3}c}{3\,{a}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x^9+e*x^6+d*x^3+c)/x^10/(b*x^3+a),x)

[Out]

-1/9*c/a/x^9-1/6/a/x^6*d+1/6/a^2/x^6*b*c-1/3/a/x^3*e+1/3/a^2/x^3*b*d-1/3/a^3/x^3
*b^2*c+1/a*ln(x)*f-1/a^2*ln(x)*b*e+1/a^3*ln(x)*b^2*d-1/a^4*ln(x)*b^3*c-1/3/a*ln(
b*x^3+a)*f+1/3/a^2*ln(b*x^3+a)*b*e-1/3/a^3*ln(b*x^3+a)*b^2*d+1/3/a^4*ln(b*x^3+a)
*b^3*c

_______________________________________________________________________________________

Maxima [A]  time = 1.43352, size = 169, normalized size = 1.32 \[ \frac{{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (b x^{3} + a\right )}{3 \, a^{4}} - \frac{{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (x^{3}\right )}{3 \, a^{4}} - \frac{6 \,{\left (b^{2} c - a b d + a^{2} e\right )} x^{6} - 3 \,{\left (a b c - a^{2} d\right )} x^{3} + 2 \, a^{2} c}{18 \, a^{3} x^{9}} \]

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^10),x, algorithm="maxima")

[Out]

1/3*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*log(b*x^3 + a)/a^4 - 1/3*(b^3*c - a*b^2*
d + a^2*b*e - a^3*f)*log(x^3)/a^4 - 1/18*(6*(b^2*c - a*b*d + a^2*e)*x^6 - 3*(a*b
*c - a^2*d)*x^3 + 2*a^2*c)/(a^3*x^9)

_______________________________________________________________________________________

Fricas [A]  time = 0.237382, size = 171, normalized size = 1.34 \[ \frac{6 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{9} \log \left (b x^{3} + a\right ) - 18 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{9} \log \left (x\right ) - 6 \,{\left (a b^{2} c - a^{2} b d + a^{3} e\right )} x^{6} - 2 \, a^{3} c + 3 \,{\left (a^{2} b c - a^{3} d\right )} x^{3}}{18 \, a^{4} x^{9}} \]

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^10),x, algorithm="fricas")

[Out]

1/18*(6*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^9*log(b*x^3 + a) - 18*(b^3*c - a*b
^2*d + a^2*b*e - a^3*f)*x^9*log(x) - 6*(a*b^2*c - a^2*b*d + a^3*e)*x^6 - 2*a^3*c
+ 3*(a^2*b*c - a^3*d)*x^3)/(a^4*x^9)

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.214182, size = 248, normalized size = 1.94 \[ -\frac{{\left (b^{3} c - a b^{2} d - a^{3} f + a^{2} b e\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{4}} + \frac{{\left (b^{4} c - a b^{3} d - a^{3} b f + a^{2} b^{2} e\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{4} b} + \frac{11 \, b^{3} c x^{9} - 11 \, a b^{2} d x^{9} - 11 \, a^{3} f x^{9} + 11 \, a^{2} b x^{9} e - 6 \, a b^{2} c x^{6} + 6 \, a^{2} b d x^{6} - 6 \, a^{3} x^{6} e + 3 \, a^{2} b c x^{3} - 3 \, a^{3} d x^{3} - 2 \, a^{3} c}{18 \, a^{4} x^{9}} \]

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^10),x, algorithm="giac")

[Out]

-(b^3*c - a*b^2*d - a^3*f + a^2*b*e)*ln(abs(x))/a^4 + 1/3*(b^4*c - a*b^3*d - a^3
*b*f + a^2*b^2*e)*ln(abs(b*x^3 + a))/(a^4*b) + 1/18*(11*b^3*c*x^9 - 11*a*b^2*d*x
^9 - 11*a^3*f*x^9 + 11*a^2*b*x^9*e - 6*a*b^2*c*x^6 + 6*a^2*b*d*x^6 - 6*a^3*x^6*e
+ 3*a^2*b*c*x^3 - 3*a^3*d*x^3 - 2*a^3*c)/(a^4*x^9)