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

Optimal. Leaf size=80 \[ -\frac{\log \left (a+b x^3\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a b^3}+\frac{x^3 (b e-a f)}{3 b^2}+\frac{c \log (x)}{a}+\frac{f x^6}{6 b} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.221115, antiderivative size = 80, 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 \[ -\frac{\log \left (a+b x^3\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a b^3}+\frac{x^3 (b e-a f)}{3 b^2}+\frac{c \log (x)}{a}+\frac{f x^6}{6 b} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \left (\frac{a f}{3} - \frac{b e}{3}\right ) \int ^{x^{3}} \frac{1}{b^{2}}\, dx + \frac{f \int ^{x^{3}} x\, dx}{3 b} + \frac{c \log{\left (x^{3} \right )}}{3 a} + \frac{\left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (a + b x^{3} \right )}}{3 a b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a*f/3 - b*e/3)*Integral(b**(-2), (x, x**3)) + f*Integral(x, (x, x**3))/(3*b) +
c*log(x**3)/(3*a) + (a**3*f - a**2*b*e + a*b**2*d - b**3*c)*log(a + b*x**3)/(3*
a*b**3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0552851, size = 75, normalized size = 0.94 \[ \frac{-2 \log \left (a+b x^3\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )+a b x^3 \left (-2 a f+2 b e+b f x^3\right )+6 b^3 c \log (x)}{6 a b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 97, normalized size = 1.2 \[{\frac{f{x}^{6}}{6\,b}}-{\frac{a{x}^{3}f}{3\,{b}^{2}}}+{\frac{e{x}^{3}}{3\,b}}+{\frac{c\ln \left ( x \right ) }{a}}+{\frac{{a}^{2}\ln \left ( b{x}^{3}+a \right ) f}{3\,{b}^{3}}}-{\frac{ae\ln \left ( b{x}^{3}+a \right ) }{3\,{b}^{2}}}+{\frac{d\ln \left ( b{x}^{3}+a \right ) }{3\,b}}-{\frac{c\ln \left ( b{x}^{3}+a \right ) }{3\,a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 7.32697, size = 104, normalized size = 1.3 \[ \frac{c \log \left (x^{3}\right )}{3 \, a} + \frac{b f x^{6} + 2 \,{\left (b e - a f\right )} x^{3}}{6 \, b^{2}} - \frac{{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (b x^{3} + a\right )}{3 \, a b^{3}} \]

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),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.246527, size = 108, normalized size = 1.35 \[ \frac{a b^{2} f x^{6} + 6 \, b^{3} c \log \left (x\right ) + 2 \,{\left (a b^{2} e - a^{2} b f\right )} x^{3} - 2 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (b x^{3} + a\right )}{6 \, a b^{3}} \]

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),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 9.6601, size = 68, normalized size = 0.85 \[ \frac{f x^{6}}{6 b} - \frac{x^{3} \left (a f - b e\right )}{3 b^{2}} + \frac{c \log{\left (x \right )}}{a} + \frac{\left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (\frac{a}{b} + x^{3} \right )}}{3 a b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.209952, size = 107, normalized size = 1.34 \[ \frac{c{\rm ln}\left ({\left | x \right |}\right )}{a} + \frac{b f x^{6} - 2 \, a f x^{3} + 2 \, b x^{3} e}{6 \, b^{2}} - \frac{{\left (b^{3} c - a b^{2} d - a^{3} f + a^{2} b e\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a b^{3}} \]

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),x, algorithm="giac")

[Out]

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