3.255 \(\int \frac{c+d x^3+e x^6+f x^9}{x \left (a+b x^3\right )^2} \, dx\)

Optimal. Leaf size=100 \[ \frac{c \log (x)}{a^2}-\frac{\log \left (a+b x^3\right ) \left (2 a^3 f-a^2 b e+b^3 c\right )}{3 a^2 b^3}+\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{3 a b^3 \left (a+b x^3\right )}+\frac{f x^3}{3 b^2} \]

[Out]

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

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Rubi [A]  time = 0.24865, antiderivative size = 100, 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 \[ \frac{c \log (x)}{a^2}-\frac{\log \left (a+b x^3\right ) \left (2 a^3 f-a^2 b e+b^3 c\right )}{3 a^2 b^3}+\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{3 a b^3 \left (a+b x^3\right )}+\frac{f x^3}{3 b^2} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^3 + e*x^6 + f*x^9)/(x*(a + b*x^3)^2),x]

[Out]

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

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

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)**2,x)

[Out]

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

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Mathematica [A]  time = 0.228831, size = 95, normalized size = 0.95 \[ \frac{\frac{\log \left (a+b x^3\right ) \left (-2 a^3 f+a^2 b e-b^3 c\right )+\frac{a \left (a^3 (-f)+a^2 b \left (e+f x^3\right )+a b^2 \left (f x^6-d\right )+b^3 c\right )}{a+b x^3}}{b^3}+3 c \log (x)}{3 a^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x*(a + b*x^3)^2),x]

[Out]

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

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Maple [A]  time = 0.023, size = 125, normalized size = 1.3 \[{\frac{f{x}^{3}}{3\,{b}^{2}}}+{\frac{c\ln \left ( x \right ) }{{a}^{2}}}-{\frac{2\,a\ln \left ( b{x}^{3}+a \right ) f}{3\,{b}^{3}}}+{\frac{\ln \left ( b{x}^{3}+a \right ) e}{3\,{b}^{2}}}-{\frac{c\ln \left ( b{x}^{3}+a \right ) }{3\,{a}^{2}}}-{\frac{{a}^{2}f}{3\,{b}^{3} \left ( b{x}^{3}+a \right ) }}+{\frac{ae}{3\,{b}^{2} \left ( b{x}^{3}+a \right ) }}-{\frac{d}{3\,b \left ( b{x}^{3}+a \right ) }}+{\frac{c}{3\,a \left ( b{x}^{3}+a \right ) }} \]

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)^2,x)

[Out]

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

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Maxima [A]  time = 1.38728, size = 135, normalized size = 1.35 \[ \frac{f x^{3}}{3 \, b^{2}} + \frac{b^{3} c - a b^{2} d + a^{2} b e - a^{3} f}{3 \,{\left (a b^{4} x^{3} + a^{2} b^{3}\right )}} + \frac{c \log \left (x^{3}\right )}{3 \, a^{2}} - \frac{{\left (b^{3} c - a^{2} b e + 2 \, a^{3} f\right )} \log \left (b x^{3} + a\right )}{3 \, a^{2} b^{3}} \]

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)^2*x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.243147, size = 196, normalized size = 1.96 \[ \frac{a^{2} b^{2} f x^{6} + a^{3} b f x^{3} + a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f -{\left (a b^{3} c - a^{3} b e + 2 \, a^{4} f +{\left (b^{4} c - a^{2} b^{2} e + 2 \, a^{3} b f\right )} x^{3}\right )} \log \left (b x^{3} + a\right ) + 3 \,{\left (b^{4} c x^{3} + a b^{3} c\right )} \log \left (x\right )}{3 \,{\left (a^{2} b^{4} x^{3} + a^{3} b^{3}\right )}} \]

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)^2*x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 54.8426, size = 95, normalized size = 0.95 \[ - \frac{a^{3} f - a^{2} b e + a b^{2} d - b^{3} c}{3 a^{2} b^{3} + 3 a b^{4} x^{3}} + \frac{f x^{3}}{3 b^{2}} + \frac{c \log{\left (x \right )}}{a^{2}} - \frac{\left (2 a^{3} f - a^{2} b e + b^{3} c\right ) \log{\left (\frac{a}{b} + x^{3} \right )}}{3 a^{2} b^{3}} \]

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)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.213549, size = 169, normalized size = 1.69 \[ \frac{f x^{3}}{3 \, b^{2}} + \frac{c{\rm ln}\left ({\left | x \right |}\right )}{a^{2}} - \frac{{\left (b^{3} c + 2 \, a^{3} f - a^{2} b e\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{2} b^{3}} + \frac{b^{4} c x^{3} + 2 \, a^{3} b f x^{3} - a^{2} b^{2} x^{3} e + 2 \, a b^{3} c - a^{2} b^{2} d + a^{4} f}{3 \,{\left (b x^{3} + a\right )} a^{2} b^{3}} \]

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)^2*x),x, algorithm="giac")

[Out]

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