3.256 \(\int \frac{c+d x^3+e x^6+f x^9}{x^4 \left (a+b x^3\right )^2} \, dx\)

Optimal. Leaf size=109 \[ \frac{\log \left (a+b x^3\right ) \left (a^3 f-a b^2 d+2 b^3 c\right )}{3 a^3 b^2}-\frac{\log (x) (2 b c-a d)}{a^3}-\frac{c}{3 a^2 x^3}-\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{3 a^2 b^2 \left (a+b x^3\right )} \]

[Out]

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

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Rubi [A]  time = 0.277588, antiderivative size = 109, 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 \[ \frac{\log \left (a+b x^3\right ) \left (a^3 f-a b^2 d+2 b^3 c\right )}{3 a^3 b^2}-\frac{\log (x) (2 b c-a d)}{a^3}-\frac{c}{3 a^2 x^3}-\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{3 a^2 b^2 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 48.4794, size = 102, normalized size = 0.94 \[ - \frac{c}{3 a^{2} x^{3}} + \frac{a^{3} f - a^{2} b e + a b^{2} d - b^{3} c}{3 a^{2} b^{2} \left (a + b x^{3}\right )} + \frac{\left (a d - 2 b c\right ) \log{\left (x^{3} \right )}}{3 a^{3}} + \frac{\left (a^{3} f - a b^{2} d + 2 b^{3} c\right ) \log{\left (a + b x^{3} \right )}}{3 a^{3} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((f*x**9+e*x**6+d*x**3+c)/x**4/(b*x**3+a)**2,x)

[Out]

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

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Mathematica [A]  time = 0.136676, size = 97, normalized size = 0.89 \[ \frac{\frac{\log \left (a+b x^3\right ) \left (a^3 f-a b^2 d+2 b^3 c\right )}{b^2}+\frac{a \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{b^2 \left (a+b x^3\right )}+3 \log (x) (a d-2 b c)-\frac{a c}{x^3}}{3 a^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.021, size = 132, normalized size = 1.2 \[ -{\frac{c}{3\,{a}^{2}{x}^{3}}}+{\frac{d\ln \left ( x \right ) }{{a}^{2}}}-2\,{\frac{bc\ln \left ( x \right ) }{{a}^{3}}}+{\frac{f\ln \left ( b{x}^{3}+a \right ) }{3\,{b}^{2}}}-{\frac{d\ln \left ( b{x}^{3}+a \right ) }{3\,{a}^{2}}}+{\frac{2\,bc\ln \left ( b{x}^{3}+a \right ) }{3\,{a}^{3}}}+{\frac{af}{3\,{b}^{2} \left ( b{x}^{3}+a \right ) }}-{\frac{e}{3\,b \left ( b{x}^{3}+a \right ) }}+{\frac{d}{3\,a \left ( b{x}^{3}+a \right ) }}-{\frac{bc}{3\,{a}^{2} \left ( b{x}^{3}+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.38097, size = 157, normalized size = 1.44 \[ -\frac{a b^{2} c +{\left (2 \, b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{3}}{3 \,{\left (a^{2} b^{3} x^{6} + a^{3} b^{2} x^{3}\right )}} - \frac{{\left (2 \, b c - a d\right )} \log \left (x^{3}\right )}{3 \, a^{3}} + \frac{{\left (2 \, b^{3} c - a b^{2} d + a^{3} f\right )} \log \left (b x^{3} + a\right )}{3 \, a^{3} b^{2}} \]

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^4),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.232838, size = 232, normalized size = 2.13 \[ -\frac{a^{2} b^{2} c +{\left (2 \, a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} x^{3} -{\left ({\left (2 \, b^{4} c - a b^{3} d + a^{3} b f\right )} x^{6} +{\left (2 \, a b^{3} c - a^{2} b^{2} d + a^{4} f\right )} x^{3}\right )} \log \left (b x^{3} + a\right ) + 3 \,{\left ({\left (2 \, b^{4} c - a b^{3} d\right )} x^{6} +{\left (2 \, a b^{3} c - a^{2} b^{2} d\right )} x^{3}\right )} \log \left (x\right )}{3 \,{\left (a^{3} b^{3} x^{6} + a^{4} b^{2} x^{3}\right )}} \]

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^4),x, algorithm="fricas")

[Out]

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

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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**4/(b*x**3+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.215592, size = 177, normalized size = 1.62 \[ -\frac{{\left (2 \, b c - a d\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{3}} + \frac{{\left (2 \, b^{3} c - a b^{2} d + a^{3} f\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{3} b^{2}} - \frac{a^{2} b f x^{6} + 4 \, b^{3} c x^{3} - 2 \, a b^{2} d x^{3} - a^{3} f x^{3} + 2 \, a^{2} b x^{3} e + 2 \, a b^{2} c}{6 \,{\left (b x^{6} + a x^{3}\right )} a^{2} b^{2}} \]

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^4),x, algorithm="giac")

[Out]

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