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

Optimal. Leaf size=103 \[ \frac{\log \left (a+b x^3\right ) \left (3 a^2 f-2 a b e+b^2 d\right )}{3 b^4}-\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{3 b^4 \left (a+b x^3\right )}+\frac{x^3 (b e-2 a f)}{3 b^3}+\frac{f x^6}{6 b^2} \]

[Out]

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

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Rubi [A]  time = 0.299313, antiderivative size = 103, 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 \[ \frac{\log \left (a+b x^3\right ) \left (3 a^2 f-2 a b e+b^2 d\right )}{3 b^4}-\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{3 b^4 \left (a+b x^3\right )}+\frac{x^3 (b e-2 a f)}{3 b^3}+\frac{f x^6}{6 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \left (\frac{2 a f}{3} - \frac{b e}{3}\right ) \int ^{x^{3}} \frac{1}{b^{3}}\, dx + \frac{f \int ^{x^{3}} x\, dx}{3 b^{2}} + \frac{\left (3 a^{2} f - 2 a b e + b^{2} d\right ) \log{\left (a + b x^{3} \right )}}{3 b^{4}} + \frac{a^{3} f - a^{2} b e + a b^{2} d - b^{3} c}{3 b^{4} \left (a + b x^{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.106698, size = 93, normalized size = 0.9 \[ \frac{2 \log \left (a+b x^3\right ) \left (3 a^2 f-2 a b e+b^2 d\right )+\frac{2 \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{a+b x^3}+2 b x^3 (b e-2 a f)+b^2 f x^6}{6 b^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.016, size = 142, normalized size = 1.4 \[{\frac{f{x}^{6}}{6\,{b}^{2}}}-{\frac{2\,a{x}^{3}f}{3\,{b}^{3}}}+{\frac{{x}^{3}e}{3\,{b}^{2}}}+{\frac{\ln \left ( b{x}^{3}+a \right ){a}^{2}f}{{b}^{4}}}-{\frac{2\,\ln \left ( b{x}^{3}+a \right ) ae}{3\,{b}^{3}}}+{\frac{\ln \left ( b{x}^{3}+a \right ) d}{3\,{b}^{2}}}+{\frac{{a}^{3}f}{3\,{b}^{4} \left ( b{x}^{3}+a \right ) }}-{\frac{{a}^{2}e}{3\,{b}^{3} \left ( b{x}^{3}+a \right ) }}+{\frac{ad}{3\,{b}^{2} \left ( b{x}^{3}+a \right ) }}-{\frac{c}{3\,b \left ( b{x}^{3}+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 9.60462, size = 132, normalized size = 1.28 \[ -\frac{b^{3} c - a b^{2} d + a^{2} b e - a^{3} f}{3 \,{\left (b^{5} x^{3} + a b^{4}\right )}} + \frac{b f x^{6} + 2 \,{\left (b e - 2 \, a f\right )} x^{3}}{6 \, b^{3}} + \frac{{\left (b^{2} d - 2 \, a b e + 3 \, a^{2} f\right )} \log \left (b x^{3} + a\right )}{3 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^2/(b*x^3 + a)^2,x, algorithm="maxima")

[Out]

-1/3*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)/(b^5*x^3 + a*b^4) + 1/6*(b*f*x^6 + 2*(b
*e - 2*a*f)*x^3)/b^3 + 1/3*(b^2*d - 2*a*b*e + 3*a^2*f)*log(b*x^3 + a)/b^4

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Fricas [A]  time = 0.215871, size = 193, normalized size = 1.87 \[ \frac{b^{3} f x^{9} +{\left (2 \, b^{3} e - 3 \, a b^{2} f\right )} x^{6} - 2 \, b^{3} c + 2 \, a b^{2} d - 2 \, a^{2} b e + 2 \, a^{3} f + 2 \,{\left (a b^{2} e - 2 \, a^{2} b f\right )} x^{3} + 2 \,{\left (a b^{2} d - 2 \, a^{2} b e + 3 \, a^{3} f +{\left (b^{3} d - 2 \, a b^{2} e + 3 \, a^{2} b f\right )} x^{3}\right )} \log \left (b x^{3} + a\right )}{6 \,{\left (b^{5} x^{3} + a b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^2/(b*x^3 + a)^2,x, algorithm="fricas")

[Out]

1/6*(b^3*f*x^9 + (2*b^3*e - 3*a*b^2*f)*x^6 - 2*b^3*c + 2*a*b^2*d - 2*a^2*b*e + 2
*a^3*f + 2*(a*b^2*e - 2*a^2*b*f)*x^3 + 2*(a*b^2*d - 2*a^2*b*e + 3*a^3*f + (b^3*d
- 2*a*b^2*e + 3*a^2*b*f)*x^3)*log(b*x^3 + a))/(b^5*x^3 + a*b^4)

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Sympy [A]  time = 16.5737, size = 97, normalized size = 0.94 \[ \frac{a^{3} f - a^{2} b e + a b^{2} d - b^{3} c}{3 a b^{4} + 3 b^{5} x^{3}} + \frac{f x^{6}}{6 b^{2}} - \frac{x^{3} \left (2 a f - b e\right )}{3 b^{3}} + \frac{\left (3 a^{2} f - 2 a b e + b^{2} d\right ) \log{\left (a + b x^{3} \right )}}{3 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a**3*f - a**2*b*e + a*b**2*d - b**3*c)/(3*a*b**4 + 3*b**5*x**3) + f*x**6/(6*b**
2) - x**3*(2*a*f - b*e)/(3*b**3) + (3*a**2*f - 2*a*b*e + b**2*d)*log(a + b*x**3)
/(3*b**4)

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GIAC/XCAS [A]  time = 0.21559, size = 278, normalized size = 2.7 \[ -\frac{1}{6} \, f{\left (\frac{{\left (b x^{3} + a\right )}^{2}{\left (\frac{6 \, a}{b x^{3} + a} - 1\right )}}{b^{4}} + \frac{6 \, a^{2}{\rm ln}\left (\frac{{\left | b x^{3} + a \right |}}{{\left (b x^{3} + a\right )}^{2}{\left | b \right |}}\right )}{b^{4}} - \frac{2 \, a^{3}}{{\left (b x^{3} + a\right )} b^{4}}\right )} + \frac{1}{3} \,{\left (\frac{2 \, a{\rm ln}\left (\frac{{\left | b x^{3} + a \right |}}{{\left (b x^{3} + a\right )}^{2}{\left | b \right |}}\right )}{b^{3}} + \frac{b x^{3} + a}{b^{3}} - \frac{a^{2}}{{\left (b x^{3} + a\right )} b^{3}}\right )} e - \frac{d{\left (\frac{{\rm ln}\left (\frac{{\left | b x^{3} + a \right |}}{{\left (b x^{3} + a\right )}^{2}{\left | b \right |}}\right )}{b} - \frac{a}{{\left (b x^{3} + a\right )} b}\right )}}{3 \, b} - \frac{c}{3 \,{\left (b x^{3} + a\right )} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^2/(b*x^3 + a)^2,x, algorithm="giac")

[Out]

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