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

Optimal. Leaf size=109 \[ -\frac{3 a^2 f-2 a b e+b^2 d}{3 b^4 \left (a+b x^3\right )}-\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{6 b^4 \left (a+b x^3\right )^2}+\frac{(b e-3 a f) \log \left (a+b x^3\right )}{3 b^4}+\frac{f x^3}{3 b^3} \]

[Out]

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

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Rubi [A]  time = 0.303931, 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 \[ -\frac{3 a^2 f-2 a b e+b^2 d}{3 b^4 \left (a+b x^3\right )}-\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{6 b^4 \left (a+b x^3\right )^2}+\frac{(b e-3 a f) \log \left (a+b x^3\right )}{3 b^4}+\frac{f x^3}{3 b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\int ^{x^{3}} f\, dx}{3 b^{3}} - \frac{\left (3 a f - b e\right ) \log{\left (a + b x^{3} \right )}}{3 b^{4}} - \frac{3 a^{2} f - 2 a b e + b^{2} d}{3 b^{4} \left (a + b x^{3}\right )} + \frac{a^{3} f - a^{2} b e + a b^{2} d - b^{3} c}{6 b^{4} \left (a + b x^{3}\right )^{2}} \]

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)**3,x)

[Out]

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

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Mathematica [A]  time = 0.101789, size = 105, normalized size = 0.96 \[ \frac{-5 a^3 f+a^2 b \left (3 e-4 f x^3\right )+a b^2 \left (-d+4 e x^3+4 f x^6\right )+2 \left (a+b x^3\right )^2 (b e-3 a f) \log \left (a+b x^3\right )-b^3 \left (c+2 d x^3-2 f x^9\right )}{6 b^4 \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.016, size = 156, normalized size = 1.4 \[{\frac{f{x}^{3}}{3\,{b}^{3}}}-{\frac{\ln \left ( b{x}^{3}+a \right ) af}{{b}^{4}}}+{\frac{\ln \left ( b{x}^{3}+a \right ) e}{3\,{b}^{3}}}+{\frac{{a}^{3}f}{6\,{b}^{4} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{{a}^{2}e}{6\,{b}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{ad}{6\,{b}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{c}{6\,b \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{{a}^{2}f}{{b}^{4} \left ( b{x}^{3}+a \right ) }}+{\frac{2\,ae}{3\,{b}^{3} \left ( b{x}^{3}+a \right ) }}-{\frac{d}{3\,{b}^{2} \left ( b{x}^{3}+a \right ) }} \]

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)^3,x)

[Out]

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

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Maxima [A]  time = 1.38247, size = 147, normalized size = 1.35 \[ \frac{f x^{3}}{3 \, b^{3}} - \frac{b^{3} c + a b^{2} d - 3 \, a^{2} b e + 5 \, a^{3} f + 2 \,{\left (b^{3} d - 2 \, a b^{2} e + 3 \, a^{2} b f\right )} x^{3}}{6 \,{\left (b^{6} x^{6} + 2 \, a b^{5} x^{3} + a^{2} b^{4}\right )}} + \frac{{\left (b e - 3 \, a f\right )} \log \left (b x^{3} + a\right )}{3 \, b^{4}} \]

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)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.202961, size = 213, normalized size = 1.95 \[ \frac{2 \, b^{3} f x^{9} + 4 \, a b^{2} f x^{6} - b^{3} c - a b^{2} d + 3 \, a^{2} b e - 5 \, a^{3} f - 2 \,{\left (b^{3} d - 2 \, a b^{2} e + 2 \, a^{2} b f\right )} x^{3} + 2 \,{\left ({\left (b^{3} e - 3 \, a b^{2} f\right )} x^{6} + a^{2} b e - 3 \, a^{3} f + 2 \,{\left (a b^{2} e - 3 \, a^{2} b f\right )} x^{3}\right )} \log \left (b x^{3} + a\right )}{6 \,{\left (b^{6} x^{6} + 2 \, a b^{5} x^{3} + a^{2} b^{4}\right )}} \]

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)^3,x, algorithm="fricas")

[Out]

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

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.215713, size = 135, normalized size = 1.24 \[ \frac{f x^{3}}{3 \, b^{3}} - \frac{{\left (3 \, a f - b e\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{4}} - \frac{b^{3} c + a b^{2} d + 5 \, a^{3} f + 2 \,{\left (b^{3} d + 3 \, a^{2} b f - 2 \, a b^{2} e\right )} x^{3} - 3 \, a^{2} b e}{6 \,{\left (b x^{3} + a\right )}^{2} b^{4}} \]

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)^3,x, algorithm="giac")

[Out]

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