3.279 \(\int \frac{x^5 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx\)

Optimal. Leaf size=146 \[ \frac{\log \left (a+b x^3\right ) \left (6 a^2 f-3 a b e+b^2 d\right )}{3 b^5}-\frac{-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c}{3 b^5 \left (a+b x^3\right )}+\frac{a \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^5 \left (a+b x^3\right )^2}+\frac{x^3 (b e-3 a f)}{3 b^4}+\frac{f x^6}{6 b^3} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.387975, antiderivative size = 146, 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 \[ \frac{\log \left (a+b x^3\right ) \left (6 a^2 f-3 a b e+b^2 d\right )}{3 b^5}-\frac{-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c}{3 b^5 \left (a+b x^3\right )}+\frac{a \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^5 \left (a+b x^3\right )^2}+\frac{x^3 (b e-3 a f)}{3 b^4}+\frac{f x^6}{6 b^3} \]

Antiderivative was successfully verified.

[In]  Int[(x^5*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{6 b^{5} \left (a + b x^{3}\right )^{2}} - \left (a f - \frac{b e}{3}\right ) \int ^{x^{3}} \frac{1}{b^{4}}\, dx + \frac{f \int ^{x^{3}} x\, dx}{3 b^{3}} + \frac{\left (6 a^{2} f - 3 a b e + b^{2} d\right ) \log{\left (a + b x^{3} \right )}}{3 b^{5}} + \frac{4 a^{3} f - 3 a^{2} b e + 2 a b^{2} d - b^{3} c}{3 b^{5} \left (a + b x^{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a)**3,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.135377, size = 145, normalized size = 0.99 \[ \frac{7 a^4 f+a^3 b \left (2 f x^3-5 e\right )+2 \left (a+b x^3\right )^2 \log \left (a+b x^3\right ) \left (6 a^2 f-3 a b e+b^2 d\right )+a^2 b^2 \left (3 d-4 e x^3-11 f x^6\right )-a b^3 \left (c-4 x^3 \left (d+e x^3-f x^6\right )\right )+b^4 x^3 \left (-2 c+2 e x^6+f x^9\right )}{6 b^5 \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^5*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]

[Out]

(7*a^4*f + a^3*b*(-5*e + 2*f*x^3) + a^2*b^2*(3*d - 4*e*x^3 - 11*f*x^6) + b^4*x^3
*(-2*c + 2*e*x^6 + f*x^9) - a*b^3*(c - 4*x^3*(d + e*x^3 - f*x^6)) + 2*(b^2*d - 3
*a*b*e + 6*a^2*f)*(a + b*x^3)^2*Log[a + b*x^3])/(6*b^5*(a + b*x^3)^2)

_______________________________________________________________________________________

Maple [A]  time = 0.017, size = 213, normalized size = 1.5 \[{\frac{f{x}^{6}}{6\,{b}^{3}}}-{\frac{a{x}^{3}f}{{b}^{4}}}+{\frac{{x}^{3}e}{3\,{b}^{3}}}+2\,{\frac{\ln \left ( b{x}^{3}+a \right ){a}^{2}f}{{b}^{5}}}-{\frac{\ln \left ( b{x}^{3}+a \right ) ae}{{b}^{4}}}+{\frac{\ln \left ( b{x}^{3}+a \right ) d}{3\,{b}^{3}}}-{\frac{{a}^{4}f}{6\,{b}^{5} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{{a}^{3}e}{6\,{b}^{4} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{{a}^{2}d}{6\,{b}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{ac}{6\,{b}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{4\,{a}^{3}f}{3\,{b}^{5} \left ( b{x}^{3}+a \right ) }}-{\frac{{a}^{2}e}{{b}^{4} \left ( b{x}^{3}+a \right ) }}+{\frac{2\,ad}{3\,{b}^{3} \left ( b{x}^{3}+a \right ) }}-{\frac{c}{3\,{b}^{2} \left ( b{x}^{3}+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.38214, size = 198, normalized size = 1.36 \[ -\frac{a b^{3} c - 3 \, a^{2} b^{2} d + 5 \, a^{3} b e - 7 \, a^{4} f + 2 \,{\left (b^{4} c - 2 \, a b^{3} d + 3 \, a^{2} b^{2} e - 4 \, a^{3} b f\right )} x^{3}}{6 \,{\left (b^{7} x^{6} + 2 \, a b^{6} x^{3} + a^{2} b^{5}\right )}} + \frac{b f x^{6} + 2 \,{\left (b e - 3 \, a f\right )} x^{3}}{6 \, b^{4}} + \frac{{\left (b^{2} d - 3 \, a b e + 6 \, a^{2} f\right )} \log \left (b x^{3} + a\right )}{3 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^5/(b*x^3 + a)^3,x, algorithm="maxima")

[Out]

-1/6*(a*b^3*c - 3*a^2*b^2*d + 5*a^3*b*e - 7*a^4*f + 2*(b^4*c - 2*a*b^3*d + 3*a^2
*b^2*e - 4*a^3*b*f)*x^3)/(b^7*x^6 + 2*a*b^6*x^3 + a^2*b^5) + 1/6*(b*f*x^6 + 2*(b
*e - 3*a*f)*x^3)/b^4 + 1/3*(b^2*d - 3*a*b*e + 6*a^2*f)*log(b*x^3 + a)/b^5

_______________________________________________________________________________________

Fricas [A]  time = 0.20742, size = 304, normalized size = 2.08 \[ \frac{b^{4} f x^{12} + 2 \,{\left (b^{4} e - 2 \, a b^{3} f\right )} x^{9} +{\left (4 \, a b^{3} e - 11 \, a^{2} b^{2} f\right )} x^{6} - a b^{3} c + 3 \, a^{2} b^{2} d - 5 \, a^{3} b e + 7 \, a^{4} f - 2 \,{\left (b^{4} c - 2 \, a b^{3} d + 2 \, a^{2} b^{2} e - a^{3} b f\right )} x^{3} + 2 \,{\left ({\left (b^{4} d - 3 \, a b^{3} e + 6 \, a^{2} b^{2} f\right )} x^{6} + a^{2} b^{2} d - 3 \, a^{3} b e + 6 \, a^{4} f + 2 \,{\left (a b^{3} d - 3 \, a^{2} b^{2} e + 6 \, a^{3} b f\right )} x^{3}\right )} \log \left (b x^{3} + a\right )}{6 \,{\left (b^{7} x^{6} + 2 \, a b^{6} x^{3} + a^{2} b^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^5/(b*x^3 + a)^3,x, algorithm="fricas")

[Out]

1/6*(b^4*f*x^12 + 2*(b^4*e - 2*a*b^3*f)*x^9 + (4*a*b^3*e - 11*a^2*b^2*f)*x^6 - a
*b^3*c + 3*a^2*b^2*d - 5*a^3*b*e + 7*a^4*f - 2*(b^4*c - 2*a*b^3*d + 2*a^2*b^2*e
- a^3*b*f)*x^3 + 2*((b^4*d - 3*a*b^3*e + 6*a^2*b^2*f)*x^6 + a^2*b^2*d - 3*a^3*b*
e + 6*a^4*f + 2*(a*b^3*d - 3*a^2*b^2*e + 6*a^3*b*f)*x^3)*log(b*x^3 + a))/(b^7*x^
6 + 2*a*b^6*x^3 + a^2*b^5)

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.21479, size = 197, normalized size = 1.35 \[ \frac{{\left (b^{2} d + 6 \, a^{2} f - 3 \, a b e\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{5}} + \frac{b^{3} f x^{6} - 6 \, a b^{2} f x^{3} + 2 \, b^{3} x^{3} e}{6 \, b^{6}} - \frac{a b^{3} c - 3 \, a^{2} b^{2} d - 7 \, a^{4} f + 5 \, a^{3} b e + 2 \,{\left (b^{4} c - 2 \, a b^{3} d - 4 \, a^{3} b f + 3 \, a^{2} b^{2} e\right )} x^{3}}{6 \,{\left (b x^{3} + a\right )}^{2} b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^5/(b*x^3 + a)^3,x, algorithm="giac")

[Out]

1/3*(b^2*d + 6*a^2*f - 3*a*b*e)*ln(abs(b*x^3 + a))/b^5 + 1/6*(b^3*f*x^6 - 6*a*b^
2*f*x^3 + 2*b^3*x^3*e)/b^6 - 1/6*(a*b^3*c - 3*a^2*b^2*d - 7*a^4*f + 5*a^3*b*e +
2*(b^4*c - 2*a*b^3*d - 4*a^3*b*f + 3*a^2*b^2*e)*x^3)/((b*x^3 + a)^2*b^5)