3.252 \(\int \frac{x^8 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx\)

Optimal. Leaf size=180 \[ \frac{x^6 \left (3 a^2 f-2 a b e+b^2 d\right )}{6 b^4}-\frac{a^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^6 \left (a+b x^3\right )}-\frac{a \log \left (a+b x^3\right ) \left (-5 a^3 f+4 a^2 b e-3 a b^2 d+2 b^3 c\right )}{3 b^6}+\frac{x^3 \left (-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right )}{3 b^5}+\frac{x^9 (b e-2 a f)}{9 b^3}+\frac{f x^{12}}{12 b^2} \]

[Out]

((b^3*c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)*x^3)/(3*b^5) + ((b^2*d - 2*a*b*e + 3*
a^2*f)*x^6)/(6*b^4) + ((b*e - 2*a*f)*x^9)/(9*b^3) + (f*x^12)/(12*b^2) - (a^2*(b^
3*c - a*b^2*d + a^2*b*e - a^3*f))/(3*b^6*(a + b*x^3)) - (a*(2*b^3*c - 3*a*b^2*d
+ 4*a^2*b*e - 5*a^3*f)*Log[a + b*x^3])/(3*b^6)

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Rubi [A]  time = 0.531902, antiderivative size = 180, 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 \[ \frac{x^6 \left (3 a^2 f-2 a b e+b^2 d\right )}{6 b^4}-\frac{a^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^6 \left (a+b x^3\right )}-\frac{a \log \left (a+b x^3\right ) \left (-5 a^3 f+4 a^2 b e-3 a b^2 d+2 b^3 c\right )}{3 b^6}+\frac{x^3 \left (-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right )}{3 b^5}+\frac{x^9 (b e-2 a f)}{9 b^3}+\frac{f x^{12}}{12 b^2} \]

Antiderivative was successfully verified.

[In]  Int[(x^8*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^2,x]

[Out]

((b^3*c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)*x^3)/(3*b^5) + ((b^2*d - 2*a*b*e + 3*
a^2*f)*x^6)/(6*b^4) + ((b*e - 2*a*f)*x^9)/(9*b^3) + (f*x^12)/(12*b^2) - (a^2*(b^
3*c - a*b^2*d + a^2*b*e - a^3*f))/(3*b^6*(a + b*x^3)) - (a*(2*b^3*c - 3*a*b^2*d
+ 4*a^2*b*e - 5*a^3*f)*Log[a + b*x^3])/(3*b^6)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{2} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{3 b^{6} \left (a + b x^{3}\right )} + \frac{a \left (5 a^{3} f - 4 a^{2} b e + 3 a b^{2} d - 2 b^{3} c\right ) \log{\left (a + b x^{3} \right )}}{3 b^{6}} - \left (\frac{4 a^{3} f}{3} - a^{2} b e + \frac{2 a b^{2} d}{3} - \frac{b^{3} c}{3}\right ) \int ^{x^{3}} \frac{1}{b^{5}}\, dx + \frac{f x^{12}}{12 b^{2}} - \frac{x^{9} \left (2 a f - b e\right )}{9 b^{3}} + \frac{\left (3 a^{2} f - 2 a b e + b^{2} d\right ) \int ^{x^{3}} x\, dx}{3 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a)**2,x)

[Out]

a**2*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)/(3*b**6*(a + b*x**3)) + a*(5*a**3*f
- 4*a**2*b*e + 3*a*b**2*d - 2*b**3*c)*log(a + b*x**3)/(3*b**6) - (4*a**3*f/3 -
a**2*b*e + 2*a*b**2*d/3 - b**3*c/3)*Integral(b**(-5), (x, x**3)) + f*x**12/(12*b
**2) - x**9*(2*a*f - b*e)/(9*b**3) + (3*a**2*f - 2*a*b*e + b**2*d)*Integral(x, (
x, x**3))/(3*b**4)

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Mathematica [A]  time = 0.236282, size = 167, normalized size = 0.93 \[ \frac{6 b^2 x^6 \left (3 a^2 f-2 a b e+b^2 d\right )+12 b x^3 \left (-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right )+\frac{12 a^2 \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{a+b x^3}+12 a \log \left (a+b x^3\right ) \left (5 a^3 f-4 a^2 b e+3 a b^2 d-2 b^3 c\right )+4 b^3 x^9 (b e-2 a f)+3 b^4 f x^{12}}{36 b^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^8*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^2,x]

[Out]

(12*b*(b^3*c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)*x^3 + 6*b^2*(b^2*d - 2*a*b*e + 3
*a^2*f)*x^6 + 4*b^3*(b*e - 2*a*f)*x^9 + 3*b^4*f*x^12 + (12*a^2*(-(b^3*c) + a*b^2
*d - a^2*b*e + a^3*f))/(a + b*x^3) + 12*a*(-2*b^3*c + 3*a*b^2*d - 4*a^2*b*e + 5*
a^3*f)*Log[a + b*x^3])/(36*b^6)

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Maple [A]  time = 0.019, size = 240, normalized size = 1.3 \[{\frac{f{x}^{12}}{12\,{b}^{2}}}-{\frac{2\,{x}^{9}af}{9\,{b}^{3}}}+{\frac{{x}^{9}e}{9\,{b}^{2}}}+{\frac{{a}^{2}f{x}^{6}}{2\,{b}^{4}}}-{\frac{ae{x}^{6}}{3\,{b}^{3}}}+{\frac{d{x}^{6}}{6\,{b}^{2}}}-{\frac{4\,{a}^{3}f{x}^{3}}{3\,{b}^{5}}}+{\frac{{a}^{2}e{x}^{3}}{{b}^{4}}}-{\frac{2\,ad{x}^{3}}{3\,{b}^{3}}}+{\frac{c{x}^{3}}{3\,{b}^{2}}}+{\frac{5\,{a}^{4}\ln \left ( b{x}^{3}+a \right ) f}{3\,{b}^{6}}}-{\frac{4\,{a}^{3}\ln \left ( b{x}^{3}+a \right ) e}{3\,{b}^{5}}}+{\frac{{a}^{2}\ln \left ( b{x}^{3}+a \right ) d}{{b}^{4}}}-{\frac{2\,a\ln \left ( b{x}^{3}+a \right ) c}{3\,{b}^{3}}}+{\frac{{a}^{5}f}{3\,{b}^{6} \left ( b{x}^{3}+a \right ) }}-{\frac{{a}^{4}e}{3\,{b}^{5} \left ( b{x}^{3}+a \right ) }}+{\frac{{a}^{3}d}{3\,{b}^{4} \left ( b{x}^{3}+a \right ) }}-{\frac{{a}^{2}c}{3\,{b}^{3} \left ( b{x}^{3}+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.40935, size = 243, normalized size = 1.35 \[ -\frac{a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f}{3 \,{\left (b^{7} x^{3} + a b^{6}\right )}} + \frac{3 \, b^{3} f x^{12} + 4 \,{\left (b^{3} e - 2 \, a b^{2} f\right )} x^{9} + 6 \,{\left (b^{3} d - 2 \, a b^{2} e + 3 \, a^{2} b f\right )} x^{6} + 12 \,{\left (b^{3} c - 2 \, a b^{2} d + 3 \, a^{2} b e - 4 \, a^{3} f\right )} x^{3}}{36 \, b^{5}} - \frac{{\left (2 \, a b^{3} c - 3 \, a^{2} b^{2} d + 4 \, a^{3} b e - 5 \, a^{4} f\right )} \log \left (b x^{3} + a\right )}{3 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^8/(b*x^3 + a)^2,x, algorithm="maxima")

[Out]

-1/3*(a^2*b^3*c - a^3*b^2*d + a^4*b*e - a^5*f)/(b^7*x^3 + a*b^6) + 1/36*(3*b^3*f
*x^12 + 4*(b^3*e - 2*a*b^2*f)*x^9 + 6*(b^3*d - 2*a*b^2*e + 3*a^2*b*f)*x^6 + 12*(
b^3*c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)*x^3)/b^5 - 1/3*(2*a*b^3*c - 3*a^2*b^2*d
+ 4*a^3*b*e - 5*a^4*f)*log(b*x^3 + a)/b^6

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Fricas [A]  time = 0.204973, size = 347, normalized size = 1.93 \[ \frac{3 \, b^{5} f x^{15} +{\left (4 \, b^{5} e - 5 \, a b^{4} f\right )} x^{12} + 2 \,{\left (3 \, b^{5} d - 4 \, a b^{4} e + 5 \, a^{2} b^{3} f\right )} x^{9} + 6 \,{\left (2 \, b^{5} c - 3 \, a b^{4} d + 4 \, a^{2} b^{3} e - 5 \, a^{3} b^{2} f\right )} x^{6} - 12 \, a^{2} b^{3} c + 12 \, a^{3} b^{2} d - 12 \, a^{4} b e + 12 \, a^{5} f + 12 \,{\left (a b^{4} c - 2 \, a^{2} b^{3} d + 3 \, a^{3} b^{2} e - 4 \, a^{4} b f\right )} x^{3} - 12 \,{\left (2 \, a^{2} b^{3} c - 3 \, a^{3} b^{2} d + 4 \, a^{4} b e - 5 \, a^{5} f +{\left (2 \, a b^{4} c - 3 \, a^{2} b^{3} d + 4 \, a^{3} b^{2} e - 5 \, a^{4} b f\right )} x^{3}\right )} \log \left (b x^{3} + a\right )}{36 \,{\left (b^{7} x^{3} + a b^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^8/(b*x^3 + a)^2,x, algorithm="fricas")

[Out]

1/36*(3*b^5*f*x^15 + (4*b^5*e - 5*a*b^4*f)*x^12 + 2*(3*b^5*d - 4*a*b^4*e + 5*a^2
*b^3*f)*x^9 + 6*(2*b^5*c - 3*a*b^4*d + 4*a^2*b^3*e - 5*a^3*b^2*f)*x^6 - 12*a^2*b
^3*c + 12*a^3*b^2*d - 12*a^4*b*e + 12*a^5*f + 12*(a*b^4*c - 2*a^2*b^3*d + 3*a^3*
b^2*e - 4*a^4*b*f)*x^3 - 12*(2*a^2*b^3*c - 3*a^3*b^2*d + 4*a^4*b*e - 5*a^5*f + (
2*a*b^4*c - 3*a^2*b^3*d + 4*a^3*b^2*e - 5*a^4*b*f)*x^3)*log(b*x^3 + a))/(b^7*x^3
+ a*b^6)

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Sympy [A]  time = 19.1881, size = 180, normalized size = 1. \[ \frac{a \left (5 a^{3} f - 4 a^{2} b e + 3 a b^{2} d - 2 b^{3} c\right ) \log{\left (a + b x^{3} \right )}}{3 b^{6}} + \frac{a^{5} f - a^{4} b e + a^{3} b^{2} d - a^{2} b^{3} c}{3 a b^{6} + 3 b^{7} x^{3}} + \frac{f x^{12}}{12 b^{2}} - \frac{x^{9} \left (2 a f - b e\right )}{9 b^{3}} + \frac{x^{6} \left (3 a^{2} f - 2 a b e + b^{2} d\right )}{6 b^{4}} - \frac{x^{3} \left (4 a^{3} f - 3 a^{2} b e + 2 a b^{2} d - b^{3} c\right )}{3 b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.215384, size = 335, normalized size = 1.86 \[ -\frac{{\left (2 \, a b^{3} c - 3 \, a^{2} b^{2} d - 5 \, a^{4} f + 4 \, a^{3} b e\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{6}} + \frac{2 \, a b^{4} c x^{3} - 3 \, a^{2} b^{3} d x^{3} - 5 \, a^{4} b f x^{3} + 4 \, a^{3} b^{2} x^{3} e + a^{2} b^{3} c - 2 \, a^{3} b^{2} d - 4 \, a^{5} f + 3 \, a^{4} b e}{3 \,{\left (b x^{3} + a\right )} b^{6}} + \frac{3 \, b^{6} f x^{12} - 8 \, a b^{5} f x^{9} + 4 \, b^{6} x^{9} e + 6 \, b^{6} d x^{6} + 18 \, a^{2} b^{4} f x^{6} - 12 \, a b^{5} x^{6} e + 12 \, b^{6} c x^{3} - 24 \, a b^{5} d x^{3} - 48 \, a^{3} b^{3} f x^{3} + 36 \, a^{2} b^{4} x^{3} e}{36 \, b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^8/(b*x^3 + a)^2,x, algorithm="giac")

[Out]

-1/3*(2*a*b^3*c - 3*a^2*b^2*d - 5*a^4*f + 4*a^3*b*e)*ln(abs(b*x^3 + a))/b^6 + 1/
3*(2*a*b^4*c*x^3 - 3*a^2*b^3*d*x^3 - 5*a^4*b*f*x^3 + 4*a^3*b^2*x^3*e + a^2*b^3*c
- 2*a^3*b^2*d - 4*a^5*f + 3*a^4*b*e)/((b*x^3 + a)*b^6) + 1/36*(3*b^6*f*x^12 - 8
*a*b^5*f*x^9 + 4*b^6*x^9*e + 6*b^6*d*x^6 + 18*a^2*b^4*f*x^6 - 12*a*b^5*x^6*e + 1
2*b^6*c*x^3 - 24*a*b^5*d*x^3 - 48*a^3*b^3*f*x^3 + 36*a^2*b^4*x^3*e)/b^8