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

Optimal. Leaf size=186 \[ \frac{x^3 \left (6 a^2 f-3 a b e+b^2 d\right )}{3 b^5}+\frac{a \left (-5 a^3 f+4 a^2 b e-3 a b^2 d+2 b^3 c\right )}{3 b^6 \left (a+b x^3\right )}-\frac{a^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^6 \left (a+b x^3\right )^2}+\frac{\log \left (a+b x^3\right ) \left (-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right )}{3 b^6}+\frac{x^6 (b e-3 a f)}{6 b^4}+\frac{f x^9}{9 b^3} \]

[Out]

((b^2*d - 3*a*b*e + 6*a^2*f)*x^3)/(3*b^5) + ((b*e - 3*a*f)*x^6)/(6*b^4) + (f*x^9
)/(9*b^3) - (a^2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f))/(6*b^6*(a + b*x^3)^2) + (a
*(2*b^3*c - 3*a*b^2*d + 4*a^2*b*e - 5*a^3*f))/(3*b^6*(a + b*x^3)) + ((b^3*c - 3*
a*b^2*d + 6*a^2*b*e - 10*a^3*f)*Log[a + b*x^3])/(3*b^6)

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Rubi [A]  time = 0.549245, antiderivative size = 186, 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 \[ \frac{x^3 \left (6 a^2 f-3 a b e+b^2 d\right )}{3 b^5}+\frac{a \left (-5 a^3 f+4 a^2 b e-3 a b^2 d+2 b^3 c\right )}{3 b^6 \left (a+b x^3\right )}-\frac{a^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^6 \left (a+b x^3\right )^2}+\frac{\log \left (a+b x^3\right ) \left (-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right )}{3 b^6}+\frac{x^6 (b e-3 a f)}{6 b^4}+\frac{f x^9}{9 b^3} \]

Antiderivative was successfully verified.

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

[Out]

((b^2*d - 3*a*b*e + 6*a^2*f)*x^3)/(3*b^5) + ((b*e - 3*a*f)*x^6)/(6*b^4) + (f*x^9
)/(9*b^3) - (a^2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f))/(6*b^6*(a + b*x^3)^2) + (a
*(2*b^3*c - 3*a*b^2*d + 4*a^2*b*e - 5*a^3*f))/(3*b^6*(a + b*x^3)) + ((b^3*c - 3*
a*b^2*d + 6*a^2*b*e - 10*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. \[ \frac{a^{2} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{6 b^{6} \left (a + b x^{3}\right )^{2}} - \frac{a \left (5 a^{3} f - 4 a^{2} b e + 3 a b^{2} d - 2 b^{3} c\right )}{3 b^{6} \left (a + b x^{3}\right )} + \left (2 a^{2} f - a b e + \frac{b^{2} d}{3}\right ) \int ^{x^{3}} \frac{1}{b^{5}}\, dx + \frac{f x^{9}}{9 b^{3}} - \frac{\left (3 a f - b e\right ) \int ^{x^{3}} x\, dx}{3 b^{4}} - \frac{\left (10 a^{3} f - 6 a^{2} b e + 3 a b^{2} d - b^{3} c\right ) \log{\left (a + b x^{3} \right )}}{3 b^{6}} \]

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)**3,x)

[Out]

a**2*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)/(6*b**6*(a + b*x**3)**2) - a*(5*a**
3*f - 4*a**2*b*e + 3*a*b**2*d - 2*b**3*c)/(3*b**6*(a + b*x**3)) + (2*a**2*f - a*
b*e + b**2*d/3)*Integral(b**(-5), (x, x**3)) + f*x**9/(9*b**3) - (3*a*f - b*e)*I
ntegral(x, (x, x**3))/(3*b**4) - (10*a**3*f - 6*a**2*b*e + 3*a*b**2*d - b**3*c)*
log(a + b*x**3)/(3*b**6)

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Mathematica [A]  time = 0.189258, size = 170, normalized size = 0.91 \[ \frac{6 b x^3 \left (6 a^2 f-3 a b e+b^2 d\right )-\frac{6 a \left (5 a^3 f-4 a^2 b e+3 a b^2 d-2 b^3 c\right )}{a+b x^3}+\frac{3 a^2 \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{\left (a+b x^3\right )^2}+6 \log \left (a+b x^3\right ) \left (-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right )+3 b^2 x^6 (b e-3 a f)+2 b^3 f x^9}{18 b^6} \]

Antiderivative was successfully verified.

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

[Out]

(6*b*(b^2*d - 3*a*b*e + 6*a^2*f)*x^3 + 3*b^2*(b*e - 3*a*f)*x^6 + 2*b^3*f*x^9 + (
3*a^2*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f))/(a + b*x^3)^2 - (6*a*(-2*b^3*c + 3
*a*b^2*d - 4*a^2*b*e + 5*a^3*f))/(a + b*x^3) + 6*(b^3*c - 3*a*b^2*d + 6*a^2*b*e
- 10*a^3*f)*Log[a + b*x^3])/(18*b^6)

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Maple [A]  time = 0.02, size = 266, normalized size = 1.4 \[{\frac{f{x}^{9}}{9\,{b}^{3}}}-{\frac{{x}^{6}af}{2\,{b}^{4}}}+{\frac{e{x}^{6}}{6\,{b}^{3}}}+2\,{\frac{{a}^{2}f{x}^{3}}{{b}^{5}}}-{\frac{ae{x}^{3}}{{b}^{4}}}+{\frac{d{x}^{3}}{3\,{b}^{3}}}-{\frac{10\,\ln \left ( b{x}^{3}+a \right ){a}^{3}f}{3\,{b}^{6}}}+2\,{\frac{\ln \left ( b{x}^{3}+a \right ){a}^{2}e}{{b}^{5}}}-{\frac{\ln \left ( b{x}^{3}+a \right ) ad}{{b}^{4}}}+{\frac{\ln \left ( b{x}^{3}+a \right ) c}{3\,{b}^{3}}}+{\frac{{a}^{5}f}{6\,{b}^{6} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{{a}^{4}e}{6\,{b}^{5} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{{a}^{3}d}{6\,{b}^{4} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{{a}^{2}c}{6\,{b}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{5\,{a}^{4}f}{3\,{b}^{6} \left ( b{x}^{3}+a \right ) }}+{\frac{4\,{a}^{3}e}{3\,{b}^{5} \left ( b{x}^{3}+a \right ) }}-{\frac{{a}^{2}d}{{b}^{4} \left ( b{x}^{3}+a \right ) }}+{\frac{2\,ac}{3\,{b}^{3} \left ( b{x}^{3}+a \right ) }} \]

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)^3,x)

[Out]

1/9*f*x^9/b^3-1/2/b^4*x^6*a*f+1/6/b^3*x^6*e+2/b^5*a^2*f*x^3-1/b^4*a*e*x^3+1/3/b^
3*d*x^3-10/3/b^6*ln(b*x^3+a)*a^3*f+2/b^5*ln(b*x^3+a)*a^2*e-1/b^4*ln(b*x^3+a)*a*d
+1/3/b^3*ln(b*x^3+a)*c+1/6/b^6*a^5/(b*x^3+a)^2*f-1/6/b^5*a^4/(b*x^3+a)^2*e+1/6/b
^4*a^3/(b*x^3+a)^2*d-1/6/b^3*a^2/(b*x^3+a)^2*c-5/3/b^6*a^4/(b*x^3+a)*f+4/3/b^5*a
^3/(b*x^3+a)*e-1/b^4*a^2/(b*x^3+a)*d+2/3/b^3*a/(b*x^3+a)*c

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Maxima [A]  time = 1.44621, size = 258, normalized size = 1.39 \[ \frac{3 \, a^{2} b^{3} c - 5 \, a^{3} b^{2} d + 7 \, a^{4} b e - 9 \, a^{5} f + 2 \,{\left (2 \, a b^{4} c - 3 \, a^{2} b^{3} d + 4 \, a^{3} b^{2} e - 5 \, a^{4} b f\right )} x^{3}}{6 \,{\left (b^{8} x^{6} + 2 \, a b^{7} x^{3} + a^{2} b^{6}\right )}} + \frac{2 \, b^{2} f x^{9} + 3 \,{\left (b^{2} e - 3 \, a b f\right )} x^{6} + 6 \,{\left (b^{2} d - 3 \, a b e + 6 \, a^{2} f\right )} x^{3}}{18 \, b^{5}} + \frac{{\left (b^{3} c - 3 \, a b^{2} d + 6 \, a^{2} b e - 10 \, a^{3} f\right )} \log \left (b x^{3} + a\right )}{3 \, b^{6}} \]

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)^3,x, algorithm="maxima")

[Out]

1/6*(3*a^2*b^3*c - 5*a^3*b^2*d + 7*a^4*b*e - 9*a^5*f + 2*(2*a*b^4*c - 3*a^2*b^3*
d + 4*a^3*b^2*e - 5*a^4*b*f)*x^3)/(b^8*x^6 + 2*a*b^7*x^3 + a^2*b^6) + 1/18*(2*b^
2*f*x^9 + 3*(b^2*e - 3*a*b*f)*x^6 + 6*(b^2*d - 3*a*b*e + 6*a^2*f)*x^3)/b^5 + 1/3
*(b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*log(b*x^3 + a)/b^6

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Fricas [A]  time = 0.206585, size = 398, normalized size = 2.14 \[ \frac{2 \, b^{5} f x^{15} +{\left (3 \, b^{5} e - 5 \, a b^{4} f\right )} x^{12} + 2 \,{\left (3 \, b^{5} d - 6 \, a b^{4} e + 10 \, a^{2} b^{3} f\right )} x^{9} + 3 \,{\left (4 \, a b^{4} d - 11 \, a^{2} b^{3} e + 21 \, a^{3} b^{2} f\right )} x^{6} + 9 \, a^{2} b^{3} c - 15 \, a^{3} b^{2} d + 21 \, a^{4} b e - 27 \, a^{5} f + 6 \,{\left (2 \, a b^{4} c - 2 \, a^{2} b^{3} d + a^{3} b^{2} e + a^{4} b f\right )} x^{3} + 6 \,{\left ({\left (b^{5} c - 3 \, a b^{4} d + 6 \, a^{2} b^{3} e - 10 \, a^{3} b^{2} f\right )} x^{6} + a^{2} b^{3} c - 3 \, a^{3} b^{2} d + 6 \, a^{4} b e - 10 \, a^{5} f + 2 \,{\left (a b^{4} c - 3 \, a^{2} b^{3} d + 6 \, a^{3} b^{2} e - 10 \, a^{4} b f\right )} x^{3}\right )} \log \left (b x^{3} + a\right )}{18 \,{\left (b^{8} x^{6} + 2 \, a b^{7} x^{3} + a^{2} b^{6}\right )}} \]

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)^3,x, algorithm="fricas")

[Out]

1/18*(2*b^5*f*x^15 + (3*b^5*e - 5*a*b^4*f)*x^12 + 2*(3*b^5*d - 6*a*b^4*e + 10*a^
2*b^3*f)*x^9 + 3*(4*a*b^4*d - 11*a^2*b^3*e + 21*a^3*b^2*f)*x^6 + 9*a^2*b^3*c - 1
5*a^3*b^2*d + 21*a^4*b*e - 27*a^5*f + 6*(2*a*b^4*c - 2*a^2*b^3*d + a^3*b^2*e + a
^4*b*f)*x^3 + 6*((b^5*c - 3*a*b^4*d + 6*a^2*b^3*e - 10*a^3*b^2*f)*x^6 + a^2*b^3*
c - 3*a^3*b^2*d + 6*a^4*b*e - 10*a^5*f + 2*(a*b^4*c - 3*a^2*b^3*d + 6*a^3*b^2*e
- 10*a^4*b*f)*x^3)*log(b*x^3 + a))/(b^8*x^6 + 2*a*b^7*x^3 + a^2*b^6)

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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**8*(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 = 0.218015, size = 319, normalized size = 1.72 \[ \frac{{\left (b^{3} c - 3 \, a b^{2} d - 10 \, a^{3} f + 6 \, a^{2} b e\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{6}} - \frac{3 \, b^{5} c x^{6} - 9 \, a b^{4} d x^{6} - 30 \, a^{3} b^{2} f x^{6} + 18 \, a^{2} b^{3} x^{6} e + 2 \, a b^{4} c x^{3} - 12 \, a^{2} b^{3} d x^{3} - 50 \, a^{4} b f x^{3} + 28 \, a^{3} b^{2} x^{3} e - 4 \, a^{3} b^{2} d - 21 \, a^{5} f + 11 \, a^{4} b e}{6 \,{\left (b x^{3} + a\right )}^{2} b^{6}} + \frac{2 \, b^{6} f x^{9} - 9 \, a b^{5} f x^{6} + 3 \, b^{6} x^{6} e + 6 \, b^{6} d x^{3} + 36 \, a^{2} b^{4} f x^{3} - 18 \, a b^{5} x^{3} e}{18 \, b^{9}} \]

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)^3,x, algorithm="giac")

[Out]

1/3*(b^3*c - 3*a*b^2*d - 10*a^3*f + 6*a^2*b*e)*ln(abs(b*x^3 + a))/b^6 - 1/6*(3*b
^5*c*x^6 - 9*a*b^4*d*x^6 - 30*a^3*b^2*f*x^6 + 18*a^2*b^3*x^6*e + 2*a*b^4*c*x^3 -
12*a^2*b^3*d*x^3 - 50*a^4*b*f*x^3 + 28*a^3*b^2*x^3*e - 4*a^3*b^2*d - 21*a^5*f +
11*a^4*b*e)/((b*x^3 + a)^2*b^6) + 1/18*(2*b^6*f*x^9 - 9*a*b^5*f*x^6 + 3*b^6*x^6
*e + 6*b^6*d*x^3 + 36*a^2*b^4*f*x^3 - 18*a*b^5*x^3*e)/b^9