3.283 \(\int \frac{c+d x^3+e x^6+f x^9}{x^7 \left (a+b x^3\right )^3} \, dx\)

Optimal. Leaf size=163 \[ \frac{3 b c-a d}{3 a^4 x^3}-\frac{c}{6 a^3 x^6}-\frac{\log \left (a+b x^3\right ) \left (a^2 e-3 a b d+6 b^2 c\right )}{3 a^5}+\frac{\log (x) \left (a^2 e-3 a b d+6 b^2 c\right )}{a^5}+\frac{a^2 e-2 a b d+3 b^2 c}{3 a^4 \left (a+b x^3\right )}+\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{6 a^3 b \left (a+b x^3\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.402378, antiderivative size = 163, 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 \[ \frac{3 b c-a d}{3 a^4 x^3}-\frac{c}{6 a^3 x^6}-\frac{\log \left (a+b x^3\right ) \left (a^2 e-3 a b d+6 b^2 c\right )}{3 a^5}+\frac{\log (x) \left (a^2 e-3 a b d+6 b^2 c\right )}{a^5}+\frac{a^2 e-2 a b d+3 b^2 c}{3 a^4 \left (a+b x^3\right )}+\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{6 a^3 b \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^7*(a + b*x^3)^3),x]

[Out]

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

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Rubi in Sympy [A]  time = 61.9752, size = 156, normalized size = 0.96 \[ - \frac{c}{6 a^{3} x^{6}} - \frac{a^{3} f - a^{2} b e + a b^{2} d - b^{3} c}{6 a^{3} b \left (a + b x^{3}\right )^{2}} + \frac{a^{2} e - 2 a b d + 3 b^{2} c}{3 a^{4} \left (a + b x^{3}\right )} - \frac{a d - 3 b c}{3 a^{4} x^{3}} + \frac{\left (a^{2} e - 3 a b d + 6 b^{2} c\right ) \log{\left (x^{3} \right )}}{3 a^{5}} - \frac{\left (a^{2} e - 3 a b d + 6 b^{2} c\right ) \log{\left (a + b x^{3} \right )}}{3 a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((f*x**9+e*x**6+d*x**3+c)/x**7/(b*x**3+a)**3,x)

[Out]

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

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Mathematica [A]  time = 0.212299, size = 149, normalized size = 0.91 \[ \frac{\frac{2 a \left (a^2 e-2 a b d+3 b^2 c\right )}{a+b x^3}-2 \log \left (a+b x^3\right ) \left (a^2 e-3 a b d+6 b^2 c\right )+6 \log (x) \left (a^2 e-3 a b d+6 b^2 c\right )-\frac{a^2 c}{x^6}+\frac{a^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{b \left (a+b x^3\right )^2}-\frac{2 a (a d-3 b c)}{x^3}}{6 a^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^7*(a + b*x^3)^3),x]

[Out]

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

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Maple [A]  time = 0.025, size = 213, normalized size = 1.3 \[ -{\frac{c}{6\,{a}^{3}{x}^{6}}}-{\frac{d}{3\,{a}^{3}{x}^{3}}}+{\frac{bc}{{a}^{4}{x}^{3}}}+{\frac{e\ln \left ( x \right ) }{{a}^{3}}}-3\,{\frac{\ln \left ( x \right ) bd}{{a}^{4}}}+6\,{\frac{\ln \left ( x \right ){b}^{2}c}{{a}^{5}}}-{\frac{e\ln \left ( b{x}^{3}+a \right ) }{3\,{a}^{3}}}+{\frac{\ln \left ( b{x}^{3}+a \right ) bd}{{a}^{4}}}-2\,{\frac{\ln \left ( b{x}^{3}+a \right ){b}^{2}c}{{a}^{5}}}-{\frac{f}{6\,b \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{e}{6\,a \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{bd}{6\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{{b}^{2}c}{6\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{e}{3\,{a}^{2} \left ( b{x}^{3}+a \right ) }}-{\frac{2\,bd}{3\,{a}^{3} \left ( b{x}^{3}+a \right ) }}+{\frac{{b}^{2}c}{{a}^{4} \left ( b{x}^{3}+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x^9+e*x^6+d*x^3+c)/x^7/(b*x^3+a)^3,x)

[Out]

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

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Maxima [A]  time = 1.38375, size = 246, normalized size = 1.51 \[ \frac{2 \,{\left (6 \, b^{4} c - 3 \, a b^{3} d + a^{2} b^{2} e\right )} x^{9} +{\left (18 \, a b^{3} c - 9 \, a^{2} b^{2} d + 3 \, a^{3} b e - a^{4} f\right )} x^{6} - a^{3} b c + 2 \,{\left (2 \, a^{2} b^{2} c - a^{3} b d\right )} x^{3}}{6 \,{\left (a^{4} b^{3} x^{12} + 2 \, a^{5} b^{2} x^{9} + a^{6} b x^{6}\right )}} - \frac{{\left (6 \, b^{2} c - 3 \, a b d + a^{2} e\right )} \log \left (b x^{3} + a\right )}{3 \, a^{5}} + \frac{{\left (6 \, b^{2} c - 3 \, a b d + a^{2} e\right )} \log \left (x^{3}\right )}{3 \, a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)/((b*x^3 + a)^3*x^7),x, algorithm="maxima")

[Out]

1/6*(2*(6*b^4*c - 3*a*b^3*d + a^2*b^2*e)*x^9 + (18*a*b^3*c - 9*a^2*b^2*d + 3*a^3
*b*e - a^4*f)*x^6 - a^3*b*c + 2*(2*a^2*b^2*c - a^3*b*d)*x^3)/(a^4*b^3*x^12 + 2*a
^5*b^2*x^9 + a^6*b*x^6) - 1/3*(6*b^2*c - 3*a*b*d + a^2*e)*log(b*x^3 + a)/a^5 + 1
/3*(6*b^2*c - 3*a*b*d + a^2*e)*log(x^3)/a^5

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Fricas [A]  time = 0.279899, size = 427, normalized size = 2.62 \[ \frac{2 \,{\left (6 \, a b^{4} c - 3 \, a^{2} b^{3} d + a^{3} b^{2} e\right )} x^{9} +{\left (18 \, a^{2} b^{3} c - 9 \, a^{3} b^{2} d + 3 \, a^{4} b e - a^{5} f\right )} x^{6} - a^{4} b c + 2 \,{\left (2 \, a^{3} b^{2} c - a^{4} b d\right )} x^{3} - 2 \,{\left ({\left (6 \, b^{5} c - 3 \, a b^{4} d + a^{2} b^{3} e\right )} x^{12} + 2 \,{\left (6 \, a b^{4} c - 3 \, a^{2} b^{3} d + a^{3} b^{2} e\right )} x^{9} +{\left (6 \, a^{2} b^{3} c - 3 \, a^{3} b^{2} d + a^{4} b e\right )} x^{6}\right )} \log \left (b x^{3} + a\right ) + 6 \,{\left ({\left (6 \, b^{5} c - 3 \, a b^{4} d + a^{2} b^{3} e\right )} x^{12} + 2 \,{\left (6 \, a b^{4} c - 3 \, a^{2} b^{3} d + a^{3} b^{2} e\right )} x^{9} +{\left (6 \, a^{2} b^{3} c - 3 \, a^{3} b^{2} d + a^{4} b e\right )} x^{6}\right )} \log \left (x\right )}{6 \,{\left (a^{5} b^{3} x^{12} + 2 \, a^{6} b^{2} x^{9} + a^{7} b x^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)/((b*x^3 + a)^3*x^7),x, algorithm="fricas")

[Out]

1/6*(2*(6*a*b^4*c - 3*a^2*b^3*d + a^3*b^2*e)*x^9 + (18*a^2*b^3*c - 9*a^3*b^2*d +
3*a^4*b*e - a^5*f)*x^6 - a^4*b*c + 2*(2*a^3*b^2*c - a^4*b*d)*x^3 - 2*((6*b^5*c
- 3*a*b^4*d + a^2*b^3*e)*x^12 + 2*(6*a*b^4*c - 3*a^2*b^3*d + a^3*b^2*e)*x^9 + (6
*a^2*b^3*c - 3*a^3*b^2*d + a^4*b*e)*x^6)*log(b*x^3 + a) + 6*((6*b^5*c - 3*a*b^4*
d + a^2*b^3*e)*x^12 + 2*(6*a*b^4*c - 3*a^2*b^3*d + a^3*b^2*e)*x^9 + (6*a^2*b^3*c
- 3*a^3*b^2*d + a^4*b*e)*x^6)*log(x))/(a^5*b^3*x^12 + 2*a^6*b^2*x^9 + a^7*b*x^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((f*x**9+e*x**6+d*x**3+c)/x**7/(b*x**3+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.216499, size = 255, normalized size = 1.56 \[ \frac{{\left (6 \, b^{2} c - 3 \, a b d + a^{2} e\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{5}} - \frac{{\left (6 \, b^{3} c - 3 \, a b^{2} d + a^{2} b e\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{5} b} + \frac{12 \, b^{4} c x^{9} - 6 \, a b^{3} d x^{9} + 2 \, a^{2} b^{2} x^{9} e + 18 \, a b^{3} c x^{6} - 9 \, a^{2} b^{2} d x^{6} - a^{4} f x^{6} + 3 \, a^{3} b x^{6} e + 4 \, a^{2} b^{2} c x^{3} - 2 \, a^{3} b d x^{3} - a^{3} b c}{6 \,{\left (b x^{6} + a x^{3}\right )}^{2} a^{4} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)/((b*x^3 + a)^3*x^7),x, algorithm="giac")

[Out]

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