3.228 \(\int \frac{c+d x^3+e x^6+f x^9}{x^4 \left (a+b x^3\right )} \, dx\)

Optimal. Leaf size=81 \[ -\frac{\log (x) (b c-a d)}{a^2}+\frac{\log \left (a+b x^3\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^2 b^2}-\frac{c}{3 a x^3}+\frac{f x^3}{3 b} \]

[Out]

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

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Rubi [A]  time = 0.210764, antiderivative size = 81, 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 \[ -\frac{\log (x) (b c-a d)}{a^2}+\frac{\log \left (a+b x^3\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^2 b^2}-\frac{c}{3 a x^3}+\frac{f x^3}{3 b} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^4*(a + b*x^3)),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((f*x**9+e*x**6+d*x**3+c)/x**4/(b*x**3+a),x)

[Out]

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

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Mathematica [A]  time = 0.0753243, size = 77, normalized size = 0.95 \[ \frac{1}{3} \left (\frac{3 \log (x) (a d-b c)}{a^2}+\frac{\log \left (a+b x^3\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a^2 b^2}-\frac{c}{a x^3}+\frac{f x^3}{b}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^4*(a + b*x^3)),x]

[Out]

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

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Maple [A]  time = 0.013, size = 94, normalized size = 1.2 \[{\frac{f{x}^{3}}{3\,b}}-{\frac{c}{3\,a{x}^{3}}}+{\frac{d\ln \left ( x \right ) }{a}}-{\frac{\ln \left ( x \right ) bc}{{a}^{2}}}-{\frac{a\ln \left ( b{x}^{3}+a \right ) f}{3\,{b}^{2}}}+{\frac{e\ln \left ( b{x}^{3}+a \right ) }{3\,b}}-{\frac{d\ln \left ( b{x}^{3}+a \right ) }{3\,a}}+{\frac{b\ln \left ( b{x}^{3}+a \right ) c}{3\,{a}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x^9+e*x^6+d*x^3+c)/x^4/(b*x^3+a),x)

[Out]

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

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Maxima [A]  time = 1.42186, size = 104, normalized size = 1.28 \[ \frac{f x^{3}}{3 \, b} - \frac{{\left (b c - a d\right )} \log \left (x^{3}\right )}{3 \, a^{2}} + \frac{{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (b x^{3} + a\right )}{3 \, a^{2} b^{2}} - \frac{c}{3 \, a x^{3}} \]

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)*x^4),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.239169, size = 115, normalized size = 1.42 \[ \frac{a^{2} b f x^{6} +{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{3} \log \left (b x^{3} + a\right ) - 3 \,{\left (b^{3} c - a b^{2} d\right )} x^{3} \log \left (x\right ) - a b^{2} c}{3 \, a^{2} b^{2} x^{3}} \]

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)*x^4),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 23.0633, size = 70, normalized size = 0.86 \[ \frac{f x^{3}}{3 b} - \frac{c}{3 a x^{3}} + \frac{\left (a d - b c\right ) \log{\left (x \right )}}{a^{2}} - \frac{\left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (\frac{a}{b} + x^{3} \right )}}{3 a^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x**9+e*x**6+d*x**3+c)/x**4/(b*x**3+a),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.213959, size = 128, normalized size = 1.58 \[ \frac{f x^{3}}{3 \, b} - \frac{{\left (b c - a d\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{2}} + \frac{{\left (b^{3} c - a b^{2} d - a^{3} f + a^{2} b e\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{2} b^{2}} + \frac{b c x^{3} - a d x^{3} - a c}{3 \, a^{2} x^{3}} \]

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)*x^4),x, algorithm="giac")

[Out]

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