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

Optimal. Leaf size=95 \[ \frac{b c-a d}{3 a^2 x^3}+\frac{\log (x) \left (a^2 e-a b d+b^2 c\right )}{a^3}-\frac{\log \left (a+b x^3\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^3 b}-\frac{c}{6 a x^6} \]

[Out]

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

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Rubi [A]  time = 0.243478, antiderivative size = 95, 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 \[ \frac{b c-a d}{3 a^2 x^3}+\frac{\log (x) \left (a^2 e-a b d+b^2 c\right )}{a^3}-\frac{\log \left (a+b x^3\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^3 b}-\frac{c}{6 a x^6} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 44.4511, size = 87, normalized size = 0.92 \[ - \frac{c}{6 a x^{6}} - \frac{a d - b c}{3 a^{2} x^{3}} + \frac{\left (a^{2} e - a b d + b^{2} c\right ) \log{\left (x^{3} \right )}}{3 a^{3}} + \frac{\left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (a + b x^{3} \right )}}{3 a^{3} b} \]

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),x)

[Out]

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

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Mathematica [A]  time = 0.138683, size = 88, normalized size = 0.93 \[ \frac{6 \log (x) \left (a^2 e-a b d+b^2 c\right )+\log \left (a+b x^3\right ) \left (\frac{2 a^3 f}{b}-2 a^2 e+2 a b d-2 b^2 c\right )-\frac{a \left (a c+2 a d x^3-2 b c x^3\right )}{x^6}}{6 a^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 116, normalized size = 1.2 \[ -{\frac{c}{6\,a{x}^{6}}}-{\frac{d}{3\,a{x}^{3}}}+{\frac{bc}{3\,{a}^{2}{x}^{3}}}+{\frac{e\ln \left ( x \right ) }{a}}-{\frac{\ln \left ( x \right ) bd}{{a}^{2}}}+{\frac{\ln \left ( x \right ){b}^{2}c}{{a}^{3}}}+{\frac{\ln \left ( b{x}^{3}+a \right ) f}{3\,b}}-{\frac{e\ln \left ( b{x}^{3}+a \right ) }{3\,a}}+{\frac{b\ln \left ( b{x}^{3}+a \right ) d}{3\,{a}^{2}}}-{\frac{{b}^{2}\ln \left ( b{x}^{3}+a \right ) c}{3\,{a}^{3}}} \]

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),x)

[Out]

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

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Maxima [A]  time = 1.38527, size = 126, normalized size = 1.33 \[ \frac{{\left (b^{2} c - a b d + a^{2} e\right )} \log \left (x^{3}\right )}{3 \, a^{3}} - \frac{{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (b x^{3} + a\right )}{3 \, a^{3} b} + \frac{2 \,{\left (b c - a d\right )} x^{3} - a c}{6 \, a^{2} x^{6}} \]

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^7),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.248078, size = 136, normalized size = 1.43 \[ -\frac{2 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{6} \log \left (b x^{3} + a\right ) - 6 \,{\left (b^{3} c - a b^{2} d + a^{2} b e\right )} x^{6} \log \left (x\right ) + a^{2} b c - 2 \,{\left (a b^{2} c - a^{2} b d\right )} x^{3}}{6 \, a^{3} b x^{6}} \]

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^7),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 92.4216, size = 85, normalized size = 0.89 \[ - \frac{a c + x^{3} \left (2 a d - 2 b c\right )}{6 a^{2} x^{6}} + \frac{\left (a^{2} e - a b d + b^{2} c\right ) \log{\left (x \right )}}{a^{3}} + \frac{\left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (\frac{a}{b} + x^{3} \right )}}{3 a^{3} b} \]

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),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.215596, size = 170, normalized size = 1.79 \[ \frac{{\left (b^{2} c - a b d + a^{2} e\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{3}} - \frac{{\left (b^{3} c - a b^{2} d - a^{3} f + a^{2} b e\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{3} b} - \frac{3 \, b^{2} c x^{6} - 3 \, a b d x^{6} + 3 \, a^{2} x^{6} e - 2 \, a b c x^{3} + 2 \, a^{2} d x^{3} + a^{2} c}{6 \, a^{3} x^{6}} \]

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^7),x, algorithm="giac")

[Out]

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