3.225 \(\int \frac{x^5 \left (c+d x^3+e x^6+f x^9\right )}{a+b x^3} \, dx\)

Optimal. Leaf size=132 \[ \frac{x^6 \left (a^2 f-a b e+b^2 d\right )}{6 b^3}-\frac{a \log \left (a+b x^3\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^5}+\frac{x^3 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^4}+\frac{x^9 (b e-a f)}{9 b^2}+\frac{f x^{12}}{12 b} \]

[Out]

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

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Rubi [A]  time = 0.347144, antiderivative size = 132, 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 \[ \frac{x^6 \left (a^2 f-a b e+b^2 d\right )}{6 b^3}-\frac{a \log \left (a+b x^3\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^5}+\frac{x^3 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^4}+\frac{x^9 (b e-a f)}{9 b^2}+\frac{f x^{12}}{12 b} \]

Antiderivative was successfully verified.

[In]  Int[(x^5*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a),x)

[Out]

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

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Mathematica [A]  time = 0.10138, size = 119, normalized size = 0.9 \[ \frac{12 a \log \left (a+b x^3\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )+b x^3 \left (-12 a^3 f+6 a^2 b \left (2 e+f x^3\right )-2 a b^2 \left (6 d+3 e x^3+2 f x^6\right )+b^3 \left (12 c+6 d x^3+4 e x^6+3 f x^9\right )\right )}{36 b^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^5*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3),x]

[Out]

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

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Maple [A]  time = 0.005, size = 170, normalized size = 1.3 \[{\frac{f{x}^{12}}{12\,b}}-{\frac{{x}^{9}af}{9\,{b}^{2}}}+{\frac{{x}^{9}e}{9\,b}}+{\frac{{a}^{2}f{x}^{6}}{6\,{b}^{3}}}-{\frac{ae{x}^{6}}{6\,{b}^{2}}}+{\frac{d{x}^{6}}{6\,b}}-{\frac{{a}^{3}f{x}^{3}}{3\,{b}^{4}}}+{\frac{{a}^{2}e{x}^{3}}{3\,{b}^{3}}}-{\frac{ad{x}^{3}}{3\,{b}^{2}}}+{\frac{c{x}^{3}}{3\,b}}+{\frac{{a}^{4}\ln \left ( b{x}^{3}+a \right ) f}{3\,{b}^{5}}}-{\frac{{a}^{3}\ln \left ( b{x}^{3}+a \right ) e}{3\,{b}^{4}}}+{\frac{{a}^{2}\ln \left ( b{x}^{3}+a \right ) d}{3\,{b}^{3}}}-{\frac{a\ln \left ( b{x}^{3}+a \right ) c}{3\,{b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a),x)

[Out]

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

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Maxima [A]  time = 1.41639, size = 174, normalized size = 1.32 \[ \frac{3 \, b^{3} f x^{12} + 4 \,{\left (b^{3} e - a b^{2} f\right )} x^{9} + 6 \,{\left (b^{3} d - a b^{2} e + a^{2} b f\right )} x^{6} + 12 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{3}}{36 \, b^{4}} - \frac{{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} \log \left (b x^{3} + a\right )}{3 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^5/(b*x^3 + a),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.225766, size = 176, normalized size = 1.33 \[ \frac{3 \, b^{4} f x^{12} + 4 \,{\left (b^{4} e - a b^{3} f\right )} x^{9} + 6 \,{\left (b^{4} d - a b^{3} e + a^{2} b^{2} f\right )} x^{6} + 12 \,{\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{3} - 12 \,{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} \log \left (b x^{3} + a\right )}{36 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^5/(b*x^3 + a),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.23357, size = 117, normalized size = 0.89 \[ \frac{a \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (a + b x^{3} \right )}}{3 b^{5}} + \frac{f x^{12}}{12 b} - \frac{x^{9} \left (a f - b e\right )}{9 b^{2}} + \frac{x^{6} \left (a^{2} f - a b e + b^{2} d\right )}{6 b^{3}} - \frac{x^{3} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{3 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.213717, size = 200, normalized size = 1.52 \[ \frac{3 \, b^{3} f x^{12} - 4 \, a b^{2} f x^{9} + 4 \, b^{3} x^{9} e + 6 \, b^{3} d x^{6} + 6 \, a^{2} b f x^{6} - 6 \, a b^{2} x^{6} e + 12 \, b^{3} c x^{3} - 12 \, a b^{2} d x^{3} - 12 \, a^{3} f x^{3} + 12 \, a^{2} b x^{3} e}{36 \, b^{4}} - \frac{{\left (a b^{3} c - a^{2} b^{2} d - a^{4} f + a^{3} b e\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^5/(b*x^3 + a),x, algorithm="giac")

[Out]

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