3.844 \(\int \frac{\left (a+b x^2+c x^4\right )^3}{x} \, dx\)

Optimal. Leaf size=85 \[ a^3 \log (x)+\frac{3}{2} a^2 b x^2+\frac{3}{8} c x^8 \left (a c+b^2\right )+\frac{1}{6} b x^6 \left (6 a c+b^2\right )+\frac{3}{4} a x^4 \left (a c+b^2\right )+\frac{3}{10} b c^2 x^{10}+\frac{c^3 x^{12}}{12} \]

[Out]

(3*a^2*b*x^2)/2 + (3*a*(b^2 + a*c)*x^4)/4 + (b*(b^2 + 6*a*c)*x^6)/6 + (3*c*(b^2
+ a*c)*x^8)/8 + (3*b*c^2*x^10)/10 + (c^3*x^12)/12 + a^3*Log[x]

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Rubi [A]  time = 0.181391, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ a^3 \log (x)+\frac{3}{2} a^2 b x^2+\frac{3}{8} c x^8 \left (a c+b^2\right )+\frac{1}{6} b x^6 \left (6 a c+b^2\right )+\frac{3}{4} a x^4 \left (a c+b^2\right )+\frac{3}{10} b c^2 x^{10}+\frac{c^3 x^{12}}{12} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2 + c*x^4)^3/x,x]

[Out]

(3*a^2*b*x^2)/2 + (3*a*(b^2 + a*c)*x^4)/4 + (b*(b^2 + 6*a*c)*x^6)/6 + (3*c*(b^2
+ a*c)*x^8)/8 + (3*b*c^2*x^10)/10 + (c^3*x^12)/12 + a^3*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{3} \log{\left (x^{2} \right )}}{2} + \frac{3 a^{2} b x^{2}}{2} + \frac{3 a \left (a c + b^{2}\right ) \int ^{x^{2}} x\, dx}{2} + \frac{3 b c^{2} x^{10}}{10} + \frac{b x^{6} \left (6 a c + b^{2}\right )}{6} + \frac{c^{3} x^{12}}{12} + \frac{3 c x^{8} \left (a c + b^{2}\right )}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**2+a)**3/x,x)

[Out]

a**3*log(x**2)/2 + 3*a**2*b*x**2/2 + 3*a*(a*c + b**2)*Integral(x, (x, x**2))/2 +
3*b*c**2*x**10/10 + b*x**6*(6*a*c + b**2)/6 + c**3*x**12/12 + 3*c*x**8*(a*c + b
**2)/8

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Mathematica [A]  time = 0.0363254, size = 85, normalized size = 1. \[ a^3 \log (x)+\frac{3}{2} a^2 b x^2+\frac{3}{8} c x^8 \left (a c+b^2\right )+\frac{1}{6} b x^6 \left (6 a c+b^2\right )+\frac{3}{4} a x^4 \left (a c+b^2\right )+\frac{3}{10} b c^2 x^{10}+\frac{c^3 x^{12}}{12} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2 + c*x^4)^3/x,x]

[Out]

(3*a^2*b*x^2)/2 + (3*a*(b^2 + a*c)*x^4)/4 + (b*(b^2 + 6*a*c)*x^6)/6 + (3*c*(b^2
+ a*c)*x^8)/8 + (3*b*c^2*x^10)/10 + (c^3*x^12)/12 + a^3*Log[x]

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Maple [A]  time = 0.004, size = 85, normalized size = 1. \[{\frac{{c}^{3}{x}^{12}}{12}}+{\frac{3\,b{c}^{2}{x}^{10}}{10}}+{\frac{3\,{x}^{8}a{c}^{2}}{8}}+{\frac{3\,{b}^{2}c{x}^{8}}{8}}+{x}^{6}abc+{\frac{{b}^{3}{x}^{6}}{6}}+{\frac{3\,{x}^{4}{a}^{2}c}{4}}+{\frac{3\,a{x}^{4}{b}^{2}}{4}}+{\frac{3\,{a}^{2}b{x}^{2}}{2}}+{a}^{3}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2+a)^3/x,x)

[Out]

1/12*c^3*x^12+3/10*b*c^2*x^10+3/8*x^8*a*c^2+3/8*b^2*c*x^8+x^6*a*b*c+1/6*b^3*x^6+
3/4*x^4*a^2*c+3/4*a*x^4*b^2+3/2*a^2*b*x^2+a^3*ln(x)

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Maxima [A]  time = 0.686222, size = 111, normalized size = 1.31 \[ \frac{1}{12} \, c^{3} x^{12} + \frac{3}{10} \, b c^{2} x^{10} + \frac{3}{8} \,{\left (b^{2} c + a c^{2}\right )} x^{8} + \frac{1}{6} \,{\left (b^{3} + 6 \, a b c\right )} x^{6} + \frac{3}{2} \, a^{2} b x^{2} + \frac{3}{4} \,{\left (a b^{2} + a^{2} c\right )} x^{4} + \frac{1}{2} \, a^{3} \log \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^3/x,x, algorithm="maxima")

[Out]

1/12*c^3*x^12 + 3/10*b*c^2*x^10 + 3/8*(b^2*c + a*c^2)*x^8 + 1/6*(b^3 + 6*a*b*c)*
x^6 + 3/2*a^2*b*x^2 + 3/4*(a*b^2 + a^2*c)*x^4 + 1/2*a^3*log(x^2)

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Fricas [A]  time = 0.254026, size = 107, normalized size = 1.26 \[ \frac{1}{12} \, c^{3} x^{12} + \frac{3}{10} \, b c^{2} x^{10} + \frac{3}{8} \,{\left (b^{2} c + a c^{2}\right )} x^{8} + \frac{1}{6} \,{\left (b^{3} + 6 \, a b c\right )} x^{6} + \frac{3}{2} \, a^{2} b x^{2} + \frac{3}{4} \,{\left (a b^{2} + a^{2} c\right )} x^{4} + a^{3} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^3/x,x, algorithm="fricas")

[Out]

1/12*c^3*x^12 + 3/10*b*c^2*x^10 + 3/8*(b^2*c + a*c^2)*x^8 + 1/6*(b^3 + 6*a*b*c)*
x^6 + 3/2*a^2*b*x^2 + 3/4*(a*b^2 + a^2*c)*x^4 + a^3*log(x)

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Sympy [A]  time = 1.23174, size = 92, normalized size = 1.08 \[ a^{3} \log{\left (x \right )} + \frac{3 a^{2} b x^{2}}{2} + \frac{3 b c^{2} x^{10}}{10} + \frac{c^{3} x^{12}}{12} + x^{8} \left (\frac{3 a c^{2}}{8} + \frac{3 b^{2} c}{8}\right ) + x^{6} \left (a b c + \frac{b^{3}}{6}\right ) + x^{4} \left (\frac{3 a^{2} c}{4} + \frac{3 a b^{2}}{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+b*x**2+a)**3/x,x)

[Out]

a**3*log(x) + 3*a**2*b*x**2/2 + 3*b*c**2*x**10/10 + c**3*x**12/12 + x**8*(3*a*c*
*2/8 + 3*b**2*c/8) + x**6*(a*b*c + b**3/6) + x**4*(3*a**2*c/4 + 3*a*b**2/4)

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GIAC/XCAS [A]  time = 0.262871, size = 117, normalized size = 1.38 \[ \frac{1}{12} \, c^{3} x^{12} + \frac{3}{10} \, b c^{2} x^{10} + \frac{3}{8} \, b^{2} c x^{8} + \frac{3}{8} \, a c^{2} x^{8} + \frac{1}{6} \, b^{3} x^{6} + a b c x^{6} + \frac{3}{4} \, a b^{2} x^{4} + \frac{3}{4} \, a^{2} c x^{4} + \frac{3}{2} \, a^{2} b x^{2} + \frac{1}{2} \, a^{3}{\rm ln}\left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^3/x,x, algorithm="giac")

[Out]

1/12*c^3*x^12 + 3/10*b*c^2*x^10 + 3/8*b^2*c*x^8 + 3/8*a*c^2*x^8 + 1/6*b^3*x^6 +
a*b*c*x^6 + 3/4*a*b^2*x^4 + 3/4*a^2*c*x^4 + 3/2*a^2*b*x^2 + 1/2*a^3*ln(x^2)