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

Optimal. Leaf size=86 \[ -\frac{a^3}{2 x^2}+3 a^2 b \log (x)+\frac{1}{2} c x^6 \left (a c+b^2\right )+\frac{1}{4} b x^4 \left (6 a c+b^2\right )+\frac{3}{2} a x^2 \left (a c+b^2\right )+\frac{3}{8} b c^2 x^8+\frac{c^3 x^{10}}{10} \]

[Out]

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

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Rubi [A]  time = 0.201523, antiderivative size = 86, 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 \[ -\frac{a^3}{2 x^2}+3 a^2 b \log (x)+\frac{1}{2} c x^6 \left (a c+b^2\right )+\frac{1}{4} b x^4 \left (6 a c+b^2\right )+\frac{3}{2} a x^2 \left (a c+b^2\right )+\frac{3}{8} b c^2 x^8+\frac{c^3 x^{10}}{10} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0569752, size = 78, normalized size = 0.91 \[ \frac{1}{40} \left (-\frac{20 a^3}{x^2}+120 a^2 b \log (x)+20 c x^6 \left (a c+b^2\right )+10 b x^4 \left (6 a c+b^2\right )+60 a x^2 \left (a c+b^2\right )+15 b c^2 x^8+4 c^3 x^{10}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-20*a^3)/x^2 + 60*a*(b^2 + a*c)*x^2 + 10*b*(b^2 + 6*a*c)*x^4 + 20*c*(b^2 + a*c
)*x^6 + 15*b*c^2*x^8 + 4*c^3*x^10 + 120*a^2*b*Log[x])/40

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Maple [A]  time = 0.009, size = 87, normalized size = 1. \[{\frac{{c}^{3}{x}^{10}}{10}}+{\frac{3\,b{c}^{2}{x}^{8}}{8}}+{\frac{{x}^{6}a{c}^{2}}{2}}+{\frac{{b}^{2}c{x}^{6}}{2}}+{\frac{3\,{x}^{4}abc}{2}}+{\frac{{b}^{3}{x}^{4}}{4}}+{\frac{3\,{x}^{2}{a}^{2}c}{2}}+{\frac{3\,a{b}^{2}{x}^{2}}{2}}+3\,{a}^{2}b\ln \left ( x \right ) -{\frac{{a}^{3}}{2\,{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2+a)^3/x^3,x)

[Out]

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

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Maxima [A]  time = 0.69536, size = 111, normalized size = 1.29 \[ \frac{1}{10} \, c^{3} x^{10} + \frac{3}{8} \, b c^{2} x^{8} + \frac{1}{2} \,{\left (b^{2} c + a c^{2}\right )} x^{6} + \frac{1}{4} \,{\left (b^{3} + 6 \, a b c\right )} x^{4} + \frac{3}{2} \, a^{2} b \log \left (x^{2}\right ) + \frac{3}{2} \,{\left (a b^{2} + a^{2} c\right )} x^{2} - \frac{a^{3}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.253357, size = 115, normalized size = 1.34 \[ \frac{4 \, c^{3} x^{12} + 15 \, b c^{2} x^{10} + 20 \,{\left (b^{2} c + a c^{2}\right )} x^{8} + 10 \,{\left (b^{3} + 6 \, a b c\right )} x^{6} + 120 \, a^{2} b x^{2} \log \left (x\right ) + 60 \,{\left (a b^{2} + a^{2} c\right )} x^{4} - 20 \, a^{3}}{40 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/40*(4*c^3*x^12 + 15*b*c^2*x^10 + 20*(b^2*c + a*c^2)*x^8 + 10*(b^3 + 6*a*b*c)*x
^6 + 120*a^2*b*x^2*log(x) + 60*(a*b^2 + a^2*c)*x^4 - 20*a^3)/x^2

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Sympy [A]  time = 1.38243, size = 92, normalized size = 1.07 \[ - \frac{a^{3}}{2 x^{2}} + 3 a^{2} b \log{\left (x \right )} + \frac{3 b c^{2} x^{8}}{8} + \frac{c^{3} x^{10}}{10} + x^{6} \left (\frac{a c^{2}}{2} + \frac{b^{2} c}{2}\right ) + x^{4} \left (\frac{3 a b c}{2} + \frac{b^{3}}{4}\right ) + x^{2} \left (\frac{3 a^{2} c}{2} + \frac{3 a b^{2}}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.264958, size = 132, normalized size = 1.53 \[ \frac{1}{10} \, c^{3} x^{10} + \frac{3}{8} \, b c^{2} x^{8} + \frac{1}{2} \, b^{2} c x^{6} + \frac{1}{2} \, a c^{2} x^{6} + \frac{1}{4} \, b^{3} x^{4} + \frac{3}{2} \, a b c x^{4} + \frac{3}{2} \, a b^{2} x^{2} + \frac{3}{2} \, a^{2} c x^{2} + \frac{3}{2} \, a^{2} b{\rm ln}\left (x^{2}\right ) - \frac{3 \, a^{2} b x^{2} + a^{3}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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