3.842 \(\int x \left (a+b x^2+c x^4\right )^3 \, dx\)

Optimal. Leaf size=89 \[ \frac{a^3 x^2}{2}+\frac{3}{4} a^2 b x^4+\frac{3}{10} c x^{10} \left (a c+b^2\right )+\frac{1}{8} b x^8 \left (6 a c+b^2\right )+\frac{1}{2} a x^6 \left (a c+b^2\right )+\frac{1}{4} b c^2 x^{12}+\frac{c^3 x^{14}}{14} \]

[Out]

(a^3*x^2)/2 + (3*a^2*b*x^4)/4 + (a*(b^2 + a*c)*x^6)/2 + (b*(b^2 + 6*a*c)*x^8)/8
+ (3*c*(b^2 + a*c)*x^10)/10 + (b*c^2*x^12)/4 + (c^3*x^14)/14

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Rubi [A]  time = 0.192922, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{a^3 x^2}{2}+\frac{3}{4} a^2 b x^4+\frac{3}{10} c x^{10} \left (a c+b^2\right )+\frac{1}{8} b x^8 \left (6 a c+b^2\right )+\frac{1}{2} a x^6 \left (a c+b^2\right )+\frac{1}{4} b c^2 x^{12}+\frac{c^3 x^{14}}{14} \]

Antiderivative was successfully verified.

[In]  Int[x*(a + b*x^2 + c*x^4)^3,x]

[Out]

(a^3*x^2)/2 + (3*a^2*b*x^4)/4 + (a*(b^2 + a*c)*x^6)/2 + (b*(b^2 + 6*a*c)*x^8)/8
+ (3*c*(b^2 + a*c)*x^10)/10 + (b*c^2*x^12)/4 + (c^3*x^14)/14

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{3 a^{2} b \int ^{x^{2}} x\, dx}{2} + \frac{a x^{6} \left (a c + b^{2}\right )}{2} + \frac{b c^{2} x^{12}}{4} + \frac{b x^{8} \left (6 a c + b^{2}\right )}{8} + \frac{c^{3} x^{14}}{14} + \frac{3 c x^{10} \left (a c + b^{2}\right )}{10} + \frac{\int ^{x^{2}} a^{3}\, dx}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*a**2*b*Integral(x, (x, x**2))/2 + a*x**6*(a*c + b**2)/2 + b*c**2*x**12/4 + b*x
**8*(6*a*c + b**2)/8 + c**3*x**14/14 + 3*c*x**10*(a*c + b**2)/10 + Integral(a**3
, (x, x**2))/2

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Mathematica [A]  time = 0.0291661, size = 79, normalized size = 0.89 \[ \frac{1}{280} x^2 \left (140 a^3+210 a^2 b x^2+84 c x^8 \left (a c+b^2\right )+35 b x^6 \left (6 a c+b^2\right )+140 a x^4 \left (a c+b^2\right )+70 b c^2 x^{10}+20 c^3 x^{12}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x^2*(140*a^3 + 210*a^2*b*x^2 + 140*a*(b^2 + a*c)*x^4 + 35*b*(b^2 + 6*a*c)*x^6 +
84*c*(b^2 + a*c)*x^8 + 70*b*c^2*x^10 + 20*c^3*x^12))/280

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Maple [A]  time = 0.001, size = 111, normalized size = 1.3 \[{\frac{{c}^{3}{x}^{14}}{14}}+{\frac{b{c}^{2}{x}^{12}}{4}}+{\frac{ \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{10}}{10}}+{\frac{ \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{8}}{8}}+{\frac{ \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,a{b}^{2}+{a}^{2}c \right ){x}^{6}}{6}}+{\frac{3\,{a}^{2}b{x}^{4}}{4}}+{\frac{{a}^{3}{x}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(c*x^4+b*x^2+a)^3,x)

[Out]

1/14*c^3*x^14+1/4*b*c^2*x^12+1/10*(a*c^2+2*b^2*c+c*(2*a*c+b^2))*x^10+1/8*(4*a*b*
c+b*(2*a*c+b^2))*x^8+1/6*(a*(2*a*c+b^2)+2*a*b^2+a^2*c)*x^6+3/4*a^2*b*x^4+1/2*a^3
*x^2

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Maxima [A]  time = 0.692803, size = 109, normalized size = 1.22 \[ \frac{1}{14} \, c^{3} x^{14} + \frac{1}{4} \, b c^{2} x^{12} + \frac{3}{10} \,{\left (b^{2} c + a c^{2}\right )} x^{10} + \frac{1}{8} \,{\left (b^{3} + 6 \, a b c\right )} x^{8} + \frac{3}{4} \, a^{2} b x^{4} + \frac{1}{2} \,{\left (a b^{2} + a^{2} c\right )} x^{6} + \frac{1}{2} \, a^{3} x^{2} \]

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]

1/14*c^3*x^14 + 1/4*b*c^2*x^12 + 3/10*(b^2*c + a*c^2)*x^10 + 1/8*(b^3 + 6*a*b*c)
*x^8 + 3/4*a^2*b*x^4 + 1/2*(a*b^2 + a^2*c)*x^6 + 1/2*a^3*x^2

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Fricas [A]  time = 0.232941, size = 1, normalized size = 0.01 \[ \frac{1}{14} x^{14} c^{3} + \frac{1}{4} x^{12} c^{2} b + \frac{3}{10} x^{10} c b^{2} + \frac{3}{10} x^{10} c^{2} a + \frac{1}{8} x^{8} b^{3} + \frac{3}{4} x^{8} c b a + \frac{1}{2} x^{6} b^{2} a + \frac{1}{2} x^{6} c a^{2} + \frac{3}{4} x^{4} b a^{2} + \frac{1}{2} x^{2} a^{3} \]

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]

1/14*x^14*c^3 + 1/4*x^12*c^2*b + 3/10*x^10*c*b^2 + 3/10*x^10*c^2*a + 1/8*x^8*b^3
+ 3/4*x^8*c*b*a + 1/2*x^6*b^2*a + 1/2*x^6*c*a^2 + 3/4*x^4*b*a^2 + 1/2*x^2*a^3

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Sympy [A]  time = 0.133142, size = 92, normalized size = 1.03 \[ \frac{a^{3} x^{2}}{2} + \frac{3 a^{2} b x^{4}}{4} + \frac{b c^{2} x^{12}}{4} + \frac{c^{3} x^{14}}{14} + x^{10} \left (\frac{3 a c^{2}}{10} + \frac{3 b^{2} c}{10}\right ) + x^{8} \left (\frac{3 a b c}{4} + \frac{b^{3}}{8}\right ) + x^{6} \left (\frac{a^{2} c}{2} + \frac{a b^{2}}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**3*x**2/2 + 3*a**2*b*x**4/4 + b*c**2*x**12/4 + c**3*x**14/14 + x**10*(3*a*c**2
/10 + 3*b**2*c/10) + x**8*(3*a*b*c/4 + b**3/8) + x**6*(a**2*c/2 + a*b**2/2)

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GIAC/XCAS [A]  time = 0.262272, size = 117, normalized size = 1.31 \[ \frac{1}{14} \, c^{3} x^{14} + \frac{1}{4} \, b c^{2} x^{12} + \frac{3}{10} \, b^{2} c x^{10} + \frac{3}{10} \, a c^{2} x^{10} + \frac{1}{8} \, b^{3} x^{8} + \frac{3}{4} \, a b c x^{8} + \frac{1}{2} \, a b^{2} x^{6} + \frac{1}{2} \, a^{2} c x^{6} + \frac{3}{4} \, a^{2} b x^{4} + \frac{1}{2} \, a^{3} x^{2} \]

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]

1/14*c^3*x^14 + 1/4*b*c^2*x^12 + 3/10*b^2*c*x^10 + 3/10*a*c^2*x^10 + 1/8*b^3*x^8
+ 3/4*a*b*c*x^8 + 1/2*a*b^2*x^6 + 1/2*a^2*c*x^6 + 3/4*a^2*b*x^4 + 1/2*a^3*x^2