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

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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.08, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1107, 611} \[ \frac {3}{4} a^2 b x^4+\frac {a^3 x^2}{2}+\frac {3}{10} c x^{10} \left (a c+b^2\right )+\frac {1}{8} b x^8 \left (6 a c+b^2\right )+\frac {1}{2} a x^6 \left (a c+b^2\right )+\frac {1}{4} b c^2 x^{12}+\frac {c^3 x^{14}}{14} \]

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

Rule 611

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x + c*x^2)^p, x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && (EqQ[a, 0] ||  !PerfectSquareQ[b^2 - 4*a*c])

Rule 1107

Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(a + b*x + c*x^2)^p, x],
x, x^2], x] /; FreeQ[{a, b, c, p}, x]

Rubi steps

\begin {align*} \int x \left (a+b x^2+c x^4\right )^3 \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \left (a+b x+c x^2\right )^3 \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (a^3+3 a^2 b x+3 a b^2 \left (1+\frac {a c}{b^2}\right ) x^2+b^3 \left (1+\frac {6 a c}{b^2}\right ) x^3+3 b^2 c \left (1+\frac {a c}{b^2}\right ) x^4+3 b c^2 x^5+c^3 x^6\right ) \, dx,x,x^2\right )\\ &=\frac {a^3 x^2}{2}+\frac {3}{4} a^2 b x^4+\frac {1}{2} a \left (b^2+a c\right ) x^6+\frac {1}{8} b \left (b^2+6 a c\right ) x^8+\frac {3}{10} c \left (b^2+a c\right ) x^{10}+\frac {1}{4} b c^2 x^{12}+\frac {c^3 x^{14}}{14}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, 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

________________________________________________________________________________________

fricas [A]  time = 0.97, size = 87, normalized size = 0.98 \[ \frac {1}{14} x^{14} c^{3} + \frac {1}{4} x^{12} c^{2} b + \frac {3}{10} x^{10} c b^{2} + \frac {3}{10} x^{10} c^{2} a + \frac {1}{8} x^{8} b^{3} + \frac {3}{4} x^{8} c b a + \frac {1}{2} x^{6} b^{2} a + \frac {1}{2} x^{6} c a^{2} + \frac {3}{4} x^{4} b a^{2} + \frac {1}{2} x^{2} a^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c*x^4+b*x^2+a)^3,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

________________________________________________________________________________________

giac [A]  time = 0.17, size = 87, normalized size = 0.98 \[ \frac {1}{14} \, c^{3} x^{14} + \frac {1}{4} \, b c^{2} x^{12} + \frac {3}{10} \, b^{2} c x^{10} + \frac {3}{10} \, a c^{2} x^{10} + \frac {1}{8} \, b^{3} x^{8} + \frac {3}{4} \, a b c x^{8} + \frac {1}{2} \, a b^{2} x^{6} + \frac {1}{2} \, a^{2} c x^{6} + \frac {3}{4} \, a^{2} b x^{4} + \frac {1}{2} \, a^{3} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c*x^4+b*x^2+a)^3,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

________________________________________________________________________________________

maple [A]  time = 0.00, size = 111, normalized size = 1.25 \[ \frac {c^{3} x^{14}}{14}+\frac {b \,c^{2} x^{12}}{4}+\frac {\left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) x^{10}}{10}+\frac {\left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) x^{8}}{8}+\frac {3 a^{2} b \,x^{4}}{4}+\frac {\left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) x^{6}}{6}+\frac {a^{3} x^{2}}{2} \]

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

________________________________________________________________________________________

maxima [A]  time = 1.37, size = 81, normalized size = 0.91 \[ \frac {1}{14} \, c^{3} x^{14} + \frac {1}{4} \, b c^{2} x^{12} + \frac {3}{10} \, {\left (b^{2} c + a c^{2}\right )} x^{10} + \frac {1}{8} \, {\left (b^{3} + 6 \, a b c\right )} x^{8} + \frac {3}{4} \, a^{2} b x^{4} + \frac {1}{2} \, {\left (a b^{2} + a^{2} c\right )} x^{6} + \frac {1}{2} \, a^{3} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c*x^4+b*x^2+a)^3,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

________________________________________________________________________________________

mupad [B]  time = 0.03, size = 76, normalized size = 0.85 \[ x^8\,\left (\frac {b^3}{8}+\frac {3\,a\,c\,b}{4}\right )+\frac {a^3\,x^2}{2}+\frac {c^3\,x^{14}}{14}+\frac {3\,a^2\,b\,x^4}{4}+\frac {b\,c^2\,x^{12}}{4}+\frac {a\,x^6\,\left (b^2+a\,c\right )}{2}+\frac {3\,c\,x^{10}\,\left (b^2+a\,c\right )}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*x^2 + c*x^4)^3,x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.10, 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)

________________________________________________________________________________________