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

Optimal. Leaf size=81 \[ a^2 b x^3+a^3 x+\frac{1}{3} c x^9 \left (a c+b^2\right )+\frac{1}{7} b x^7 \left (6 a c+b^2\right )+\frac{3}{5} a x^5 \left (a c+b^2\right )+\frac{3}{11} b c^2 x^{11}+\frac{c^3 x^{13}}{13} \]

[Out]

a^3*x + a^2*b*x^3 + (3*a*(b^2 + a*c)*x^5)/5 + (b*(b^2 + 6*a*c)*x^7)/7 + (c*(b^2 + a*c)*x^9)/3 + (3*b*c^2*x^11)
/11 + (c^3*x^13)/13

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Rubi [A]  time = 0.0447497, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {1090} \[ a^2 b x^3+a^3 x+\frac{1}{3} c x^9 \left (a c+b^2\right )+\frac{1}{7} b x^7 \left (6 a c+b^2\right )+\frac{3}{5} a x^5 \left (a c+b^2\right )+\frac{3}{11} b c^2 x^{11}+\frac{c^3 x^{13}}{13} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)^3,x]

[Out]

a^3*x + a^2*b*x^3 + (3*a*(b^2 + a*c)*x^5)/5 + (b*(b^2 + 6*a*c)*x^7)/7 + (c*(b^2 + a*c)*x^9)/3 + (3*b*c^2*x^11)
/11 + (c^3*x^13)/13

Rule 1090

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

Rubi steps

\begin{align*} \int \left (a+b x^2+c x^4\right )^3 \, dx &=\int \left (a^3+3 a^2 b x^2+3 a b^2 \left (1+\frac{a c}{b^2}\right ) x^4+b^3 \left (1+\frac{6 a c}{b^2}\right ) x^6+3 b^2 c \left (1+\frac{a c}{b^2}\right ) x^8+3 b c^2 x^{10}+c^3 x^{12}\right ) \, dx\\ &=a^3 x+a^2 b x^3+\frac{3}{5} a \left (b^2+a c\right ) x^5+\frac{1}{7} b \left (b^2+6 a c\right ) x^7+\frac{1}{3} c \left (b^2+a c\right ) x^9+\frac{3}{11} b c^2 x^{11}+\frac{c^3 x^{13}}{13}\\ \end{align*}

Mathematica [A]  time = 0.0097713, size = 81, normalized size = 1. \[ a^2 b x^3+a^3 x+\frac{1}{3} c x^9 \left (a c+b^2\right )+\frac{1}{7} b x^7 \left (6 a c+b^2\right )+\frac{3}{5} a x^5 \left (a c+b^2\right )+\frac{3}{11} b c^2 x^{11}+\frac{c^3 x^{13}}{13} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)^3,x]

[Out]

a^3*x + a^2*b*x^3 + (3*a*(b^2 + a*c)*x^5)/5 + (b*(b^2 + 6*a*c)*x^7)/7 + (c*(b^2 + a*c)*x^9)/3 + (3*b*c^2*x^11)
/11 + (c^3*x^13)/13

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Maple [A]  time = 0.043, size = 107, normalized size = 1.3 \begin{align*}{\frac{{c}^{3}{x}^{13}}{13}}+{\frac{3\,b{c}^{2}{x}^{11}}{11}}+{\frac{ \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{9}}{9}}+{\frac{ \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{7}}{7}}+{\frac{ \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+{a}^{2}c \right ){x}^{5}}{5}}+{a}^{2}b{x}^{3}+x{a}^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*c^3*x^13+3/11*b*c^2*x^11+1/9*(a*c^2+2*b^2*c+c*(2*a*c+b^2))*x^9+1/7*(4*a*b*c+b*(2*a*c+b^2))*x^7+1/5*(a*(2*
a*c+b^2)+2*b^2*a+a^2*c)*x^5+a^2*b*x^3+x*a^3

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Maxima [A]  time = 0.963325, size = 115, normalized size = 1.42 \begin{align*} \frac{1}{13} \, c^{3} x^{13} + \frac{3}{11} \, b c^{2} x^{11} + \frac{1}{3} \, b^{2} c x^{9} + \frac{1}{7} \, b^{3} x^{7} + a^{3} x + \frac{1}{5} \,{\left (3 \, c x^{5} + 5 \, b x^{3}\right )} a^{2} + \frac{1}{105} \,{\left (35 \, c^{2} x^{9} + 90 \, b c x^{7} + 63 \, b^{2} x^{5}\right )} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*c^3*x^13 + 3/11*b*c^2*x^11 + 1/3*b^2*c*x^9 + 1/7*b^3*x^7 + a^3*x + 1/5*(3*c*x^5 + 5*b*x^3)*a^2 + 1/105*(3
5*c^2*x^9 + 90*b*c*x^7 + 63*b^2*x^5)*a

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Fricas [A]  time = 1.54389, size = 198, normalized size = 2.44 \begin{align*} \frac{1}{13} x^{13} c^{3} + \frac{3}{11} x^{11} c^{2} b + \frac{1}{3} x^{9} c b^{2} + \frac{1}{3} x^{9} c^{2} a + \frac{1}{7} x^{7} b^{3} + \frac{6}{7} x^{7} c b a + \frac{3}{5} x^{5} b^{2} a + \frac{3}{5} x^{5} c a^{2} + x^{3} b a^{2} + x a^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*x^13*c^3 + 3/11*x^11*c^2*b + 1/3*x^9*c*b^2 + 1/3*x^9*c^2*a + 1/7*x^7*b^3 + 6/7*x^7*c*b*a + 3/5*x^5*b^2*a
+ 3/5*x^5*c*a^2 + x^3*b*a^2 + x*a^3

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Sympy [A]  time = 0.080683, size = 87, normalized size = 1.07 \begin{align*} a^{3} x + a^{2} b x^{3} + \frac{3 b c^{2} x^{11}}{11} + \frac{c^{3} x^{13}}{13} + x^{9} \left (\frac{a c^{2}}{3} + \frac{b^{2} c}{3}\right ) + x^{7} \left (\frac{6 a b c}{7} + \frac{b^{3}}{7}\right ) + x^{5} \left (\frac{3 a^{2} c}{5} + \frac{3 a b^{2}}{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*x + a**2*b*x**3 + 3*b*c**2*x**11/11 + c**3*x**13/13 + x**9*(a*c**2/3 + b**2*c/3) + x**7*(6*a*b*c/7 + b**3
/7) + x**5*(3*a**2*c/5 + 3*a*b**2/5)

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Giac [A]  time = 1.11074, size = 112, normalized size = 1.38 \begin{align*} \frac{1}{13} \, c^{3} x^{13} + \frac{3}{11} \, b c^{2} x^{11} + \frac{1}{3} \, b^{2} c x^{9} + \frac{1}{3} \, a c^{2} x^{9} + \frac{1}{7} \, b^{3} x^{7} + \frac{6}{7} \, a b c x^{7} + \frac{3}{5} \, a b^{2} x^{5} + \frac{3}{5} \, a^{2} c x^{5} + a^{2} b x^{3} + a^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*c^3*x^13 + 3/11*b*c^2*x^11 + 1/3*b^2*c*x^9 + 1/3*a*c^2*x^9 + 1/7*b^3*x^7 + 6/7*a*b*c*x^7 + 3/5*a*b^2*x^5
+ 3/5*a^2*c*x^5 + a^2*b*x^3 + a^3*x