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

Optimal. Leaf size=89 \[ \frac{3}{5} a^2 b x^5+\frac{a^3 x^3}{3}+\frac{3}{11} c x^{11} \left (a c+b^2\right )+\frac{1}{9} b x^9 \left (6 a c+b^2\right )+\frac{3}{7} a x^7 \left (a c+b^2\right )+\frac{3}{13} b c^2 x^{13}+\frac{c^3 x^{15}}{15} \]

[Out]

(a^3*x^3)/3 + (3*a^2*b*x^5)/5 + (3*a*(b^2 + a*c)*x^7)/7 + (b*(b^2 + 6*a*c)*x^9)/9 + (3*c*(b^2 + a*c)*x^11)/11
+ (3*b*c^2*x^13)/13 + (c^3*x^15)/15

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Rubi [A]  time = 0.0594182, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {1108} \[ \frac{3}{5} a^2 b x^5+\frac{a^3 x^3}{3}+\frac{3}{11} c x^{11} \left (a c+b^2\right )+\frac{1}{9} b x^9 \left (6 a c+b^2\right )+\frac{3}{7} a x^7 \left (a c+b^2\right )+\frac{3}{13} b c^2 x^{13}+\frac{c^3 x^{15}}{15} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x^2 + c*x^4)^3,x]

[Out]

(a^3*x^3)/3 + (3*a^2*b*x^5)/5 + (3*a*(b^2 + a*c)*x^7)/7 + (b*(b^2 + 6*a*c)*x^9)/9 + (3*c*(b^2 + a*c)*x^11)/11
+ (3*b*c^2*x^13)/13 + (c^3*x^15)/15

Rule 1108

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d*x)^m*(a
+ b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] &&  !IntegerQ[(m + 1)/2]

Rubi steps

\begin{align*} \int x^2 \left (a+b x^2+c x^4\right )^3 \, dx &=\int \left (a^3 x^2+3 a^2 b x^4+3 a \left (b^2+a c\right ) x^6+b \left (b^2+6 a c\right ) x^8+3 c \left (b^2+a c\right ) x^{10}+3 b c^2 x^{12}+c^3 x^{14}\right ) \, dx\\ &=\frac{a^3 x^3}{3}+\frac{3}{5} a^2 b x^5+\frac{3}{7} a \left (b^2+a c\right ) x^7+\frac{1}{9} b \left (b^2+6 a c\right ) x^9+\frac{3}{11} c \left (b^2+a c\right ) x^{11}+\frac{3}{13} b c^2 x^{13}+\frac{c^3 x^{15}}{15}\\ \end{align*}

Mathematica [A]  time = 0.0131496, size = 89, normalized size = 1. \[ \frac{3}{5} a^2 b x^5+\frac{a^3 x^3}{3}+\frac{3}{11} c x^{11} \left (a c+b^2\right )+\frac{1}{9} b x^9 \left (6 a c+b^2\right )+\frac{3}{7} a x^7 \left (a c+b^2\right )+\frac{3}{13} b c^2 x^{13}+\frac{c^3 x^{15}}{15} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*x^3)/3 + (3*a^2*b*x^5)/5 + (3*a*(b^2 + a*c)*x^7)/7 + (b*(b^2 + 6*a*c)*x^9)/9 + (3*c*(b^2 + a*c)*x^11)/11
+ (3*b*c^2*x^13)/13 + (c^3*x^15)/15

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Maple [A]  time = 0.041, size = 111, normalized size = 1.3 \begin{align*}{\frac{{c}^{3}{x}^{15}}{15}}+{\frac{3\,b{c}^{2}{x}^{13}}{13}}+{\frac{ \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{11}}{11}}+{\frac{ \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{9}}{9}}+{\frac{ \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+{a}^{2}c \right ){x}^{7}}{7}}+{\frac{3\,{a}^{2}b{x}^{5}}{5}}+{\frac{{a}^{3}{x}^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*c^3*x^15+3/13*b*c^2*x^13+1/11*(a*c^2+2*b^2*c+c*(2*a*c+b^2))*x^11+1/9*(4*a*b*c+b*(2*a*c+b^2))*x^9+1/7*(a*(
2*a*c+b^2)+2*b^2*a+a^2*c)*x^7+3/5*a^2*b*x^5+1/3*a^3*x^3

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Maxima [A]  time = 0.952867, size = 109, normalized size = 1.22 \begin{align*} \frac{1}{15} \, c^{3} x^{15} + \frac{3}{13} \, b c^{2} x^{13} + \frac{3}{11} \,{\left (b^{2} c + a c^{2}\right )} x^{11} + \frac{1}{9} \,{\left (b^{3} + 6 \, a b c\right )} x^{9} + \frac{3}{5} \, a^{2} b x^{5} + \frac{3}{7} \,{\left (a b^{2} + a^{2} c\right )} x^{7} + \frac{1}{3} \, a^{3} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*c^3*x^15 + 3/13*b*c^2*x^13 + 3/11*(b^2*c + a*c^2)*x^11 + 1/9*(b^3 + 6*a*b*c)*x^9 + 3/5*a^2*b*x^5 + 3/7*(a
*b^2 + a^2*c)*x^7 + 1/3*a^3*x^3

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Fricas [A]  time = 1.52763, size = 217, normalized size = 2.44 \begin{align*} \frac{1}{15} x^{15} c^{3} + \frac{3}{13} x^{13} c^{2} b + \frac{3}{11} x^{11} c b^{2} + \frac{3}{11} x^{11} c^{2} a + \frac{1}{9} x^{9} b^{3} + \frac{2}{3} x^{9} c b a + \frac{3}{7} x^{7} b^{2} a + \frac{3}{7} x^{7} c a^{2} + \frac{3}{5} x^{5} b a^{2} + \frac{1}{3} x^{3} a^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*x^15*c^3 + 3/13*x^13*c^2*b + 3/11*x^11*c*b^2 + 3/11*x^11*c^2*a + 1/9*x^9*b^3 + 2/3*x^9*c*b*a + 3/7*x^7*b^
2*a + 3/7*x^7*c*a^2 + 3/5*x^5*b*a^2 + 1/3*x^3*a^3

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Sympy [A]  time = 0.088396, size = 97, normalized size = 1.09 \begin{align*} \frac{a^{3} x^{3}}{3} + \frac{3 a^{2} b x^{5}}{5} + \frac{3 b c^{2} x^{13}}{13} + \frac{c^{3} x^{15}}{15} + x^{11} \left (\frac{3 a c^{2}}{11} + \frac{3 b^{2} c}{11}\right ) + x^{9} \left (\frac{2 a b c}{3} + \frac{b^{3}}{9}\right ) + x^{7} \left (\frac{3 a^{2} c}{7} + \frac{3 a b^{2}}{7}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*x**3/3 + 3*a**2*b*x**5/5 + 3*b*c**2*x**13/13 + c**3*x**15/15 + x**11*(3*a*c**2/11 + 3*b**2*c/11) + x**9*(
2*a*b*c/3 + b**3/9) + x**7*(3*a**2*c/7 + 3*a*b**2/7)

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Giac [A]  time = 1.11659, size = 117, normalized size = 1.31 \begin{align*} \frac{1}{15} \, c^{3} x^{15} + \frac{3}{13} \, b c^{2} x^{13} + \frac{3}{11} \, b^{2} c x^{11} + \frac{3}{11} \, a c^{2} x^{11} + \frac{1}{9} \, b^{3} x^{9} + \frac{2}{3} \, a b c x^{9} + \frac{3}{7} \, a b^{2} x^{7} + \frac{3}{7} \, a^{2} c x^{7} + \frac{3}{5} \, a^{2} b x^{5} + \frac{1}{3} \, a^{3} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*c^3*x^15 + 3/13*b*c^2*x^13 + 3/11*b^2*c*x^11 + 3/11*a*c^2*x^11 + 1/9*b^3*x^9 + 2/3*a*b*c*x^9 + 3/7*a*b^2*
x^7 + 3/7*a^2*c*x^7 + 3/5*a^2*b*x^5 + 1/3*a^3*x^3