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

Optimal. Leaf size=54 \[ \frac{a^2 x^3}{3}+\frac{1}{7} x^7 \left (2 a c+b^2\right )+\frac{2}{5} a b x^5+\frac{2}{9} b c x^9+\frac{c^2 x^{11}}{11} \]

[Out]

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

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Rubi [A]  time = 0.0301173, antiderivative size = 54, 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} \[ \frac{a^2 x^3}{3}+\frac{1}{7} x^7 \left (2 a c+b^2\right )+\frac{2}{5} a b x^5+\frac{2}{9} b c x^9+\frac{c^2 x^{11}}{11} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x^2 + c*x^4)^2,x]

[Out]

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

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

Mathematica [A]  time = 0.0079476, size = 54, normalized size = 1. \[ \frac{a^2 x^3}{3}+\frac{1}{7} x^7 \left (2 a c+b^2\right )+\frac{2}{5} a b x^5+\frac{2}{9} b c x^9+\frac{c^2 x^{11}}{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.044, size = 45, normalized size = 0.8 \begin{align*}{\frac{{a}^{2}{x}^{3}}{3}}+{\frac{2\,ab{x}^{5}}{5}}+{\frac{ \left ( 2\,ac+{b}^{2} \right ){x}^{7}}{7}}+{\frac{2\,bc{x}^{9}}{9}}+{\frac{{c}^{2}{x}^{11}}{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.956985, size = 59, normalized size = 1.09 \begin{align*} \frac{1}{11} \, c^{2} x^{11} + \frac{2}{9} \, b c x^{9} + \frac{1}{7} \,{\left (b^{2} + 2 \, a c\right )} x^{7} + \frac{2}{5} \, a b x^{5} + \frac{1}{3} \, a^{2} 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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.28016, size = 115, normalized size = 2.13 \begin{align*} \frac{1}{11} x^{11} c^{2} + \frac{2}{9} x^{9} c b + \frac{1}{7} x^{7} b^{2} + \frac{2}{7} x^{7} c a + \frac{2}{5} x^{5} b a + \frac{1}{3} x^{3} a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.073569, size = 51, normalized size = 0.94 \begin{align*} \frac{a^{2} x^{3}}{3} + \frac{2 a b x^{5}}{5} + \frac{2 b c x^{9}}{9} + \frac{c^{2} x^{11}}{11} + x^{7} \left (\frac{2 a c}{7} + \frac{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)**2,x)

[Out]

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

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Giac [A]  time = 1.28443, size = 62, normalized size = 1.15 \begin{align*} \frac{1}{11} \, c^{2} x^{11} + \frac{2}{9} \, b c x^{9} + \frac{1}{7} \, b^{2} x^{7} + \frac{2}{7} \, a c x^{7} + \frac{2}{5} \, a b x^{5} + \frac{1}{3} \, a^{2} 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)^2,x, algorithm="giac")

[Out]

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