3.73 \(\int x^2 (a x+b x^3+c x^5)^2 \, dx\)

Optimal. Leaf size=54 \[ \frac{a^2 x^5}{5}+\frac{1}{9} x^9 \left (2 a c+b^2\right )+\frac{2}{7} a b x^7+\frac{2}{11} b c x^{11}+\frac{c^2 x^{13}}{13} \]

[Out]

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

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Rubi [A]  time = 0.0360609, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {1585, 1108} \[ \frac{a^2 x^5}{5}+\frac{1}{9} x^9 \left (2 a c+b^2\right )+\frac{2}{7} a b x^7+\frac{2}{11} b c x^{11}+\frac{c^2 x^{13}}{13} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a*x + b*x^3 + c*x^5)^2,x]

[Out]

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

Rule 1585

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(m +
n*p)*(a + b*x^(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, m, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] &
& PosQ[r - p]

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

Mathematica [A]  time = 0.0070494, size = 54, normalized size = 1. \[ \frac{a^2 x^5}{5}+\frac{1}{9} x^9 \left (2 a c+b^2\right )+\frac{2}{7} a b x^7+\frac{2}{11} b c x^{11}+\frac{c^2 x^{13}}{13} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a*x + b*x^3 + c*x^5)^2,x]

[Out]

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

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Maple [A]  time = 0.001, size = 45, normalized size = 0.8 \begin{align*}{\frac{{a}^{2}{x}^{5}}{5}}+{\frac{2\,ab{x}^{7}}{7}}+{\frac{ \left ( 2\,ac+{b}^{2} \right ){x}^{9}}{9}}+{\frac{2\,bc{x}^{11}}{11}}+{\frac{{c}^{2}{x}^{13}}{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(c*x^5+b*x^3+a*x)^2,x)

[Out]

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

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Maxima [A]  time = 1.11908, size = 59, normalized size = 1.09 \begin{align*} \frac{1}{13} \, c^{2} x^{13} + \frac{2}{11} \, b c x^{11} + \frac{1}{9} \,{\left (b^{2} + 2 \, a c\right )} x^{9} + \frac{2}{7} \, a b x^{7} + \frac{1}{5} \, a^{2} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(c*x^5+b*x^3+a*x)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.01494, size = 117, normalized size = 2.17 \begin{align*} \frac{1}{13} x^{13} c^{2} + \frac{2}{11} x^{11} c b + \frac{1}{9} x^{9} b^{2} + \frac{2}{9} x^{9} c a + \frac{2}{7} x^{7} b a + \frac{1}{5} x^{5} a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(c*x^5+b*x^3+a*x)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.069509, size = 51, normalized size = 0.94 \begin{align*} \frac{a^{2} x^{5}}{5} + \frac{2 a b x^{7}}{7} + \frac{2 b c x^{11}}{11} + \frac{c^{2} x^{13}}{13} + x^{9} \left (\frac{2 a c}{9} + \frac{b^{2}}{9}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(c*x**5+b*x**3+a*x)**2,x)

[Out]

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

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Giac [A]  time = 1.08607, size = 62, normalized size = 1.15 \begin{align*} \frac{1}{13} \, c^{2} x^{13} + \frac{2}{11} \, b c x^{11} + \frac{1}{9} \, b^{2} x^{9} + \frac{2}{9} \, a c x^{9} + \frac{2}{7} \, a b x^{7} + \frac{1}{5} \, a^{2} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(c*x^5+b*x^3+a*x)^2,x, algorithm="giac")

[Out]

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