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

Optimal. Leaf size=54 \[ \frac{a^2 x^4}{4}+\frac{1}{8} x^8 \left (2 a c+b^2\right )+\frac{1}{3} a b x^6+\frac{1}{5} b c x^{10}+\frac{c^2 x^{12}}{12} \]

[Out]

(a^2*x^4)/4 + (a*b*x^6)/3 + ((b^2 + 2*a*c)*x^8)/8 + (b*c*x^10)/5 + (c^2*x^12)/12

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Rubi [A]  time = 0.0544296, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1585, 1114, 631} \[ \frac{a^2 x^4}{4}+\frac{1}{8} x^8 \left (2 a c+b^2\right )+\frac{1}{3} a b x^6+\frac{1}{5} b c x^{10}+\frac{c^2 x^{12}}{12} \]

Antiderivative was successfully verified.

[In]

Int[x*(a*x + b*x^3 + c*x^5)^2,x]

[Out]

(a^2*x^4)/4 + (a*b*x^6)/3 + ((b^2 + 2*a*c)*x^8)/8 + (b*c*x^10)/5 + (c^2*x^12)/12

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 1114

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

Rule 631

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)
*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0]
|| EqQ[a, 0])

Rubi steps

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

Mathematica [A]  time = 0.0092113, size = 48, normalized size = 0.89 \[ \frac{1}{120} x^4 \left (30 a^2+15 x^4 \left (2 a c+b^2\right )+40 a b x^2+24 b c x^6+10 c^2 x^8\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^4*(30*a^2 + 40*a*b*x^2 + 15*(b^2 + 2*a*c)*x^4 + 24*b*c*x^6 + 10*c^2*x^8))/120

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Maple [A]  time = 0.001, size = 45, normalized size = 0.8 \begin{align*}{\frac{{a}^{2}{x}^{4}}{4}}+{\frac{ab{x}^{6}}{3}}+{\frac{ \left ( 2\,ac+{b}^{2} \right ){x}^{8}}{8}}+{\frac{bc{x}^{10}}{5}}+{\frac{{c}^{2}{x}^{12}}{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a^2*x^4+1/3*a*b*x^6+1/8*(2*a*c+b^2)*x^8+1/5*b*c*x^10+1/12*c^2*x^12

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Maxima [A]  time = 1.0896, size = 59, normalized size = 1.09 \begin{align*} \frac{1}{12} \, c^{2} x^{12} + \frac{1}{5} \, b c x^{10} + \frac{1}{8} \,{\left (b^{2} + 2 \, a c\right )} x^{8} + \frac{1}{3} \, a b x^{6} + \frac{1}{4} \, a^{2} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*c^2*x^12 + 1/5*b*c*x^10 + 1/8*(b^2 + 2*a*c)*x^8 + 1/3*a*b*x^6 + 1/4*a^2*x^4

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Fricas [A]  time = 1.1233, size = 116, normalized size = 2.15 \begin{align*} \frac{1}{12} x^{12} c^{2} + \frac{1}{5} x^{10} c b + \frac{1}{8} x^{8} b^{2} + \frac{1}{4} x^{8} c a + \frac{1}{3} x^{6} b a + \frac{1}{4} x^{4} a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*x^12*c^2 + 1/5*x^10*c*b + 1/8*x^8*b^2 + 1/4*x^8*c*a + 1/3*x^6*b*a + 1/4*x^4*a^2

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Sympy [A]  time = 0.069945, size = 46, normalized size = 0.85 \begin{align*} \frac{a^{2} x^{4}}{4} + \frac{a b x^{6}}{3} + \frac{b c x^{10}}{5} + \frac{c^{2} x^{12}}{12} + x^{8} \left (\frac{a c}{4} + \frac{b^{2}}{8}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*x**4/4 + a*b*x**6/3 + b*c*x**10/5 + c**2*x**12/12 + x**8*(a*c/4 + b**2/8)

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Giac [A]  time = 1.08953, size = 62, normalized size = 1.15 \begin{align*} \frac{1}{12} \, c^{2} x^{12} + \frac{1}{5} \, b c x^{10} + \frac{1}{8} \, b^{2} x^{8} + \frac{1}{4} \, a c x^{8} + \frac{1}{3} \, a b x^{6} + \frac{1}{4} \, a^{2} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*c^2*x^12 + 1/5*b*c*x^10 + 1/8*b^2*x^8 + 1/4*a*c*x^8 + 1/3*a*b*x^6 + 1/4*a^2*x^4