3.77 \(\int \frac{(a x+b x^3+c x^5)^2}{x^2} \, dx\)

Optimal. Leaf size=49 \[ a^2 x+\frac{1}{5} x^5 \left (2 a c+b^2\right )+\frac{2}{3} a b x^3+\frac{2}{7} b c x^7+\frac{c^2 x^9}{9} \]

[Out]

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

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Rubi [A]  time = 0.0272706, antiderivative size = 49, 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, 1090} \[ a^2 x+\frac{1}{5} x^5 \left (2 a c+b^2\right )+\frac{2}{3} a b x^3+\frac{2}{7} b c x^7+\frac{c^2 x^9}{9} \]

Antiderivative was successfully verified.

[In]

Int[(a*x + b*x^3 + c*x^5)^2/x^2,x]

[Out]

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

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

Mathematica [A]  time = 0.0053894, size = 49, normalized size = 1. \[ a^2 x+\frac{1}{5} x^5 \left (2 a c+b^2\right )+\frac{2}{3} a b x^3+\frac{2}{7} b c x^7+\frac{c^2 x^9}{9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.001, size = 42, normalized size = 0.9 \begin{align*} x{a}^{2}+{\frac{2\,ab{x}^{3}}{3}}+{\frac{ \left ( 2\,ac+{b}^{2} \right ){x}^{5}}{5}}+{\frac{2\,bc{x}^{7}}{7}}+{\frac{{c}^{2}{x}^{9}}{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.12796, size = 55, normalized size = 1.12 \begin{align*} \frac{1}{9} \, c^{2} x^{9} + \frac{2}{7} \, b c x^{7} + \frac{1}{5} \,{\left (b^{2} + 2 \, a c\right )} x^{5} + \frac{2}{3} \, a b x^{3} + a^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.09929, size = 99, normalized size = 2.02 \begin{align*} \frac{1}{9} \, c^{2} x^{9} + \frac{2}{7} \, b c x^{7} + \frac{1}{5} \,{\left (b^{2} + 2 \, a c\right )} x^{5} + \frac{2}{3} \, a b x^{3} + a^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.072256, size = 48, normalized size = 0.98 \begin{align*} a^{2} x + \frac{2 a b x^{3}}{3} + \frac{2 b c x^{7}}{7} + \frac{c^{2} x^{9}}{9} + x^{5} \left (\frac{2 a c}{5} + \frac{b^{2}}{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.07832, size = 58, normalized size = 1.18 \begin{align*} \frac{1}{9} \, c^{2} x^{9} + \frac{2}{7} \, b c x^{7} + \frac{1}{5} \, b^{2} x^{5} + \frac{2}{5} \, a c x^{5} + \frac{2}{3} \, a b x^{3} + a^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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