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

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

[Out]

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

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Rubi [A]  time = 0.0456916, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {1585, 1107, 611} \[ \frac{a^2 x^2}{2}+\frac{1}{6} x^6 \left (2 a c+b^2\right )+\frac{1}{2} a b x^4+\frac{1}{4} b c x^8+\frac{c^2 x^{10}}{10} \]

Antiderivative was successfully verified.

[In]

Int[(a*x + b*x^3 + c*x^5)^2/x,x]

[Out]

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

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 1107

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

Rule 611

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x + c*x^2)^p, x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && (EqQ[a, 0] ||  !PerfectSquareQ[b^2 - 4*a*c])

Rubi steps

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

Mathematica [A]  time = 0.0086665, size = 48, normalized size = 0.89 \[ \frac{1}{60} x^2 \left (30 a^2+10 x^4 \left (2 a c+b^2\right )+30 a b x^2+15 b c x^6+6 c^2 x^8\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*(30*a^2 + 30*a*b*x^2 + 10*(b^2 + 2*a*c)*x^4 + 15*b*c*x^6 + 6*c^2*x^8))/60

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Maple [A]  time = 0.002, size = 45, normalized size = 0.8 \begin{align*}{\frac{{a}^{2}{x}^{2}}{2}}+{\frac{ab{x}^{4}}{2}}+{\frac{ \left ( 2\,ac+{b}^{2} \right ){x}^{6}}{6}}+{\frac{bc{x}^{8}}{4}}+{\frac{{c}^{2}{x}^{10}}{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02237, size = 59, normalized size = 1.09 \begin{align*} \frac{1}{10} \, c^{2} x^{10} + \frac{1}{4} \, b c x^{8} + \frac{1}{6} \,{\left (b^{2} + 2 \, a c\right )} x^{6} + \frac{1}{2} \, a b x^{4} + \frac{1}{2} \, a^{2} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.30918, size = 109, normalized size = 2.02 \begin{align*} \frac{1}{10} \, c^{2} x^{10} + \frac{1}{4} \, b c x^{8} + \frac{1}{6} \,{\left (b^{2} + 2 \, a c\right )} x^{6} + \frac{1}{2} \, a b x^{4} + \frac{1}{2} \, a^{2} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.069357, size = 46, normalized size = 0.85 \begin{align*} \frac{a^{2} x^{2}}{2} + \frac{a b x^{4}}{2} + \frac{b c x^{8}}{4} + \frac{c^{2} x^{10}}{10} + x^{6} \left (\frac{a c}{3} + \frac{b^{2}}{6}\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,x)

[Out]

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

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Giac [A]  time = 1.08928, size = 62, normalized size = 1.15 \begin{align*} \frac{1}{10} \, c^{2} x^{10} + \frac{1}{4} \, b c x^{8} + \frac{1}{6} \, b^{2} x^{6} + \frac{1}{3} \, a c x^{6} + \frac{1}{2} \, a b x^{4} + \frac{1}{2} \, a^{2} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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