3.828 \(\int \frac{(a+b x^2+c x^4)^2}{x} \, dx\)

Optimal. Leaf size=47 \[ a^2 \log (x)+\frac{1}{4} x^4 \left (2 a c+b^2\right )+a b x^2+\frac{1}{3} b c x^6+\frac{c^2 x^8}{8} \]

[Out]

a*b*x^2 + ((b^2 + 2*a*c)*x^4)/4 + (b*c*x^6)/3 + (c^2*x^8)/8 + a^2*Log[x]

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Rubi [A]  time = 0.0413996, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {1114, 698} \[ a^2 \log (x)+\frac{1}{4} x^4 \left (2 a c+b^2\right )+a b x^2+\frac{1}{3} b c x^6+\frac{c^2 x^8}{8} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)^2/x,x]

[Out]

a*b*x^2 + ((b^2 + 2*a*c)*x^4)/4 + (b*c*x^6)/3 + (c^2*x^8)/8 + a^2*Log[x]

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 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

Mathematica [A]  time = 0.0113947, size = 47, normalized size = 1. \[ a^2 \log (x)+\frac{1}{4} x^4 \left (2 a c+b^2\right )+a b x^2+\frac{1}{3} b c x^6+\frac{c^2 x^8}{8} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)^2/x,x]

[Out]

a*b*x^2 + ((b^2 + 2*a*c)*x^4)/4 + (b*c*x^6)/3 + (c^2*x^8)/8 + a^2*Log[x]

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Maple [A]  time = 0.045, size = 44, normalized size = 0.9 \begin{align*}{\frac{{c}^{2}{x}^{8}}{8}}+{\frac{bc{x}^{6}}{3}}+{\frac{{x}^{4}ac}{2}}+{\frac{{b}^{2}{x}^{4}}{4}}+ab{x}^{2}+{a}^{2}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.965898, size = 59, normalized size = 1.26 \begin{align*} \frac{1}{8} \, c^{2} x^{8} + \frac{1}{3} \, b c x^{6} + \frac{1}{4} \,{\left (b^{2} + 2 \, a c\right )} x^{4} + a b x^{2} + \frac{1}{2} \, a^{2} \log \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.46607, size = 100, normalized size = 2.13 \begin{align*} \frac{1}{8} \, c^{2} x^{8} + \frac{1}{3} \, b c x^{6} + \frac{1}{4} \,{\left (b^{2} + 2 \, a c\right )} x^{4} + a b x^{2} + a^{2} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.315114, size = 42, normalized size = 0.89 \begin{align*} a^{2} \log{\left (x \right )} + a b x^{2} + \frac{b c x^{6}}{3} + \frac{c^{2} x^{8}}{8} + x^{4} \left (\frac{a c}{2} + \frac{b^{2}}{4}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2+a)**2/x,x)

[Out]

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

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Giac [A]  time = 1.12332, size = 62, normalized size = 1.32 \begin{align*} \frac{1}{8} \, c^{2} x^{8} + \frac{1}{3} \, b c x^{6} + \frac{1}{4} \, b^{2} x^{4} + \frac{1}{2} \, a c x^{4} + a b x^{2} + \frac{1}{2} \, a^{2} \log \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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