3.836 \(\int \frac{(a+b x^2+c x^4)^2}{x^9} \, dx\)

Optimal. Leaf size=48 \[ -\frac{a^2}{8 x^8}-\frac{2 a c+b^2}{4 x^4}-\frac{a b}{3 x^6}-\frac{b c}{x^2}+c^2 \log (x) \]

[Out]

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

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Rubi [A]  time = 0.0350476, antiderivative size = 48, 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} \[ -\frac{a^2}{8 x^8}-\frac{2 a c+b^2}{4 x^4}-\frac{a b}{3 x^6}-\frac{b c}{x^2}+c^2 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)^2/x^9,x]

[Out]

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

Mathematica [A]  time = 0.0265462, size = 50, normalized size = 1.04 \[ -\frac{a^2}{8 x^8}+\frac{-2 a c-b^2}{4 x^4}-\frac{a b}{3 x^6}-\frac{b c}{x^2}+c^2 \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)^2/x^9,x]

[Out]

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

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Maple [A]  time = 0.049, size = 45, normalized size = 0.9 \begin{align*}{c}^{2}\ln \left ( x \right ) -{\frac{ac}{2\,{x}^{4}}}-{\frac{{b}^{2}}{4\,{x}^{4}}}-{\frac{bc}{{x}^{2}}}-{\frac{{a}^{2}}{8\,{x}^{8}}}-{\frac{ab}{3\,{x}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2+a)^2/x^9,x)

[Out]

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

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Maxima [A]  time = 0.967694, size = 65, normalized size = 1.35 \begin{align*} \frac{1}{2} \, c^{2} \log \left (x^{2}\right ) - \frac{24 \, b c x^{6} + 6 \,{\left (b^{2} + 2 \, a c\right )} x^{4} + 8 \, a b x^{2} + 3 \, a^{2}}{24 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^2/x^9,x, algorithm="maxima")

[Out]

1/2*c^2*log(x^2) - 1/24*(24*b*c*x^6 + 6*(b^2 + 2*a*c)*x^4 + 8*a*b*x^2 + 3*a^2)/x^8

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Fricas [A]  time = 1.4253, size = 115, normalized size = 2.4 \begin{align*} \frac{24 \, c^{2} x^{8} \log \left (x\right ) - 24 \, b c x^{6} - 6 \,{\left (b^{2} + 2 \, a c\right )} x^{4} - 8 \, a b x^{2} - 3 \, a^{2}}{24 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^2/x^9,x, algorithm="fricas")

[Out]

1/24*(24*c^2*x^8*log(x) - 24*b*c*x^6 - 6*(b^2 + 2*a*c)*x^4 - 8*a*b*x^2 - 3*a^2)/x^8

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Sympy [A]  time = 1.46377, size = 46, normalized size = 0.96 \begin{align*} c^{2} \log{\left (x \right )} - \frac{3 a^{2} + 8 a b x^{2} + 24 b c x^{6} + x^{4} \left (12 a c + 6 b^{2}\right )}{24 x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2+a)**2/x**9,x)

[Out]

c**2*log(x) - (3*a**2 + 8*a*b*x**2 + 24*b*c*x**6 + x**4*(12*a*c + 6*b**2))/(24*x**8)

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Giac [A]  time = 1.13102, size = 78, normalized size = 1.62 \begin{align*} \frac{1}{2} \, c^{2} \log \left (x^{2}\right ) - \frac{25 \, c^{2} x^{8} + 24 \, b c x^{6} + 6 \, b^{2} x^{4} + 12 \, a c x^{4} + 8 \, a b x^{2} + 3 \, a^{2}}{24 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^2/x^9,x, algorithm="giac")

[Out]

1/2*c^2*log(x^2) - 1/24*(25*c^2*x^8 + 24*b*c*x^6 + 6*b^2*x^4 + 12*a*c*x^4 + 8*a*b*x^2 + 3*a^2)/x^8