3.840 \(\int \frac{(a+b x^2+c x^4)^2}{x^{13}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0353243, antiderivative size = 54, 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} \[ -\frac{a^2}{12 x^{12}}-\frac{2 a c+b^2}{8 x^8}-\frac{a b}{5 x^{10}}-\frac{b c}{3 x^6}-\frac{c^2}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)^2/x^13,x]

[Out]

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

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^{13}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\left (a+b x+c x^2\right )^2}{x^7} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a^2}{x^7}+\frac{2 a b}{x^6}+\frac{b^2+2 a c}{x^5}+\frac{2 b c}{x^4}+\frac{c^2}{x^3}\right ) \, dx,x,x^2\right )\\ &=-\frac{a^2}{12 x^{12}}-\frac{a b}{5 x^{10}}-\frac{b^2+2 a c}{8 x^8}-\frac{b c}{3 x^6}-\frac{c^2}{4 x^4}\\ \end{align*}

Mathematica [A]  time = 0.0153282, size = 50, normalized size = 0.93 \[ -\frac{10 a^2+24 a b x^2+30 a c x^4+15 b^2 x^4+40 b c x^6+30 c^2 x^8}{120 x^{12}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)^2/x^13,x]

[Out]

-(10*a^2 + 24*a*b*x^2 + 15*b^2*x^4 + 30*a*c*x^4 + 40*b*c*x^6 + 30*c^2*x^8)/(120*x^12)

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Maple [A]  time = 0.048, size = 45, normalized size = 0.8 \begin{align*} -{\frac{ab}{5\,{x}^{10}}}-{\frac{{c}^{2}}{4\,{x}^{4}}}-{\frac{2\,ac+{b}^{2}}{8\,{x}^{8}}}-{\frac{bc}{3\,{x}^{6}}}-{\frac{{a}^{2}}{12\,{x}^{12}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2+a)^2/x^13,x)

[Out]

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

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Maxima [A]  time = 0.956137, size = 62, normalized size = 1.15 \begin{align*} -\frac{30 \, c^{2} x^{8} + 40 \, b c x^{6} + 15 \,{\left (b^{2} + 2 \, a c\right )} x^{4} + 24 \, a b x^{2} + 10 \, a^{2}}{120 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^2/x^13,x, algorithm="maxima")

[Out]

-1/120*(30*c^2*x^8 + 40*b*c*x^6 + 15*(b^2 + 2*a*c)*x^4 + 24*a*b*x^2 + 10*a^2)/x^12

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Fricas [A]  time = 1.58857, size = 113, normalized size = 2.09 \begin{align*} -\frac{30 \, c^{2} x^{8} + 40 \, b c x^{6} + 15 \,{\left (b^{2} + 2 \, a c\right )} x^{4} + 24 \, a b x^{2} + 10 \, a^{2}}{120 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^2/x^13,x, algorithm="fricas")

[Out]

-1/120*(30*c^2*x^8 + 40*b*c*x^6 + 15*(b^2 + 2*a*c)*x^4 + 24*a*b*x^2 + 10*a^2)/x^12

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Sympy [A]  time = 2.80806, size = 49, normalized size = 0.91 \begin{align*} - \frac{10 a^{2} + 24 a b x^{2} + 40 b c x^{6} + 30 c^{2} x^{8} + x^{4} \left (30 a c + 15 b^{2}\right )}{120 x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2+a)**2/x**13,x)

[Out]

-(10*a**2 + 24*a*b*x**2 + 40*b*c*x**6 + 30*c**2*x**8 + x**4*(30*a*c + 15*b**2))/(120*x**12)

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Giac [A]  time = 1.1095, size = 65, normalized size = 1.2 \begin{align*} -\frac{30 \, c^{2} x^{8} + 40 \, b c x^{6} + 15 \, b^{2} x^{4} + 30 \, a c x^{4} + 24 \, a b x^{2} + 10 \, a^{2}}{120 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^2/x^13,x, algorithm="giac")

[Out]

-1/120*(30*c^2*x^8 + 40*b*c*x^6 + 15*b^2*x^4 + 30*a*c*x^4 + 24*a*b*x^2 + 10*a^2)/x^12