3.7.38 \(\int \frac {(a+b x^2+c x^4)^2}{x^8} \, dx\)

Optimal. Leaf size=47 \[ -\frac {a^2}{7 x^7}-\frac {2 a c+b^2}{3 x^3}-\frac {2 a b}{5 x^5}-\frac {2 b c}{x}+c^2 x \]

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Rubi [A]  time = 0.02, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {1108} \begin {gather*} -\frac {a^2}{7 x^7}-\frac {2 a c+b^2}{3 x^3}-\frac {2 a b}{5 x^5}-\frac {2 b c}{x}+c^2 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)^2/x^8,x]

[Out]

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

Rule 1108

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d*x)^m*(a
+ b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] &&  !IntegerQ[(m + 1)/2]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 49, normalized size = 1.04 \begin {gather*} -\frac {a^2}{7 x^7}+\frac {-2 a c-b^2}{3 x^3}-\frac {2 a b}{5 x^5}-\frac {2 b c}{x}+c^2 x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)^2/x^8,x]

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b x^2+c x^4\right )^2}{x^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x^2 + c*x^4)^2/x^8,x]

[Out]

IntegrateAlgebraic[(a + b*x^2 + c*x^4)^2/x^8, x]

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fricas [A]  time = 1.53, size = 46, normalized size = 0.98 \begin {gather*} \frac {105 \, c^{2} x^{8} - 210 \, b c x^{6} - 35 \, {\left (b^{2} + 2 \, a c\right )} x^{4} - 42 \, a b x^{2} - 15 \, a^{2}}{105 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^2/x^8,x, algorithm="fricas")

[Out]

1/105*(105*c^2*x^8 - 210*b*c*x^6 - 35*(b^2 + 2*a*c)*x^4 - 42*a*b*x^2 - 15*a^2)/x^7

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giac [A]  time = 0.17, size = 46, normalized size = 0.98 \begin {gather*} c^{2} x - \frac {210 \, b c x^{6} + 35 \, b^{2} x^{4} + 70 \, a c x^{4} + 42 \, a b x^{2} + 15 \, a^{2}}{105 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^2/x^8,x, algorithm="giac")

[Out]

c^2*x - 1/105*(210*b*c*x^6 + 35*b^2*x^4 + 70*a*c*x^4 + 42*a*b*x^2 + 15*a^2)/x^7

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maple [A]  time = 0.01, size = 42, normalized size = 0.89 \begin {gather*} c^{2} x -\frac {2 b c}{x}-\frac {2 a b}{5 x^{5}}-\frac {2 a c +b^{2}}{3 x^{3}}-\frac {a^{2}}{7 x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2+a)^2/x^8,x)

[Out]

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

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maxima [A]  time = 1.34, size = 44, normalized size = 0.94 \begin {gather*} c^{2} x - \frac {210 \, b c x^{6} + 35 \, {\left (b^{2} + 2 \, a c\right )} x^{4} + 42 \, a b x^{2} + 15 \, a^{2}}{105 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^2/x^8,x, algorithm="maxima")

[Out]

c^2*x - 1/105*(210*b*c*x^6 + 35*(b^2 + 2*a*c)*x^4 + 42*a*b*x^2 + 15*a^2)/x^7

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mupad [B]  time = 4.17, size = 45, normalized size = 0.96 \begin {gather*} c^2\,x-\frac {\frac {a^2}{7}+x^4\,\left (\frac {b^2}{3}+\frac {2\,a\,c}{3}\right )+\frac {2\,a\,b\,x^2}{5}+2\,b\,c\,x^6}{x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2 + c*x^4)^2/x^8,x)

[Out]

c^2*x - (a^2/7 + x^4*((2*a*c)/3 + b^2/3) + (2*a*b*x^2)/5 + 2*b*c*x^6)/x^7

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sympy [A]  time = 0.76, size = 46, normalized size = 0.98 \begin {gather*} c^{2} x + \frac {- 15 a^{2} - 42 a b x^{2} - 210 b c x^{6} + x^{4} \left (- 70 a c - 35 b^{2}\right )}{105 x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2+a)**2/x**8,x)

[Out]

c**2*x + (-15*a**2 - 42*a*b*x**2 - 210*b*c*x**6 + x**4*(-70*a*c - 35*b**2))/(105*x**7)

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