3.7.40 \(\int \frac {(a+b x^2+c x^4)^2}{x^{10}} \, dx\)

Optimal. Leaf size=52 \[ -\frac {a^2}{9 x^9}-\frac {2 a c+b^2}{5 x^5}-\frac {2 a b}{7 x^7}-\frac {2 b c}{3 x^3}-\frac {c^2}{x} \]

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Rubi [A]  time = 0.02, antiderivative size = 52, 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}{9 x^9}-\frac {2 a c+b^2}{5 x^5}-\frac {2 a b}{7 x^7}-\frac {2 b c}{3 x^3}-\frac {c^2}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)^2/x^10,x]

[Out]

-a^2/(9*x^9) - (2*a*b)/(7*x^7) - (b^2 + 2*a*c)/(5*x^5) - (2*b*c)/(3*x^3) - 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^{10}} \, dx &=\int \left (\frac {a^2}{x^{10}}+\frac {2 a b}{x^8}+\frac {b^2+2 a c}{x^6}+\frac {2 b c}{x^4}+\frac {c^2}{x^2}\right ) \, dx\\ &=-\frac {a^2}{9 x^9}-\frac {2 a b}{7 x^7}-\frac {b^2+2 a c}{5 x^5}-\frac {2 b c}{3 x^3}-\frac {c^2}{x}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 50, normalized size = 0.96 \begin {gather*} -\frac {35 a^2+90 a b x^2+126 a c x^4+63 b^2 x^4+210 b c x^6+315 c^2 x^8}{315 x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)^2/x^10,x]

[Out]

-1/315*(35*a^2 + 90*a*b*x^2 + 63*b^2*x^4 + 126*a*c*x^4 + 210*b*c*x^6 + 315*c^2*x^8)/x^9

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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^{10}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x^2 + c*x^4)^2/x^10,x]

[Out]

IntegrateAlgebraic[(a + b*x^2 + c*x^4)^2/x^10, x]

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fricas [A]  time = 2.10, size = 46, normalized size = 0.88 \begin {gather*} -\frac {315 \, c^{2} x^{8} + 210 \, b c x^{6} + 63 \, {\left (b^{2} + 2 \, a c\right )} x^{4} + 90 \, a b x^{2} + 35 \, a^{2}}{315 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^2/x^10,x, algorithm="fricas")

[Out]

-1/315*(315*c^2*x^8 + 210*b*c*x^6 + 63*(b^2 + 2*a*c)*x^4 + 90*a*b*x^2 + 35*a^2)/x^9

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giac [A]  time = 0.15, size = 48, normalized size = 0.92 \begin {gather*} -\frac {315 \, c^{2} x^{8} + 210 \, b c x^{6} + 63 \, b^{2} x^{4} + 126 \, a c x^{4} + 90 \, a b x^{2} + 35 \, a^{2}}{315 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^2/x^10,x, algorithm="giac")

[Out]

-1/315*(315*c^2*x^8 + 210*b*c*x^6 + 63*b^2*x^4 + 126*a*c*x^4 + 90*a*b*x^2 + 35*a^2)/x^9

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maple [A]  time = 0.00, size = 45, normalized size = 0.87 \begin {gather*} -\frac {c^{2}}{x}-\frac {2 b c}{3 x^{3}}-\frac {2 a b}{7 x^{7}}-\frac {2 a c +b^{2}}{5 x^{5}}-\frac {a^{2}}{9 x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2+a)^2/x^10,x)

[Out]

-2/7*a*b/x^7-1/9*a^2/x^9-1/5*(2*a*c+b^2)/x^5-c^2/x-2/3*b*c/x^3

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maxima [A]  time = 1.33, size = 46, normalized size = 0.88 \begin {gather*} -\frac {315 \, c^{2} x^{8} + 210 \, b c x^{6} + 63 \, {\left (b^{2} + 2 \, a c\right )} x^{4} + 90 \, a b x^{2} + 35 \, a^{2}}{315 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^2/x^10,x, algorithm="maxima")

[Out]

-1/315*(315*c^2*x^8 + 210*b*c*x^6 + 63*(b^2 + 2*a*c)*x^4 + 90*a*b*x^2 + 35*a^2)/x^9

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mupad [B]  time = 0.03, size = 46, normalized size = 0.88 \begin {gather*} -\frac {\frac {a^2}{9}+x^4\,\left (\frac {b^2}{5}+\frac {2\,a\,c}{5}\right )+c^2\,x^8+\frac {2\,a\,b\,x^2}{7}+\frac {2\,b\,c\,x^6}{3}}{x^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2 + c*x^4)^2/x^10,x)

[Out]

-(a^2/9 + x^4*((2*a*c)/5 + b^2/5) + c^2*x^8 + (2*a*b*x^2)/7 + (2*b*c*x^6)/3)/x^9

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sympy [A]  time = 1.47, size = 49, normalized size = 0.94 \begin {gather*} \frac {- 35 a^{2} - 90 a b x^{2} - 210 b c x^{6} - 315 c^{2} x^{8} + x^{4} \left (- 126 a c - 63 b^{2}\right )}{315 x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2+a)**2/x**10,x)

[Out]

(-35*a**2 - 90*a*b*x**2 - 210*b*c*x**6 - 315*c**2*x**8 + x**4*(-126*a*c - 63*b**2))/(315*x**9)

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