3.33.73 \(\int \frac {8-240 x^2+7800 x^4-64000 x^6+390625 x^9+1600000 x^{10}-4875000 x^{12}+3750000 x^{14}-3125000 x^{16}}{390625 x^9} \, dx\)

Optimal. Leaf size=30 \[ -5-\frac {e}{3}+e^3-\left (1+\left (\frac {1}{5 x}-x\right )^4\right )^2+x \]

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Rubi [A]  time = 0.03, antiderivative size = 56, normalized size of antiderivative = 1.87, number of steps used = 3, number of rules used = 2, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.041, Rules used = {12, 14} \begin {gather*} -x^8-\frac {1}{390625 x^8}+\frac {8 x^6}{5}+\frac {8}{78125 x^6}-\frac {78 x^4}{25}-\frac {78}{15625 x^4}+\frac {256 x^2}{125}+\frac {256}{3125 x^2}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(8 - 240*x^2 + 7800*x^4 - 64000*x^6 + 390625*x^9 + 1600000*x^10 - 4875000*x^12 + 3750000*x^14 - 3125000*x^
16)/(390625*x^9),x]

[Out]

-1/390625*1/x^8 + 8/(78125*x^6) - 78/(15625*x^4) + 256/(3125*x^2) + x + (256*x^2)/125 - (78*x^4)/25 + (8*x^6)/
5 - x^8

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {8-240 x^2+7800 x^4-64000 x^6+390625 x^9+1600000 x^{10}-4875000 x^{12}+3750000 x^{14}-3125000 x^{16}}{x^9} \, dx}{390625}\\ &=\frac {\int \left (390625+\frac {8}{x^9}-\frac {240}{x^7}+\frac {7800}{x^5}-\frac {64000}{x^3}+1600000 x-4875000 x^3+3750000 x^5-3125000 x^7\right ) \, dx}{390625}\\ &=-\frac {1}{390625 x^8}+\frac {8}{78125 x^6}-\frac {78}{15625 x^4}+\frac {256}{3125 x^2}+x+\frac {256 x^2}{125}-\frac {78 x^4}{25}+\frac {8 x^6}{5}-x^8\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 56, normalized size = 1.87 \begin {gather*} -\frac {1}{390625 x^8}+\frac {8}{78125 x^6}-\frac {78}{15625 x^4}+\frac {256}{3125 x^2}+x+\frac {256 x^2}{125}-\frac {78 x^4}{25}+\frac {8 x^6}{5}-x^8 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(8 - 240*x^2 + 7800*x^4 - 64000*x^6 + 390625*x^9 + 1600000*x^10 - 4875000*x^12 + 3750000*x^14 - 3125
000*x^16)/(390625*x^9),x]

[Out]

-1/390625*1/x^8 + 8/(78125*x^6) - 78/(15625*x^4) + 256/(3125*x^2) + x + (256*x^2)/125 - (78*x^4)/25 + (8*x^6)/
5 - x^8

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fricas [A]  time = 0.65, size = 47, normalized size = 1.57 \begin {gather*} -\frac {390625 \, x^{16} - 625000 \, x^{14} + 1218750 \, x^{12} - 800000 \, x^{10} - 390625 \, x^{9} - 32000 \, x^{6} + 1950 \, x^{4} - 40 \, x^{2} + 1}{390625 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/390625*(-3125000*x^16+3750000*x^14-4875000*x^12+1600000*x^10+390625*x^9-64000*x^6+7800*x^4-240*x^2
+8)/x^9,x, algorithm="fricas")

[Out]

-1/390625*(390625*x^16 - 625000*x^14 + 1218750*x^12 - 800000*x^10 - 390625*x^9 - 32000*x^6 + 1950*x^4 - 40*x^2
+ 1)/x^8

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giac [A]  time = 0.24, size = 44, normalized size = 1.47 \begin {gather*} -x^{8} + \frac {8}{5} \, x^{6} - \frac {78}{25} \, x^{4} + \frac {256}{125} \, x^{2} + x + \frac {32000 \, x^{6} - 1950 \, x^{4} + 40 \, x^{2} - 1}{390625 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/390625*(-3125000*x^16+3750000*x^14-4875000*x^12+1600000*x^10+390625*x^9-64000*x^6+7800*x^4-240*x^2
+8)/x^9,x, algorithm="giac")

[Out]

-x^8 + 8/5*x^6 - 78/25*x^4 + 256/125*x^2 + x + 1/390625*(32000*x^6 - 1950*x^4 + 40*x^2 - 1)/x^8

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maple [A]  time = 0.02, size = 43, normalized size = 1.43

method	result	size
default	\(-x^{8}+\frac {8 x^{6}}{5}-\frac {78 x^{4}}{25}+\frac {256 x^{2}}{125}+x -\frac {1}{390625 x^{8}}+\frac {8}{78125 x^{6}}+\frac {256}{3125 x^{2}}-\frac {78}{15625 x^{4}}\)	\(43\)
norman	\(\frac {-\frac {1}{390625}+x^{9}+\frac {8}{78125} x^{2}-\frac {78}{15625} x^{4}+\frac {256}{3125} x^{6}+\frac {256}{125} x^{10}-\frac {78}{25} x^{12}+\frac {8}{5} x^{14}-x^{16}}{x^{8}}\)	\(45\)
risch	\(-x^{8}+\frac {8 x^{6}}{5}-\frac {78 x^{4}}{25}+\frac {256 x^{2}}{125}+x +\frac {32000 x^{6}-1950 x^{4}+40 x^{2}-1}{390625 x^{8}}\)	\(45\)
gosper	\(-\frac {390625 x^{16}-625000 x^{14}+1218750 x^{12}-800000 x^{10}-390625 x^{9}-32000 x^{6}+1950 x^{4}-40 x^{2}+1}{390625 x^{8}}\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/390625*(-3125000*x^16+3750000*x^14-4875000*x^12+1600000*x^10+390625*x^9-64000*x^6+7800*x^4-240*x^2+8)/x^
9,x,method=_RETURNVERBOSE)

[Out]

-x^8+8/5*x^6-78/25*x^4+256/125*x^2+x-1/390625/x^8+8/78125/x^6+256/3125/x^2-78/15625/x^4

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maxima [A]  time = 0.40, size = 44, normalized size = 1.47 \begin {gather*} -x^{8} + \frac {8}{5} \, x^{6} - \frac {78}{25} \, x^{4} + \frac {256}{125} \, x^{2} + x + \frac {32000 \, x^{6} - 1950 \, x^{4} + 40 \, x^{2} - 1}{390625 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/390625*(-3125000*x^16+3750000*x^14-4875000*x^12+1600000*x^10+390625*x^9-64000*x^6+7800*x^4-240*x^2
+8)/x^9,x, algorithm="maxima")

[Out]

-x^8 + 8/5*x^6 - 78/25*x^4 + 256/125*x^2 + x + 1/390625*(32000*x^6 - 1950*x^4 + 40*x^2 - 1)/x^8

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mupad [B]  time = 1.98, size = 43, normalized size = 1.43 \begin {gather*} x+\frac {\frac {256\,x^6}{3125}-\frac {78\,x^4}{15625}+\frac {8\,x^2}{78125}-\frac {1}{390625}}{x^8}+\frac {256\,x^2}{125}-\frac {78\,x^4}{25}+\frac {8\,x^6}{5}-x^8 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((312*x^4)/15625 - (48*x^2)/78125 - (512*x^6)/3125 + x^9 + (512*x^10)/125 - (312*x^12)/25 + (48*x^14)/5 -
8*x^16 + 8/390625)/x^9,x)

[Out]

x + ((8*x^2)/78125 - (78*x^4)/15625 + (256*x^6)/3125 - 1/390625)/x^8 + (256*x^2)/125 - (78*x^4)/25 + (8*x^6)/5
- x^8

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sympy [B]  time = 0.13, size = 46, normalized size = 1.53 \begin {gather*} - x^{8} + \frac {8 x^{6}}{5} - \frac {78 x^{4}}{25} + \frac {256 x^{2}}{125} + x - \frac {- 32000 x^{6} + 1950 x^{4} - 40 x^{2} + 1}{390625 x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/390625*(-3125000*x**16+3750000*x**14-4875000*x**12+1600000*x**10+390625*x**9-64000*x**6+7800*x**4-
240*x**2+8)/x**9,x)

[Out]

-x**8 + 8*x**6/5 - 78*x**4/25 + 256*x**2/125 + x - (-32000*x**6 + 1950*x**4 - 40*x**2 + 1)/(390625*x**8)

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