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

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

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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} $$\begin{array}{c}-x^8-\frac{1}{390625x^8}+\frac{8x^6}{5}+\frac{8}{78125x^6}-\frac{78x^4}{25}-\frac{78}{15625x^4}+\frac{256x^2}{125}+\frac{256}{3125x^2}+x\end{array}$$

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

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

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

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 $$\begin{array}{c}-\frac{390625x^{16}-625000x^{14}+1218750x^{12}-800000x^{10}-390625x^9-32000x^6+1950x^4-40x^2+1}{390625x^8}\end{array}$$

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 $$\begin{array}{c}-x^8+\frac{8}{5}{x}^6-\frac{78}{25}{x}^4+\frac{256}{125}{x}^2+x+\frac{32000x^6-1950x^4+40x^2-1}{390625x^8}\end{array}$$

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

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 $$\begin{array}{c}-x^8+\frac{8}{5}{x}^6-\frac{78}{25}{x}^4+\frac{256}{125}{x}^2+x+\frac{32000x^6-1950x^4+40x^2-1}{390625x^8}\end{array}$$

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 $$\begin{array}{c}x+\frac{\frac{256x^6}{3125}-\frac{78x^4}{15625}+\frac{8x^2}{78125}-\frac{1}{390625}}{x^8}+\frac{256x^2}{125}-\frac{78x^4}{25}+\frac{8x^6}{5}-x^8\end{array}$$

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 $$\begin{array}{c}-x^8+\frac{8x^6}{5}-\frac{78x^4}{25}+\frac{256x^2}{125}+x-\frac{-32000x^6+1950x^4-40x^2+1}{390625x^8}\end{array}$$

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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