3.50.92 \(\int \frac {-10000-30000 x-19600 x^2+200 x^3+792 x^5-24 x^6+48 x^7+64 x^8+e^4 (16 x^5+32 x^6)+e^2 (400 x^2+800 x^3+8 x^5-64 x^6-96 x^7)}{x^5} \, dx\)

Optimal. Leaf size=23 \[ \left (-2-\left (e^2-\frac {25}{x^2}-x\right ) (2+4 x)\right )^2 \]

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Rubi [B]  time = 0.06, antiderivative size = 82, normalized size of antiderivative = 3.57, number of steps used = 2, number of rules used = 1, integrand size = 84, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.012, Rules used = {14} \begin {gather*} 16 x^4+\frac {2500}{x^4}+16 \left (1-2 e^2\right ) x^3+\frac {10000}{x^3}-4 \left (3+8 e^2-4 e^4\right ) x^2+\frac {200 \left (49-e^2\right )}{x^2}+8 \left (99+e^2+2 e^4\right ) x-\frac {200 \left (1+4 e^2\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-10000 - 30000*x - 19600*x^2 + 200*x^3 + 792*x^5 - 24*x^6 + 48*x^7 + 64*x^8 + E^4*(16*x^5 + 32*x^6) + E^2
*(400*x^2 + 800*x^3 + 8*x^5 - 64*x^6 - 96*x^7))/x^5,x]

[Out]

2500/x^4 + 10000/x^3 + (200*(49 - E^2))/x^2 - (200*(1 + 4*E^2))/x + 8*(99 + E^2 + 2*E^4)*x - 4*(3 + 8*E^2 - 4*
E^4)*x^2 + 16*(1 - 2*E^2)*x^3 + 16*x^4

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} &=\int \left (8 \left (99+e^2+2 e^4\right )-\frac {10000}{x^5}-\frac {30000}{x^4}+\frac {400 \left (-49+e^2\right )}{x^3}+\frac {200 \left (1+4 e^2\right )}{x^2}+8 \left (-3-8 e^2+4 e^4\right ) x-48 \left (-1+2 e^2\right ) x^2+64 x^3\right ) \, dx\\ &=\frac {2500}{x^4}+\frac {10000}{x^3}+\frac {200 \left (49-e^2\right )}{x^2}-\frac {200 \left (1+4 e^2\right )}{x}+8 \left (99+e^2+2 e^4\right ) x-4 \left (3+8 e^2-4 e^4\right ) x^2+16 \left (1-2 e^2\right ) x^3+16 x^4\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.03, size = 85, normalized size = 3.70 \begin {gather*} 8 \left (\frac {625}{2 x^4}+\frac {1250}{x^3}-\frac {25 \left (-49+e^2\right )}{x^2}-\frac {25 \left (1+4 e^2\right )}{x}+\left (99+e^2+2 e^4\right ) x+\frac {1}{2} \left (-3-8 e^2+4 e^4\right ) x^2-2 \left (-1+2 e^2\right ) x^3+2 x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-10000 - 30000*x - 19600*x^2 + 200*x^3 + 792*x^5 - 24*x^6 + 48*x^7 + 64*x^8 + E^4*(16*x^5 + 32*x^6)
+ E^2*(400*x^2 + 800*x^3 + 8*x^5 - 64*x^6 - 96*x^7))/x^5,x]

[Out]

8*(625/(2*x^4) + 1250/x^3 - (25*(-49 + E^2))/x^2 - (25*(1 + 4*E^2))/x + (99 + E^2 + 2*E^4)*x + ((-3 - 8*E^2 +
4*E^4)*x^2)/2 - 2*(-1 + 2*E^2)*x^3 + 2*x^4)

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fricas [B]  time = 0.79, size = 81, normalized size = 3.52 \begin {gather*} \frac {4 \, {\left (4 \, x^{8} + 4 \, x^{7} - 3 \, x^{6} + 198 \, x^{5} - 50 \, x^{3} + 2450 \, x^{2} + 4 \, {\left (x^{6} + x^{5}\right )} e^{4} - 2 \, {\left (4 \, x^{7} + 4 \, x^{6} - x^{5} + 100 \, x^{3} + 25 \, x^{2}\right )} e^{2} + 2500 \, x + 625\right )}}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((32*x^6+16*x^5)*exp(2)^2+(-96*x^7-64*x^6+8*x^5+800*x^3+400*x^2)*exp(2)+64*x^8+48*x^7-24*x^6+792*x^5
+200*x^3-19600*x^2-30000*x-10000)/x^5,x, algorithm="fricas")

[Out]

4*(4*x^8 + 4*x^7 - 3*x^6 + 198*x^5 - 50*x^3 + 2450*x^2 + 4*(x^6 + x^5)*e^4 - 2*(4*x^7 + 4*x^6 - x^5 + 100*x^3
+ 25*x^2)*e^2 + 2500*x + 625)/x^4

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giac [B]  time = 0.17, size = 84, normalized size = 3.65 \begin {gather*} 16 \, x^{4} - 32 \, x^{3} e^{2} + 16 \, x^{3} + 16 \, x^{2} e^{4} - 32 \, x^{2} e^{2} - 12 \, x^{2} + 16 \, x e^{4} + 8 \, x e^{2} + 792 \, x - \frac {100 \, {\left (8 \, x^{3} e^{2} + 2 \, x^{3} + 2 \, x^{2} e^{2} - 98 \, x^{2} - 100 \, x - 25\right )}}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((32*x^6+16*x^5)*exp(2)^2+(-96*x^7-64*x^6+8*x^5+800*x^3+400*x^2)*exp(2)+64*x^8+48*x^7-24*x^6+792*x^5
+200*x^3-19600*x^2-30000*x-10000)/x^5,x, algorithm="giac")

[Out]

16*x^4 - 32*x^3*e^2 + 16*x^3 + 16*x^2*e^4 - 32*x^2*e^2 - 12*x^2 + 16*x*e^4 + 8*x*e^2 + 792*x - 100*(8*x^3*e^2
+ 2*x^3 + 2*x^2*e^2 - 98*x^2 - 100*x - 25)/x^4

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maple [B]  time = 0.06, size = 77, normalized size = 3.35

method	result	size
norman	\(\frac {2500+\left (-800 \,{\mathrm e}^{2}-200\right ) x^{3}+\left (-200 \,{\mathrm e}^{2}+9800\right ) x^{2}+\left (-32 \,{\mathrm e}^{2}+16\right ) x^{7}+\left (16 \,{\mathrm e}^{4}-32 \,{\mathrm e}^{2}-12\right ) x^{6}+\left (16 \,{\mathrm e}^{4}+8 \,{\mathrm e}^{2}+792\right ) x^{5}+10000 x +16 x^{8}}{x^{4}}\)	\(77\)
risch	\(16 x^{2} {\mathrm e}^{4}-32 x^{3} {\mathrm e}^{2}+16 x^{4}+16 x \,{\mathrm e}^{4}-32 x^{2} {\mathrm e}^{2}+16 x^{3}+8 \,{\mathrm e}^{2} x -12 x^{2}+792 x +\frac {\left (-800 \,{\mathrm e}^{2}-200\right ) x^{3}+\left (-200 \,{\mathrm e}^{2}+9800\right ) x^{2}+10000 x +2500}{x^{4}}\)	\(80\)
default	\(-32 x^{3} {\mathrm e}^{2}+16 x^{4}-32 x^{2} {\mathrm e}^{2}+16 x^{2} {\mathrm e}^{4}+16 x^{3}+8 \,{\mathrm e}^{2} x +16 x \,{\mathrm e}^{4}-12 x^{2}+792 x +\frac {2500}{x^{4}}-\frac {8 \left (100 \,{\mathrm e}^{2}+25\right )}{x}-\frac {4 \left (-2450+50 \,{\mathrm e}^{2}\right )}{x^{2}}+\frac {10000}{x^{3}}\)	\(83\)
gosper	\(\frac {16 x^{6} {\mathrm e}^{4}-32 \,{\mathrm e}^{2} x^{7}+16 x^{8}+16 x^{5} {\mathrm e}^{4}-32 x^{6} {\mathrm e}^{2}+16 x^{7}+8 \,{\mathrm e}^{2} x^{5}-12 x^{6}+792 x^{5}-800 x^{3} {\mathrm e}^{2}-200 x^{2} {\mathrm e}^{2}-200 x^{3}+9800 x^{2}+10000 x +2500}{x^{4}}\)	\(94\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((32*x^6+16*x^5)*exp(2)^2+(-96*x^7-64*x^6+8*x^5+800*x^3+400*x^2)*exp(2)+64*x^8+48*x^7-24*x^6+792*x^5+200*x
^3-19600*x^2-30000*x-10000)/x^5,x,method=_RETURNVERBOSE)

[Out]

(2500+(-800*exp(2)-200)*x^3+(-200*exp(2)+9800)*x^2+(-32*exp(2)+16)*x^7+(16*exp(2)^2-32*exp(2)-12)*x^6+(16*exp(
2)^2+8*exp(2)+792)*x^5+10000*x+16*x^8)/x^4

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maxima [B]  time = 0.35, size = 73, normalized size = 3.17 \begin {gather*} 16 \, x^{4} - 16 \, x^{3} {\left (2 \, e^{2} - 1\right )} + 4 \, x^{2} {\left (4 \, e^{4} - 8 \, e^{2} - 3\right )} + 8 \, x {\left (2 \, e^{4} + e^{2} + 99\right )} - \frac {100 \, {\left (2 \, x^{3} {\left (4 \, e^{2} + 1\right )} + 2 \, x^{2} {\left (e^{2} - 49\right )} - 100 \, x - 25\right )}}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((32*x^6+16*x^5)*exp(2)^2+(-96*x^7-64*x^6+8*x^5+800*x^3+400*x^2)*exp(2)+64*x^8+48*x^7-24*x^6+792*x^5
+200*x^3-19600*x^2-30000*x-10000)/x^5,x, algorithm="maxima")

[Out]

16*x^4 - 16*x^3*(2*e^2 - 1) + 4*x^2*(4*e^4 - 8*e^2 - 3) + 8*x*(2*e^4 + e^2 + 99) - 100*(2*x^3*(4*e^2 + 1) + 2*
x^2*(e^2 - 49) - 100*x - 25)/x^4

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mupad [B]  time = 0.09, size = 75, normalized size = 3.26 \begin {gather*} \frac {\left (-800\,{\mathrm {e}}^2-200\right )\,x^3+\left (9800-200\,{\mathrm {e}}^2\right )\,x^2+10000\,x+2500}{x^4}-x^2\,\left (32\,{\mathrm {e}}^2-16\,{\mathrm {e}}^4+12\right )-x^3\,\left (32\,{\mathrm {e}}^2-16\right )+x\,\left (8\,{\mathrm {e}}^2+16\,{\mathrm {e}}^4+792\right )+16\,x^4 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2)*(400*x^2 + 800*x^3 + 8*x^5 - 64*x^6 - 96*x^7) - 30000*x + exp(4)*(16*x^5 + 32*x^6) - 19600*x^2 + 2
00*x^3 + 792*x^5 - 24*x^6 + 48*x^7 + 64*x^8 - 10000)/x^5,x)

[Out]

(10000*x - x^3*(800*exp(2) + 200) - x^2*(200*exp(2) - 9800) + 2500)/x^4 - x^2*(32*exp(2) - 16*exp(4) + 12) - x
^3*(32*exp(2) - 16) + x*(8*exp(2) + 16*exp(4) + 792) + 16*x^4

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sympy [B]  time = 0.46, size = 73, normalized size = 3.17 \begin {gather*} 16 x^{4} + x^{3} \left (16 - 32 e^{2}\right ) + x^{2} \left (- 32 e^{2} - 12 + 16 e^{4}\right ) + x \left (8 e^{2} + 792 + 16 e^{4}\right ) + \frac {x^{3} \left (- 800 e^{2} - 200\right ) + x^{2} \left (9800 - 200 e^{2}\right ) + 10000 x + 2500}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((32*x**6+16*x**5)*exp(2)**2+(-96*x**7-64*x**6+8*x**5+800*x**3+400*x**2)*exp(2)+64*x**8+48*x**7-24*x
**6+792*x**5+200*x**3-19600*x**2-30000*x-10000)/x**5,x)

[Out]

16*x**4 + x**3*(16 - 32*exp(2)) + x**2*(-32*exp(2) - 12 + 16*exp(4)) + x*(8*exp(2) + 792 + 16*exp(4)) + (x**3*
(-800*exp(2) - 200) + x**2*(9800 - 200*exp(2)) + 10000*x + 2500)/x**4

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