3.46.21 \(\int \frac {-200000000 x^2-125000000 x^3+2500000 x^5+120000 x^6+37500 x^7+1600 x^8+6 x^{10}+e^{32} (-10000000000-200000000 x^2+20000 x^6+100 x^8)+e^{16} (-3000000000 x-1250000000 x^2-40000000 x^3+600000 x^5+125000 x^6+12000 x^7+50 x^9)}{9765625 x^5} \, dx\)

Optimal. Leaf size=34 \[ \frac {\left (2 x+\frac {\left (e^{16}+\frac {x}{5}\right ) \left (4+\frac {x^2}{25}\right )^2}{x}\right )^2}{x^2} \]

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Rubi [B]  time = 0.11, antiderivative size = 126, normalized size of antiderivative = 3.71, number of steps used = 3, number of rules used = 2, integrand size = 102, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {12, 14} \begin {gather*} \frac {x^6}{9765625}+\frac {2 e^{16} x^5}{1953125}+\frac {\left (16+e^{32}\right ) x^4}{390625}+\frac {256 e^{32}}{x^4}+\frac {4 \left (25+8 e^{16}\right ) x^3}{78125}+\frac {512 e^{16}}{5 x^3}+\frac {4 \left (24+25 e^{16}+4 e^{32}\right ) x^2}{15625}+\frac {64 \left (4+25 e^{16}+4 e^{32}\right )}{25 x^2}+\frac {32 \left (25+6 e^{16}\right ) x}{3125}+\frac {64 \left (25+8 e^{16}\right )}{125 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-200000000*x^2 - 125000000*x^3 + 2500000*x^5 + 120000*x^6 + 37500*x^7 + 1600*x^8 + 6*x^10 + E^32*(-100000
00000 - 200000000*x^2 + 20000*x^6 + 100*x^8) + E^16*(-3000000000*x - 1250000000*x^2 - 40000000*x^3 + 600000*x^
5 + 125000*x^6 + 12000*x^7 + 50*x^9))/(9765625*x^5),x]

[Out]

(256*E^32)/x^4 + (512*E^16)/(5*x^3) + (64*(4 + 25*E^16 + 4*E^32))/(25*x^2) + (64*(25 + 8*E^16))/(125*x) + (32*
(25 + 6*E^16)*x)/3125 + (4*(24 + 25*E^16 + 4*E^32)*x^2)/15625 + (4*(25 + 8*E^16)*x^3)/78125 + ((16 + E^32)*x^4
)/390625 + (2*E^16*x^5)/1953125 + x^6/9765625

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 {-200000000 x^2-125000000 x^3+2500000 x^5+120000 x^6+37500 x^7+1600 x^8+6 x^{10}+e^{32} \left (-10000000000-200000000 x^2+20000 x^6+100 x^8\right )+e^{16} \left (-3000000000 x-1250000000 x^2-40000000 x^3+600000 x^5+125000 x^6+12000 x^7+50 x^9\right )}{x^5} \, dx}{9765625}\\ &=\frac {\int \left (100000 \left (25+6 e^{16}\right )-\frac {10000000000 e^{32}}{x^5}-\frac {3000000000 e^{16}}{x^4}-\frac {50000000 \left (4+25 e^{16}+4 e^{32}\right )}{x^3}-\frac {5000000 \left (25+8 e^{16}\right )}{x^2}+5000 \left (24+25 e^{16}+4 e^{32}\right ) x+1500 \left (25+8 e^{16}\right ) x^2+100 \left (16+e^{32}\right ) x^3+50 e^{16} x^4+6 x^5\right ) \, dx}{9765625}\\ &=\frac {256 e^{32}}{x^4}+\frac {512 e^{16}}{5 x^3}+\frac {64 \left (4+25 e^{16}+4 e^{32}\right )}{25 x^2}+\frac {64 \left (25+8 e^{16}\right )}{125 x}+\frac {32 \left (25+6 e^{16}\right ) x}{3125}+\frac {4 \left (24+25 e^{16}+4 e^{32}\right ) x^2}{15625}+\frac {4 \left (25+8 e^{16}\right ) x^3}{78125}+\frac {\left (16+e^{32}\right ) x^4}{390625}+\frac {2 e^{16} x^5}{1953125}+\frac {x^6}{9765625}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.02, size = 152, normalized size = 4.47 \begin {gather*} \frac {256 e^{32}}{x^4}+\frac {512 e^{16}}{5 x^3}+\frac {256}{25 x^2}+\frac {64 e^{16}}{x^2}+\frac {256 e^{32}}{25 x^2}+\frac {64}{5 x}+\frac {512 e^{16}}{125 x}+\frac {32 x}{125}+\frac {192 e^{16} x}{3125}+\frac {96 x^2}{15625}+\frac {4 e^{16} x^2}{625}+\frac {16 e^{32} x^2}{15625}+\frac {4 x^3}{3125}+\frac {32 e^{16} x^3}{78125}+\frac {16 x^4}{390625}+\frac {e^{32} x^4}{390625}+\frac {2 e^{16} x^5}{1953125}+\frac {x^6}{9765625} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-200000000*x^2 - 125000000*x^3 + 2500000*x^5 + 120000*x^6 + 37500*x^7 + 1600*x^8 + 6*x^10 + E^32*(-
10000000000 - 200000000*x^2 + 20000*x^6 + 100*x^8) + E^16*(-3000000000*x - 1250000000*x^2 - 40000000*x^3 + 600
000*x^5 + 125000*x^6 + 12000*x^7 + 50*x^9))/(9765625*x^5),x]

[Out]

(256*E^32)/x^4 + (512*E^16)/(5*x^3) + 256/(25*x^2) + (64*E^16)/x^2 + (256*E^32)/(25*x^2) + 64/(5*x) + (512*E^1
6)/(125*x) + (32*x)/125 + (192*E^16*x)/3125 + (96*x^2)/15625 + (4*E^16*x^2)/625 + (16*E^32*x^2)/15625 + (4*x^3
)/3125 + (32*E^16*x^3)/78125 + (16*x^4)/390625 + (E^32*x^4)/390625 + (2*E^16*x^5)/1953125 + x^6/9765625

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fricas [B]  time = 0.58, size = 94, normalized size = 2.76 \begin {gather*} \frac {x^{10} + 400 \, x^{8} + 12500 \, x^{7} + 60000 \, x^{6} + 2500000 \, x^{5} + 125000000 \, x^{3} + 100000000 \, x^{2} + 25 \, {\left (x^{8} + 400 \, x^{6} + 4000000 \, x^{2} + 100000000\right )} e^{32} + 10 \, {\left (x^{9} + 400 \, x^{7} + 6250 \, x^{6} + 60000 \, x^{5} + 4000000 \, x^{3} + 62500000 \, x^{2} + 100000000 \, x\right )} e^{16}}{9765625 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9765625*((100*x^8+20000*x^6-200000000*x^2-10000000000)*exp(4)^8+(50*x^9+12000*x^7+125000*x^6+60000
0*x^5-40000000*x^3-1250000000*x^2-3000000000*x)*exp(4)^4+6*x^10+1600*x^8+37500*x^7+120000*x^6+2500000*x^5-1250
00000*x^3-200000000*x^2)/x^5,x, algorithm="fricas")

[Out]

1/9765625*(x^10 + 400*x^8 + 12500*x^7 + 60000*x^6 + 2500000*x^5 + 125000000*x^3 + 100000000*x^2 + 25*(x^8 + 40
0*x^6 + 4000000*x^2 + 100000000)*e^32 + 10*(x^9 + 400*x^7 + 6250*x^6 + 60000*x^5 + 4000000*x^3 + 62500000*x^2
+ 100000000*x)*e^16)/x^4

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giac [B]  time = 0.15, size = 110, normalized size = 3.24 \begin {gather*} \frac {1}{9765625} \, x^{6} + \frac {2}{1953125} \, x^{5} e^{16} + \frac {1}{390625} \, x^{4} e^{32} + \frac {16}{390625} \, x^{4} + \frac {32}{78125} \, x^{3} e^{16} + \frac {4}{3125} \, x^{3} + \frac {16}{15625} \, x^{2} e^{32} + \frac {4}{625} \, x^{2} e^{16} + \frac {96}{15625} \, x^{2} + \frac {192}{3125} \, x e^{16} + \frac {32}{125} \, x + \frac {64 \, {\left (8 \, x^{3} e^{16} + 25 \, x^{3} + 20 \, x^{2} e^{32} + 125 \, x^{2} e^{16} + 20 \, x^{2} + 200 \, x e^{16} + 500 \, e^{32}\right )}}{125 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9765625*((100*x^8+20000*x^6-200000000*x^2-10000000000)*exp(4)^8+(50*x^9+12000*x^7+125000*x^6+60000
0*x^5-40000000*x^3-1250000000*x^2-3000000000*x)*exp(4)^4+6*x^10+1600*x^8+37500*x^7+120000*x^6+2500000*x^5-1250
00000*x^3-200000000*x^2)/x^5,x, algorithm="giac")

[Out]

1/9765625*x^6 + 2/1953125*x^5*e^16 + 1/390625*x^4*e^32 + 16/390625*x^4 + 32/78125*x^3*e^16 + 4/3125*x^3 + 16/1
5625*x^2*e^32 + 4/625*x^2*e^16 + 96/15625*x^2 + 192/3125*x*e^16 + 32/125*x + 64/125*(8*x^3*e^16 + 25*x^3 + 20*
x^2*e^32 + 125*x^2*e^16 + 20*x^2 + 200*x*e^16 + 500*e^32)/x^4

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maple [B]  time = 0.10, size = 104, normalized size = 3.06

method	result	size
risch	\(\frac {x^{4} {\mathrm e}^{32}}{390625}+\frac {16 \,{\mathrm e}^{32} x^{2}}{15625}+\frac {2 x^{5} {\mathrm e}^{16}}{1953125}+\frac {32 x^{3} {\mathrm e}^{16}}{78125}+\frac {4 x^{2} {\mathrm e}^{16}}{625}+\frac {x^{6}}{9765625}+\frac {192 x \,{\mathrm e}^{16}}{3125}+\frac {16 x^{4}}{390625}+\frac {4 x^{3}}{3125}+\frac {96 x^{2}}{15625}+\frac {32 x}{125}+\frac {\left (40000000 \,{\mathrm e}^{16}[phone]\right ) x^{3}+\left (100000000 \,{\mathrm e}^{32}[phone] \,{\mathrm e}^{16}[phone]\right ) x^{2}[phone] x \,{\mathrm e}^{16}[phone] \,{\mathrm e}^{32}}{9765625 x^{4}}\)	\(104\)
default	\(\frac {2 x^{5} {\mathrm e}^{16}}{1953125}+\frac {x^{6}}{9765625}+\frac {x^{4} {\mathrm e}^{32}}{390625}+\frac {32 x^{3} {\mathrm e}^{16}}{78125}+\frac {16 x^{4}}{390625}+\frac {16 \,{\mathrm e}^{32} x^{2}}{15625}+\frac {4 x^{2} {\mathrm e}^{16}}{625}+\frac {4 x^{3}}{3125}+\frac {192 x \,{\mathrm e}^{16}}{3125}+\frac {96 x^{2}}{15625}+\frac {32 x}{125}+\frac {512 \,{\mathrm e}^{16}}{5 x^{3}}-\frac {-100000000 \,{\mathrm e}^{32}-625000000 \,{\mathrm e}^{16}-100000000}{9765625 x^{2}}-\frac {2 \left (-20000000 \,{\mathrm e}^{16}-62500000\right )}{9765625 x}+\frac {256 \,{\mathrm e}^{32}}{x^{4}}\)	\(105\)
norman	\(\frac {\left (\frac {32 \,{\mathrm e}^{16}}{78125}+\frac {4}{3125}\right ) x^{7}+\left (\frac {192 \,{\mathrm e}^{16}}{3125}+\frac {32}{125}\right ) x^{5}+\left (\frac {512 \,{\mathrm e}^{16}}{125}+\frac {64}{5}\right ) x^{3}+\left (\frac {{\mathrm e}^{32}}{390625}+\frac {16}{390625}\right ) x^{8}+\left (\frac {16 \,{\mathrm e}^{32}}{15625}+\frac {4 \,{\mathrm e}^{16}}{625}+\frac {96}{15625}\right ) x^{6}+\left (\frac {256 \,{\mathrm e}^{32}}{25}+64 \,{\mathrm e}^{16}+\frac {256}{25}\right ) x^{2}+\frac {x^{10}}{9765625}+256 \,{\mathrm e}^{32}+\frac {512 x \,{\mathrm e}^{16}}{5}+\frac {2 x^{9} {\mathrm e}^{16}}{1953125}}{x^{4}}\)	\(117\)
gosper	\(\frac {25 x^{8} {\mathrm e}^{32}+10000 \,{\mathrm e}^{32} x^{6}+10 x^{9} {\mathrm e}^{16}+4000 \,{\mathrm e}^{16} x^{7}[phone] \,{\mathrm e}^{32} x^{2}+62500 \,{\mathrm e}^{16} x^{6}+x^{10}+600000 x^{5} {\mathrm e}^{16}[phone] \,{\mathrm e}^{32}+400 x^{8}[phone] x^{3} {\mathrm e}^{16}+12500 x^{7}[phone] x^{2} {\mathrm e}^{16}+60000 x^{6}[phone] x \,{\mathrm e}^{16}+2500000 x^{5}[phone] x^{3}[phone] x^{2}}{9765625 x^{4}}\)	\(134\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/9765625*((100*x^8+20000*x^6-200000000*x^2-10000000000)*exp(4)^8+(50*x^9+12000*x^7+125000*x^6+600000*x^5-
40000000*x^3-1250000000*x^2-3000000000*x)*exp(4)^4+6*x^10+1600*x^8+37500*x^7+120000*x^6+2500000*x^5-125000000*
x^3-200000000*x^2)/x^5,x,method=_RETURNVERBOSE)

[Out]

1/390625*x^4*exp(32)+16/15625*exp(32)*x^2+2/1953125*x^5*exp(16)+32/78125*x^3*exp(16)+4/625*x^2*exp(16)+1/97656
25*x^6+192/3125*x*exp(16)+16/390625*x^4+4/3125*x^3+96/15625*x^2+32/125*x+1/9765625*((40000000*exp(16)[phone]
0)*x^3+(100000000*exp(32)[phone]*exp(16)[phone])*x^2+1000000000*x*exp(16)[phone]*exp(32))/x^4

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maxima [B]  time = 0.38, size = 97, normalized size = 2.85 \begin {gather*} \frac {1}{9765625} \, x^{6} + \frac {2}{1953125} \, x^{5} e^{16} + \frac {1}{390625} \, x^{4} {\left (e^{32} + 16\right )} + \frac {4}{78125} \, x^{3} {\left (8 \, e^{16} + 25\right )} + \frac {4}{15625} \, x^{2} {\left (4 \, e^{32} + 25 \, e^{16} + 24\right )} + \frac {32}{3125} \, x {\left (6 \, e^{16} + 25\right )} + \frac {64 \, {\left (x^{3} {\left (8 \, e^{16} + 25\right )} + 5 \, x^{2} {\left (4 \, e^{32} + 25 \, e^{16} + 4\right )} + 200 \, x e^{16} + 500 \, e^{32}\right )}}{125 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9765625*((100*x^8+20000*x^6-200000000*x^2-10000000000)*exp(4)^8+(50*x^9+12000*x^7+125000*x^6+60000
0*x^5-40000000*x^3-1250000000*x^2-3000000000*x)*exp(4)^4+6*x^10+1600*x^8+37500*x^7+120000*x^6+2500000*x^5-1250
00000*x^3-200000000*x^2)/x^5,x, algorithm="maxima")

[Out]

1/9765625*x^6 + 2/1953125*x^5*e^16 + 1/390625*x^4*(e^32 + 16) + 4/78125*x^3*(8*e^16 + 25) + 4/15625*x^2*(4*e^3
2 + 25*e^16 + 24) + 32/3125*x*(6*e^16 + 25) + 64/125*(x^3*(8*e^16 + 25) + 5*x^2*(4*e^32 + 25*e^16 + 4) + 200*x
*e^16 + 500*e^32)/x^4

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mupad [B]  time = 3.32, size = 94, normalized size = 2.76 \begin {gather*} \frac {\left (512\,{\mathrm {e}}^{16}+1600\right )\,x^3+\left (8000\,{\mathrm {e}}^{16}+1280\,{\mathrm {e}}^{32}+1280\right )\,x^2+12800\,{\mathrm {e}}^{16}\,x+32000\,{\mathrm {e}}^{32}}{125\,x^4}+x^2\,\left (\frac {4\,{\mathrm {e}}^{16}}{625}+\frac {16\,{\mathrm {e}}^{32}}{15625}+\frac {96}{15625}\right )+x^3\,\left (\frac {32\,{\mathrm {e}}^{16}}{78125}+\frac {4}{3125}\right )+x^4\,\left (\frac {{\mathrm {e}}^{32}}{390625}+\frac {16}{390625}\right )+\frac {2\,x^5\,{\mathrm {e}}^{16}}{1953125}+\frac {x^6}{9765625}+x\,\left (\frac {192\,{\mathrm {e}}^{16}}{3125}+\frac {32}{125}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((exp(16)*(600000*x^5 - 1250000000*x^2 - 40000000*x^3 - 3000000000*x + 125000*x^6 + 12000*x^7 + 50*x^9))/9
765625 - (exp(32)*(200000000*x^2 - 20000*x^6 - 100*x^8 + 10000000000))/9765625 - (512*x^2)/25 - (64*x^3)/5 + (
32*x^5)/125 + (192*x^6)/15625 + (12*x^7)/3125 + (64*x^8)/390625 + (6*x^10)/9765625)/x^5,x)

[Out]

(32000*exp(32) + x^2*(8000*exp(16) + 1280*exp(32) + 1280) + 12800*x*exp(16) + x^3*(512*exp(16) + 1600))/(125*x
^4) + x^2*((4*exp(16))/625 + (16*exp(32))/15625 + 96/15625) + x^3*((32*exp(16))/78125 + 4/3125) + x^4*(exp(32)
/390625 + 16/390625) + (2*x^5*exp(16))/1953125 + x^6/9765625 + x*((192*exp(16))/3125 + 32/125)

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sympy [B]  time = 0.78, size = 107, normalized size = 3.15 \begin {gather*} \frac {x^{6}}{9765625} + \frac {2 x^{5} e^{16}}{1953125} + \frac {x^{4} \left (400 + 25 e^{32}\right )}{9765625} + \frac {x^{3} \left (12500 + 4000 e^{16}\right )}{9765625} + \frac {x^{2} \left (60000 + 62500 e^{16} + 10000 e^{32}\right )}{9765625} + \frac {x \left (2500000 + 600000 e^{16}\right )}{9765625} + \frac {x^{3} \left (125000000 + 40000000 e^{16}\right ) + x^{2} \left (100000000 + 625000000 e^{16} + 100000000 e^{32}\right ) + 1000000000 x e^{16} + 2500000000 e^{32}}{9765625 x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9765625*((100*x**8+20000*x**6-200000000*x**2-10000000000)*exp(4)**8+(50*x**9+12000*x**7+125000*x**
6+600000*x**5-40000000*x**3-1250000000*x**2-3000000000*x)*exp(4)**4+6*x**10+1600*x**8+37500*x**7+120000*x**6+2
500000*x**5-125000000*x**3-200000000*x**2)/x**5,x)

[Out]

x**6/9765625 + 2*x**5*exp(16)/1953125 + x**4*(400 + 25*exp(32))/9765625 + x**3*(12500 + 4000*exp(16))/9765625
+ x**2*(60000 + 62500*exp(16) + 10000*exp(32))/9765625 + x*(2500000 + 600000*exp(16))/9765625 + (x**3*(1250000
00 + 40000000*exp(16)) + x**2*(100000000 + 625000000*exp(16) + 100000000*exp(32)) + 1000000000*x*exp(16) + 250
0000000*exp(32))/(9765625*x**4)

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