3.60.47 \(\int \frac {e^{2 x} (e^4 (48-96 x)-168 x+168 x^2+e^{\frac {1}{3} (2-2 x)} (-48+128 x))}{48 e^8 x^2+48 e^{\frac {2}{3} (2-2 x)} x^2-168 e^4 x^3+147 x^4+e^{\frac {1}{3} (2-2 x)} (-96 e^4 x^2+168 x^3)} \, dx\)

Optimal. Leaf size=33 \[ \frac {e^{2 x}}{x \left (-e^4+e^{\frac {2 (1-x)}{3}}+\frac {7 x}{4}\right )} \]

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Rubi [F]  time = 6.96, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {e^{2 x} \left (e^4 (48-96 x)-168 x+168 x^2+e^{\frac {1}{3} (2-2 x)} (-48+128 x)\right )}{48 e^8 x^2+48 e^{\frac {2}{3} (2-2 x)} x^2-168 e^4 x^3+147 x^4+e^{\frac {1}{3} (2-2 x)} \left (-96 e^4 x^2+168 x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(2*x)*(E^4*(48 - 96*x) - 168*x + 168*x^2 + E^((2 - 2*x)/3)*(-48 + 128*x)))/(48*E^8*x^2 + 48*E^((2*(2 -
2*x))/3)*x^2 - 168*E^4*x^3 + 147*x^4 + E^((2 - 2*x)/3)*(-96*E^4*x^2 + 168*x^3)),x]

[Out]

E^(-2/3 + (8*x)/3)/x - (56*Defer[Int][E^((10*x)/3)/(-4*E^(2/3) + 4*E^(4 + (2*x)/3) - 7*E^((2*x)/3)*x)^2, x])/3
- (4*(21 - 8*E^4)*Defer[Int][E^((10*x)/3)/(x*(-4*E^(2/3) + 4*E^(4 + (2*x)/3) - 7*E^((2*x)/3)*x)^2), x])/3 + (
56*Defer[Int][E^(-2/3 + (10*x)/3)/(-4*E^(2/3) + 4*E^(4 + (2*x)/3) - 7*E^((2*x)/3)*x), x])/3 + 4*Defer[Int][E^(
10/3 + (10*x)/3)/(x^2*(-4*E^(2/3) + 4*E^(4 + (2*x)/3) - 7*E^((2*x)/3)*x)), x] - ((21 + 32*E^4)*Defer[Int][E^(-
2/3 + (10*x)/3)/(x*(-4*E^(2/3) + 4*E^(4 + (2*x)/3) - 7*E^((2*x)/3)*x)), x])/3

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{10 x/3} \left (e^4 (48-96 x)-168 x+168 x^2+e^{\frac {1}{3} (2-2 x)} (-48+128 x)\right )}{3 x^2 \left (4 e^{2/3}-4 e^{4+\frac {2 x}{3}}+7 e^{2 x/3} x\right )^2} \, dx\\ &=\frac {1}{3} \int \frac {e^{10 x/3} \left (e^4 (48-96 x)-168 x+168 x^2+e^{\frac {1}{3} (2-2 x)} (-48+128 x)\right )}{x^2 \left (4 e^{2/3}-4 e^{4+\frac {2 x}{3}}+7 e^{2 x/3} x\right )^2} \, dx\\ &=\frac {1}{3} \int \left (\frac {e^{-\frac {2}{3}+\frac {8 x}{3}} (-3+8 x)}{x^2}+\frac {4 e^{10 x/3} \left (-21+8 e^4-14 x\right )}{x \left (-4 e^{2/3}+4 e^{4+\frac {2 x}{3}}-7 e^{2 x/3} x\right )^2}-\frac {e^{-\frac {2}{3}+\frac {10 x}{3}} \left (4 e^4-7 x\right ) (-3+8 x)}{x^2 \left (-4 e^{2/3}+4 e^{4+\frac {2 x}{3}}-7 e^{2 x/3} x\right )}\right ) \, dx\\ &=\frac {1}{3} \int \frac {e^{-\frac {2}{3}+\frac {8 x}{3}} (-3+8 x)}{x^2} \, dx-\frac {1}{3} \int \frac {e^{-\frac {2}{3}+\frac {10 x}{3}} \left (4 e^4-7 x\right ) (-3+8 x)}{x^2 \left (-4 e^{2/3}+4 e^{4+\frac {2 x}{3}}-7 e^{2 x/3} x\right )} \, dx+\frac {4}{3} \int \frac {e^{10 x/3} \left (-21+8 e^4-14 x\right )}{x \left (-4 e^{2/3}+4 e^{4+\frac {2 x}{3}}-7 e^{2 x/3} x\right )^2} \, dx\\ &=\frac {e^{-\frac {2}{3}+\frac {8 x}{3}}}{x}-\frac {1}{3} \int \left (-\frac {56 e^{-\frac {2}{3}+\frac {10 x}{3}}}{-4 e^{2/3}+4 e^{4+\frac {2 x}{3}}-7 e^{2 x/3} x}-\frac {12 e^{\frac {10}{3}+\frac {10 x}{3}}}{x^2 \left (-4 e^{2/3}+4 e^{4+\frac {2 x}{3}}-7 e^{2 x/3} x\right )}+\frac {e^{-\frac {2}{3}+\frac {10 x}{3}} \left (21+32 e^4\right )}{x \left (-4 e^{2/3}+4 e^{4+\frac {2 x}{3}}-7 e^{2 x/3} x\right )}\right ) \, dx+\frac {4}{3} \int \left (-\frac {14 e^{10 x/3}}{\left (-4 e^{2/3}+4 e^{4+\frac {2 x}{3}}-7 e^{2 x/3} x\right )^2}+\frac {e^{10 x/3} \left (-21+8 e^4\right )}{x \left (-4 e^{2/3}+4 e^{4+\frac {2 x}{3}}-7 e^{2 x/3} x\right )^2}\right ) \, dx\\ &=\frac {e^{-\frac {2}{3}+\frac {8 x}{3}}}{x}+4 \int \frac {e^{\frac {10}{3}+\frac {10 x}{3}}}{x^2 \left (-4 e^{2/3}+4 e^{4+\frac {2 x}{3}}-7 e^{2 x/3} x\right )} \, dx-\frac {56}{3} \int \frac {e^{10 x/3}}{\left (-4 e^{2/3}+4 e^{4+\frac {2 x}{3}}-7 e^{2 x/3} x\right )^2} \, dx+\frac {56}{3} \int \frac {e^{-\frac {2}{3}+\frac {10 x}{3}}}{-4 e^{2/3}+4 e^{4+\frac {2 x}{3}}-7 e^{2 x/3} x} \, dx-\frac {1}{3} \left (4 \left (21-8 e^4\right )\right ) \int \frac {e^{10 x/3}}{x \left (-4 e^{2/3}+4 e^{4+\frac {2 x}{3}}-7 e^{2 x/3} x\right )^2} \, dx-\frac {1}{3} \left (21+32 e^4\right ) \int \frac {e^{-\frac {2}{3}+\frac {10 x}{3}}}{x \left (-4 e^{2/3}+4 e^{4+\frac {2 x}{3}}-7 e^{2 x/3} x\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.14, size = 44, normalized size = 1.33 \begin {gather*} \frac {8 e^{8 x/3}}{8 e^{2/3} x-8 e^{4+\frac {2 x}{3}} x+14 e^{2 x/3} x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*x)*(E^4*(48 - 96*x) - 168*x + 168*x^2 + E^((2 - 2*x)/3)*(-48 + 128*x)))/(48*E^8*x^2 + 48*E^((2
*(2 - 2*x))/3)*x^2 - 168*E^4*x^3 + 147*x^4 + E^((2 - 2*x)/3)*(-96*E^4*x^2 + 168*x^3)),x]

[Out]

(8*E^((8*x)/3))/(8*E^(2/3)*x - 8*E^(4 + (2*x)/3)*x + 14*E^((2*x)/3)*x^2)

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fricas [A]  time = 0.65, size = 34, normalized size = 1.03 \begin {gather*} \frac {4 \, e^{2}}{{\left (7 \, x^{2} - 4 \, x e^{4}\right )} e^{\left (-2 \, x + 2\right )} + 4 \, x e^{\left (-\frac {8}{3} \, x + \frac {8}{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((128*x-48)*exp(-2/3*x+2/3)+(-96*x+48)*exp(4)+168*x^2-168*x)*exp(x)^2/(48*x^2*exp(-2/3*x+2/3)^2+(-96
*x^2*exp(4)+168*x^3)*exp(-2/3*x+2/3)+48*x^2*exp(4)^2-168*x^3*exp(4)+147*x^4),x, algorithm="fricas")

[Out]

4*e^2/((7*x^2 - 4*x*e^4)*e^(-2*x + 2) + 4*x*e^(-8/3*x + 8/3))

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giac [B]  time = 0.36, size = 529, normalized size = 16.03 \begin {gather*} \frac {4 \, {\left (33614 \, x^{6} e^{\left (\frac {8}{3} \, x\right )} + 50421 \, x^{5} e^{\left (\frac {8}{3} \, x\right )} - 96040 \, x^{5} e^{\left (\frac {8}{3} \, x + 4\right )} + 19208 \, x^{5} e^{\left (2 \, x + \frac {2}{3}\right )} + 109760 \, x^{4} e^{\left (\frac {8}{3} \, x + 8\right )} - 115248 \, x^{4} e^{\left (\frac {8}{3} \, x + 4\right )} - 43904 \, x^{4} e^{\left (2 \, x + \frac {14}{3}\right )} + 28812 \, x^{4} e^{\left (2 \, x + \frac {2}{3}\right )} - 62720 \, x^{3} e^{\left (\frac {8}{3} \, x + 12\right )} + 98784 \, x^{3} e^{\left (\frac {8}{3} \, x + 8\right )} + 37632 \, x^{3} e^{\left (2 \, x + \frac {26}{3}\right )} - 49392 \, x^{3} e^{\left (2 \, x + \frac {14}{3}\right )} + 17920 \, x^{2} e^{\left (\frac {8}{3} \, x + 16\right )} - 37632 \, x^{2} e^{\left (\frac {8}{3} \, x + 12\right )} - 14336 \, x^{2} e^{\left (2 \, x + \frac {38}{3}\right )} + 28224 \, x^{2} e^{\left (2 \, x + \frac {26}{3}\right )} - 2048 \, x e^{\left (\frac {8}{3} \, x + 20\right )} + 5376 \, x e^{\left (\frac {8}{3} \, x + 16\right )} + 2048 \, x e^{\left (2 \, x + \frac {50}{3}\right )} - 5376 \, x e^{\left (2 \, x + \frac {38}{3}\right )}\right )}}{235298 \, x^{8} e^{\left (\frac {2}{3} \, x\right )} + 268912 \, x^{7} e^{\frac {2}{3}} + 352947 \, x^{7} e^{\left (\frac {2}{3} \, x\right )} - 806736 \, x^{7} e^{\left (\frac {2}{3} \, x + 4\right )} - 768320 \, x^{6} e^{\frac {14}{3}} + 403368 \, x^{6} e^{\frac {2}{3}} + 1152480 \, x^{6} e^{\left (\frac {2}{3} \, x + 8\right )} - 1008420 \, x^{6} e^{\left (\frac {2}{3} \, x + 4\right )} + 76832 \, x^{6} e^{\left (-\frac {2}{3} \, x + \frac {4}{3}\right )} + 878080 \, x^{5} e^{\frac {26}{3}} - 921984 \, x^{5} e^{\frac {14}{3}} - 878080 \, x^{5} e^{\left (\frac {2}{3} \, x + 12\right )} + 1152480 \, x^{5} e^{\left (\frac {2}{3} \, x + 8\right )} - 175616 \, x^{5} e^{\left (-\frac {2}{3} \, x + \frac {16}{3}\right )} + 115248 \, x^{5} e^{\left (-\frac {2}{3} \, x + \frac {4}{3}\right )} - 501760 \, x^{4} e^{\frac {38}{3}} + 790272 \, x^{4} e^{\frac {26}{3}} + 376320 \, x^{4} e^{\left (\frac {2}{3} \, x + 16\right )} - 658560 \, x^{4} e^{\left (\frac {2}{3} \, x + 12\right )} + 150528 \, x^{4} e^{\left (-\frac {2}{3} \, x + \frac {28}{3}\right )} - 197568 \, x^{4} e^{\left (-\frac {2}{3} \, x + \frac {16}{3}\right )} + 143360 \, x^{3} e^{\frac {50}{3}} - 301056 \, x^{3} e^{\frac {38}{3}} - 86016 \, x^{3} e^{\left (\frac {2}{3} \, x + 20\right )} + 188160 \, x^{3} e^{\left (\frac {2}{3} \, x + 16\right )} - 57344 \, x^{3} e^{\left (-\frac {2}{3} \, x + \frac {40}{3}\right )} + 112896 \, x^{3} e^{\left (-\frac {2}{3} \, x + \frac {28}{3}\right )} - 16384 \, x^{2} e^{\frac {62}{3}} + 43008 \, x^{2} e^{\frac {50}{3}} + 8192 \, x^{2} e^{\left (\frac {2}{3} \, x + 24\right )} - 21504 \, x^{2} e^{\left (\frac {2}{3} \, x + 20\right )} + 8192 \, x^{2} e^{\left (-\frac {2}{3} \, x + \frac {52}{3}\right )} - 21504 \, x^{2} e^{\left (-\frac {2}{3} \, x + \frac {40}{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((128*x-48)*exp(-2/3*x+2/3)+(-96*x+48)*exp(4)+168*x^2-168*x)*exp(x)^2/(48*x^2*exp(-2/3*x+2/3)^2+(-96
*x^2*exp(4)+168*x^3)*exp(-2/3*x+2/3)+48*x^2*exp(4)^2-168*x^3*exp(4)+147*x^4),x, algorithm="giac")

[Out]

4*(33614*x^6*e^(8/3*x) + 50421*x^5*e^(8/3*x) - 96040*x^5*e^(8/3*x + 4) + 19208*x^5*e^(2*x + 2/3) + 109760*x^4*
e^(8/3*x + 8) - 115248*x^4*e^(8/3*x + 4) - 43904*x^4*e^(2*x + 14/3) + 28812*x^4*e^(2*x + 2/3) - 62720*x^3*e^(8
/3*x + 12) + 98784*x^3*e^(8/3*x + 8) + 37632*x^3*e^(2*x + 26/3) - 49392*x^3*e^(2*x + 14/3) + 17920*x^2*e^(8/3*
x + 16) - 37632*x^2*e^(8/3*x + 12) - 14336*x^2*e^(2*x + 38/3) + 28224*x^2*e^(2*x + 26/3) - 2048*x*e^(8/3*x + 2
0) + 5376*x*e^(8/3*x + 16) + 2048*x*e^(2*x + 50/3) - 5376*x*e^(2*x + 38/3))/(235298*x^8*e^(2/3*x) + 268912*x^7
*e^(2/3) + 352947*x^7*e^(2/3*x) - 806736*x^7*e^(2/3*x + 4) - 768320*x^6*e^(14/3) + 403368*x^6*e^(2/3) + 115248
0*x^6*e^(2/3*x + 8) - 1008420*x^6*e^(2/3*x + 4) + 76832*x^6*e^(-2/3*x + 4/3) + 878080*x^5*e^(26/3) - 921984*x^
5*e^(14/3) - 878080*x^5*e^(2/3*x + 12) + 1152480*x^5*e^(2/3*x + 8) - 175616*x^5*e^(-2/3*x + 16/3) + 115248*x^5
*e^(-2/3*x + 4/3) - 501760*x^4*e^(38/3) + 790272*x^4*e^(26/3) + 376320*x^4*e^(2/3*x + 16) - 658560*x^4*e^(2/3*
x + 12) + 150528*x^4*e^(-2/3*x + 28/3) - 197568*x^4*e^(-2/3*x + 16/3) + 143360*x^3*e^(50/3) - 301056*x^3*e^(38
/3) - 86016*x^3*e^(2/3*x + 20) + 188160*x^3*e^(2/3*x + 16) - 57344*x^3*e^(-2/3*x + 40/3) + 112896*x^3*e^(-2/3*
x + 28/3) - 16384*x^2*e^(62/3) + 43008*x^2*e^(50/3) + 8192*x^2*e^(2/3*x + 24) - 21504*x^2*e^(2/3*x + 20) + 819
2*x^2*e^(-2/3*x + 52/3) - 21504*x^2*e^(-2/3*x + 40/3))

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maple [F]  time = 0.21, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (128 x -48\right ) {\mathrm e}^{-\frac {2 x}{3}+\frac {2}{3}}+\left (-96 x +48\right ) {\mathrm e}^{4}+168 x^{2}-168 x \right ) {\mathrm e}^{2 x}}{48 x^{2} {\mathrm e}^{-\frac {4 x}{3}+\frac {4}{3}}+\left (-96 x^{2} {\mathrm e}^{4}+168 x^{3}\right ) {\mathrm e}^{-\frac {2 x}{3}+\frac {2}{3}}+48 x^{2} {\mathrm e}^{8}-168 x^{3} {\mathrm e}^{4}+147 x^{4}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((128*x-48)*exp(-2/3*x+2/3)+(-96*x+48)*exp(4)+168*x^2-168*x)*exp(x)^2/(48*x^2*exp(-2/3*x+2/3)^2+(-96*x^2*e
xp(4)+168*x^3)*exp(-2/3*x+2/3)+48*x^2*exp(4)^2-168*x^3*exp(4)+147*x^4),x)

[Out]

int(((128*x-48)*exp(-2/3*x+2/3)+(-96*x+48)*exp(4)+168*x^2-168*x)*exp(x)^2/(48*x^2*exp(-2/3*x+2/3)^2+(-96*x^2*e
xp(4)+168*x^3)*exp(-2/3*x+2/3)+48*x^2*exp(4)^2-168*x^3*exp(4)+147*x^4),x)

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maxima [A]  time = 0.46, size = 34, normalized size = 1.03 \begin {gather*} \frac {4 \, e^{\left (\frac {8}{3} \, x + \frac {1}{3}\right )}}{4 \, x e + {\left (7 \, x^{2} e^{\frac {1}{3}} - 4 \, x e^{\frac {13}{3}}\right )} e^{\left (\frac {2}{3} \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((128*x-48)*exp(-2/3*x+2/3)+(-96*x+48)*exp(4)+168*x^2-168*x)*exp(x)^2/(48*x^2*exp(-2/3*x+2/3)^2+(-96
*x^2*exp(4)+168*x^3)*exp(-2/3*x+2/3)+48*x^2*exp(4)^2-168*x^3*exp(4)+147*x^4),x, algorithm="maxima")

[Out]

4*e^(8/3*x + 1/3)/(4*x*e + (7*x^2*e^(1/3) - 4*x*e^(13/3))*e^(2/3*x))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} -\int \frac {{\mathrm {e}}^{2\,x}\,\left (168\,x-{\mathrm {e}}^{\frac {2}{3}-\frac {2\,x}{3}}\,\left (128\,x-48\right )-168\,x^2+{\mathrm {e}}^4\,\left (96\,x-48\right )\right )}{48\,x^2\,{\mathrm {e}}^8-168\,x^3\,{\mathrm {e}}^4-{\mathrm {e}}^{\frac {2}{3}-\frac {2\,x}{3}}\,\left (96\,x^2\,{\mathrm {e}}^4-168\,x^3\right )+48\,x^2\,{\mathrm {e}}^{\frac {4}{3}-\frac {4\,x}{3}}+147\,x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(2*x)*(168*x - exp(2/3 - (2*x)/3)*(128*x - 48) - 168*x^2 + exp(4)*(96*x - 48)))/(48*x^2*exp(8) - 168*
x^3*exp(4) - exp(2/3 - (2*x)/3)*(96*x^2*exp(4) - 168*x^3) + 48*x^2*exp(4/3 - (4*x)/3) + 147*x^4),x)

[Out]

-int((exp(2*x)*(168*x - exp(2/3 - (2*x)/3)*(128*x - 48) - 168*x^2 + exp(4)*(96*x - 48)))/(48*x^2*exp(8) - 168*
x^3*exp(4) - exp(2/3 - (2*x)/3)*(96*x^2*exp(4) - 168*x^3) + 48*x^2*exp(4/3 - (4*x)/3) + 147*x^4), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((128*x-48)*exp(-2/3*x+2/3)+(-96*x+48)*exp(4)+168*x**2-168*x)*exp(x)**2/(48*x**2*exp(-2/3*x+2/3)**2+
(-96*x**2*exp(4)+168*x**3)*exp(-2/3*x+2/3)+48*x**2*exp(4)**2-168*x**3*exp(4)+147*x**4),x)

[Out]

Timed out

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