3.73.62 \(\int \frac {-69177612+5618340 x-172284 x^2+2352 x^3-12 x^4+e^x (103766418+60706884 x-5387844 x^2+169344 x^3-2334 x^4+12 x^5)}{23059204 x^2-1872780 x^3+57233 x^4-780 x^5+4 x^6+e^x (-23059204 x-44240826 x^2+3697736 x^3-114074 x^4+1560 x^5-8 x^6)+e^{2 x} (5764801+22588608 x+21191226 x^2-1824956 x^3+56841 x^4-780 x^5+4 x^6)} \, dx\)

Optimal. Leaf size=33 \[ \frac {3}{x-e^x x+\frac {1}{2} \left (-e^x+\frac {x^2}{(49-x)^2}\right )} \]

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Rubi [F]  time = 4.56, 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 {-69177612+5618340 x-172284 x^2+2352 x^3-12 x^4+e^x \left (103766418+60706884 x-5387844 x^2+169344 x^3-2334 x^4+12 x^5\right )}{23059204 x^2-1872780 x^3+57233 x^4-780 x^5+4 x^6+e^x \left (-23059204 x-44240826 x^2+3697736 x^3-114074 x^4+1560 x^5-8 x^6\right )+e^{2 x} \left (5764801+22588608 x+21191226 x^2-1824956 x^3+56841 x^4-780 x^5+4 x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-69177612 + 5618340*x - 172284*x^2 + 2352*x^3 - 12*x^4 + E^x*(103766418 + 60706884*x - 5387844*x^2 + 1693
44*x^3 - 2334*x^4 + 12*x^5))/(23059204*x^2 - 1872780*x^3 + 57233*x^4 - 780*x^5 + 4*x^6 + E^x*(-23059204*x - 44
240826*x^2 + 3697736*x^3 - 114074*x^4 + 1560*x^5 - 8*x^6) + E^(2*x)*(5764801 + 22588608*x + 21191226*x^2 - 182
4956*x^3 + 56841*x^4 - 780*x^5 + 4*x^6)),x]

[Out]

2859738*Defer[Int][(E^x*(-49 + x)^2*(1 + 2*x) + x*(-4802 + 195*x - 2*x^2))^(-2), x] + 69076476*Defer[Int][x/(E
^x*(-49 + x)^2*(1 + 2*x) + x*(-4802 + 195*x - 2*x^2))^2, x] - 5631570*Defer[Int][x^2/(E^x*(-49 + x)^2*(1 + 2*x
) + x*(-4802 + 195*x - 2*x^2))^2, x] + 172284*Defer[Int][x^3/(E^x*(-49 + x)^2*(1 + 2*x) + x*(-4802 + 195*x - 2
*x^2))^2, x] - 2346*Defer[Int][x^4/(E^x*(-49 + x)^2*(1 + 2*x) + x*(-4802 + 195*x - 2*x^2))^2, x] + 12*Defer[In
t][x^5/(E^x*(-49 + x)^2*(1 + 2*x) + x*(-4802 + 195*x - 2*x^2))^2, x] - 72037350*Defer[Int][1/((1 + 2*x)*(E^x*(
-49 + x)^2*(1 + 2*x) + x*(-4802 + 195*x - 2*x^2))^2), x] + 13815*Defer[Int][(E^x*(-49 + x)^2*(1 + 2*x) + x*(-4
802 + 195*x - 2*x^2))^(-1), x] - 582*Defer[Int][x/(E^x*(-49 + x)^2*(1 + 2*x) + x*(-4802 + 195*x - 2*x^2)), x]
+ 6*Defer[Int][x^2/(E^x*(-49 + x)^2*(1 + 2*x) + x*(-4802 + 195*x - 2*x^2)), x] + 29403*Defer[Int][1/((1 + 2*x)
*(E^x*(-49 + x)^2*(1 + 2*x) + x*(-4802 + 195*x - 2*x^2))), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {6 (49-x) \left (-e^x (-49+x)^3 (3+2 x)+2 \left (-117649+7154 x-147 x^2+x^3\right )\right )}{\left (e^x (-49+x)^2 (1+2 x)+x \left (-4802+195 x-2 x^2\right )\right )^2} \, dx\\ &=6 \int \frac {(49-x) \left (-e^x (-49+x)^3 (3+2 x)+2 \left (-117649+7154 x-147 x^2+x^3\right )\right )}{\left (e^x (-49+x)^2 (1+2 x)+x \left (-4802+195 x-2 x^2\right )\right )^2} \, dx\\ &=6 \int \left (\frac {(-49+x)^2 (3+2 x)}{(1+2 x) \left (2401 e^x-4802 x+4704 e^x x+195 x^2-195 e^x x^2-2 x^3+2 e^x x^3\right )}+\frac {-11529602+12465992 x+22086897 x^2-1848476 x^3+57037 x^4-780 x^5+4 x^6}{(1+2 x) \left (2401 e^x-4802 x+4704 e^x x+195 x^2-195 e^x x^2-2 x^3+2 e^x x^3\right )^2}\right ) \, dx\\ &=6 \int \frac {(-49+x)^2 (3+2 x)}{(1+2 x) \left (2401 e^x-4802 x+4704 e^x x+195 x^2-195 e^x x^2-2 x^3+2 e^x x^3\right )} \, dx+6 \int \frac {-11529602+12465992 x+22086897 x^2-1848476 x^3+57037 x^4-780 x^5+4 x^6}{(1+2 x) \left (2401 e^x-4802 x+4704 e^x x+195 x^2-195 e^x x^2-2 x^3+2 e^x x^3\right )^2} \, dx\\ &=6 \int \frac {-11529602+12465992 x+22086897 x^2-1848476 x^3+57037 x^4-780 x^5+4 x^6}{(1+2 x) \left (e^x (-49+x)^2 (1+2 x)+x \left (-4802+195 x-2 x^2\right )\right )^2} \, dx+6 \int \left (\frac {4605}{2 \left (2401 e^x-4802 x+4704 e^x x+195 x^2-195 e^x x^2-2 x^3+2 e^x x^3\right )}-\frac {97 x}{2401 e^x-4802 x+4704 e^x x+195 x^2-195 e^x x^2-2 x^3+2 e^x x^3}+\frac {x^2}{2401 e^x-4802 x+4704 e^x x+195 x^2-195 e^x x^2-2 x^3+2 e^x x^3}+\frac {9801}{2 (1+2 x) \left (2401 e^x-4802 x+4704 e^x x+195 x^2-195 e^x x^2-2 x^3+2 e^x x^3\right )}\right ) \, dx\\ &=6 \int \frac {x^2}{2401 e^x-4802 x+4704 e^x x+195 x^2-195 e^x x^2-2 x^3+2 e^x x^3} \, dx+6 \int \left (\frac {476623}{\left (2401 e^x-4802 x+4704 e^x x+195 x^2-195 e^x x^2-2 x^3+2 e^x x^3\right )^2}+\frac {11512746 x}{\left (2401 e^x-4802 x+4704 e^x x+195 x^2-195 e^x x^2-2 x^3+2 e^x x^3\right )^2}-\frac {938595 x^2}{\left (2401 e^x-4802 x+4704 e^x x+195 x^2-195 e^x x^2-2 x^3+2 e^x x^3\right )^2}+\frac {28714 x^3}{\left (2401 e^x-4802 x+4704 e^x x+195 x^2-195 e^x x^2-2 x^3+2 e^x x^3\right )^2}-\frac {391 x^4}{\left (2401 e^x-4802 x+4704 e^x x+195 x^2-195 e^x x^2-2 x^3+2 e^x x^3\right )^2}+\frac {2 x^5}{\left (2401 e^x-4802 x+4704 e^x x+195 x^2-195 e^x x^2-2 x^3+2 e^x x^3\right )^2}-\frac {12006225}{(1+2 x) \left (2401 e^x-4802 x+4704 e^x x+195 x^2-195 e^x x^2-2 x^3+2 e^x x^3\right )^2}\right ) \, dx-582 \int \frac {x}{2401 e^x-4802 x+4704 e^x x+195 x^2-195 e^x x^2-2 x^3+2 e^x x^3} \, dx+13815 \int \frac {1}{2401 e^x-4802 x+4704 e^x x+195 x^2-195 e^x x^2-2 x^3+2 e^x x^3} \, dx+29403 \int \frac {1}{(1+2 x) \left (2401 e^x-4802 x+4704 e^x x+195 x^2-195 e^x x^2-2 x^3+2 e^x x^3\right )} \, dx\\ &=6 \int \frac {x^2}{e^x (-49+x)^2 (1+2 x)+x \left (-4802+195 x-2 x^2\right )} \, dx+12 \int \frac {x^5}{\left (2401 e^x-4802 x+4704 e^x x+195 x^2-195 e^x x^2-2 x^3+2 e^x x^3\right )^2} \, dx-582 \int \frac {x}{e^x (-49+x)^2 (1+2 x)+x \left (-4802+195 x-2 x^2\right )} \, dx-2346 \int \frac {x^4}{\left (2401 e^x-4802 x+4704 e^x x+195 x^2-195 e^x x^2-2 x^3+2 e^x x^3\right )^2} \, dx+13815 \int \frac {1}{e^x (-49+x)^2 (1+2 x)+x \left (-4802+195 x-2 x^2\right )} \, dx+29403 \int \frac {1}{(1+2 x) \left (e^x (-49+x)^2 (1+2 x)+x \left (-4802+195 x-2 x^2\right )\right )} \, dx+172284 \int \frac {x^3}{\left (2401 e^x-4802 x+4704 e^x x+195 x^2-195 e^x x^2-2 x^3+2 e^x x^3\right )^2} \, dx+2859738 \int \frac {1}{\left (2401 e^x-4802 x+4704 e^x x+195 x^2-195 e^x x^2-2 x^3+2 e^x x^3\right )^2} \, dx-5631570 \int \frac {x^2}{\left (2401 e^x-4802 x+4704 e^x x+195 x^2-195 e^x x^2-2 x^3+2 e^x x^3\right )^2} \, dx+69076476 \int \frac {x}{\left (2401 e^x-4802 x+4704 e^x x+195 x^2-195 e^x x^2-2 x^3+2 e^x x^3\right )^2} \, dx-72037350 \int \frac {1}{(1+2 x) \left (2401 e^x-4802 x+4704 e^x x+195 x^2-195 e^x x^2-2 x^3+2 e^x x^3\right )^2} \, dx\\ &=6 \int \frac {x^2}{e^x (-49+x)^2 (1+2 x)+x \left (-4802+195 x-2 x^2\right )} \, dx+12 \int \frac {x^5}{\left (e^x (-49+x)^2 (1+2 x)+x \left (-4802+195 x-2 x^2\right )\right )^2} \, dx-582 \int \frac {x}{e^x (-49+x)^2 (1+2 x)+x \left (-4802+195 x-2 x^2\right )} \, dx-2346 \int \frac {x^4}{\left (e^x (-49+x)^2 (1+2 x)+x \left (-4802+195 x-2 x^2\right )\right )^2} \, dx+13815 \int \frac {1}{e^x (-49+x)^2 (1+2 x)+x \left (-4802+195 x-2 x^2\right )} \, dx+29403 \int \frac {1}{(1+2 x) \left (e^x (-49+x)^2 (1+2 x)+x \left (-4802+195 x-2 x^2\right )\right )} \, dx+172284 \int \frac {x^3}{\left (e^x (-49+x)^2 (1+2 x)+x \left (-4802+195 x-2 x^2\right )\right )^2} \, dx+2859738 \int \frac {1}{\left (e^x (-49+x)^2 (1+2 x)+x \left (-4802+195 x-2 x^2\right )\right )^2} \, dx-5631570 \int \frac {x^2}{\left (e^x (-49+x)^2 (1+2 x)+x \left (-4802+195 x-2 x^2\right )\right )^2} \, dx+69076476 \int \frac {x}{\left (e^x (-49+x)^2 (1+2 x)+x \left (-4802+195 x-2 x^2\right )\right )^2} \, dx-72037350 \int \frac {1}{(1+2 x) \left (e^x (-49+x)^2 (1+2 x)+x \left (-4802+195 x-2 x^2\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.77, size = 36, normalized size = 1.09 \begin {gather*} -\frac {6 (-49+x)^2}{e^x (-49+x)^2 (1+2 x)+x \left (-4802+195 x-2 x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-69177612 + 5618340*x - 172284*x^2 + 2352*x^3 - 12*x^4 + E^x*(103766418 + 60706884*x - 5387844*x^2
+ 169344*x^3 - 2334*x^4 + 12*x^5))/(23059204*x^2 - 1872780*x^3 + 57233*x^4 - 780*x^5 + 4*x^6 + E^x*(-23059204*
x - 44240826*x^2 + 3697736*x^3 - 114074*x^4 + 1560*x^5 - 8*x^6) + E^(2*x)*(5764801 + 22588608*x + 21191226*x^2
- 1824956*x^3 + 56841*x^4 - 780*x^5 + 4*x^6)),x]

[Out]

(-6*(-49 + x)^2)/(E^x*(-49 + x)^2*(1 + 2*x) + x*(-4802 + 195*x - 2*x^2))

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fricas [A]  time = 0.87, size = 45, normalized size = 1.36 \begin {gather*} \frac {6 \, {\left (x^{2} - 98 \, x + 2401\right )}}{2 \, x^{3} - 195 \, x^{2} - {\left (2 \, x^{3} - 195 \, x^{2} + 4704 \, x + 2401\right )} e^{x} + 4802 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x^5-2334*x^4+169344*x^3-5387844*x^2+60706884*x+103766418)*exp(x)-12*x^4+2352*x^3-172284*x^2+561
8340*x-69177612)/((4*x^6-780*x^5+56841*x^4-1824956*x^3+21191226*x^2+22588608*x+5764801)*exp(x)^2+(-8*x^6+1560*
x^5-114074*x^4+3697736*x^3-44240826*x^2-23059204*x)*exp(x)+4*x^6-780*x^5+57233*x^4-1872780*x^3+23059204*x^2),x
, algorithm="fricas")

[Out]

6*(x^2 - 98*x + 2401)/(2*x^3 - 195*x^2 - (2*x^3 - 195*x^2 + 4704*x + 2401)*e^x + 4802*x)

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giac [A]  time = 0.30, size = 49, normalized size = 1.48 \begin {gather*} -\frac {6 \, {\left (x^{2} - 98 \, x + 2401\right )}}{2 \, x^{3} e^{x} - 2 \, x^{3} - 195 \, x^{2} e^{x} + 195 \, x^{2} + 4704 \, x e^{x} - 4802 \, x + 2401 \, e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x^5-2334*x^4+169344*x^3-5387844*x^2+60706884*x+103766418)*exp(x)-12*x^4+2352*x^3-172284*x^2+561
8340*x-69177612)/((4*x^6-780*x^5+56841*x^4-1824956*x^3+21191226*x^2+22588608*x+5764801)*exp(x)^2+(-8*x^6+1560*
x^5-114074*x^4+3697736*x^3-44240826*x^2-23059204*x)*exp(x)+4*x^6-780*x^5+57233*x^4-1872780*x^3+23059204*x^2),x
, algorithm="giac")

[Out]

-6*(x^2 - 98*x + 2401)/(2*x^3*e^x - 2*x^3 - 195*x^2*e^x + 195*x^2 + 4704*x*e^x - 4802*x + 2401*e^x)

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maple [A]  time = 0.10, size = 47, normalized size = 1.42

method	result	size
risch	\(-\frac {6 \left (x -49\right )^{2}}{2 \,{\mathrm e}^{x} x^{3}-195 \,{\mathrm e}^{x} x^{2}-2 x^{3}+4704 \,{\mathrm e}^{x} x +195 x^{2}+2401 \,{\mathrm e}^{x}-4802 x}\)	\(47\)
norman	\(\frac {-6 x^{2}+588 x -14406}{2 \,{\mathrm e}^{x} x^{3}-195 \,{\mathrm e}^{x} x^{2}-2 x^{3}+4704 \,{\mathrm e}^{x} x +195 x^{2}+2401 \,{\mathrm e}^{x}-4802 x}\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((12*x^5-2334*x^4+169344*x^3-5387844*x^2+60706884*x+103766418)*exp(x)-12*x^4+2352*x^3-172284*x^2+5618340*x
-69177612)/((4*x^6-780*x^5+56841*x^4-1824956*x^3+21191226*x^2+22588608*x+5764801)*exp(x)^2+(-8*x^6+1560*x^5-11
4074*x^4+3697736*x^3-44240826*x^2-23059204*x)*exp(x)+4*x^6-780*x^5+57233*x^4-1872780*x^3+23059204*x^2),x,metho
d=_RETURNVERBOSE)

[Out]

-6*(x-49)^2/(2*exp(x)*x^3-195*exp(x)*x^2-2*x^3+4704*exp(x)*x+195*x^2+2401*exp(x)-4802*x)

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maxima [A]  time = 0.45, size = 45, normalized size = 1.36 \begin {gather*} \frac {6 \, {\left (x^{2} - 98 \, x + 2401\right )}}{2 \, x^{3} - 195 \, x^{2} - {\left (2 \, x^{3} - 195 \, x^{2} + 4704 \, x + 2401\right )} e^{x} + 4802 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x^5-2334*x^4+169344*x^3-5387844*x^2+60706884*x+103766418)*exp(x)-12*x^4+2352*x^3-172284*x^2+561
8340*x-69177612)/((4*x^6-780*x^5+56841*x^4-1824956*x^3+21191226*x^2+22588608*x+5764801)*exp(x)^2+(-8*x^6+1560*
x^5-114074*x^4+3697736*x^3-44240826*x^2-23059204*x)*exp(x)+4*x^6-780*x^5+57233*x^4-1872780*x^3+23059204*x^2),x
, algorithm="maxima")

[Out]

6*(x^2 - 98*x + 2401)/(2*x^3 - 195*x^2 - (2*x^3 - 195*x^2 + 4704*x + 2401)*e^x + 4802*x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {5618340\,x+{\mathrm {e}}^x\,\left (12\,x^5-2334\,x^4+169344\,x^3-5387844\,x^2+60706884\,x+103766418\right )-172284\,x^2+2352\,x^3-12\,x^4-69177612}{23059204\,x^2-{\mathrm {e}}^x\,\left (8\,x^6-1560\,x^5+114074\,x^4-3697736\,x^3+44240826\,x^2+23059204\,x\right )-1872780\,x^3+57233\,x^4-780\,x^5+4\,x^6+{\mathrm {e}}^{2\,x}\,\left (4\,x^6-780\,x^5+56841\,x^4-1824956\,x^3+21191226\,x^2+22588608\,x+5764801\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5618340*x + exp(x)*(60706884*x - 5387844*x^2 + 169344*x^3 - 2334*x^4 + 12*x^5 + 103766418) - 172284*x^2 +
2352*x^3 - 12*x^4 - 69177612)/(23059204*x^2 - exp(x)*(23059204*x + 44240826*x^2 - 3697736*x^3 + 114074*x^4 -
1560*x^5 + 8*x^6) - 1872780*x^3 + 57233*x^4 - 780*x^5 + 4*x^6 + exp(2*x)*(22588608*x + 21191226*x^2 - 1824956*
x^3 + 56841*x^4 - 780*x^5 + 4*x^6 + 5764801)),x)

[Out]

int((5618340*x + exp(x)*(60706884*x - 5387844*x^2 + 169344*x^3 - 2334*x^4 + 12*x^5 + 103766418) - 172284*x^2 +
2352*x^3 - 12*x^4 - 69177612)/(23059204*x^2 - exp(x)*(23059204*x + 44240826*x^2 - 3697736*x^3 + 114074*x^4 -
1560*x^5 + 8*x^6) - 1872780*x^3 + 57233*x^4 - 780*x^5 + 4*x^6 + exp(2*x)*(22588608*x + 21191226*x^2 - 1824956*
x^3 + 56841*x^4 - 780*x^5 + 4*x^6 + 5764801)), x)

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sympy [A]  time = 0.59, size = 41, normalized size = 1.24 \begin {gather*} \frac {- 6 x^{2} + 588 x - 14406}{- 2 x^{3} + 195 x^{2} - 4802 x + \left (2 x^{3} - 195 x^{2} + 4704 x + 2401\right ) e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x**5-2334*x**4+169344*x**3-5387844*x**2+60706884*x+103766418)*exp(x)-12*x**4+2352*x**3-172284*x
**2+5618340*x-69177612)/((4*x**6-780*x**5+56841*x**4-1824956*x**3+21191226*x**2+22588608*x+5764801)*exp(x)**2+
(-8*x**6+1560*x**5-114074*x**4+3697736*x**3-44240826*x**2-23059204*x)*exp(x)+4*x**6-780*x**5+57233*x**4-187278
0*x**3+23059204*x**2),x)

[Out]

(-6*x**2 + 588*x - 14406)/(-2*x**3 + 195*x**2 - 4802*x + (2*x**3 - 195*x**2 + 4704*x + 2401)*exp(x))

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