3.66.4 \(\int \frac {e^{-x} (-5600+525 x+20500 x^5-750 x^6-12500 x^{10})}{3675-630 x+27 x^2-10500 x^5+900 x^6+7500 x^{10}} \, dx\)

Optimal. Leaf size=25 \[ \frac {5 e^{-x}}{3-\frac {9 x}{5 \left (7-10 x^5\right )}} \]

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Rubi [F]  time = 0.75, 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^{-x} \left (-5600+525 x+20500 x^5-750 x^6-12500 x^{10}\right )}{3675-630 x+27 x^2-10500 x^5+900 x^6+7500 x^{10}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-5600 + 525*x + 20500*x^5 - 750*x^6 - 12500*x^10)/(E^x*(3675 - 630*x + 27*x^2 - 10500*x^5 + 900*x^6 + 750
0*x^10)),x]

[Out]

5/(3*E^x) + 875*Defer[Int][1/(E^x*(-35 + 3*x + 50*x^5)^2), x] - 60*Defer[Int][x/(E^x*(-35 + 3*x + 50*x^5)^2),
x] + 20*Defer[Int][1/(E^x*(-35 + 3*x + 50*x^5)), x] + 5*Defer[Int][x/(E^x*(-35 + 3*x + 50*x^5)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {25 e^{-x} \left (-224+21 x+820 x^5-30 x^6-500 x^{10}\right )}{3 \left (35-3 x-50 x^5\right )^2} \, dx\\ &=\frac {25}{3} \int \frac {e^{-x} \left (-224+21 x+820 x^5-30 x^6-500 x^{10}\right )}{\left (35-3 x-50 x^5\right )^2} \, dx\\ &=\frac {25}{3} \int \left (-\frac {e^{-x}}{5}-\frac {3 e^{-x} (-175+12 x)}{5 \left (-35+3 x+50 x^5\right )^2}+\frac {3 e^{-x} (4+x)}{5 \left (-35+3 x+50 x^5\right )}\right ) \, dx\\ &=-\left (\frac {5}{3} \int e^{-x} \, dx\right )-5 \int \frac {e^{-x} (-175+12 x)}{\left (-35+3 x+50 x^5\right )^2} \, dx+5 \int \frac {e^{-x} (4+x)}{-35+3 x+50 x^5} \, dx\\ &=\frac {5 e^{-x}}{3}-5 \int \left (-\frac {175 e^{-x}}{\left (-35+3 x+50 x^5\right )^2}+\frac {12 e^{-x} x}{\left (-35+3 x+50 x^5\right )^2}\right ) \, dx+5 \int \left (\frac {4 e^{-x}}{-35+3 x+50 x^5}+\frac {e^{-x} x}{-35+3 x+50 x^5}\right ) \, dx\\ &=\frac {5 e^{-x}}{3}+5 \int \frac {e^{-x} x}{-35+3 x+50 x^5} \, dx+20 \int \frac {e^{-x}}{-35+3 x+50 x^5} \, dx-60 \int \frac {e^{-x} x}{\left (-35+3 x+50 x^5\right )^2} \, dx+875 \int \frac {e^{-x}}{\left (-35+3 x+50 x^5\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.18, size = 30, normalized size = 1.20 \begin {gather*} -\frac {25}{3} e^{-x} \left (-\frac {1}{5}+\frac {3 x}{5 \left (-35+3 x+50 x^5\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-5600 + 525*x + 20500*x^5 - 750*x^6 - 12500*x^10)/(E^x*(3675 - 630*x + 27*x^2 - 10500*x^5 + 900*x^6
+ 7500*x^10)),x]

[Out]

(-25*(-1/5 + (3*x)/(5*(-35 + 3*x + 50*x^5))))/(3*E^x)

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fricas [A]  time = 0.78, size = 25, normalized size = 1.00 \begin {gather*} \frac {25 \, {\left (10 \, x^{5} - 7\right )} e^{\left (-x\right )}}{3 \, {\left (50 \, x^{5} + 3 \, x - 35\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12500*x^10-750*x^6+20500*x^5+525*x-5600)/(7500*x^10+900*x^6-10500*x^5+27*x^2-630*x+3675)/exp(x),x,
algorithm="fricas")

[Out]

25/3*(10*x^5 - 7)*e^(-x)/(50*x^5 + 3*x - 35)

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giac [A]  time = 0.16, size = 30, normalized size = 1.20 \begin {gather*} \frac {25 \, {\left (10 \, x^{5} e^{\left (-x\right )} - 7 \, e^{\left (-x\right )}\right )}}{3 \, {\left (50 \, x^{5} + 3 \, x - 35\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12500*x^10-750*x^6+20500*x^5+525*x-5600)/(7500*x^10+900*x^6-10500*x^5+27*x^2-630*x+3675)/exp(x),x,
algorithm="giac")

[Out]

25/3*(10*x^5*e^(-x) - 7*e^(-x))/(50*x^5 + 3*x - 35)

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maple [A]  time = 0.12, size = 25, normalized size = 1.00

method	result	size
norman	\(\frac {\left (-\frac {175}{3}+\frac {250 x^{5}}{3}\right ) {\mathrm e}^{-x}}{50 x^{5}+3 x -35}\)	\(25\)
gosper	\(\frac {25 \left (10 x^{5}-7\right ) {\mathrm e}^{-x}}{3 \left (50 x^{5}+3 x -35\right )}\)	\(26\)
risch	\(\frac {25 \left (10 x^{5}-7\right ) {\mathrm e}^{-x}}{3 \left (50 x^{5}+3 x -35\right )}\)	\(26\)
default	\(\frac {5600 \,{\mathrm e}^{-x} \left (216000 x^{4}+3150000 x^{3}[phone] x^{2}[phone] x +10368\right )}{351709077687 \left (50 x^{5}+3 x -35\right )}-\frac {5600 \left (\munderset {\textit {\_R1} =\RootOf \left (50 \textit {\_Z}^{5}+3 \textit {\_Z} -35\right )}{\sum }\frac {\left (216000 \textit {\_R1}^{4}+3366000 \textit {\_R1}^{3}[phone] \textit {\_R1}^{2}[phone] \textit {\_R1} [phone]\right ) {\mathrm e}^{-\textit {\_R1}} \expIntegralEi \left (1, x -\textit {\_R1} \right )}{250 \textit {\_R1}^{4}+3}\right )}{351709077687}-\frac {175 \,{\mathrm e}^{-x} \left (3150000 x^{4}[phone] x^{3}[phone] x^{2}-2592 x +151200\right )}{117236359229 \left (50 x^{5}+3 x -35\right )}+\frac {175 \left (\munderset {\textit {\_R1} =\RootOf \left (50 \textit {\_Z}^{5}+3 \textit {\_Z} -35\right )}{\sum }\frac {\left (3150000 \textit {\_R1}^{4}[phone] \textit {\_R1}^{3}[phone] \textit {\_R1}^{2}[phone] \textit {\_R1} +140832\right ) {\mathrm e}^{-\textit {\_R1}} \expIntegralEi \left (1, x -\textit {\_R1} \right )}{250 \textit {\_R1}^{4}+3}\right )}{117236359229}+\frac {410 \,{\mathrm e}^{-x} \left (1890000 x^{4}[phone] x^{3}[phone] x^{2}-23447273401 x +90720\right )}{351709077687 \left (50 x^{5}+3 x -35\right )}-\frac {410 \left (\munderset {\textit {\_R1} =\RootOf \left (50 \textit {\_Z}^{5}+3 \textit {\_Z} -35\right )}{\sum }\frac {\left (1890000 \textit {\_R1}^{4}[phone] \textit {\_R1}^{3}[phone] \textit {\_R1}^{2}-22241414026 \textit {\_R1} [phone]\right ) {\mathrm e}^{-\textit {\_R1}} \expIntegralEi \left (1, x -\textit {\_R1} \right )}{250 \textit {\_R1}^{4}+3}\right )}{351709077687}-\frac {5 \,{\mathrm e}^{-x} \left (27562500 x^{4}[phone] x^{3}-23447273401 x^{2}-22680 x +1323000\right )}{117236359229 \left (50 x^{5}+3 x -35\right )}+\frac {5 \left (\munderset {\textit {\_R1} =\RootOf \left (50 \textit {\_Z}^{5}+3 \textit {\_Z} -35\right )}{\sum }\frac {\left (27562500 \textit {\_R1}^{4}[phone] \textit {\_R1}^{3}-22643367151 \textit {\_R1}^{2}[phone] \textit {\_R1} +1232280\right ) {\mathrm e}^{-\textit {\_R1}} \expIntegralEi \left (1, x -\textit {\_R1} \right )}{250 \textit {\_R1}^{4}+3}\right )}{117236359229}+\frac {5 \,{\mathrm e}^{-x}}{3}+\frac {5 \,{\mathrm e}^{-x} \left (16537500 x^{4}[phone] x^{3}-84410179578 x^{2}[phone] x +793800\right )}{351709077687 \left (50 x^{5}+3 x -35\right )}-\frac {5 \left (\munderset {\textit {\_R1} =\RootOf \left (50 \textit {\_Z}^{5}+3 \textit {\_Z} -35\right )}{\sum }\frac {\left (16537500 \textit {\_R1}^{4}[phone] \textit {\_R1}^{3}-83927835828 \textit {\_R1}^{2}[phone] \textit {\_R1} -4923926348250\right ) {\mathrm e}^{-\textit {\_R1}} \expIntegralEi \left (1, x -\textit {\_R1} \right )}{250 \textit {\_R1}^{4}+3}\right )}{351709077687}\)	\(483\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-12500*x^10-750*x^6+20500*x^5+525*x-5600)/(7500*x^10+900*x^6-10500*x^5+27*x^2-630*x+3675)/exp(x),x,method
=_RETURNVERBOSE)

[Out]

(-175/3+250/3*x^5)/(50*x^5+3*x-35)/exp(x)

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maxima [A]  time = 0.41, size = 25, normalized size = 1.00 \begin {gather*} \frac {25 \, {\left (10 \, x^{5} - 7\right )} e^{\left (-x\right )}}{3 \, {\left (50 \, x^{5} + 3 \, x - 35\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12500*x^10-750*x^6+20500*x^5+525*x-5600)/(7500*x^10+900*x^6-10500*x^5+27*x^2-630*x+3675)/exp(x),x,
algorithm="maxima")

[Out]

25/3*(10*x^5 - 7)*e^(-x)/(50*x^5 + 3*x - 35)

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mupad [B]  time = 4.23, size = 22, normalized size = 0.88 \begin {gather*} \frac {{\mathrm {e}}^{-x}\,\left (\frac {5\,x^5}{3}-\frac {7}{6}\right )}{x^5+\frac {3\,x}{50}-\frac {7}{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-x)*(750*x^6 - 20500*x^5 - 525*x + 12500*x^10 + 5600))/(27*x^2 - 630*x - 10500*x^5 + 900*x^6 + 7500*
x^10 + 3675),x)

[Out]

(exp(-x)*((5*x^5)/3 - 7/6))/((3*x)/50 + x^5 - 7/10)

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sympy [A]  time = 0.12, size = 19, normalized size = 0.76 \begin {gather*} \frac {\left (250 x^{5} - 175\right ) e^{- x}}{150 x^{5} + 9 x - 105} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12500*x**10-750*x**6+20500*x**5+525*x-5600)/(7500*x**10+900*x**6-10500*x**5+27*x**2-630*x+3675)/ex
p(x),x)

[Out]

(250*x**5 - 175)*exp(-x)/(150*x**5 + 9*x - 105)

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