3.53.11 \(\int \frac {e^x (125+220 x-90 x^2+100 x^3+e^2 (-10 x+20 x^2))+e^x (125+125 x-20 x^2+10 e^2 x^2+50 x^3) \log (x)}{2500+1200 x+2144 x^2+16 e^4 x^2+480 x^3+400 x^4+e^2 (400 x+96 x^2+160 x^3)+(2500+1200 x+2144 x^2+16 e^4 x^2+480 x^3+400 x^4+e^2 (400 x+96 x^2+160 x^3)) \log (x)+(625+300 x+536 x^2+4 e^4 x^2+120 x^3+100 x^4+e^2 (100 x+24 x^2+40 x^3)) \log ^2(x)} \, dx\)

Optimal. Leaf size=33 \[ \frac {e^x}{2 \left (\frac {1}{5} \left (3+e^2\right )+\frac {5}{2 x}+x\right ) (2+\log (x))} \]

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Rubi [F]  time = 4.72, 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 (125+220 x-90 x^2+100 x^3+e^2 \left (-10 x+20 x^2\right )\right )+e^x \left (125+125 x-20 x^2+10 e^2 x^2+50 x^3\right ) \log (x)}{2500+1200 x+2144 x^2+16 e^4 x^2+480 x^3+400 x^4+e^2 \left (400 x+96 x^2+160 x^3\right )+\left (2500+1200 x+2144 x^2+16 e^4 x^2+480 x^3+400 x^4+e^2 \left (400 x+96 x^2+160 x^3\right )\right ) \log (x)+\left (625+300 x+536 x^2+4 e^4 x^2+120 x^3+100 x^4+e^2 \left (100 x+24 x^2+40 x^3\right )\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^x*(125 + 220*x - 90*x^2 + 100*x^3 + E^2*(-10*x + 20*x^2)) + E^x*(125 + 125*x - 20*x^2 + 10*E^2*x^2 + 50
*x^3)*Log[x])/(2500 + 1200*x + 2144*x^2 + 16*E^4*x^2 + 480*x^3 + 400*x^4 + E^2*(400*x + 96*x^2 + 160*x^3) + (2
500 + 1200*x + 2144*x^2 + 16*E^4*x^2 + 480*x^3 + 400*x^4 + E^2*(400*x + 96*x^2 + 160*x^3))*Log[x] + (625 + 300
*x + 536*x^2 + 4*E^4*x^2 + 120*x^3 + 100*x^4 + E^2*(100*x + 24*x^2 + 40*x^3))*Log[x]^2),x]

[Out]

((-50*I)*Defer[Int][E^x/((-6 - 2*E^2 + (2*I)*Sqrt[241 - 6*E^2 - E^4] - 20*x)*(2 + Log[x])^2), x])/Sqrt[241 - 6
*E^2 - E^4] - ((50*I)*Defer[Int][E^x/((6 + 2*E^2 + (2*I)*Sqrt[241 - 6*E^2 - E^4] + 20*x)*(2 + Log[x])^2), x])/
Sqrt[241 - 6*E^2 - E^4] - ((50*I)*Defer[Int][E^x/((-6 - 2*E^2 + (2*I)*Sqrt[241 - 6*E^2 - E^4] - 20*x)*(2 + Log
[x])), x])/Sqrt[241 - 6*E^2 - E^4] + 5*(1 + (I*(3 + E^2))/Sqrt[241 - 6*E^2 - E^4])*Defer[Int][E^x/((2*(3 + E^2
) - (2*I)*Sqrt[241 - 6*E^2 - E^4] + 20*x)*(2 + Log[x])), x] - ((50*I)*Defer[Int][E^x/((6 + 2*E^2 + (2*I)*Sqrt[
241 - 6*E^2 - E^4] + 20*x)*(2 + Log[x])), x])/Sqrt[241 - 6*E^2 - E^4] + 5*(1 - (I*(3 + E^2))/Sqrt[241 - 6*E^2
- E^4])*Defer[Int][E^x/((2*(3 + E^2) + (2*I)*Sqrt[241 - 6*E^2 - E^4] + 20*x)*(2 + Log[x])), x] + 250*Defer[Int
][E^x/((25 + 2*(3 + E^2)*x + 10*x^2)^2*(2 + Log[x])), x] + 10*(3 + E^2)*Defer[Int][(E^x*x)/((25 + 2*(3 + E^2)*
x + 10*x^2)^2*(2 + Log[x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^x \left (125+220 x-90 x^2+100 x^3+e^2 \left (-10 x+20 x^2\right )\right )+e^x \left (125+125 x-20 x^2+10 e^2 x^2+50 x^3\right ) \log (x)}{2500+1200 x+\left (2144+16 e^4\right ) x^2+480 x^3+400 x^4+e^2 \left (400 x+96 x^2+160 x^3\right )+\left (2500+1200 x+2144 x^2+16 e^4 x^2+480 x^3+400 x^4+e^2 \left (400 x+96 x^2+160 x^3\right )\right ) \log (x)+\left (625+300 x+536 x^2+4 e^4 x^2+120 x^3+100 x^4+e^2 \left (100 x+24 x^2+40 x^3\right )\right ) \log ^2(x)} \, dx\\ &=\int \frac {5 e^x \left (25-2 \left (-22+e^2\right ) x+2 \left (-9+2 e^2\right ) x^2+20 x^3+\left (25+25 x+2 \left (-2+e^2\right ) x^2+10 x^3\right ) \log (x)\right )}{\left (25+2 \left (3+e^2\right ) x+10 x^2\right )^2 (2+\log (x))^2} \, dx\\ &=5 \int \frac {e^x \left (25-2 \left (-22+e^2\right ) x+2 \left (-9+2 e^2\right ) x^2+20 x^3+\left (25+25 x+2 \left (-2+e^2\right ) x^2+10 x^3\right ) \log (x)\right )}{\left (25+2 \left (3+e^2\right ) x+10 x^2\right )^2 (2+\log (x))^2} \, dx\\ &=5 \int \left (\frac {e^x}{\left (-25-2 \left (3+e^2\right ) x-10 x^2\right ) (2+\log (x))^2}+\frac {e^x \left (25+25 x-2 \left (2-e^2\right ) x^2+10 x^3\right )}{\left (25+2 \left (3+e^2\right ) x+10 x^2\right )^2 (2+\log (x))}\right ) \, dx\\ &=5 \int \frac {e^x}{\left (-25-2 \left (3+e^2\right ) x-10 x^2\right ) (2+\log (x))^2} \, dx+5 \int \frac {e^x \left (25+25 x-2 \left (2-e^2\right ) x^2+10 x^3\right )}{\left (25+2 \left (3+e^2\right ) x+10 x^2\right )^2 (2+\log (x))} \, dx\\ &=5 \int \left (-\frac {10 i e^x}{\sqrt {241-6 e^2-e^4} \left (-6-2 e^2+2 i \sqrt {241-6 e^2-e^4}-20 x\right ) (2+\log (x))^2}-\frac {10 i e^x}{\sqrt {241-6 e^2-e^4} \left (6+2 e^2+2 i \sqrt {241-6 e^2-e^4}+20 x\right ) (2+\log (x))^2}\right ) \, dx+5 \int \left (\frac {2 e^x \left (25+\left (3+e^2\right ) x\right )}{\left (25+2 \left (3+e^2\right ) x+10 x^2\right )^2 (2+\log (x))}+\frac {e^x (-1+x)}{\left (25+2 \left (3+e^2\right ) x+10 x^2\right ) (2+\log (x))}\right ) \, dx\\ &=5 \int \frac {e^x (-1+x)}{\left (25+2 \left (3+e^2\right ) x+10 x^2\right ) (2+\log (x))} \, dx+10 \int \frac {e^x \left (25+\left (3+e^2\right ) x\right )}{\left (25+2 \left (3+e^2\right ) x+10 x^2\right )^2 (2+\log (x))} \, dx-\frac {(50 i) \int \frac {e^x}{\left (-6-2 e^2+2 i \sqrt {241-6 e^2-e^4}-20 x\right ) (2+\log (x))^2} \, dx}{\sqrt {241-6 e^2-e^4}}-\frac {(50 i) \int \frac {e^x}{\left (6+2 e^2+2 i \sqrt {241-6 e^2-e^4}+20 x\right ) (2+\log (x))^2} \, dx}{\sqrt {241-6 e^2-e^4}}\\ &=5 \int \left (\frac {e^x}{\left (-25-2 \left (3+e^2\right ) x-10 x^2\right ) (2+\log (x))}+\frac {e^x x}{\left (25+2 \left (3+e^2\right ) x+10 x^2\right ) (2+\log (x))}\right ) \, dx+10 \int \left (\frac {25 e^x}{\left (25+2 \left (3+e^2\right ) x+10 x^2\right )^2 (2+\log (x))}+\frac {e^x \left (3+e^2\right ) x}{\left (25+2 \left (3+e^2\right ) x+10 x^2\right )^2 (2+\log (x))}\right ) \, dx-\frac {(50 i) \int \frac {e^x}{\left (-6-2 e^2+2 i \sqrt {241-6 e^2-e^4}-20 x\right ) (2+\log (x))^2} \, dx}{\sqrt {241-6 e^2-e^4}}-\frac {(50 i) \int \frac {e^x}{\left (6+2 e^2+2 i \sqrt {241-6 e^2-e^4}+20 x\right ) (2+\log (x))^2} \, dx}{\sqrt {241-6 e^2-e^4}}\\ &=5 \int \frac {e^x}{\left (-25-2 \left (3+e^2\right ) x-10 x^2\right ) (2+\log (x))} \, dx+5 \int \frac {e^x x}{\left (25+2 \left (3+e^2\right ) x+10 x^2\right ) (2+\log (x))} \, dx+250 \int \frac {e^x}{\left (25+2 \left (3+e^2\right ) x+10 x^2\right )^2 (2+\log (x))} \, dx+\left (10 \left (3+e^2\right )\right ) \int \frac {e^x x}{\left (25+2 \left (3+e^2\right ) x+10 x^2\right )^2 (2+\log (x))} \, dx-\frac {(50 i) \int \frac {e^x}{\left (-6-2 e^2+2 i \sqrt {241-6 e^2-e^4}-20 x\right ) (2+\log (x))^2} \, dx}{\sqrt {241-6 e^2-e^4}}-\frac {(50 i) \int \frac {e^x}{\left (6+2 e^2+2 i \sqrt {241-6 e^2-e^4}+20 x\right ) (2+\log (x))^2} \, dx}{\sqrt {241-6 e^2-e^4}}\\ &=5 \int \left (-\frac {10 i e^x}{\sqrt {241-6 e^2-e^4} \left (-6-2 e^2+2 i \sqrt {241-6 e^2-e^4}-20 x\right ) (2+\log (x))}-\frac {10 i e^x}{\sqrt {241-6 e^2-e^4} \left (6+2 e^2+2 i \sqrt {241-6 e^2-e^4}+20 x\right ) (2+\log (x))}\right ) \, dx+5 \int \left (\frac {e^x \left (1+\frac {i \left (3+e^2\right )}{\sqrt {241-6 e^2-e^4}}\right )}{\left (2 \left (3+e^2\right )-2 i \sqrt {241-6 e^2-e^4}+20 x\right ) (2+\log (x))}+\frac {e^x \left (1-\frac {i \left (3+e^2\right )}{\sqrt {241-6 e^2-e^4}}\right )}{\left (2 \left (3+e^2\right )+2 i \sqrt {241-6 e^2-e^4}+20 x\right ) (2+\log (x))}\right ) \, dx+250 \int \frac {e^x}{\left (25+2 \left (3+e^2\right ) x+10 x^2\right )^2 (2+\log (x))} \, dx+\left (10 \left (3+e^2\right )\right ) \int \frac {e^x x}{\left (25+2 \left (3+e^2\right ) x+10 x^2\right )^2 (2+\log (x))} \, dx-\frac {(50 i) \int \frac {e^x}{\left (-6-2 e^2+2 i \sqrt {241-6 e^2-e^4}-20 x\right ) (2+\log (x))^2} \, dx}{\sqrt {241-6 e^2-e^4}}-\frac {(50 i) \int \frac {e^x}{\left (6+2 e^2+2 i \sqrt {241-6 e^2-e^4}+20 x\right ) (2+\log (x))^2} \, dx}{\sqrt {241-6 e^2-e^4}}\\ &=250 \int \frac {e^x}{\left (25+2 \left (3+e^2\right ) x+10 x^2\right )^2 (2+\log (x))} \, dx+\left (10 \left (3+e^2\right )\right ) \int \frac {e^x x}{\left (25+2 \left (3+e^2\right ) x+10 x^2\right )^2 (2+\log (x))} \, dx-\frac {(50 i) \int \frac {e^x}{\left (-6-2 e^2+2 i \sqrt {241-6 e^2-e^4}-20 x\right ) (2+\log (x))^2} \, dx}{\sqrt {241-6 e^2-e^4}}-\frac {(50 i) \int \frac {e^x}{\left (6+2 e^2+2 i \sqrt {241-6 e^2-e^4}+20 x\right ) (2+\log (x))^2} \, dx}{\sqrt {241-6 e^2-e^4}}-\frac {(50 i) \int \frac {e^x}{\left (-6-2 e^2+2 i \sqrt {241-6 e^2-e^4}-20 x\right ) (2+\log (x))} \, dx}{\sqrt {241-6 e^2-e^4}}-\frac {(50 i) \int \frac {e^x}{\left (6+2 e^2+2 i \sqrt {241-6 e^2-e^4}+20 x\right ) (2+\log (x))} \, dx}{\sqrt {241-6 e^2-e^4}}+\left (5 \left (1-\frac {i \left (3+e^2\right )}{\sqrt {241-6 e^2-e^4}}\right )\right ) \int \frac {e^x}{\left (2 \left (3+e^2\right )+2 i \sqrt {241-6 e^2-e^4}+20 x\right ) (2+\log (x))} \, dx+\left (5 \left (1+\frac {i \left (3+e^2\right )}{\sqrt {241-6 e^2-e^4}}\right )\right ) \int \frac {e^x}{\left (2 \left (3+e^2\right )-2 i \sqrt {241-6 e^2-e^4}+20 x\right ) (2+\log (x))} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.98, size = 30, normalized size = 0.91 \begin {gather*} \frac {5 e^x x}{\left (25+6 x+2 e^2 x+10 x^2\right ) (2+\log (x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x*(125 + 220*x - 90*x^2 + 100*x^3 + E^2*(-10*x + 20*x^2)) + E^x*(125 + 125*x - 20*x^2 + 10*E^2*x^
2 + 50*x^3)*Log[x])/(2500 + 1200*x + 2144*x^2 + 16*E^4*x^2 + 480*x^3 + 400*x^4 + E^2*(400*x + 96*x^2 + 160*x^3
) + (2500 + 1200*x + 2144*x^2 + 16*E^4*x^2 + 480*x^3 + 400*x^4 + E^2*(400*x + 96*x^2 + 160*x^3))*Log[x] + (625
+ 300*x + 536*x^2 + 4*E^4*x^2 + 120*x^3 + 100*x^4 + E^2*(100*x + 24*x^2 + 40*x^3))*Log[x]^2),x]

[Out]

(5*E^x*x)/((25 + 6*x + 2*E^2*x + 10*x^2)*(2 + Log[x]))

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fricas [A]  time = 0.53, size = 40, normalized size = 1.21 \begin {gather*} \frac {5 \, x e^{x}}{20 \, x^{2} + 4 \, x e^{2} + {\left (10 \, x^{2} + 2 \, x e^{2} + 6 \, x + 25\right )} \log \relax (x) + 12 \, x + 50} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x^2*exp(2)+50*x^3-20*x^2+125*x+125)*exp(x)*log(x)+((20*x^2-10*x)*exp(2)+100*x^3-90*x^2+220*x+12
5)*exp(x))/((4*x^2*exp(2)^2+(40*x^3+24*x^2+100*x)*exp(2)+100*x^4+120*x^3+536*x^2+300*x+625)*log(x)^2+(16*x^2*e
xp(2)^2+(160*x^3+96*x^2+400*x)*exp(2)+400*x^4+480*x^3+2144*x^2+1200*x+2500)*log(x)+16*x^2*exp(2)^2+(160*x^3+96
*x^2+400*x)*exp(2)+400*x^4+480*x^3+2144*x^2+1200*x+2500),x, algorithm="fricas")

[Out]

5*x*e^x/(20*x^2 + 4*x*e^2 + (10*x^2 + 2*x*e^2 + 6*x + 25)*log(x) + 12*x + 50)

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giac [A]  time = 0.62, size = 45, normalized size = 1.36 \begin {gather*} \frac {5 \, x e^{x}}{10 \, x^{2} \log \relax (x) + 2 \, x e^{2} \log \relax (x) + 20 \, x^{2} + 4 \, x e^{2} + 6 \, x \log \relax (x) + 12 \, x + 25 \, \log \relax (x) + 50} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x^2*exp(2)+50*x^3-20*x^2+125*x+125)*exp(x)*log(x)+((20*x^2-10*x)*exp(2)+100*x^3-90*x^2+220*x+12
5)*exp(x))/((4*x^2*exp(2)^2+(40*x^3+24*x^2+100*x)*exp(2)+100*x^4+120*x^3+536*x^2+300*x+625)*log(x)^2+(16*x^2*e
xp(2)^2+(160*x^3+96*x^2+400*x)*exp(2)+400*x^4+480*x^3+2144*x^2+1200*x+2500)*log(x)+16*x^2*exp(2)^2+(160*x^3+96
*x^2+400*x)*exp(2)+400*x^4+480*x^3+2144*x^2+1200*x+2500),x, algorithm="giac")

[Out]

5*x*e^x/(10*x^2*log(x) + 2*x*e^2*log(x) + 20*x^2 + 4*x*e^2 + 6*x*log(x) + 12*x + 25*log(x) + 50)

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maple [A]  time = 0.06, size = 29, normalized size = 0.88

method	result	size
risch	\(\frac {5 x \,{\mathrm e}^{x}}{\left (2 \,{\mathrm e}^{2} x +10 x^{2}+6 x +25\right ) \left (\ln \relax (x )+2\right )}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((10*x^2*exp(2)+50*x^3-20*x^2+125*x+125)*exp(x)*ln(x)+((20*x^2-10*x)*exp(2)+100*x^3-90*x^2+220*x+125)*exp(
x))/((4*x^2*exp(2)^2+(40*x^3+24*x^2+100*x)*exp(2)+100*x^4+120*x^3+536*x^2+300*x+625)*ln(x)^2+(16*x^2*exp(2)^2+
(160*x^3+96*x^2+400*x)*exp(2)+400*x^4+480*x^3+2144*x^2+1200*x+2500)*ln(x)+16*x^2*exp(2)^2+(160*x^3+96*x^2+400*
x)*exp(2)+400*x^4+480*x^3+2144*x^2+1200*x+2500),x,method=_RETURNVERBOSE)

[Out]

5*x*exp(x)/(2*exp(2)*x+10*x^2+6*x+25)/(ln(x)+2)

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maxima [A]  time = 0.42, size = 38, normalized size = 1.15 \begin {gather*} \frac {5 \, x e^{x}}{20 \, x^{2} + 4 \, x {\left (e^{2} + 3\right )} + {\left (10 \, x^{2} + 2 \, x {\left (e^{2} + 3\right )} + 25\right )} \log \relax (x) + 50} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x^2*exp(2)+50*x^3-20*x^2+125*x+125)*exp(x)*log(x)+((20*x^2-10*x)*exp(2)+100*x^3-90*x^2+220*x+12
5)*exp(x))/((4*x^2*exp(2)^2+(40*x^3+24*x^2+100*x)*exp(2)+100*x^4+120*x^3+536*x^2+300*x+625)*log(x)^2+(16*x^2*e
xp(2)^2+(160*x^3+96*x^2+400*x)*exp(2)+400*x^4+480*x^3+2144*x^2+1200*x+2500)*log(x)+16*x^2*exp(2)^2+(160*x^3+96
*x^2+400*x)*exp(2)+400*x^4+480*x^3+2144*x^2+1200*x+2500),x, algorithm="maxima")

[Out]

5*x*e^x/(20*x^2 + 4*x*(e^2 + 3) + (10*x^2 + 2*x*(e^2 + 3) + 25)*log(x) + 50)

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mupad [B]  time = 4.42, size = 28, normalized size = 0.85 \begin {gather*} \frac {5\,x\,{\mathrm {e}}^x}{\left (\ln \relax (x)+2\right )\,\left (6\,x+2\,x\,{\mathrm {e}}^2+10\,x^2+25\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)*(220*x - exp(2)*(10*x - 20*x^2) - 90*x^2 + 100*x^3 + 125) + exp(x)*log(x)*(125*x + 10*x^2*exp(2) -
20*x^2 + 50*x^3 + 125))/(1200*x + log(x)*(1200*x + exp(2)*(400*x + 96*x^2 + 160*x^3) + 16*x^2*exp(4) + 2144*x
^2 + 480*x^3 + 400*x^4 + 2500) + exp(2)*(400*x + 96*x^2 + 160*x^3) + log(x)^2*(300*x + exp(2)*(100*x + 24*x^2
+ 40*x^3) + 4*x^2*exp(4) + 536*x^2 + 120*x^3 + 100*x^4 + 625) + 16*x^2*exp(4) + 2144*x^2 + 480*x^3 + 400*x^4 +
2500),x)

[Out]

(5*x*exp(x))/((log(x) + 2)*(6*x + 2*x*exp(2) + 10*x^2 + 25))

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sympy [B]  time = 0.51, size = 53, normalized size = 1.61 \begin {gather*} \frac {5 x e^{x}}{10 x^{2} \log {\relax (x )} + 20 x^{2} + 6 x \log {\relax (x )} + 2 x e^{2} \log {\relax (x )} + 12 x + 4 x e^{2} + 25 \log {\relax (x )} + 50} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x**2*exp(2)+50*x**3-20*x**2+125*x+125)*exp(x)*ln(x)+((20*x**2-10*x)*exp(2)+100*x**3-90*x**2+220
*x+125)*exp(x))/((4*x**2*exp(2)**2+(40*x**3+24*x**2+100*x)*exp(2)+100*x**4+120*x**3+536*x**2+300*x+625)*ln(x)*
*2+(16*x**2*exp(2)**2+(160*x**3+96*x**2+400*x)*exp(2)+400*x**4+480*x**3+2144*x**2+1200*x+2500)*ln(x)+16*x**2*e
xp(2)**2+(160*x**3+96*x**2+400*x)*exp(2)+400*x**4+480*x**3+2144*x**2+1200*x+2500),x)

[Out]

5*x*exp(x)/(10*x**2*log(x) + 20*x**2 + 6*x*log(x) + 2*x*exp(2)*log(x) + 12*x + 4*x*exp(2) + 25*log(x) + 50)

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