3.1.39 \(\int \frac {25 x-1249 x^2-99 x^3-102 x^4-4 x^5-2 x^6+e^{3 x} (625 x+50 x^2+51 x^3+2 x^4+x^5)+e^{2 x} (-1250-100 x+1148 x^2+96 x^3+100 x^4+4 x^5+2 x^6)+e^x (25-2499 x-199 x^2+421 x^3+42 x^4+47 x^5+2 x^6+x^7)+(-25 x-2 x^2-3 x^3+e^x (-26 x-3 x^2-x^3)) \log (x)}{625 x^3+50 x^4+51 x^5+2 x^6+x^7+e^{2 x} (625 x+50 x^2+51 x^3+2 x^4+x^5)+e^x (1250 x^2+100 x^3+102 x^4+4 x^5+2 x^6)} \, dx\)

Optimal. Leaf size=29 \[ 5+e^x+\frac {\log (x)}{\left (e^x+x\right ) \left (25+x+x^2\right )}-\log \left (x^2\right ) \]

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Rubi [F]  time = 18.07, 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 {25 x-1249 x^2-99 x^3-102 x^4-4 x^5-2 x^6+e^{3 x} \left (625 x+50 x^2+51 x^3+2 x^4+x^5\right )+e^{2 x} \left (-1250-100 x+1148 x^2+96 x^3+100 x^4+4 x^5+2 x^6\right )+e^x \left (25-2499 x-199 x^2+421 x^3+42 x^4+47 x^5+2 x^6+x^7\right )+\left (-25 x-2 x^2-3 x^3+e^x \left (-26 x-3 x^2-x^3\right )\right ) \log (x)}{625 x^3+50 x^4+51 x^5+2 x^6+x^7+e^{2 x} \left (625 x+50 x^2+51 x^3+2 x^4+x^5\right )+e^x \left (1250 x^2+100 x^3+102 x^4+4 x^5+2 x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(25*x - 1249*x^2 - 99*x^3 - 102*x^4 - 4*x^5 - 2*x^6 + E^(3*x)*(625*x + 50*x^2 + 51*x^3 + 2*x^4 + x^5) + E^
(2*x)*(-1250 - 100*x + 1148*x^2 + 96*x^3 + 100*x^4 + 4*x^5 + 2*x^6) + E^x*(25 - 2499*x - 199*x^2 + 421*x^3 + 4
2*x^4 + 47*x^5 + 2*x^6 + x^7) + (-25*x - 2*x^2 - 3*x^3 + E^x*(-26*x - 3*x^2 - x^3))*Log[x])/(625*x^3 + 50*x^4
+ 51*x^5 + 2*x^6 + x^7 + E^(2*x)*(625*x + 50*x^2 + 51*x^3 + 2*x^4 + x^5) + E^x*(1250*x^2 + 100*x^3 + 102*x^4 +
4*x^5 + 2*x^6)),x]

[Out]

E^x - 2*Log[x] - (((2*I)/3)*Log[x]*Defer[Int][1/((-1 + (3*I)*Sqrt[11] - 2*x)*(E^x + x)^2), x])/Sqrt[11] - (((2
*I)/75)*Defer[Int][1/((-1 + (3*I)*Sqrt[11] - 2*x)*(E^x + x)), x])/Sqrt[11] - (((2*I)/3)*Log[x]*Defer[Int][1/((
-1 + (3*I)*Sqrt[11] - 2*x)*(E^x + x)), x])/Sqrt[11] + Defer[Int][1/(x*(E^x + x)), x]/25 + ((33 + I*Sqrt[11])*L
og[x]*Defer[Int][1/((E^x + x)^2*(1 - (3*I)*Sqrt[11] + 2*x)), x])/33 - ((33 + I*Sqrt[11])*Defer[Int][1/((E^x +
x)*(1 - (3*I)*Sqrt[11] + 2*x)), x])/825 - (((2*I)/3)*Log[x]*Defer[Int][1/((E^x + x)^2*(1 + (3*I)*Sqrt[11] + 2*
x)), x])/Sqrt[11] + ((33 - I*Sqrt[11])*Log[x]*Defer[Int][1/((E^x + x)^2*(1 + (3*I)*Sqrt[11] + 2*x)), x])/33 -
(((2*I)/75)*Defer[Int][1/((E^x + x)*(1 + (3*I)*Sqrt[11] + 2*x)), x])/Sqrt[11] - ((33 - I*Sqrt[11])*Defer[Int][
1/((E^x + x)*(1 + (3*I)*Sqrt[11] + 2*x)), x])/825 - (((2*I)/3)*Log[x]*Defer[Int][1/((E^x + x)*(1 + (3*I)*Sqrt[
11] + 2*x)), x])/Sqrt[11] - Log[x]*Defer[Int][1/((E^x + x)*(25 + x + x^2)^2), x] - 2*Log[x]*Defer[Int][x/((E^x
+ x)*(25 + x + x^2)^2), x] + (((2*I)/3)*Defer[Int][Defer[Int][1/((-1 + (3*I)*Sqrt[11] - 2*x)*(E^x + x)^2), x]
/x, x])/Sqrt[11] + (((2*I)/3)*Defer[Int][Defer[Int][1/((-1 + (3*I)*Sqrt[11] - 2*x)*(E^x + x)), x]/x, x])/Sqrt[
11] - ((33 + I*Sqrt[11])*Defer[Int][Defer[Int][1/((E^x + x)^2*(1 - (3*I)*Sqrt[11] + 2*x)), x]/x, x])/33 - ((11
- I*Sqrt[11])*Defer[Int][Defer[Int][1/((E^x + x)^2*(1 + (3*I)*Sqrt[11] + 2*x)), x]/x, x])/11 - (((58*I)/75)*D
efer[Int][Defer[Int][1/((E^x + x)*(1 + (3*I)*Sqrt[11] + 2*x)), x]/x, x])/Sqrt[11] - (4*(11 - I*Sqrt[11])*Defer
[Int][Defer[Int][1/((E^x + x)*(1 + (3*I)*Sqrt[11] + 2*x)), x]/x, x])/275 - (26*(33 - I*Sqrt[11])*Defer[Int][De
fer[Int][1/((E^x + x)*(1 + (3*I)*Sqrt[11] + 2*x)), x]/x, x])/20625 + (2*(11 + (16*I)*Sqrt[11])*Defer[Int][Defe
r[Int][1/((E^x + x)*(1 + (3*I)*Sqrt[11] + 2*x)), x]/x, x])/275 + (8*(396 - (37*I)*Sqrt[11])*Defer[Int][Defer[I
nt][1/((E^x + x)*(1 + (3*I)*Sqrt[11] + 2*x)), x]/x, x])/20625 + (29*(33 + (49*I)*Sqrt[11])*Defer[Int][Defer[In
t][1/((E^x + x)*(1 + (3*I)*Sqrt[11] + 2*x)), x]/x, x])/20625 - ((1617 + (1151*I)*Sqrt[11])*Defer[Int][Defer[In
t][1/((E^x + x)*(1 + (3*I)*Sqrt[11] + 2*x)), x]/x, x])/20625 + Defer[Int][Defer[Int][1/((E^x + x)*(25 + x + x^
2)^2), x]/x, x] + 2*Defer[Int][Defer[Int][x/((E^x + x)*(25 + x + x^2)^2), x]/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (25+x+x^2\right ) \left (e^{3 x} x \left (25+x+x^2\right )-x \left (-1+50 x+2 x^2+2 x^3\right )+2 e^{2 x} \left (-25-x+24 x^2+x^3+x^4\right )+e^x \left (1-100 x-4 x^2+21 x^3+x^4+x^5\right )\right )-x \left (25+2 x+3 x^2+e^x \left (26+3 x+x^2\right )\right ) \log (x)}{x \left (e^x+x\right )^2 \left (25+x+x^2\right )^2} \, dx\\ &=\int \left (e^x-\frac {2}{x}+\frac {(-1+x) \log (x)}{\left (e^x+x\right )^2 \left (25+x+x^2\right )}-\frac {-25-x-x^2+26 x \log (x)+3 x^2 \log (x)+x^3 \log (x)}{x \left (e^x+x\right ) \left (25+x+x^2\right )^2}\right ) \, dx\\ &=-2 \log (x)+\int e^x \, dx+\int \frac {(-1+x) \log (x)}{\left (e^x+x\right )^2 \left (25+x+x^2\right )} \, dx-\int \frac {-25-x-x^2+26 x \log (x)+3 x^2 \log (x)+x^3 \log (x)}{x \left (e^x+x\right ) \left (25+x+x^2\right )^2} \, dx\\ &=e^x-2 \log (x)-\frac {(2 i \log (x)) \int \frac {1}{\left (-1+3 i \sqrt {11}-2 x\right ) \left (e^x+x\right )^2} \, dx}{3 \sqrt {11}}-\frac {(2 i \log (x)) \int \frac {1}{\left (e^x+x\right )^2 \left (1+3 i \sqrt {11}+2 x\right )} \, dx}{3 \sqrt {11}}+\frac {1}{33} \left (\left (33-i \sqrt {11}\right ) \log (x)\right ) \int \frac {1}{\left (e^x+x\right )^2 \left (1+3 i \sqrt {11}+2 x\right )} \, dx+\frac {1}{33} \left (\left (33+i \sqrt {11}\right ) \log (x)\right ) \int \frac {1}{\left (e^x+x\right )^2 \left (1-3 i \sqrt {11}+2 x\right )} \, dx-\int \left (\frac {-25-x-x^2+26 x \log (x)+3 x^2 \log (x)+x^3 \log (x)}{625 x \left (e^x+x\right )}-\frac {(1+x) \left (-25-x-x^2+26 x \log (x)+3 x^2 \log (x)+x^3 \log (x)\right )}{25 \left (e^x+x\right ) \left (25+x+x^2\right )^2}-\frac {(1+x) \left (-25-x-x^2+26 x \log (x)+3 x^2 \log (x)+x^3 \log (x)\right )}{625 \left (e^x+x\right ) \left (25+x+x^2\right )}\right ) \, dx-\int \frac {-2 i \sqrt {11} \int \frac {1}{\left (-1+3 i \sqrt {11}-2 x\right ) \left (e^x+x\right )^2} \, dx+\left (33+i \sqrt {11}\right ) \int \frac {1}{\left (e^x+x\right )^2 \left (1-3 i \sqrt {11}+2 x\right )} \, dx+3 \left (11-i \sqrt {11}\right ) \int \frac {1}{\left (e^x+x\right )^2 \left (1+3 i \sqrt {11}+2 x\right )} \, dx}{33 x} \, dx\\ &=e^x-2 \log (x)-\frac {1}{625} \int \frac {-25-x-x^2+26 x \log (x)+3 x^2 \log (x)+x^3 \log (x)}{x \left (e^x+x\right )} \, dx+\frac {1}{625} \int \frac {(1+x) \left (-25-x-x^2+26 x \log (x)+3 x^2 \log (x)+x^3 \log (x)\right )}{\left (e^x+x\right ) \left (25+x+x^2\right )} \, dx-\frac {1}{33} \int \frac {-2 i \sqrt {11} \int \frac {1}{\left (-1+3 i \sqrt {11}-2 x\right ) \left (e^x+x\right )^2} \, dx+\left (33+i \sqrt {11}\right ) \int \frac {1}{\left (e^x+x\right )^2 \left (1-3 i \sqrt {11}+2 x\right )} \, dx+3 \left (11-i \sqrt {11}\right ) \int \frac {1}{\left (e^x+x\right )^2 \left (1+3 i \sqrt {11}+2 x\right )} \, dx}{x} \, dx+\frac {1}{25} \int \frac {(1+x) \left (-25-x-x^2+26 x \log (x)+3 x^2 \log (x)+x^3 \log (x)\right )}{\left (e^x+x\right ) \left (25+x+x^2\right )^2} \, dx-\frac {(2 i \log (x)) \int \frac {1}{\left (-1+3 i \sqrt {11}-2 x\right ) \left (e^x+x\right )^2} \, dx}{3 \sqrt {11}}-\frac {(2 i \log (x)) \int \frac {1}{\left (e^x+x\right )^2 \left (1+3 i \sqrt {11}+2 x\right )} \, dx}{3 \sqrt {11}}+\frac {1}{33} \left (\left (33-i \sqrt {11}\right ) \log (x)\right ) \int \frac {1}{\left (e^x+x\right )^2 \left (1+3 i \sqrt {11}+2 x\right )} \, dx+\frac {1}{33} \left (\left (33+i \sqrt {11}\right ) \log (x)\right ) \int \frac {1}{\left (e^x+x\right )^2 \left (1-3 i \sqrt {11}+2 x\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 26, normalized size = 0.90 \begin {gather*} e^x-2 \log (x)+\frac {\log (x)}{\left (e^x+x\right ) \left (25+x+x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(25*x - 1249*x^2 - 99*x^3 - 102*x^4 - 4*x^5 - 2*x^6 + E^(3*x)*(625*x + 50*x^2 + 51*x^3 + 2*x^4 + x^5
) + E^(2*x)*(-1250 - 100*x + 1148*x^2 + 96*x^3 + 100*x^4 + 4*x^5 + 2*x^6) + E^x*(25 - 2499*x - 199*x^2 + 421*x
^3 + 42*x^4 + 47*x^5 + 2*x^6 + x^7) + (-25*x - 2*x^2 - 3*x^3 + E^x*(-26*x - 3*x^2 - x^3))*Log[x])/(625*x^3 + 5
0*x^4 + 51*x^5 + 2*x^6 + x^7 + E^(2*x)*(625*x + 50*x^2 + 51*x^3 + 2*x^4 + x^5) + E^x*(1250*x^2 + 100*x^3 + 102
*x^4 + 4*x^5 + 2*x^6)),x]

[Out]

E^x - 2*Log[x] + Log[x]/((E^x + x)*(25 + x + x^2))

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fricas [B]  time = 0.77, size = 76, normalized size = 2.62 \begin {gather*} \frac {{\left (x^{2} + x + 25\right )} e^{\left (2 \, x\right )} + {\left (x^{3} + x^{2} + 25 \, x\right )} e^{x} - {\left (2 \, x^{3} + 2 \, x^{2} + 2 \, {\left (x^{2} + x + 25\right )} e^{x} + 50 \, x - 1\right )} \log \relax (x)}{x^{3} + x^{2} + {\left (x^{2} + x + 25\right )} e^{x} + 25 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^3-3*x^2-26*x)*exp(x)-3*x^3-2*x^2-25*x)*log(x)+(x^5+2*x^4+51*x^3+50*x^2+625*x)*exp(x)^3+(2*x^6+
4*x^5+100*x^4+96*x^3+1148*x^2-100*x-1250)*exp(x)^2+(x^7+2*x^6+47*x^5+42*x^4+421*x^3-199*x^2-2499*x+25)*exp(x)-
2*x^6-4*x^5-102*x^4-99*x^3-1249*x^2+25*x)/((x^5+2*x^4+51*x^3+50*x^2+625*x)*exp(x)^2+(2*x^6+4*x^5+102*x^4+100*x
^3+1250*x^2)*exp(x)+x^7+2*x^6+51*x^5+50*x^4+625*x^3),x, algorithm="fricas")

[Out]

((x^2 + x + 25)*e^(2*x) + (x^3 + x^2 + 25*x)*e^x - (2*x^3 + 2*x^2 + 2*(x^2 + x + 25)*e^x + 50*x - 1)*log(x))/(
x^3 + x^2 + (x^2 + x + 25)*e^x + 25*x)

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giac [B]  time = 0.58, size = 108, normalized size = 3.72 \begin {gather*} \frac {x^{3} e^{x} - 2 \, x^{3} \log \relax (x) - 2 \, x^{2} e^{x} \log \relax (x) + x^{2} e^{\left (2 \, x\right )} + x^{2} e^{x} - 2 \, x^{2} \log \relax (x) - 2 \, x e^{x} \log \relax (x) + x e^{\left (2 \, x\right )} + 25 \, x e^{x} - 50 \, x \log \relax (x) - 50 \, e^{x} \log \relax (x) + 25 \, e^{\left (2 \, x\right )} + \log \relax (x)}{x^{3} + x^{2} e^{x} + x^{2} + x e^{x} + 25 \, x + 25 \, e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^3-3*x^2-26*x)*exp(x)-3*x^3-2*x^2-25*x)*log(x)+(x^5+2*x^4+51*x^3+50*x^2+625*x)*exp(x)^3+(2*x^6+
4*x^5+100*x^4+96*x^3+1148*x^2-100*x-1250)*exp(x)^2+(x^7+2*x^6+47*x^5+42*x^4+421*x^3-199*x^2-2499*x+25)*exp(x)-
2*x^6-4*x^5-102*x^4-99*x^3-1249*x^2+25*x)/((x^5+2*x^4+51*x^3+50*x^2+625*x)*exp(x)^2+(2*x^6+4*x^5+102*x^4+100*x
^3+1250*x^2)*exp(x)+x^7+2*x^6+51*x^5+50*x^4+625*x^3),x, algorithm="giac")

[Out]

(x^3*e^x - 2*x^3*log(x) - 2*x^2*e^x*log(x) + x^2*e^(2*x) + x^2*e^x - 2*x^2*log(x) - 2*x*e^x*log(x) + x*e^(2*x)
+ 25*x*e^x - 50*x*log(x) - 50*e^x*log(x) + 25*e^(2*x) + log(x))/(x^3 + x^2*e^x + x^2 + x*e^x + 25*x + 25*e^x)

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maple [A]  time = 0.04, size = 25, normalized size = 0.86

method	result	size
risch	\(\frac {\ln \relax (x )}{\left (x^{2}+x +25\right ) \left ({\mathrm e}^{x}+x \right )}-2 \ln \relax (x )+{\mathrm e}^{x}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-x^3-3*x^2-26*x)*exp(x)-3*x^3-2*x^2-25*x)*ln(x)+(x^5+2*x^4+51*x^3+50*x^2+625*x)*exp(x)^3+(2*x^6+4*x^5+1
00*x^4+96*x^3+1148*x^2-100*x-1250)*exp(x)^2+(x^7+2*x^6+47*x^5+42*x^4+421*x^3-199*x^2-2499*x+25)*exp(x)-2*x^6-4
*x^5-102*x^4-99*x^3-1249*x^2+25*x)/((x^5+2*x^4+51*x^3+50*x^2+625*x)*exp(x)^2+(2*x^6+4*x^5+102*x^4+100*x^3+1250
*x^2)*exp(x)+x^7+2*x^6+51*x^5+50*x^4+625*x^3),x,method=_RETURNVERBOSE)

[Out]

ln(x)/(x^2+x+25)/(exp(x)+x)-2*ln(x)+exp(x)

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maxima [A]  time = 0.86, size = 54, normalized size = 1.86 \begin {gather*} \frac {{\left (x^{2} + x + 25\right )} e^{\left (2 \, x\right )} + {\left (x^{3} + x^{2} + 25 \, x\right )} e^{x} + \log \relax (x)}{x^{3} + x^{2} + {\left (x^{2} + x + 25\right )} e^{x} + 25 \, x} - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^3-3*x^2-26*x)*exp(x)-3*x^3-2*x^2-25*x)*log(x)+(x^5+2*x^4+51*x^3+50*x^2+625*x)*exp(x)^3+(2*x^6+
4*x^5+100*x^4+96*x^3+1148*x^2-100*x-1250)*exp(x)^2+(x^7+2*x^6+47*x^5+42*x^4+421*x^3-199*x^2-2499*x+25)*exp(x)-
2*x^6-4*x^5-102*x^4-99*x^3-1249*x^2+25*x)/((x^5+2*x^4+51*x^3+50*x^2+625*x)*exp(x)^2+(2*x^6+4*x^5+102*x^4+100*x
^3+1250*x^2)*exp(x)+x^7+2*x^6+51*x^5+50*x^4+625*x^3),x, algorithm="maxima")

[Out]

((x^2 + x + 25)*e^(2*x) + (x^3 + x^2 + 25*x)*e^x + log(x))/(x^3 + x^2 + (x^2 + x + 25)*e^x + 25*x) - 2*log(x)

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mupad [B]  time = 0.50, size = 24, normalized size = 0.83 \begin {gather*} {\mathrm {e}}^x-2\,\ln \relax (x)+\frac {\ln \relax (x)}{\left (x+{\mathrm {e}}^x\right )\,\left (x^2+x+25\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x)*(25*x + exp(x)*(26*x + 3*x^2 + x^3) + 2*x^2 + 3*x^3) - exp(3*x)*(625*x + 50*x^2 + 51*x^3 + 2*x^4
+ x^5) - exp(x)*(421*x^3 - 199*x^2 - 2499*x + 42*x^4 + 47*x^5 + 2*x^6 + x^7 + 25) - 25*x + 1249*x^2 + 99*x^3 +
102*x^4 + 4*x^5 + 2*x^6 - exp(2*x)*(1148*x^2 - 100*x + 96*x^3 + 100*x^4 + 4*x^5 + 2*x^6 - 1250))/(exp(2*x)*(6
25*x + 50*x^2 + 51*x^3 + 2*x^4 + x^5) + exp(x)*(1250*x^2 + 100*x^3 + 102*x^4 + 4*x^5 + 2*x^6) + 625*x^3 + 50*x
^4 + 51*x^5 + 2*x^6 + x^7),x)

[Out]

exp(x) - 2*log(x) + log(x)/((x + exp(x))*(x + x^2 + 25))

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sympy [A]  time = 0.40, size = 31, normalized size = 1.07 \begin {gather*} e^{x} - 2 \log {\relax (x )} + \frac {\log {\relax (x )}}{x^{3} + x^{2} + 25 x + \left (x^{2} + x + 25\right ) e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x**3-3*x**2-26*x)*exp(x)-3*x**3-2*x**2-25*x)*ln(x)+(x**5+2*x**4+51*x**3+50*x**2+625*x)*exp(x)**3
+(2*x**6+4*x**5+100*x**4+96*x**3+1148*x**2-100*x-1250)*exp(x)**2+(x**7+2*x**6+47*x**5+42*x**4+421*x**3-199*x**
2-2499*x+25)*exp(x)-2*x**6-4*x**5-102*x**4-99*x**3-1249*x**2+25*x)/((x**5+2*x**4+51*x**3+50*x**2+625*x)*exp(x)
**2+(2*x**6+4*x**5+102*x**4+100*x**3+1250*x**2)*exp(x)+x**7+2*x**6+51*x**5+50*x**4+625*x**3),x)

[Out]

exp(x) - 2*log(x) + log(x)/(x**3 + x**2 + 25*x + (x**2 + x + 25)*exp(x))

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