3.78.1 \(\int \frac {-e^9-2100 x-275 x^2+(-12250 x-1500 x^2) \log (x)+(-8750 x-625 x^2) \log ^2(x)}{e^{18}-50 e^9 x^2+625 x^4+(-500 e^9 x^2+12500 x^4) \log (x)+(-1250 e^9 x^2+93750 x^4) \log ^2(x)+312500 x^4 \log ^3(x)+390625 x^4 \log ^4(x)} \, dx\)

Optimal. Leaf size=23 \[ \frac {7+x}{-e^9+25 (x+5 x \log (x))^2} \]

________________________________________________________________________________________

Rubi [F]  time = 2.07, 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 {-e^9-2100 x-275 x^2+\left (-12250 x-1500 x^2\right ) \log (x)+\left (-8750 x-625 x^2\right ) \log ^2(x)}{e^{18}-50 e^9 x^2+625 x^4+\left (-500 e^9 x^2+12500 x^4\right ) \log (x)+\left (-1250 e^9 x^2+93750 x^4\right ) \log ^2(x)+312500 x^4 \log ^3(x)+390625 x^4 \log ^4(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-E^9 - 2100*x - 275*x^2 + (-12250*x - 1500*x^2)*Log[x] + (-8750*x - 625*x^2)*Log[x]^2)/(E^18 - 50*E^9*x^2
+ 625*x^4 + (-500*E^9*x^2 + 12500*x^4)*Log[x] + (-1250*E^9*x^2 + 93750*x^4)*Log[x]^2 + 312500*x^4*Log[x]^3 +
390625*x^4*Log[x]^4),x]

[Out]

-2*E^9*Defer[Int][(E^9 - 25*x^2 - 250*x^2*Log[x] - 625*x^2*Log[x]^2)^(-2), x] - 1750*Defer[Int][x/(E^9 - 25*x^
2 - 250*x^2*Log[x] - 625*x^2*Log[x]^2)^2, x] - 250*Defer[Int][x^2/(E^9 - 25*x^2 - 250*x^2*Log[x] - 625*x^2*Log
[x]^2)^2, x] - 8750*Defer[Int][(x*Log[x])/(E^9 - 25*x^2 - 250*x^2*Log[x] - 625*x^2*Log[x]^2)^2, x] - 1250*Defe
r[Int][(x^2*Log[x])/(E^9 - 25*x^2 - 250*x^2*Log[x] - 625*x^2*Log[x]^2)^2, x] + Defer[Int][(E^9 - 25*x^2 - 250*
x^2*Log[x] - 625*x^2*Log[x]^2)^(-1), x] + 14*Defer[Int][1/(x*(E^9 - 25*x^2 - 250*x^2*Log[x] - 625*x^2*Log[x]^2
)), x] - 14*E^9*Defer[Int][1/(x*(-E^9 + 25*x^2 + 250*x^2*Log[x] + 625*x^2*Log[x]^2)^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-e^9-25 x (84+11 x)-250 x (49+6 x) \log (x)-625 x (14+x) \log ^2(x)}{\left (e^9-25 x^2-250 x^2 \log (x)-625 x^2 \log ^2(x)\right )^2} \, dx\\ &=\int \left (\frac {14+x}{x \left (e^9-25 x^2-250 x^2 \log (x)-625 x^2 \log ^2(x)\right )}-\frac {2 (7+x) \left (e^9+125 x^2+625 x^2 \log (x)\right )}{x \left (-e^9+25 x^2+250 x^2 \log (x)+625 x^2 \log ^2(x)\right )^2}\right ) \, dx\\ &=-\left (2 \int \frac {(7+x) \left (e^9+125 x^2+625 x^2 \log (x)\right )}{x \left (-e^9+25 x^2+250 x^2 \log (x)+625 x^2 \log ^2(x)\right )^2} \, dx\right )+\int \frac {14+x}{x \left (e^9-25 x^2-250 x^2 \log (x)-625 x^2 \log ^2(x)\right )} \, dx\\ &=-\left (2 \int \left (\frac {e^9+125 x^2+625 x^2 \log (x)}{\left (e^9-25 x^2-250 x^2 \log (x)-625 x^2 \log ^2(x)\right )^2}+\frac {7 \left (e^9+125 x^2+625 x^2 \log (x)\right )}{x \left (-e^9+25 x^2+250 x^2 \log (x)+625 x^2 \log ^2(x)\right )^2}\right ) \, dx\right )+\int \left (\frac {1}{e^9-25 x^2-250 x^2 \log (x)-625 x^2 \log ^2(x)}+\frac {14}{x \left (e^9-25 x^2-250 x^2 \log (x)-625 x^2 \log ^2(x)\right )}\right ) \, dx\\ &=-\left (2 \int \frac {e^9+125 x^2+625 x^2 \log (x)}{\left (e^9-25 x^2-250 x^2 \log (x)-625 x^2 \log ^2(x)\right )^2} \, dx\right )+14 \int \frac {1}{x \left (e^9-25 x^2-250 x^2 \log (x)-625 x^2 \log ^2(x)\right )} \, dx-14 \int \frac {e^9+125 x^2+625 x^2 \log (x)}{x \left (-e^9+25 x^2+250 x^2 \log (x)+625 x^2 \log ^2(x)\right )^2} \, dx+\int \frac {1}{e^9-25 x^2-250 x^2 \log (x)-625 x^2 \log ^2(x)} \, dx\\ &=-\left (2 \int \left (\frac {e^9}{\left (e^9-25 x^2-250 x^2 \log (x)-625 x^2 \log ^2(x)\right )^2}+\frac {125 x^2}{\left (e^9-25 x^2-250 x^2 \log (x)-625 x^2 \log ^2(x)\right )^2}+\frac {625 x^2 \log (x)}{\left (e^9-25 x^2-250 x^2 \log (x)-625 x^2 \log ^2(x)\right )^2}\right ) \, dx\right )+14 \int \frac {1}{x \left (e^9-25 x^2-250 x^2 \log (x)-625 x^2 \log ^2(x)\right )} \, dx-14 \int \left (\frac {125 x}{\left (e^9-25 x^2-250 x^2 \log (x)-625 x^2 \log ^2(x)\right )^2}+\frac {625 x \log (x)}{\left (e^9-25 x^2-250 x^2 \log (x)-625 x^2 \log ^2(x)\right )^2}+\frac {e^9}{x \left (-e^9+25 x^2+250 x^2 \log (x)+625 x^2 \log ^2(x)\right )^2}\right ) \, dx+\int \frac {1}{e^9-25 x^2-250 x^2 \log (x)-625 x^2 \log ^2(x)} \, dx\\ &=14 \int \frac {1}{x \left (e^9-25 x^2-250 x^2 \log (x)-625 x^2 \log ^2(x)\right )} \, dx-250 \int \frac {x^2}{\left (e^9-25 x^2-250 x^2 \log (x)-625 x^2 \log ^2(x)\right )^2} \, dx-1250 \int \frac {x^2 \log (x)}{\left (e^9-25 x^2-250 x^2 \log (x)-625 x^2 \log ^2(x)\right )^2} \, dx-1750 \int \frac {x}{\left (e^9-25 x^2-250 x^2 \log (x)-625 x^2 \log ^2(x)\right )^2} \, dx-8750 \int \frac {x \log (x)}{\left (e^9-25 x^2-250 x^2 \log (x)-625 x^2 \log ^2(x)\right )^2} \, dx-\left (2 e^9\right ) \int \frac {1}{\left (e^9-25 x^2-250 x^2 \log (x)-625 x^2 \log ^2(x)\right )^2} \, dx-\left (14 e^9\right ) \int \frac {1}{x \left (-e^9+25 x^2+250 x^2 \log (x)+625 x^2 \log ^2(x)\right )^2} \, dx+\int \frac {1}{e^9-25 x^2-250 x^2 \log (x)-625 x^2 \log ^2(x)} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.50, size = 32, normalized size = 1.39 \begin {gather*} -\frac {7+x}{e^9-25 x^2-250 x^2 \log (x)-625 x^2 \log ^2(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-E^9 - 2100*x - 275*x^2 + (-12250*x - 1500*x^2)*Log[x] + (-8750*x - 625*x^2)*Log[x]^2)/(E^18 - 50*E
^9*x^2 + 625*x^4 + (-500*E^9*x^2 + 12500*x^4)*Log[x] + (-1250*E^9*x^2 + 93750*x^4)*Log[x]^2 + 312500*x^4*Log[x
]^3 + 390625*x^4*Log[x]^4),x]

[Out]

-((7 + x)/(E^9 - 25*x^2 - 250*x^2*Log[x] - 625*x^2*Log[x]^2))

________________________________________________________________________________________

fricas [A]  time = 0.59, size = 32, normalized size = 1.39 \begin {gather*} \frac {x + 7}{625 \, x^{2} \log \relax (x)^{2} + 250 \, x^{2} \log \relax (x) + 25 \, x^{2} - e^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-625*x^2-8750*x)*log(x)^2+(-1500*x^2-12250*x)*log(x)-exp(9)-275*x^2-2100*x)/(390625*x^4*log(x)^4+3
12500*x^4*log(x)^3+(-1250*x^2*exp(9)+93750*x^4)*log(x)^2+(-500*x^2*exp(9)+12500*x^4)*log(x)+exp(9)^2-50*x^2*ex
p(9)+625*x^4),x, algorithm="fricas")

[Out]

(x + 7)/(625*x^2*log(x)^2 + 250*x^2*log(x) + 25*x^2 - e^9)

________________________________________________________________________________________

giac [A]  time = 0.36, size = 33, normalized size = 1.43 \begin {gather*} \frac {2 \, {\left (x + 7\right )}}{625 \, x^{2} \log \relax (x)^{2} + 250 \, x^{2} \log \relax (x) + 25 \, x^{2} - e^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-625*x^2-8750*x)*log(x)^2+(-1500*x^2-12250*x)*log(x)-exp(9)-275*x^2-2100*x)/(390625*x^4*log(x)^4+3
12500*x^4*log(x)^3+(-1250*x^2*exp(9)+93750*x^4)*log(x)^2+(-500*x^2*exp(9)+12500*x^4)*log(x)+exp(9)^2-50*x^2*ex
p(9)+625*x^4),x, algorithm="giac")

[Out]

2*(x + 7)/(625*x^2*log(x)^2 + 250*x^2*log(x) + 25*x^2 - e^9)

________________________________________________________________________________________

maple [A]  time = 0.19, size = 32, normalized size = 1.39

method	result	size
risch	\(-\frac {x +7}{-625 x^{2} \ln \relax (x )^{2}-250 x^{2} \ln \relax (x )-25 x^{2}+{\mathrm e}^{9}}\)	\(32\)
norman	\(\frac {-x -7}{-625 x^{2} \ln \relax (x )^{2}-250 x^{2} \ln \relax (x )-25 x^{2}+{\mathrm e}^{9}}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-625*x^2-8750*x)*ln(x)^2+(-1500*x^2-12250*x)*ln(x)-exp(9)-275*x^2-2100*x)/(390625*x^4*ln(x)^4+312500*x^4
*ln(x)^3+(-1250*x^2*exp(9)+93750*x^4)*ln(x)^2+(-500*x^2*exp(9)+12500*x^4)*ln(x)+exp(9)^2-50*x^2*exp(9)+625*x^4
),x,method=_RETURNVERBOSE)

[Out]

-(x+7)/(-625*x^2*ln(x)^2-250*x^2*ln(x)-25*x^2+exp(9))

________________________________________________________________________________________

maxima [A]  time = 0.41, size = 32, normalized size = 1.39 \begin {gather*} \frac {x + 7}{625 \, x^{2} \log \relax (x)^{2} + 250 \, x^{2} \log \relax (x) + 25 \, x^{2} - e^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-625*x^2-8750*x)*log(x)^2+(-1500*x^2-12250*x)*log(x)-exp(9)-275*x^2-2100*x)/(390625*x^4*log(x)^4+3
12500*x^4*log(x)^3+(-1250*x^2*exp(9)+93750*x^4)*log(x)^2+(-500*x^2*exp(9)+12500*x^4)*log(x)+exp(9)^2-50*x^2*ex
p(9)+625*x^4),x, algorithm="maxima")

[Out]

(x + 7)/(625*x^2*log(x)^2 + 250*x^2*log(x) + 25*x^2 - e^9)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {2100\,x+{\mathrm {e}}^9+{\ln \relax (x)}^2\,\left (625\,x^2+8750\,x\right )+\ln \relax (x)\,\left (1500\,x^2+12250\,x\right )+275\,x^2}{{\mathrm {e}}^{18}-\ln \relax (x)\,\left (500\,x^2\,{\mathrm {e}}^9-12500\,x^4\right )+312500\,x^4\,{\ln \relax (x)}^3+390625\,x^4\,{\ln \relax (x)}^4-50\,x^2\,{\mathrm {e}}^9-{\ln \relax (x)}^2\,\left (1250\,x^2\,{\mathrm {e}}^9-93750\,x^4\right )+625\,x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2100*x + exp(9) + log(x)^2*(8750*x + 625*x^2) + log(x)*(12250*x + 1500*x^2) + 275*x^2)/(exp(18) - log(x)
*(500*x^2*exp(9) - 12500*x^4) + 312500*x^4*log(x)^3 + 390625*x^4*log(x)^4 - 50*x^2*exp(9) - log(x)^2*(1250*x^2
*exp(9) - 93750*x^4) + 625*x^4),x)

[Out]

int(-(2100*x + exp(9) + log(x)^2*(8750*x + 625*x^2) + log(x)*(12250*x + 1500*x^2) + 275*x^2)/(exp(18) - log(x)
*(500*x^2*exp(9) - 12500*x^4) + 312500*x^4*log(x)^3 + 390625*x^4*log(x)^4 - 50*x^2*exp(9) - log(x)^2*(1250*x^2
*exp(9) - 93750*x^4) + 625*x^4), x)

________________________________________________________________________________________

sympy [A]  time = 0.24, size = 29, normalized size = 1.26 \begin {gather*} \frac {x + 7}{625 x^{2} \log {\relax (x )}^{2} + 250 x^{2} \log {\relax (x )} + 25 x^{2} - e^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-625*x**2-8750*x)*ln(x)**2+(-1500*x**2-12250*x)*ln(x)-exp(9)-275*x**2-2100*x)/(390625*x**4*ln(x)**
4+312500*x**4*ln(x)**3+(-1250*x**2*exp(9)+93750*x**4)*ln(x)**2+(-500*x**2*exp(9)+12500*x**4)*ln(x)+exp(9)**2-5
0*x**2*exp(9)+625*x**4),x)

[Out]

(x + 7)/(625*x**2*log(x)**2 + 250*x**2*log(x) + 25*x**2 - exp(9))

________________________________________________________________________________________