3.95.30 \(\int \frac {-10 e^x x^2+25 x^3-2 e^{2 x} x^3+(9325 x^2-750 e^{2 x} x^2+e^x (-1250 x-10 x^3)) \log (x)+(1171875 x-93750 e^{2 x} x+e^x (-1250 x-2500 x^2)) \log ^2(x)+(48828125-3906250 e^{2 x}+e^x (-156250-156250 x)-6250 x) \log ^3(x)}{25 x^3+9375 x^2 \log (x)+1171875 x \log ^2(x)[phone] \log ^3(x)} \, dx\)

Optimal. Leaf size=28 \[ -1+x-\frac {1}{25} \left (e^x+\frac {x}{25+\frac {x}{5 \log (x)}}\right )^2 \]

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Rubi [F]  time = 1.62, 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 {-10 e^x x^2+25 x^3-2 e^{2 x} x^3+\left (9325 x^2-750 e^{2 x} x^2+e^x \left (-1250 x-10 x^3\right )\right ) \log (x)+\left (1171875 x-93750 e^{2 x} x+e^x \left (-1250 x-2500 x^2\right )\right ) \log ^2(x)+\left (48828125-3906250 e^{2 x}+e^x (-156250-156250 x)-6250 x\right ) \log ^3(x)}{25 x^3+9375 x^2 \log (x)+1171875 x \log ^2(x)[phone] \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-10*E^x*x^2 + 25*x^3 - 2*E^(2*x)*x^3 + (9325*x^2 - 750*E^(2*x)*x^2 + E^x*(-1250*x - 10*x^3))*Log[x] + (11
71875*x - 93750*E^(2*x)*x + E^x*(-1250*x - 2500*x^2))*Log[x]^2 + (48828125 - 3906250*E^(2*x) + E^x*(-156250 -
156250*x) - 6250*x)*Log[x]^3)/(25*x^3 + 9375*x^2*Log[x] + 1171875*x*Log[x]^2 + 48828125*Log[x]^3),x]

[Out]

-1/25*E^(2*x) + x - x^2/15625 - (2*E^x*(x^2*Log[x] + 125*x*Log[x]^2))/(5*(x + 125*Log[x])^2) + (2*Defer[Int][x
^3/(x + 125*Log[x])^3, x])/125 + (2*Defer[Int][x^4/(x + 125*Log[x])^3, x])/15625 - (2*Defer[Int][x^2/(x + 125*
Log[x])^2, x])/125 - (6*Defer[Int][x^3/(x + 125*Log[x])^2, x])/15625 + (6*Defer[Int][x^2/(x + 125*Log[x]), x])
/15625

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-10 e^x x^2+25 x^3-2 e^{2 x} x^3+\left (9325 x^2-750 e^{2 x} x^2+e^x \left (-1250 x-10 x^3\right )\right ) \log (x)+\left (1171875 x-93750 e^{2 x} x+e^x \left (-1250 x-2500 x^2\right )\right ) \log ^2(x)+\left (48828125-3906250 e^{2 x}+e^x (-156250-156250 x)-6250 x\right ) \log ^3(x)}{25 (x+125 \log (x))^3} \, dx\\ &=\frac {1}{25} \int \frac {-10 e^x x^2+25 x^3-2 e^{2 x} x^3+\left (9325 x^2-750 e^{2 x} x^2+e^x \left (-1250 x-10 x^3\right )\right ) \log (x)+\left (1171875 x-93750 e^{2 x} x+e^x \left (-1250 x-2500 x^2\right )\right ) \log ^2(x)+\left (48828125-3906250 e^{2 x}+e^x (-156250-156250 x)-6250 x\right ) \log ^3(x)}{(x+125 \log (x))^3} \, dx\\ &=\frac {1}{25} \int \left (-2 e^{2 x}+\frac {25 x^3}{(x+125 \log (x))^3}+\frac {9325 x^2 \log (x)}{(x+125 \log (x))^3}+\frac {1171875 x \log ^2(x)}{(x+125 \log (x))^3}+\frac {48828125 \log ^3(x)}{(x+125 \log (x))^3}-\frac {6250 x \log ^3(x)}{(x+125 \log (x))^3}-\frac {10 e^x \left (x+x^2 \log (x)+125 \log ^2(x)+125 x \log ^2(x)\right )}{(x+125 \log (x))^2}\right ) \, dx\\ &=-\left (\frac {2}{25} \int e^{2 x} \, dx\right )-\frac {2}{5} \int \frac {e^x \left (x+x^2 \log (x)+125 \log ^2(x)+125 x \log ^2(x)\right )}{(x+125 \log (x))^2} \, dx-250 \int \frac {x \log ^3(x)}{(x+125 \log (x))^3} \, dx+373 \int \frac {x^2 \log (x)}{(x+125 \log (x))^3} \, dx+46875 \int \frac {x \log ^2(x)}{(x+125 \log (x))^3} \, dx+1953125 \int \frac {\log ^3(x)}{(x+125 \log (x))^3} \, dx+\int \frac {x^3}{(x+125 \log (x))^3} \, dx\\ &=-\frac {e^{2 x}}{25}-\frac {2 e^x \left (x^2 \log (x)+125 x \log ^2(x)\right )}{5 (x+125 \log (x))^2}-250 \int \left (\frac {x}{1953125}-\frac {x^4}{1953125 (x+125 \log (x))^3}+\frac {3 x^3}{1953125 (x+125 \log (x))^2}-\frac {3 x^2}{1953125 (x+125 \log (x))}\right ) \, dx+373 \int \left (-\frac {x^3}{125 (x+125 \log (x))^3}+\frac {x^2}{125 (x+125 \log (x))^2}\right ) \, dx+46875 \int \left (\frac {x^3}{15625 (x+125 \log (x))^3}-\frac {2 x^2}{15625 (x+125 \log (x))^2}+\frac {x}{15625 (x+125 \log (x))}\right ) \, dx+1953125 \int \left (\frac {1}{1953125}-\frac {x^3}{1953125 (x+125 \log (x))^3}+\frac {3 x^2}{1953125 (x+125 \log (x))^2}-\frac {3 x}{1953125 (x+125 \log (x))}\right ) \, dx+\int \frac {x^3}{(x+125 \log (x))^3} \, dx\\ &=-\frac {e^{2 x}}{25}+x-\frac {x^2}{15625}-\frac {2 e^x \left (x^2 \log (x)+125 x \log ^2(x)\right )}{5 (x+125 \log (x))^2}+\frac {2 \int \frac {x^4}{(x+125 \log (x))^3} \, dx}{15625}-\frac {6 \int \frac {x^3}{(x+125 \log (x))^2} \, dx}{15625}+\frac {6 \int \frac {x^2}{x+125 \log (x)} \, dx}{15625}-\frac {373}{125} \int \frac {x^3}{(x+125 \log (x))^3} \, dx+\frac {373}{125} \int \frac {x^2}{(x+125 \log (x))^2} \, dx+3 \int \frac {x^3}{(x+125 \log (x))^3} \, dx+3 \int \frac {x^2}{(x+125 \log (x))^2} \, dx-6 \int \frac {x^2}{(x+125 \log (x))^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.15, size = 59, normalized size = 2.11 \begin {gather*} \frac {-625 e^{2 x}+15625 x-50 e^x x-x^2-\frac {x^4}{(x+125 \log (x))^2}+\frac {2 x^2 \left (25 e^x+x\right )}{x+125 \log (x)}}{15625} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-10*E^x*x^2 + 25*x^3 - 2*E^(2*x)*x^3 + (9325*x^2 - 750*E^(2*x)*x^2 + E^x*(-1250*x - 10*x^3))*Log[x]
+ (1171875*x - 93750*E^(2*x)*x + E^x*(-1250*x - 2500*x^2))*Log[x]^2 + (48828125 - 3906250*E^(2*x) + E^x*(-156
250 - 156250*x) - 6250*x)*Log[x]^3)/(25*x^3 + 9375*x^2*Log[x] + 1171875*x*Log[x]^2 + 48828125*Log[x]^3),x]

[Out]

(-625*E^(2*x) + 15625*x - 50*E^x*x - x^2 - x^4/(x + 125*Log[x])^2 + (2*x^2*(25*E^x + x))/(x + 125*Log[x]))/156
25

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fricas [B]  time = 0.69, size = 81, normalized size = 2.89 \begin {gather*} \frac {25 \, x^{3} - x^{2} e^{\left (2 \, x\right )} - 25 \, {\left (x^{2} + 50 \, x e^{x} - 15625 \, x + 625 \, e^{\left (2 \, x\right )}\right )} \log \relax (x)^{2} - 10 \, {\left (x^{2} e^{x} - 625 \, x^{2} + 25 \, x e^{\left (2 \, x\right )}\right )} \log \relax (x)}{25 \, {\left (x^{2} + 250 \, x \log \relax (x) + 15625 \, \log \relax (x)^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3906250*exp(x)^2+(-156250*x-156250)*exp(x)-6250*x+48828125)*log(x)^3+(-93750*x*exp(x)^2+(-2500*x^
2-1250*x)*exp(x)+1171875*x)*log(x)^2+(-750*exp(x)^2*x^2+(-10*x^3-1250*x)*exp(x)+9325*x^2)*log(x)-2*exp(x)^2*x^
3-10*exp(x)*x^2+25*x^3)/(48828125*log(x)^3+1171875*x*log(x)^2+9375*x^2*log(x)+25*x^3),x, algorithm="fricas")

[Out]

1/25*(25*x^3 - x^2*e^(2*x) - 25*(x^2 + 50*x*e^x - 15625*x + 625*e^(2*x))*log(x)^2 - 10*(x^2*e^x - 625*x^2 + 25
*x*e^(2*x))*log(x))/(x^2 + 250*x*log(x) + 15625*log(x)^2)

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giac [B]  time = 0.23, size = 93, normalized size = 3.32 \begin {gather*} -\frac {10 \, x^{2} e^{x} \log \relax (x) + 25 \, x^{2} \log \relax (x)^{2} + 1250 \, x e^{x} \log \relax (x)^{2} - 25 \, x^{3} + x^{2} e^{\left (2 \, x\right )} - 6250 \, x^{2} \log \relax (x) + 250 \, x e^{\left (2 \, x\right )} \log \relax (x) - 390625 \, x \log \relax (x)^{2} + 15625 \, e^{\left (2 \, x\right )} \log \relax (x)^{2}}{25 \, {\left (x^{2} + 250 \, x \log \relax (x) + 15625 \, \log \relax (x)^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3906250*exp(x)^2+(-156250*x-156250)*exp(x)-6250*x+48828125)*log(x)^3+(-93750*x*exp(x)^2+(-2500*x^
2-1250*x)*exp(x)+1171875*x)*log(x)^2+(-750*exp(x)^2*x^2+(-10*x^3-1250*x)*exp(x)+9325*x^2)*log(x)-2*exp(x)^2*x^
3-10*exp(x)*x^2+25*x^3)/(48828125*log(x)^3+1171875*x*log(x)^2+9375*x^2*log(x)+25*x^3),x, algorithm="giac")

[Out]

-1/25*(10*x^2*e^x*log(x) + 25*x^2*log(x)^2 + 1250*x*e^x*log(x)^2 - 25*x^3 + x^2*e^(2*x) - 6250*x^2*log(x) + 25
0*x*e^(2*x)*log(x) - 390625*x*log(x)^2 + 15625*e^(2*x)*log(x)^2)/(x^2 + 250*x*log(x) + 15625*log(x)^2)

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maple [A]  time = 0.08, size = 52, normalized size = 1.86

method	result	size
risch	\(-\frac {x^{2}}{15625}+x -\frac {{\mathrm e}^{2 x}}{25}-\frac {2 \,{\mathrm e}^{x} x}{625}+\frac {\left (x^{2}+50 \,{\mathrm e}^{x} x +250 x \ln \relax (x )+6250 \,{\mathrm e}^{x} \ln \relax (x )\right ) x^{2}}{15625 \left (125 \ln \relax (x )+x \right )^{2}}\)	\(52\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-3906250*exp(x)^2+(-156250*x-156250)*exp(x)-6250*x+48828125)*ln(x)^3+(-93750*x*exp(x)^2+(-2500*x^2-1250*
x)*exp(x)+1171875*x)*ln(x)^2+(-750*exp(x)^2*x^2+(-10*x^3-1250*x)*exp(x)+9325*x^2)*ln(x)-2*exp(x)^2*x^3-10*exp(
x)*x^2+25*x^3)/(48828125*ln(x)^3+1171875*x*ln(x)^2+9375*x^2*ln(x)+25*x^3),x,method=_RETURNVERBOSE)

[Out]

-1/15625*x^2+x-1/25*exp(2*x)-2/625*exp(x)*x+1/15625*(x^2+50*exp(x)*x+250*x*ln(x)+6250*exp(x)*ln(x))*x^2/(125*l
n(x)+x)^2

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maxima [B]  time = 0.41, size = 84, normalized size = 3.00 \begin {gather*} \frac {25 \, x^{3} + 6250 \, x^{2} \log \relax (x) - 25 \, {\left (x^{2} - 15625 \, x\right )} \log \relax (x)^{2} - {\left (x^{2} + 250 \, x \log \relax (x) + 15625 \, \log \relax (x)^{2}\right )} e^{\left (2 \, x\right )} - 10 \, {\left (x^{2} \log \relax (x) + 125 \, x \log \relax (x)^{2}\right )} e^{x}}{25 \, {\left (x^{2} + 250 \, x \log \relax (x) + 15625 \, \log \relax (x)^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3906250*exp(x)^2+(-156250*x-156250)*exp(x)-6250*x+48828125)*log(x)^3+(-93750*x*exp(x)^2+(-2500*x^
2-1250*x)*exp(x)+1171875*x)*log(x)^2+(-750*exp(x)^2*x^2+(-10*x^3-1250*x)*exp(x)+9325*x^2)*log(x)-2*exp(x)^2*x^
3-10*exp(x)*x^2+25*x^3)/(48828125*log(x)^3+1171875*x*log(x)^2+9375*x^2*log(x)+25*x^3),x, algorithm="maxima")

[Out]

1/25*(25*x^3 + 6250*x^2*log(x) - 25*(x^2 - 15625*x)*log(x)^2 - (x^2 + 250*x*log(x) + 15625*log(x)^2)*e^(2*x) -
10*(x^2*log(x) + 125*x*log(x)^2)*e^x)/(x^2 + 250*x*log(x) + 15625*log(x)^2)

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mupad [B]  time = 8.22, size = 547, normalized size = 19.54 \begin {gather*} \frac {24\,x}{25}-\frac {{\mathrm {e}}^{2\,x}}{25}+9\,\ln \relax (x)+\ln \relax (x)\,\left ({\mathrm {e}}^x\,\left (\frac {\frac {x^5}{5}+\frac {129\,x^4}{5}+\frac {627\,x^3}{5}+1534525\,x^2+335175000\,x+17969531250}{x^3+375\,x^2+46875\,x+1953125}-\frac {1534425\,x^2+335175000\,x+17969531250}{x^3+375\,x^2+46875\,x+1953125}\right )+\frac {\frac {24\,x^4}{125}+72\,x^3+9000\,x^2+375000\,x}{x^3+375\,x^2+46875\,x+1953125}-\frac {\frac {18\,x^4}{125}+72\,x^3+12375\,x^2+796875\,x+17578125}{x^3+375\,x^2+46875\,x+1953125}\right )-\frac {62500\,x+5859375}{x^3+375\,x^2+46875\,x+1953125}-\frac {\frac {x\,\left (25\,x^3\,{\mathrm {e}}^x-3125\,x^2\,{\mathrm {e}}^x+25\,x^4\,{\mathrm {e}}^x+x^4\right )}{15625\,\left (x+125\right )}+\frac {x\,\ln \relax (x)\,\left (75\,x^2\,{\mathrm {e}}^x+50\,x^3\,{\mathrm {e}}^x-3125\,x\,{\mathrm {e}}^x-125\,x^2+3\,x^3\right )}{125\,\left (x+125\right )}+\frac {x\,{\ln \relax (x)}^2\,\left (25\,x^2\,{\mathrm {e}}^x+50\,x\,{\mathrm {e}}^x+3\,x^2\right )}{x+125}}{x^2+250\,x\,\ln \relax (x)+15625\,{\ln \relax (x)}^2}-\frac {\frac {x\,\left (406250\,x^3\,{\mathrm {e}}^x-390625\,x^2\,{\mathrm {e}}^x+21950\,x^4\,{\mathrm {e}}^x+3250\,x^5\,{\mathrm {e}}^x+25\,x^6\,{\mathrm {e}}^x-48828125\,x\,{\mathrm {e}}^x-1953125\,x^2+31250\,x^3+1000\,x^4+4\,x^5\right )}{15625\,{\left (x+125\right )}^3}+\frac {x\,{\ln \relax (x)}^2\,\left (15675\,x^2\,{\mathrm {e}}^x+3225\,x^3\,{\mathrm {e}}^x+25\,x^4\,{\mathrm {e}}^x+12500\,x\,{\mathrm {e}}^x+1125\,x^2+6\,x^3\right )}{{\left (x+125\right )}^3}+\frac {x\,\ln \relax (x)\,\left (428125\,x^2\,{\mathrm {e}}^x+37650\,x^3\,{\mathrm {e}}^x+6475\,x^4\,{\mathrm {e}}^x+50\,x^5\,{\mathrm {e}}^x+781250\,x\,{\mathrm {e}}^x+46875\,x^2+2000\,x^3+9\,x^4\right )}{125\,{\left (x+125\right )}^3}}{x+125\,\ln \relax (x)}+\frac {x^2}{3125}+\frac {{\mathrm {e}}^x\,\left (\frac {x^6}{625}+\frac {131\,x^5}{625}+\frac {1129\,x^4}{625}+\frac {256\,x^3}{5}-50\,x^2-6250\,x\right )}{x^3+375\,x^2+46875\,x+1953125} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(10*x^2*exp(x) + 2*x^3*exp(2*x) + log(x)^2*(93750*x*exp(2*x) - 1171875*x + exp(x)*(1250*x + 2500*x^2)) -
25*x^3 + log(x)^3*(6250*x + 3906250*exp(2*x) + exp(x)*(156250*x + 156250) - 48828125) + log(x)*(750*x^2*exp(2*
x) + exp(x)*(1250*x + 10*x^3) - 9325*x^2))/(1171875*x*log(x)^2 + 9375*x^2*log(x) + 48828125*log(x)^3 + 25*x^3)
,x)

[Out]

(24*x)/25 - exp(2*x)/25 + 9*log(x) + log(x)*(exp(x)*((335175000*x + 1534525*x^2 + (627*x^3)/5 + (129*x^4)/5 +
x^5/5 + 17969531250)/(46875*x + 375*x^2 + x^3 + 1953125) - (335175000*x + 1534425*x^2 + 17969531250)/(46875*x
+ 375*x^2 + x^3 + 1953125)) + (375000*x + 9000*x^2 + 72*x^3 + (24*x^4)/125)/(46875*x + 375*x^2 + x^3 + 1953125
) - (796875*x + 12375*x^2 + 72*x^3 + (18*x^4)/125 + 17578125)/(46875*x + 375*x^2 + x^3 + 1953125)) - (62500*x
+ 5859375)/(46875*x + 375*x^2 + x^3 + 1953125) - ((x*(25*x^3*exp(x) - 3125*x^2*exp(x) + 25*x^4*exp(x) + x^4))/
(15625*(x + 125)) + (x*log(x)*(75*x^2*exp(x) + 50*x^3*exp(x) - 3125*x*exp(x) - 125*x^2 + 3*x^3))/(125*(x + 125
)) + (x*log(x)^2*(25*x^2*exp(x) + 50*x*exp(x) + 3*x^2))/(x + 125))/(15625*log(x)^2 + 250*x*log(x) + x^2) - ((x
*(406250*x^3*exp(x) - 390625*x^2*exp(x) + 21950*x^4*exp(x) + 3250*x^5*exp(x) + 25*x^6*exp(x) - 48828125*x*exp(
x) - 1953125*x^2 + 31250*x^3 + 1000*x^4 + 4*x^5))/(15625*(x + 125)^3) + (x*log(x)^2*(15675*x^2*exp(x) + 3225*x
^3*exp(x) + 25*x^4*exp(x) + 12500*x*exp(x) + 1125*x^2 + 6*x^3))/(x + 125)^3 + (x*log(x)*(428125*x^2*exp(x) + 3
7650*x^3*exp(x) + 6475*x^4*exp(x) + 50*x^5*exp(x) + 781250*x*exp(x) + 46875*x^2 + 2000*x^3 + 9*x^4))/(125*(x +
125)^3))/(x + 125*log(x)) + x^2/3125 + (exp(x)*((256*x^3)/5 - 50*x^2 - 6250*x + (1129*x^4)/625 + (131*x^5)/62
5 + x^6/625))/(46875*x + 375*x^2 + x^3 + 1953125)

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sympy [B]  time = 0.45, size = 70, normalized size = 2.50 \begin {gather*} - \frac {x^{2}}{15625} + x + \frac {x^{4} + 250 x^{3} \log {\relax (x )}}{15625 x^{2} + 3906250 x \log {\relax (x )} + 244140625 \log {\relax (x )}^{2}} + \frac {- 50 x e^{x} \log {\relax (x )} + \left (- 5 x - 625 \log {\relax (x )}\right ) e^{2 x}}{125 x + 15625 \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3906250*exp(x)**2+(-156250*x-156250)*exp(x)-6250*x+48828125)*ln(x)**3+(-93750*x*exp(x)**2+(-2500*
x**2-1250*x)*exp(x)+1171875*x)*ln(x)**2+(-750*exp(x)**2*x**2+(-10*x**3-1250*x)*exp(x)+9325*x**2)*ln(x)-2*exp(x
)**2*x**3-10*exp(x)*x**2+25*x**3)/(48828125*ln(x)**3+1171875*x*ln(x)**2+9375*x**2*ln(x)+25*x**3),x)

[Out]

-x**2/15625 + x + (x**4 + 250*x**3*log(x))/(15625*x**2 + 3906250*x*log(x) + 244140625*log(x)**2) + (-50*x*exp(
x)*log(x) + (-5*x - 625*log(x))*exp(2*x))/(125*x + 15625*log(x))

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