3.83.88 \(\int \frac {-4687500-7500 x-3 x^2+e^{\frac {3750+3 x+625 x^3}{1250+x}} (7031250 x^3+3750 x^4)}{1562500 x+2500 x^2+x^3} \, dx\)

Optimal. Leaf size=27 \[ 3 \left (1+e^{3+\frac {x^2}{\frac {1}{625}+\frac {2}{x}}}-\log (x)\right ) \]

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Rubi [F]  time = 1.03, 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 {-4687500-7500 x-3 x^2+e^{\frac {3750+3 x+625 x^3}{1250+x}} \left (7031250 x^3+3750 x^4\right )}{1562500 x+2500 x^2+x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-4687500 - 7500*x - 3*x^2 + E^((3750 + 3*x + 625*x^3)/(1250 + x))*(7031250*x^3 + 3750*x^4))/(1562500*x +
2500*x^2 + x^3),x]

[Out]

-3*Log[x] - 2343750*Defer[Int][E^((3750 + 3*x + 625*x^3)/(1250 + x)), x] + 3750*Defer[Int][E^((3750 + 3*x + 62
5*x^3)/(1250 + x))*x, x] + 3662109375000*Defer[Int][E^((3750 + 3*x + 625*x^3)/(1250 + x))/(1250 + x)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-4687500-7500 x-3 x^2+e^{\frac {3750+3 x+625 x^3}{1250+x}} \left (7031250 x^3+3750 x^4\right )}{x \left (1562500+2500 x+x^2\right )} \, dx\\ &=\int \frac {-4687500-7500 x-3 x^2+e^{\frac {3750+3 x+625 x^3}{1250+x}} \left (7031250 x^3+3750 x^4\right )}{x (1250+x)^2} \, dx\\ &=\int \left (-\frac {3}{x}+\frac {3750 e^{\frac {3750+3 x+625 x^3}{1250+x}} x^2 (1875+x)}{(1250+x)^2}\right ) \, dx\\ &=-3 \log (x)+3750 \int \frac {e^{\frac {3750+3 x+625 x^3}{1250+x}} x^2 (1875+x)}{(1250+x)^2} \, dx\\ &=-3 \log (x)+3750 \int \left (-625 e^{\frac {3750+3 x+625 x^3}{1250+x}}+e^{\frac {3750+3 x+625 x^3}{1250+x}} x+\frac {976562500 e^{\frac {3750+3 x+625 x^3}{1250+x}}}{(1250+x)^2}\right ) \, dx\\ &=-3 \log (x)+3750 \int e^{\frac {3750+3 x+625 x^3}{1250+x}} x \, dx-2343750 \int e^{\frac {3750+3 x+625 x^3}{1250+x}} \, dx+3662109375000 \int \frac {e^{\frac {3750+3 x+625 x^3}{1250+x}}}{(1250+x)^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 0.25, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-4687500-7500 x-3 x^2+e^{\frac {3750+3 x+625 x^3}{1250+x}} \left (7031250 x^3+3750 x^4\right )}{1562500 x+2500 x^2+x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(-4687500 - 7500*x - 3*x^2 + E^((3750 + 3*x + 625*x^3)/(1250 + x))*(7031250*x^3 + 3750*x^4))/(156250
0*x + 2500*x^2 + x^3),x]

[Out]

Integrate[(-4687500 - 7500*x - 3*x^2 + E^((3750 + 3*x + 625*x^3)/(1250 + x))*(7031250*x^3 + 3750*x^4))/(156250
0*x + 2500*x^2 + x^3), x]

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fricas [A]  time = 0.60, size = 24, normalized size = 0.89 \begin {gather*} 3 \, e^{\left (\frac {625 \, x^{3} + 3 \, x + 3750}{x + 1250}\right )} - 3 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3750*x^4+7031250*x^3)*exp((625*x^3+3*x+3750)/(x+1250))-3*x^2-7500*x-4687500)/(x^3+2500*x^2+1562500
*x),x, algorithm="fricas")

[Out]

3*e^((625*x^3 + 3*x + 3750)/(x + 1250)) - 3*log(x)

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giac [A]  time = 0.22, size = 20, normalized size = 0.74 \begin {gather*} 3 \, e^{\left (\frac {625 \, x^{3}}{x + 1250} + 3\right )} - 3 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3750*x^4+7031250*x^3)*exp((625*x^3+3*x+3750)/(x+1250))-3*x^2-7500*x-4687500)/(x^3+2500*x^2+1562500
*x),x, algorithm="giac")

[Out]

3*e^(625*x^3/(x + 1250) + 3) - 3*log(x)

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maple [A]  time = 49.17, size = 25, normalized size = 0.93

method	result	size
risch	\(-3 \ln \relax (x )+3 \,{\mathrm e}^{\frac {625 x^{3}+3 x +3750}{x +1250}}\)	\(25\)
norman	\(\frac {3 x \,{\mathrm e}^{\frac {625 x^{3}+3 x +3750}{x +1250}}+3750 \,{\mathrm e}^{\frac {625 x^{3}+3 x +3750}{x +1250}}}{x +1250}-3 \ln \relax (x )\)	\(52\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3750*x^4+7031250*x^3)*exp((625*x^3+3*x+3750)/(x+1250))-3*x^2-7500*x-4687500)/(x^3+2500*x^2+1562500*x),x,
method=_RETURNVERBOSE)

[Out]

-3*ln(x)+3*exp((625*x^3+3*x+3750)/(x+1250))

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maxima [A]  time = 0.49, size = 25, normalized size = 0.93 \begin {gather*} 3 \, e^{\left (625 \, x^{2} - 781250 \, x - \frac {1220703125000}{x + 1250} + 976562503\right )} - 3 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3750*x^4+7031250*x^3)*exp((625*x^3+3*x+3750)/(x+1250))-3*x^2-7500*x-4687500)/(x^3+2500*x^2+1562500
*x),x, algorithm="maxima")

[Out]

3*e^(625*x^2 - 781250*x - 1220703125000/(x + 1250) + 976562503) - 3*log(x)

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mupad [B]  time = 5.15, size = 35, normalized size = 1.30 \begin {gather*} 3\,{\mathrm {e}}^{\frac {3\,x}{x+1250}}\,{\mathrm {e}}^{\frac {625\,x^3}{x+1250}}\,{\mathrm {e}}^{\frac {3750}{x+1250}}-3\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(7500*x - exp((3*x + 625*x^3 + 3750)/(x + 1250))*(7031250*x^3 + 3750*x^4) + 3*x^2 + 4687500)/(1562500*x +
2500*x^2 + x^3),x)

[Out]

3*exp((3*x)/(x + 1250))*exp((625*x^3)/(x + 1250))*exp(3750/(x + 1250)) - 3*log(x)

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sympy [A]  time = 0.18, size = 20, normalized size = 0.74 \begin {gather*} 3 e^{\frac {625 x^{3} + 3 x + 3750}{x + 1250}} - 3 \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3750*x**4+7031250*x**3)*exp((625*x**3+3*x+3750)/(x+1250))-3*x**2-7500*x-4687500)/(x**3+2500*x**2+1
562500*x),x)

[Out]

3*exp((625*x**3 + 3*x + 3750)/(x + 1250)) - 3*log(x)

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