3.30.45 \(\int \frac {25 x^3+35 x^4+11 x^5+x^6+e^{\frac {-25-10 x^3+5 x^4+e (-10 x^3-2 x^4)}{25 x^3+5 x^4}} (75+20 x+7 x^4)}{25 x^4+10 x^5+x^6} \, dx\)

Optimal. Leaf size=26 \[ e^{-\frac {2 e}{5}+\frac {-2-\frac {5}{x^3}+x}{5+x}}+x+\log (x) \]

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Rubi [F]  time = 4.35, 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^3+35 x^4+11 x^5+x^6+\exp \left (\frac {-25-10 x^3+5 x^4+e \left (-10 x^3-2 x^4\right )}{25 x^3+5 x^4}\right ) \left (75+20 x+7 x^4\right )}{25 x^4+10 x^5+x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(25*x^3 + 35*x^4 + 11*x^5 + x^6 + E^((-25 - 10*x^3 + 5*x^4 + E*(-10*x^3 - 2*x^4))/(25*x^3 + 5*x^4))*(75 +
20*x + 7*x^4))/(25*x^4 + 10*x^5 + x^6),x]

[Out]

x + Log[x] + 3*Defer[Int][E^((-25 - 10*(1 + E)*x^3 + (5 - 2*E)*x^4)/(5*x^3*(5 + x)))/x^4, x] - (2*Defer[Int][E
^((-25 - 10*(1 + E)*x^3 + (5 - 2*E)*x^4)/(5*x^3*(5 + x)))/x^3, x])/5 + Defer[Int][E^((-25 - 10*(1 + E)*x^3 + (
5 - 2*E)*x^4)/(5*x^3*(5 + x)))/x^2, x]/25 + (174*Defer[Int][E^((-25 - 10*(1 + E)*x^3 + (5 - 2*E)*x^4)/(5*x^3*(
5 + x)))/(5 + x)^2, x])/25

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {25 x^3+35 x^4+11 x^5+x^6+\exp \left (\frac {-25-10 x^3+5 x^4+e \left (-10 x^3-2 x^4\right )}{25 x^3+5 x^4}\right ) \left (75+20 x+7 x^4\right )}{x^4 \left (25+10 x+x^2\right )} \, dx\\ &=\int \frac {25 x^3+35 x^4+11 x^5+x^6+\exp \left (\frac {-25-10 x^3+5 x^4+e \left (-10 x^3-2 x^4\right )}{25 x^3+5 x^4}\right ) \left (75+20 x+7 x^4\right )}{x^4 (5+x)^2} \, dx\\ &=\int \left (\frac {35}{(5+x)^2}+\frac {25}{x (5+x)^2}+\frac {11 x}{(5+x)^2}+\frac {x^2}{(5+x)^2}+\frac {\exp \left (\frac {-25-10 (1+e) x^3+(5-2 e) x^4}{5 x^3 (5+x)}\right ) \left (75+20 x+7 x^4\right )}{x^4 (5+x)^2}\right ) \, dx\\ &=-\frac {35}{5+x}+11 \int \frac {x}{(5+x)^2} \, dx+25 \int \frac {1}{x (5+x)^2} \, dx+\int \frac {x^2}{(5+x)^2} \, dx+\int \frac {\exp \left (\frac {-25-10 (1+e) x^3+(5-2 e) x^4}{5 x^3 (5+x)}\right ) \left (75+20 x+7 x^4\right )}{x^4 (5+x)^2} \, dx\\ &=-\frac {35}{5+x}+11 \int \left (-\frac {5}{(5+x)^2}+\frac {1}{5+x}\right ) \, dx+25 \int \left (\frac {1}{25 x}-\frac {1}{5 (5+x)^2}-\frac {1}{25 (5+x)}\right ) \, dx+\int \left (\frac {3 \exp \left (\frac {-25-10 (1+e) x^3+(5-2 e) x^4}{5 x^3 (5+x)}\right )}{x^4}-\frac {2 \exp \left (\frac {-25-10 (1+e) x^3+(5-2 e) x^4}{5 x^3 (5+x)}\right )}{5 x^3}+\frac {\exp \left (\frac {-25-10 (1+e) x^3+(5-2 e) x^4}{5 x^3 (5+x)}\right )}{25 x^2}+\frac {174 \exp \left (\frac {-25-10 (1+e) x^3+(5-2 e) x^4}{5 x^3 (5+x)}\right )}{25 (5+x)^2}\right ) \, dx+\int \left (1+\frac {25}{(5+x)^2}-\frac {10}{5+x}\right ) \, dx\\ &=x+\log (x)+\frac {1}{25} \int \frac {\exp \left (\frac {-25-10 (1+e) x^3+(5-2 e) x^4}{5 x^3 (5+x)}\right )}{x^2} \, dx-\frac {2}{5} \int \frac {\exp \left (\frac {-25-10 (1+e) x^3+(5-2 e) x^4}{5 x^3 (5+x)}\right )}{x^3} \, dx+3 \int \frac {\exp \left (\frac {-25-10 (1+e) x^3+(5-2 e) x^4}{5 x^3 (5+x)}\right )}{x^4} \, dx+\frac {174}{25} \int \frac {\exp \left (\frac {-25-10 (1+e) x^3+(5-2 e) x^4}{5 x^3 (5+x)}\right )}{(5+x)^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.70, size = 37, normalized size = 1.42 \begin {gather*} e^{-\frac {25+10 (1+e) x^3+(-5+2 e) x^4}{5 x^3 (5+x)}}+x+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(25*x^3 + 35*x^4 + 11*x^5 + x^6 + E^((-25 - 10*x^3 + 5*x^4 + E*(-10*x^3 - 2*x^4))/(25*x^3 + 5*x^4))*
(75 + 20*x + 7*x^4))/(25*x^4 + 10*x^5 + x^6),x]

[Out]

E^(-1/5*(25 + 10*(1 + E)*x^3 + (-5 + 2*E)*x^4)/(x^3*(5 + x))) + x + Log[x]

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fricas [A]  time = 0.46, size = 43, normalized size = 1.65 \begin {gather*} x + e^{\left (\frac {5 \, x^{4} - 10 \, x^{3} - 2 \, {\left (x^{4} + 5 \, x^{3}\right )} e - 25}{5 \, {\left (x^{4} + 5 \, x^{3}\right )}}\right )} + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((7*x^4+20*x+75)*exp(((-2*x^4-10*x^3)*exp(1)+5*x^4-10*x^3-25)/(5*x^4+25*x^3))+x^6+11*x^5+35*x^4+25*x
^3)/(x^6+10*x^5+25*x^4),x, algorithm="fricas")

[Out]

x + e^(1/5*(5*x^4 - 10*x^3 - 2*(x^4 + 5*x^3)*e - 25)/(x^4 + 5*x^3)) + log(x)

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giac [B]  time = 0.40, size = 86, normalized size = 3.31 \begin {gather*} x + e^{\left (-\frac {2 \, x^{4} e}{5 \, {\left (x^{4} + 5 \, x^{3}\right )}} + \frac {x^{4}}{x^{4} + 5 \, x^{3}} - \frac {2 \, x^{3} e}{x^{4} + 5 \, x^{3}} - \frac {2 \, x^{3}}{x^{4} + 5 \, x^{3}} - \frac {5}{x^{4} + 5 \, x^{3}}\right )} + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((7*x^4+20*x+75)*exp(((-2*x^4-10*x^3)*exp(1)+5*x^4-10*x^3-25)/(5*x^4+25*x^3))+x^6+11*x^5+35*x^4+25*x
^3)/(x^6+10*x^5+25*x^4),x, algorithm="giac")

[Out]

x + e^(-2/5*x^4*e/(x^4 + 5*x^3) + x^4/(x^4 + 5*x^3) - 2*x^3*e/(x^4 + 5*x^3) - 2*x^3/(x^4 + 5*x^3) - 5/(x^4 + 5
*x^3)) + log(x)

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maple [A]  time = 0.22, size = 42, normalized size = 1.62

method	result	size
risch	\(x +\ln \relax (x )+{\mathrm e}^{-\frac {2 x^{4} {\mathrm e}+10 x^{3} {\mathrm e}-5 x^{4}+10 x^{3}+25}{5 x^{3} \left (5+x \right )}}\)	\(42\)
norman	\(\frac {x^{5}+x^{4} {\mathrm e}^{\frac {\left (-2 x^{4}-10 x^{3}\right ) {\mathrm e}+5 x^{4}-10 x^{3}-25}{5 x^{4}+25 x^{3}}}-25 x^{3}+5 x^{3} {\mathrm e}^{\frac {\left (-2 x^{4}-10 x^{3}\right ) {\mathrm e}+5 x^{4}-10 x^{3}-25}{5 x^{4}+25 x^{3}}}}{x^{3} \left (5+x \right )}+\ln \relax (x )\)	\(113\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((7*x^4+20*x+75)*exp(((-2*x^4-10*x^3)*exp(1)+5*x^4-10*x^3-25)/(5*x^4+25*x^3))+x^6+11*x^5+35*x^4+25*x^3)/(x
^6+10*x^5+25*x^4),x,method=_RETURNVERBOSE)

[Out]

x+ln(x)+exp(-1/5*(2*x^4*exp(1)+10*x^3*exp(1)-5*x^4+10*x^3+25)/x^3/(5+x))

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maxima [A]  time = 0.86, size = 33, normalized size = 1.27 \begin {gather*} x + e^{\left (-\frac {174}{25 \, {\left (x + 5\right )}} - \frac {1}{25 \, x} + \frac {1}{5 \, x^{2}} - \frac {1}{x^{3}} - \frac {2}{5} \, e + 1\right )} + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((7*x^4+20*x+75)*exp(((-2*x^4-10*x^3)*exp(1)+5*x^4-10*x^3-25)/(5*x^4+25*x^3))+x^6+11*x^5+35*x^4+25*x
^3)/(x^6+10*x^5+25*x^4),x, algorithm="maxima")

[Out]

x + e^(-174/25/(x + 5) - 1/25/x + 1/5/x^2 - 1/x^3 - 2/5*e + 1) + log(x)

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mupad [B]  time = 1.91, size = 101, normalized size = 3.88 \begin {gather*} x+\ln \relax (x)+{\mathrm {e}}^{-\frac {2\,x^4\,\mathrm {e}}{5\,x^4+25\,x^3}}\,{\mathrm {e}}^{-\frac {10\,x^3\,\mathrm {e}}{5\,x^4+25\,x^3}}\,{\mathrm {e}}^{\frac {5\,x^4}{5\,x^4+25\,x^3}}\,{\mathrm {e}}^{-\frac {10\,x^3}{5\,x^4+25\,x^3}}\,{\mathrm {e}}^{-\frac {25}{5\,x^4+25\,x^3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-(exp(1)*(10*x^3 + 2*x^4) + 10*x^3 - 5*x^4 + 25)/(25*x^3 + 5*x^4))*(20*x + 7*x^4 + 75) + 25*x^3 + 35*
x^4 + 11*x^5 + x^6)/(25*x^4 + 10*x^5 + x^6),x)

[Out]

x + log(x) + exp(-(2*x^4*exp(1))/(25*x^3 + 5*x^4))*exp(-(10*x^3*exp(1))/(25*x^3 + 5*x^4))*exp((5*x^4)/(25*x^3
+ 5*x^4))*exp(-(10*x^3)/(25*x^3 + 5*x^4))*exp(-25/(25*x^3 + 5*x^4))

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sympy [A]  time = 0.26, size = 42, normalized size = 1.62 \begin {gather*} x + e^{\frac {5 x^{4} - 10 x^{3} + e \left (- 2 x^{4} - 10 x^{3}\right ) - 25}{5 x^{4} + 25 x^{3}}} + \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((7*x**4+20*x+75)*exp(((-2*x**4-10*x**3)*exp(1)+5*x**4-10*x**3-25)/(5*x**4+25*x**3))+x**6+11*x**5+35
*x**4+25*x**3)/(x**6+10*x**5+25*x**4),x)

[Out]

x + exp((5*x**4 - 10*x**3 + E*(-2*x**4 - 10*x**3) - 25)/(5*x**4 + 25*x**3)) + log(x)

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