3.75.24 \(\int \frac {25+e^{\frac {1}{5} (-7+5 x-2 x^3)} (-25+25 x-30 x^3)}{6-12 e^{\frac {1}{5} (-7+5 x-2 x^3)}+6 e^{\frac {2}{5} (-7+5 x-2 x^3)}} \, dx\)

Optimal. Leaf size=26 \[ \frac {25 x}{6 \left (1-e^{x+\frac {1}{5} \left (-7-2 x^3\right )}\right )} \]

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Rubi [F]  time = 3.28, 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+e^{\frac {1}{5} \left (-7+5 x-2 x^3\right )} \left (-25+25 x-30 x^3\right )}{6-12 e^{\frac {1}{5} \left (-7+5 x-2 x^3\right )}+6 e^{\frac {2}{5} \left (-7+5 x-2 x^3\right )}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(25 + E^((-7 + 5*x - 2*x^3)/5)*(-25 + 25*x - 30*x^3))/(6 - 12*E^((-7 + 5*x - 2*x^3)/5) + 6*E^((2*(-7 + 5*x
- 2*x^3))/5)),x]

[Out]

(-25*E^(7/5 - x + (2*x^3)/5)*(5*x - 6*x^3))/(6*(5 - 6*x^2)) - (25*Defer[Int][E^(14/5 - x + (4*x^3)/5)/(E^x - E
^(7/5 + (2*x^3)/5)), x])/6 + (25*Defer[Int][(E^(14/5 + (4*x^3)/5)*x)/(E^x - E^(7/5 + (2*x^3)/5))^2, x])/6 + (2
5*Defer[Int][(E^(14/5 - x + (4*x^3)/5)*x)/(E^x - E^(7/5 + (2*x^3)/5)), x])/6 - 5*Defer[Int][(E^(14/5 + (4*x^3)
/5)*x^3)/(E^x - E^(7/5 + (2*x^3)/5))^2, x] - 5*Defer[Int][(E^(14/5 - x + (4*x^3)/5)*x^3)/(E^x - E^(7/5 + (2*x^
3)/5)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {14}{5}+\frac {4 x^3}{5}} \left (25+e^{\frac {1}{5} \left (-7+5 x-2 x^3\right )} \left (-25+25 x-30 x^3\right )\right )}{6 \left (e^x-e^{\frac {7}{5}+\frac {2 x^3}{5}}\right )^2} \, dx\\ &=\frac {1}{6} \int \frac {e^{\frac {14}{5}+\frac {4 x^3}{5}} \left (25+e^{\frac {1}{5} \left (-7+5 x-2 x^3\right )} \left (-25+25 x-30 x^3\right )\right )}{\left (e^x-e^{\frac {7}{5}+\frac {2 x^3}{5}}\right )^2} \, dx\\ &=\frac {1}{6} \int \left (-\frac {5 e^{\frac {14}{5}+\frac {4 x^3}{5}} x \left (-5+6 x^2\right )}{\left (e^x-e^{\frac {7}{5}+\frac {2 x^3}{5}}\right )^2}-5 e^{\frac {7}{5}-x+\frac {2 x^3}{5}} \left (5-5 x+6 x^3\right )-\frac {5 e^{\frac {14}{5}-x+\frac {4 x^3}{5}} \left (5-5 x+6 x^3\right )}{e^x-e^{\frac {7}{5}+\frac {2 x^3}{5}}}\right ) \, dx\\ &=-\left (\frac {5}{6} \int \frac {e^{\frac {14}{5}+\frac {4 x^3}{5}} x \left (-5+6 x^2\right )}{\left (e^x-e^{\frac {7}{5}+\frac {2 x^3}{5}}\right )^2} \, dx\right )-\frac {5}{6} \int e^{\frac {7}{5}-x+\frac {2 x^3}{5}} \left (5-5 x+6 x^3\right ) \, dx-\frac {5}{6} \int \frac {e^{\frac {14}{5}-x+\frac {4 x^3}{5}} \left (5-5 x+6 x^3\right )}{e^x-e^{\frac {7}{5}+\frac {2 x^3}{5}}} \, dx\\ &=-\frac {25 e^{\frac {7}{5}-x+\frac {2 x^3}{5}} \left (5 x-6 x^3\right )}{6 \left (5-6 x^2\right )}-\frac {5}{6} \int \left (-\frac {5 e^{\frac {14}{5}+\frac {4 x^3}{5}} x}{\left (e^x-e^{\frac {7}{5}+\frac {2 x^3}{5}}\right )^2}+\frac {6 e^{\frac {14}{5}+\frac {4 x^3}{5}} x^3}{\left (e^x-e^{\frac {7}{5}+\frac {2 x^3}{5}}\right )^2}\right ) \, dx-\frac {5}{6} \int \left (\frac {5 e^{\frac {14}{5}-x+\frac {4 x^3}{5}}}{e^x-e^{\frac {7}{5}+\frac {2 x^3}{5}}}-\frac {5 e^{\frac {14}{5}-x+\frac {4 x^3}{5}} x}{e^x-e^{\frac {7}{5}+\frac {2 x^3}{5}}}+\frac {6 e^{\frac {14}{5}-x+\frac {4 x^3}{5}} x^3}{e^x-e^{\frac {7}{5}+\frac {2 x^3}{5}}}\right ) \, dx\\ &=-\frac {25 e^{\frac {7}{5}-x+\frac {2 x^3}{5}} \left (5 x-6 x^3\right )}{6 \left (5-6 x^2\right )}-\frac {25}{6} \int \frac {e^{\frac {14}{5}-x+\frac {4 x^3}{5}}}{e^x-e^{\frac {7}{5}+\frac {2 x^3}{5}}} \, dx+\frac {25}{6} \int \frac {e^{\frac {14}{5}+\frac {4 x^3}{5}} x}{\left (e^x-e^{\frac {7}{5}+\frac {2 x^3}{5}}\right )^2} \, dx+\frac {25}{6} \int \frac {e^{\frac {14}{5}-x+\frac {4 x^3}{5}} x}{e^x-e^{\frac {7}{5}+\frac {2 x^3}{5}}} \, dx-5 \int \frac {e^{\frac {14}{5}+\frac {4 x^3}{5}} x^3}{\left (e^x-e^{\frac {7}{5}+\frac {2 x^3}{5}}\right )^2} \, dx-5 \int \frac {e^{\frac {14}{5}-x+\frac {4 x^3}{5}} x^3}{e^x-e^{\frac {7}{5}+\frac {2 x^3}{5}}} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.69, size = 39, normalized size = 1.50 \begin {gather*} \frac {25 e^{\frac {7}{5}+\frac {2 x^3}{5}} x}{6 \left (-e^x+e^{\frac {7}{5}+\frac {2 x^3}{5}}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(25 + E^((-7 + 5*x - 2*x^3)/5)*(-25 + 25*x - 30*x^3))/(6 - 12*E^((-7 + 5*x - 2*x^3)/5) + 6*E^((2*(-7
+ 5*x - 2*x^3))/5)),x]

[Out]

(25*E^(7/5 + (2*x^3)/5)*x)/(6*(-E^x + E^(7/5 + (2*x^3)/5)))

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fricas [A]  time = 0.60, size = 16, normalized size = 0.62 \begin {gather*} -\frac {25 \, x}{6 \, {\left (e^{\left (-\frac {2}{5} \, x^{3} + x - \frac {7}{5}\right )} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-30*x^3+25*x-25)*exp(-2/5*x^3+x-7/5)+25)/(6*exp(-2/5*x^3+x-7/5)^2-12*exp(-2/5*x^3+x-7/5)+6),x, alg
orithm="fricas")

[Out]

-25/6*x/(e^(-2/5*x^3 + x - 7/5) - 1)

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giac [A]  time = 0.27, size = 16, normalized size = 0.62 \begin {gather*} -\frac {25 \, x}{6 \, {\left (e^{\left (-\frac {2}{5} \, x^{3} + x - \frac {7}{5}\right )} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-30*x^3+25*x-25)*exp(-2/5*x^3+x-7/5)+25)/(6*exp(-2/5*x^3+x-7/5)^2-12*exp(-2/5*x^3+x-7/5)+6),x, alg
orithm="giac")

[Out]

-25/6*x/(e^(-2/5*x^3 + x - 7/5) - 1)

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maple [A]  time = 0.09, size = 17, normalized size = 0.65

method	result	size
norman	\(-\frac {25 x}{6 \left ({\mathrm e}^{-\frac {2}{5} x^{3}+x -\frac {7}{5}}-1\right )}\)	\(17\)
risch	\(-\frac {25 x}{6 \left ({\mathrm e}^{-\frac {2}{5} x^{3}+x -\frac {7}{5}}-1\right )}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-30*x^3+25*x-25)*exp(-2/5*x^3+x-7/5)+25)/(6*exp(-2/5*x^3+x-7/5)^2-12*exp(-2/5*x^3+x-7/5)+6),x,method=_RE
TURNVERBOSE)

[Out]

-25/6*x/(exp(-2/5*x^3+x-7/5)-1)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-30*x^3+25*x-25)*exp(-2/5*x^3+x-7/5)+25)/(6*exp(-2/5*x^3+x-7/5)^2-12*exp(-2/5*x^3+x-7/5)+6),x, alg
orithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

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mupad [B]  time = 5.31, size = 18, normalized size = 0.69 \begin {gather*} -\frac {25\,x}{6\,\left ({\mathrm {e}}^{-\frac {2\,x^3}{5}+x-\frac {7}{5}}-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x - (2*x^3)/5 - 7/5)*(30*x^3 - 25*x + 25) - 25)/(6*exp(2*x - (4*x^3)/5 - 14/5) - 12*exp(x - (2*x^3)/
5 - 7/5) + 6),x)

[Out]

-(25*x)/(6*(exp(x - (2*x^3)/5 - 7/5) - 1))

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sympy [A]  time = 0.18, size = 20, normalized size = 0.77 \begin {gather*} - \frac {25 x}{6 e^{- \frac {2 x^{3}}{5} + x - \frac {7}{5}} - 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-30*x**3+25*x-25)*exp(-2/5*x**3+x-7/5)+25)/(6*exp(-2/5*x**3+x-7/5)**2-12*exp(-2/5*x**3+x-7/5)+6),x
)

[Out]

-25*x/(6*exp(-2*x**3/5 + x - 7/5) - 6)

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