3.64.72 \(\int \frac {-5 e^{10 x}-5 x^2+18 x^3-4 x^4-5 x^6+e^{5 x} (9-32 x-15 x^2+10 x^3)}{15 x^2+5 x^3+30 x^4+10 x^5+15 x^6+5 x^7+e^{10 x} (15+5 x)+e^{5 x} (-30 x-10 x^2-30 x^3-10 x^4)} \, dx\)

Optimal. Leaf size=29 \[ 7-\frac {3}{5 \left (1-\frac {e^{5 x}}{x}+x^2\right )}-\log (3+x) \]

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Rubi [F]  time = 1.21, 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 {-5 e^{10 x}-5 x^2+18 x^3-4 x^4-5 x^6+e^{5 x} \left (9-32 x-15 x^2+10 x^3\right )}{15 x^2+5 x^3+30 x^4+10 x^5+15 x^6+5 x^7+e^{10 x} (15+5 x)+e^{5 x} \left (-30 x-10 x^2-30 x^3-10 x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-5*E^(10*x) - 5*x^2 + 18*x^3 - 4*x^4 - 5*x^6 + E^(5*x)*(9 - 32*x - 15*x^2 + 10*x^3))/(15*x^2 + 5*x^3 + 30
*x^4 + 10*x^5 + 15*x^6 + 5*x^7 + E^(10*x)*(15 + 5*x) + E^(5*x)*(-30*x - 10*x^2 - 30*x^3 - 10*x^4)),x]

[Out]

-Log[3 + x] + (3*Defer[Int][(E^(5*x) - x - x^3)^(-1), x])/5 + (3*Defer[Int][x/(-E^(5*x) + x + x^3)^2, x])/5 -
3*Defer[Int][x^2/(-E^(5*x) + x + x^3)^2, x] + (9*Defer[Int][x^3/(-E^(5*x) + x + x^3)^2, x])/5 - 3*Defer[Int][x
^4/(-E^(5*x) + x + x^3)^2, x] + 3*Defer[Int][x/(-E^(5*x) + x + x^3), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-5 e^{10 x}+e^{5 x} \left (9-32 x-15 x^2+10 x^3\right )-x^2 \left (5-18 x+4 x^2+5 x^4\right )}{5 (3+x) \left (e^{5 x}-x-x^3\right )^2} \, dx\\ &=\frac {1}{5} \int \frac {-5 e^{10 x}+e^{5 x} \left (9-32 x-15 x^2+10 x^3\right )-x^2 \left (5-18 x+4 x^2+5 x^4\right )}{(3+x) \left (e^{5 x}-x-x^3\right )^2} \, dx\\ &=\frac {1}{5} \int \left (-\frac {5}{3+x}+\frac {3 (-1+5 x)}{-e^{5 x}+x+x^3}-\frac {3 x \left (-1+5 x-3 x^2+5 x^3\right )}{\left (-e^{5 x}+x+x^3\right )^2}\right ) \, dx\\ &=-\log (3+x)+\frac {3}{5} \int \frac {-1+5 x}{-e^{5 x}+x+x^3} \, dx-\frac {3}{5} \int \frac {x \left (-1+5 x-3 x^2+5 x^3\right )}{\left (-e^{5 x}+x+x^3\right )^2} \, dx\\ &=-\log (3+x)-\frac {3}{5} \int \left (-\frac {x}{\left (-e^{5 x}+x+x^3\right )^2}+\frac {5 x^2}{\left (-e^{5 x}+x+x^3\right )^2}-\frac {3 x^3}{\left (-e^{5 x}+x+x^3\right )^2}+\frac {5 x^4}{\left (-e^{5 x}+x+x^3\right )^2}\right ) \, dx+\frac {3}{5} \int \left (\frac {1}{e^{5 x}-x-x^3}+\frac {5 x}{-e^{5 x}+x+x^3}\right ) \, dx\\ &=-\log (3+x)+\frac {3}{5} \int \frac {1}{e^{5 x}-x-x^3} \, dx+\frac {3}{5} \int \frac {x}{\left (-e^{5 x}+x+x^3\right )^2} \, dx+\frac {9}{5} \int \frac {x^3}{\left (-e^{5 x}+x+x^3\right )^2} \, dx-3 \int \frac {x^2}{\left (-e^{5 x}+x+x^3\right )^2} \, dx-3 \int \frac {x^4}{\left (-e^{5 x}+x+x^3\right )^2} \, dx+3 \int \frac {x}{-e^{5 x}+x+x^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.30, size = 30, normalized size = 1.03 \begin {gather*} \frac {1}{5} \left (\frac {3 x}{e^{5 x}-x-x^3}-5 \log (3+x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-5*E^(10*x) - 5*x^2 + 18*x^3 - 4*x^4 - 5*x^6 + E^(5*x)*(9 - 32*x - 15*x^2 + 10*x^3))/(15*x^2 + 5*x^
3 + 30*x^4 + 10*x^5 + 15*x^6 + 5*x^7 + E^(10*x)*(15 + 5*x) + E^(5*x)*(-30*x - 10*x^2 - 30*x^3 - 10*x^4)),x]

[Out]

((3*x)/(E^(5*x) - x - x^3) - 5*Log[3 + x])/5

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fricas [A]  time = 0.69, size = 36, normalized size = 1.24 \begin {gather*} -\frac {5 \, {\left (x^{3} + x - e^{\left (5 \, x\right )}\right )} \log \left (x + 3\right ) + 3 \, x}{5 \, {\left (x^{3} + x - e^{\left (5 \, x\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5*exp(5*x)^2+(10*x^3-15*x^2-32*x+9)*exp(5*x)-5*x^6-4*x^4+18*x^3-5*x^2)/((5*x+15)*exp(5*x)^2+(-10*x
^4-30*x^3-10*x^2-30*x)*exp(5*x)+5*x^7+15*x^6+10*x^5+30*x^4+5*x^3+15*x^2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.15, size = 45, normalized size = 1.55 \begin {gather*} -\frac {5 \, x^{3} \log \left (x + 3\right ) + 5 \, x \log \left (x + 3\right ) - 5 \, e^{\left (5 \, x\right )} \log \left (x + 3\right ) + 3 \, x}{5 \, {\left (x^{3} + x - e^{\left (5 \, x\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5*exp(5*x)^2+(10*x^3-15*x^2-32*x+9)*exp(5*x)-5*x^6-4*x^4+18*x^3-5*x^2)/((5*x+15)*exp(5*x)^2+(-10*x
^4-30*x^3-10*x^2-30*x)*exp(5*x)+5*x^7+15*x^6+10*x^5+30*x^4+5*x^3+15*x^2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.22, size = 24, normalized size = 0.83

method	result	size
norman	\(-\frac {3 x}{5 \left (x^{3}+x -{\mathrm e}^{5 x}\right )}-\ln \left (3+x \right )\)	\(24\)
risch	\(-\frac {3 x}{5 \left (x^{3}+x -{\mathrm e}^{5 x}\right )}-\ln \left (3+x \right )\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-5*exp(5*x)^2+(10*x^3-15*x^2-32*x+9)*exp(5*x)-5*x^6-4*x^4+18*x^3-5*x^2)/((5*x+15)*exp(5*x)^2+(-10*x^4-30*
x^3-10*x^2-30*x)*exp(5*x)+5*x^7+15*x^6+10*x^5+30*x^4+5*x^3+15*x^2),x,method=_RETURNVERBOSE)

[Out]

-3/5*x/(x^3+x-exp(5*x))-ln(3+x)

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maxima [A]  time = 0.53, size = 23, normalized size = 0.79 \begin {gather*} -\frac {3 \, x}{5 \, {\left (x^{3} + x - e^{\left (5 \, x\right )}\right )}} - \log \left (x + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5*exp(5*x)^2+(10*x^3-15*x^2-32*x+9)*exp(5*x)-5*x^6-4*x^4+18*x^3-5*x^2)/((5*x+15)*exp(5*x)^2+(-10*x
^4-30*x^3-10*x^2-30*x)*exp(5*x)+5*x^7+15*x^6+10*x^5+30*x^4+5*x^3+15*x^2),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {5\,{\mathrm {e}}^{10\,x}+{\mathrm {e}}^{5\,x}\,\left (-10\,x^3+15\,x^2+32\,x-9\right )+5\,x^2-18\,x^3+4\,x^4+5\,x^6}{{\mathrm {e}}^{10\,x}\,\left (5\,x+15\right )-{\mathrm {e}}^{5\,x}\,\left (10\,x^4+30\,x^3+10\,x^2+30\,x\right )+15\,x^2+5\,x^3+30\,x^4+10\,x^5+15\,x^6+5\,x^7} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(5*exp(10*x) + exp(5*x)*(32*x + 15*x^2 - 10*x^3 - 9) + 5*x^2 - 18*x^3 + 4*x^4 + 5*x^6)/(exp(10*x)*(5*x +
15) - exp(5*x)*(30*x + 10*x^2 + 30*x^3 + 10*x^4) + 15*x^2 + 5*x^3 + 30*x^4 + 10*x^5 + 15*x^6 + 5*x^7),x)

[Out]

int(-(5*exp(10*x) + exp(5*x)*(32*x + 15*x^2 - 10*x^3 - 9) + 5*x^2 - 18*x^3 + 4*x^4 + 5*x^6)/(exp(10*x)*(5*x +
15) - exp(5*x)*(30*x + 10*x^2 + 30*x^3 + 10*x^4) + 15*x^2 + 5*x^3 + 30*x^4 + 10*x^5 + 15*x^6 + 5*x^7), x)

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sympy [A]  time = 0.16, size = 22, normalized size = 0.76 \begin {gather*} \frac {3 x}{- 5 x^{3} - 5 x + 5 e^{5 x}} - \log {\left (x + 3 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5*exp(5*x)**2+(10*x**3-15*x**2-32*x+9)*exp(5*x)-5*x**6-4*x**4+18*x**3-5*x**2)/((5*x+15)*exp(5*x)**
2+(-10*x**4-30*x**3-10*x**2-30*x)*exp(5*x)+5*x**7+15*x**6+10*x**5+30*x**4+5*x**3+15*x**2),x)

[Out]

3*x/(-5*x**3 - 5*x + 5*exp(5*x)) - log(x + 3)

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