3.74.83 \(\int \frac {e^x (-5-15 x-10 x^2)+e^{8 x} (3 x+17 x^2+2 x^3)}{-5 e^x x+e^{8 x} x^2} \, dx\)

Optimal. Leaf size=25 \[ 5+x+x^2+\log \left (x \left (e^x-\frac {1}{5} e^{8 x} x\right )^2\right ) \]

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Rubi [F]  time = 0.88, 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 {e^x \left (-5-15 x-10 x^2\right )+e^{8 x} \left (3 x+17 x^2+2 x^3\right )}{-5 e^x x+e^{8 x} x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^x*(-5 - 15*x - 10*x^2) + E^(8*x)*(3*x + 17*x^2 + 2*x^3))/(-5*E^x*x + E^(8*x)*x^2),x]

[Out]

17*x + x^2 + 3*Log[x] + 70*Defer[Int][(-5 + E^(7*x)*x)^(-1), x] + 10*Defer[Int][1/(x*(-5 + E^(7*x)*x)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-x} \left (-e^x \left (-5-15 x-10 x^2\right )-e^{8 x} \left (3 x+17 x^2+2 x^3\right )\right )}{x \left (5-e^{7 x} x\right )} \, dx\\ &=\int \left (\frac {10 (1+7 x)}{x \left (-5+e^{7 x} x\right )}+\frac {3+17 x+2 x^2}{x}\right ) \, dx\\ &=10 \int \frac {1+7 x}{x \left (-5+e^{7 x} x\right )} \, dx+\int \frac {3+17 x+2 x^2}{x} \, dx\\ &=10 \int \left (\frac {7}{-5+e^{7 x} x}+\frac {1}{x \left (-5+e^{7 x} x\right )}\right ) \, dx+\int \left (17+\frac {3}{x}+2 x\right ) \, dx\\ &=17 x+x^2+3 \log (x)+10 \int \frac {1}{x \left (-5+e^{7 x} x\right )} \, dx+70 \int \frac {1}{-5+e^{7 x} x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 22, normalized size = 0.88 \begin {gather*} 3 x+x^2+\log (x)+2 \log \left (5-e^{7 x} x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x*(-5 - 15*x - 10*x^2) + E^(8*x)*(3*x + 17*x^2 + 2*x^3))/(-5*E^x*x + E^(8*x)*x^2),x]

[Out]

3*x + x^2 + Log[x] + 2*Log[5 - E^(7*x)*x]

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fricas [A]  time = 0.68, size = 26, normalized size = 1.04 \begin {gather*} x^{2} + 3 \, x + 3 \, \log \relax (x) + 2 \, \log \left (\frac {x e^{\left (7 \, x\right )} - 5}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3+17*x^2+3*x)*exp(4*x)^2+(-10*x^2-15*x-5)*exp(x))/(x^2*exp(4*x)^2-5*exp(x)*x),x, algorithm="fr
icas")

[Out]

x^2 + 3*x + 3*log(x) + 2*log((x*e^(7*x) - 5)/x)

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giac [A]  time = 0.14, size = 20, normalized size = 0.80 \begin {gather*} x^{2} + 3 \, x + 2 \, \log \left (x e^{\left (7 \, x\right )} - 5\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3+17*x^2+3*x)*exp(4*x)^2+(-10*x^2-15*x-5)*exp(x))/(x^2*exp(4*x)^2-5*exp(x)*x),x, algorithm="gi
ac")

[Out]

x^2 + 3*x + 2*log(x*e^(7*x) - 5) + log(x)

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maple [A]  time = 0.04, size = 25, normalized size = 1.00

method	result	size
risch	\(x^{2}+3 x +3 \ln \relax (x )+2 \ln \left ({\mathrm e}^{7 x}-\frac {5}{x}\right )\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^3+17*x^2+3*x)*exp(4*x)^2+(-10*x^2-15*x-5)*exp(x))/(x^2*exp(4*x)^2-5*exp(x)*x),x,method=_RETURNVERBOS
E)

[Out]

x^2+3*x+3*ln(x)+2*ln(exp(7*x)-5/x)

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maxima [A]  time = 0.48, size = 26, normalized size = 1.04 \begin {gather*} x^{2} + 3 \, x + 3 \, \log \relax (x) + 2 \, \log \left (\frac {x e^{\left (7 \, x\right )} - 5}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3+17*x^2+3*x)*exp(4*x)^2+(-10*x^2-15*x-5)*exp(x))/(x^2*exp(4*x)^2-5*exp(x)*x),x, algorithm="ma
xima")

[Out]

x^2 + 3*x + 3*log(x) + 2*log((x*e^(7*x) - 5)/x)

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mupad [B]  time = 0.11, size = 20, normalized size = 0.80 \begin {gather*} 3\,x+2\,\ln \left (x\,{\mathrm {e}}^{7\,x}-5\right )+\ln \relax (x)+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(8*x)*(3*x + 17*x^2 + 2*x^3) - exp(x)*(15*x + 10*x^2 + 5))/(x^2*exp(8*x) - 5*x*exp(x)),x)

[Out]

3*x + 2*log(x*exp(7*x) - 5) + log(x) + x^2

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sympy [A]  time = 0.30, size = 22, normalized size = 0.88 \begin {gather*} x^{2} + 3 x + 3 \log {\relax (x )} + 2 \log {\left (e^{7 x} - \frac {5}{x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**3+17*x**2+3*x)*exp(4*x)**2+(-10*x**2-15*x-5)*exp(x))/(x**2*exp(4*x)**2-5*exp(x)*x),x)

[Out]

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

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