3.85.58 \(\int \frac {e^{5 x} (1+x+x^2)+e^{5 x} (6 x+7 x^2+5 x^3) \log (\frac {1}{x})}{x \log (5) \log ^2(\frac {1}{x})} \, dx\)

Optimal. Leaf size=28 \[ \frac {e^{5 x} \left (x+x \left (x+x^2\right )\right )}{x \log (5) \log \left (\frac {1}{x}\right )} \]

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Rubi [F]  time = 1.28, 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 {e^{5 x} \left (1+x+x^2\right )+e^{5 x} \left (6 x+7 x^2+5 x^3\right ) \log \left (\frac {1}{x}\right )}{x \log (5) \log ^2\left (\frac {1}{x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(5*x)*(1 + x + x^2) + E^(5*x)*(6*x + 7*x^2 + 5*x^3)*Log[x^(-1)])/(x*Log[5]*Log[x^(-1)]^2),x]

[Out]

Defer[Int][E^(5*x)/Log[x^(-1)]^2, x]/Log[5] + Defer[Int][E^(5*x)/(x*Log[x^(-1)]^2), x]/Log[5] + Defer[Int][(E^
(5*x)*x)/Log[x^(-1)]^2, x]/Log[5] + (6*Defer[Int][E^(5*x)/Log[x^(-1)], x])/Log[5] + (7*Defer[Int][(E^(5*x)*x)/
Log[x^(-1)], x])/Log[5] + (5*Defer[Int][(E^(5*x)*x^2)/Log[x^(-1)], x])/Log[5]

Rubi steps

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

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Mathematica [A]  time = 0.15, size = 22, normalized size = 0.79 \begin {gather*} \frac {e^{5 x} \left (1+x+x^2\right )}{\log (5) \log \left (\frac {1}{x}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(5*x)*(1 + x + x^2) + E^(5*x)*(6*x + 7*x^2 + 5*x^3)*Log[x^(-1)])/(x*Log[5]*Log[x^(-1)]^2),x]

[Out]

(E^(5*x)*(1 + x + x^2))/(Log[5]*Log[x^(-1)])

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fricas [A]  time = 0.69, size = 21, normalized size = 0.75 \begin {gather*} \frac {{\left (x^{2} + x + 1\right )} e^{\left (5 \, x\right )}}{\log \relax (5) \log \left (\frac {1}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^3+7*x^2+6*x)*exp(5*x)*log(1/x)+(x^2+x+1)*exp(5*x))/x/log(5)/log(1/x)^2,x, algorithm="fricas")

[Out]

(x^2 + x + 1)*e^(5*x)/(log(5)*log(1/x))

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giac [A]  time = 0.21, size = 29, normalized size = 1.04 \begin {gather*} -\frac {x^{2} e^{\left (5 \, x\right )} + x e^{\left (5 \, x\right )} + e^{\left (5 \, x\right )}}{\log \relax (5) \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^3+7*x^2+6*x)*exp(5*x)*log(1/x)+(x^2+x+1)*exp(5*x))/x/log(5)/log(1/x)^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.05, size = 21, normalized size = 0.75

method	result	size
risch	\(-\frac {\left (x^{2}+x +1\right ) {\mathrm e}^{5 x}}{\ln \relax (5) \ln \relax (x )}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((5*x^3+7*x^2+6*x)*exp(5*x)*ln(1/x)+(x^2+x+1)*exp(5*x))/x/ln(5)/ln(1/x)^2,x,method=_RETURNVERBOSE)

[Out]

-1/ln(5)*(x^2+x+1)*exp(5*x)/ln(x)

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maxima [A]  time = 0.53, size = 20, normalized size = 0.71 \begin {gather*} -\frac {{\left (x^{2} + x + 1\right )} e^{\left (5 \, x\right )}}{\log \relax (5) \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^3+7*x^2+6*x)*exp(5*x)*log(1/x)+(x^2+x+1)*exp(5*x))/x/log(5)/log(1/x)^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 5.26, size = 21, normalized size = 0.75 \begin {gather*} \frac {{\mathrm {e}}^{5\,x}\,\left (x^2+x+1\right )}{\ln \left (\frac {1}{x}\right )\,\ln \relax (5)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(5*x)*(x + x^2 + 1) + log(1/x)*exp(5*x)*(6*x + 7*x^2 + 5*x^3))/(x*log(1/x)^2*log(5)),x)

[Out]

(exp(5*x)*(x + x^2 + 1))/(log(1/x)*log(5))

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sympy [A]  time = 0.27, size = 19, normalized size = 0.68 \begin {gather*} \frac {\left (x^{2} + x + 1\right ) e^{5 x}}{\log {\relax (5 )} \log {\left (\frac {1}{x} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x**3+7*x**2+6*x)*exp(5*x)*ln(1/x)+(x**2+x+1)*exp(5*x))/x/ln(5)/ln(1/x)**2,x)

[Out]

(x**2 + x + 1)*exp(5*x)/(log(5)*log(1/x))

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