3.28.61 \(\int \frac {-6+10 e^{x+\frac {1}{2} (20+11 x)}+e^{x+\frac {1}{2} (20+11 x)} (-10+65 x) \log (x)}{6 x+10 e^{x+\frac {1}{2} (20+11 x)} x \log (x)} \, dx\)

Optimal. Leaf size=24 \[ \log \left (\frac {3+5 e^{4 x+\frac {5 (4+x)}{2}} \log (x)}{x}\right ) \]

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Rubi [F]  time = 1.55, 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 {-6+10 e^{x+\frac {1}{2} (20+11 x)}+e^{x+\frac {1}{2} (20+11 x)} (-10+65 x) \log (x)}{6 x+10 e^{x+\frac {1}{2} (20+11 x)} x \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-6 + 10*E^(x + (20 + 11*x)/2) + E^(x + (20 + 11*x)/2)*(-10 + 65*x)*Log[x])/(6*x + 10*E^(x + (20 + 11*x)/2
)*x*Log[x]),x]

[Out]

(13*x)/2 - Log[x] + Log[Log[x]] - 3*Defer[Int][1/(x*Log[x]*(3 + 5*E^(10 + (13*x)/2)*Log[x])), x] - 39*Defer[Su
bst][Defer[Int][(3 + 5*E^(10 + 13*x)*Log[2*x])^(-1), x], x, x/2]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-6+10 e^{x+\frac {1}{2} (20+11 x)}+e^{x+\frac {1}{2} (20+11 x)} (-10+65 x) \log (x)}{2 x \left (3+5 e^{10+\frac {13 x}{2}} \log (x)\right )} \, dx\\ &=\frac {1}{2} \int \frac {-6+10 e^{x+\frac {1}{2} (20+11 x)}+e^{x+\frac {1}{2} (20+11 x)} (-10+65 x) \log (x)}{x \left (3+5 e^{10+\frac {13 x}{2}} \log (x)\right )} \, dx\\ &=\frac {1}{2} \int \left (-\frac {3 (2+13 x \log (x))}{x \log (x) \left (3+5 e^{10+\frac {13 x}{2}} \log (x)\right )}+\frac {2-2 \log (x)+13 x \log (x)}{x \log (x)}\right ) \, dx\\ &=\frac {1}{2} \int \frac {2-2 \log (x)+13 x \log (x)}{x \log (x)} \, dx-\frac {3}{2} \int \frac {2+13 x \log (x)}{x \log (x) \left (3+5 e^{10+\frac {13 x}{2}} \log (x)\right )} \, dx\\ &=\frac {1}{2} \int \left (\frac {-2+13 x}{x}+\frac {2}{x \log (x)}\right ) \, dx-\frac {3}{2} \int \left (\frac {13}{3+5 e^{10+\frac {13 x}{2}} \log (x)}+\frac {2}{x \log (x) \left (3+5 e^{10+\frac {13 x}{2}} \log (x)\right )}\right ) \, dx\\ &=\frac {1}{2} \int \frac {-2+13 x}{x} \, dx-3 \int \frac {1}{x \log (x) \left (3+5 e^{10+\frac {13 x}{2}} \log (x)\right )} \, dx-\frac {39}{2} \int \frac {1}{3+5 e^{10+\frac {13 x}{2}} \log (x)} \, dx+\int \frac {1}{x \log (x)} \, dx\\ &=\frac {1}{2} \int \left (13-\frac {2}{x}\right ) \, dx-3 \int \frac {1}{x \log (x) \left (3+5 e^{10+\frac {13 x}{2}} \log (x)\right )} \, dx-39 \operatorname {Subst}\left (\int \frac {1}{3+5 e^{10+13 x} \log (2 x)} \, dx,x,\frac {x}{2}\right )+\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right )\\ &=\frac {13 x}{2}-\log (x)+\log (\log (x))-3 \int \frac {1}{x \log (x) \left (3+5 e^{10+\frac {13 x}{2}} \log (x)\right )} \, dx-39 \operatorname {Subst}\left (\int \frac {1}{3+5 e^{10+13 x} \log (2 x)} \, dx,x,\frac {x}{2}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.21, size = 27, normalized size = 1.12 \begin {gather*} \frac {1}{2} \left (-2 \log (x)+2 \log \left (3+5 e^{10+\frac {13 x}{2}} \log (x)\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-6 + 10*E^(x + (20 + 11*x)/2) + E^(x + (20 + 11*x)/2)*(-10 + 65*x)*Log[x])/(6*x + 10*E^(x + (20 + 1
1*x)/2)*x*Log[x]),x]

[Out]

(-2*Log[x] + 2*Log[3 + 5*E^(10 + (13*x)/2)*Log[x]])/2

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fricas [A]  time = 0.49, size = 28, normalized size = 1.17 \begin {gather*} \frac {13}{2} \, x + \log \left ({\left (5 \, e^{\left (\frac {13}{2} \, x + 10\right )} \log \relax (x) + 3\right )} e^{\left (-\frac {13}{2} \, x - 10\right )}\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((65*x-10)*exp(x)*exp(11/2*x+10)*log(x)+10*exp(x)*exp(11/2*x+10)-6)/(10*x*exp(x)*exp(11/2*x+10)*log(
x)+6*x),x, algorithm="fricas")

[Out]

13/2*x + log((5*e^(13/2*x + 10)*log(x) + 3)*e^(-13/2*x - 10)) - log(x)

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giac [A]  time = 0.28, size = 18, normalized size = 0.75 \begin {gather*} \log \left (5 \, e^{\left (\frac {13}{2} \, x + 10\right )} \log \relax (x) + 3\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((65*x-10)*exp(x)*exp(11/2*x+10)*log(x)+10*exp(x)*exp(11/2*x+10)-6)/(10*x*exp(x)*exp(11/2*x+10)*log(
x)+6*x),x, algorithm="giac")

[Out]

log(5*e^(13/2*x + 10)*log(x) + 3) - log(x)

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maple [A]  time = 0.03, size = 21, normalized size = 0.88

method	result	size
risch	\(\frac {13 x}{2}-\ln \relax (x )+\ln \left (\ln \relax (x )+\frac {3 \,{\mathrm e}^{-\frac {13 x}{2}-10}}{5}\right )\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((65*x-10)*exp(x)*exp(11/2*x+10)*ln(x)+10*exp(x)*exp(11/2*x+10)-6)/(10*x*exp(x)*exp(11/2*x+10)*ln(x)+6*x),
x,method=_RETURNVERBOSE)

[Out]

13/2*x-ln(x)+ln(ln(x)+3/5*exp(-13/2*x-10))

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maxima [A]  time = 0.52, size = 29, normalized size = 1.21 \begin {gather*} -\log \relax (x) + \log \left (\frac {{\left (5 \, e^{\left (\frac {13}{2} \, x + 10\right )} \log \relax (x) + 3\right )} e^{\left (-10\right )}}{5 \, \log \relax (x)}\right ) + \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((65*x-10)*exp(x)*exp(11/2*x+10)*log(x)+10*exp(x)*exp(11/2*x+10)-6)/(10*x*exp(x)*exp(11/2*x+10)*log(
x)+6*x),x, algorithm="maxima")

[Out]

-log(x) + log(1/5*(5*e^(13/2*x + 10)*log(x) + 3)*e^(-10)/log(x)) + log(log(x))

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mupad [B]  time = 2.00, size = 19, normalized size = 0.79 \begin {gather*} \ln \left (3\,{\mathrm {e}}^{-10}+5\,{\left ({\mathrm {e}}^x\right )}^{13/2}\,\ln \relax (x)\right )-\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((10*exp((11*x)/2 + 10)*exp(x) + exp((11*x)/2 + 10)*exp(x)*log(x)*(65*x - 10) - 6)/(6*x + 10*x*exp((11*x)/2
+ 10)*exp(x)*log(x)),x)

[Out]

log(3*exp(-10) + 5*exp(x)^(13/2)*log(x)) - log(x)

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sympy [A]  time = 0.38, size = 26, normalized size = 1.08 \begin {gather*} \frac {13 x}{2} - \log {\relax (x )} + \log {\left (\log {\relax (x )} + \frac {3}{5 e^{10} \left (e^{x}\right )^{\frac {13}{2}}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((65*x-10)*exp(x)*exp(11/2*x+10)*ln(x)+10*exp(x)*exp(11/2*x+10)-6)/(10*x*exp(x)*exp(11/2*x+10)*ln(x)
+6*x),x)

[Out]

13*x/2 - log(x) + log(log(x) + 3*exp(-10)/(5*exp(x)**(13/2)))

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