3.3.62 \(\int \frac {-1080 e^4+(25+10 x+x^2) \log (2)}{25+10 x+x^2} \, dx\)

Optimal. Leaf size=20 \[ \frac {120 e^4 (4-x)}{5+x}+x \log (2) \]

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Rubi [A]  time = 0.02, antiderivative size = 15, normalized size of antiderivative = 0.75, number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {27, 1850} \begin {gather*} \frac {1080 e^4}{x+5}+x \log (2) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1080*E^4 + (25 + 10*x + x^2)*Log[2])/(25 + 10*x + x^2),x]

[Out]

(1080*E^4)/(5 + x) + x*Log[2]

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-1080 e^4+\left (25+10 x+x^2\right ) \log (2)}{(5+x)^2} \, dx\\ &=\int \left (-\frac {1080 e^4}{(5+x)^2}+\log (2)\right ) \, dx\\ &=\frac {1080 e^4}{5+x}+x \log (2)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 17, normalized size = 0.85 \begin {gather*} \frac {1080 e^4}{5+x}+(5+x) \log (2) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1080*E^4 + (25 + 10*x + x^2)*Log[2])/(25 + 10*x + x^2),x]

[Out]

(1080*E^4)/(5 + x) + (5 + x)*Log[2]

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fricas [A]  time = 0.50, size = 21, normalized size = 1.05 \begin {gather*} \frac {{\left (x^{2} + 5 \, x\right )} \log \relax (2) + 1080 \, e^{4}}{x + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2+10*x+25)*log(2)-1080*exp(4))/(x^2+10*x+25),x, algorithm="fricas")

[Out]

((x^2 + 5*x)*log(2) + 1080*e^4)/(x + 5)

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giac [A]  time = 0.25, size = 14, normalized size = 0.70 \begin {gather*} x \log \relax (2) + \frac {1080 \, e^{4}}{x + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2+10*x+25)*log(2)-1080*exp(4))/(x^2+10*x+25),x, algorithm="giac")

[Out]

x*log(2) + 1080*e^4/(x + 5)

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maple [A]  time = 0.50, size = 15, normalized size = 0.75

method	result	size
default	\(x \ln \relax (2)+\frac {1080 \,{\mathrm e}^{4}}{5+x}\)	\(15\)
risch	\(x \ln \relax (2)+\frac {1080 \,{\mathrm e}^{4}}{5+x}\)	\(15\)
gosper	\(\frac {x^{2} \ln \relax (2)+1080 \,{\mathrm e}^{4}-25 \ln \relax (2)}{5+x}\)	\(22\)
norman	\(\frac {x^{2} \ln \relax (2)+1080 \,{\mathrm e}^{4}-25 \ln \relax (2)}{5+x}\)	\(22\)
meijerg	\(5 \ln \relax (2) \left (\frac {x \left (\frac {3 x}{5}+6\right )}{15+3 x}-2 \ln \left (1+\frac {x}{5}\right )\right )+10 \ln \relax (2) \left (-\frac {x}{5 \left (1+\frac {x}{5}\right )}+\ln \left (1+\frac {x}{5}\right )\right )+\frac {\ln \relax (2) x}{1+\frac {x}{5}}-\frac {216 \,{\mathrm e}^{4} x}{5 \left (1+\frac {x}{5}\right )}\)	\(74\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2+10*x+25)*ln(2)-1080*exp(4))/(x^2+10*x+25),x,method=_RETURNVERBOSE)

[Out]

x*ln(2)+1080*exp(4)/(5+x)

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maxima [A]  time = 0.61, size = 14, normalized size = 0.70 \begin {gather*} x \log \relax (2) + \frac {1080 \, e^{4}}{x + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2+10*x+25)*log(2)-1080*exp(4))/(x^2+10*x+25),x, algorithm="maxima")

[Out]

x*log(2) + 1080*e^4/(x + 5)

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mupad [B]  time = 0.32, size = 14, normalized size = 0.70 \begin {gather*} x\,\ln \relax (2)+\frac {1080\,{\mathrm {e}}^4}{x+5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(1080*exp(4) - log(2)*(10*x + x^2 + 25))/(10*x + x^2 + 25),x)

[Out]

x*log(2) + (1080*exp(4))/(x + 5)

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sympy [A]  time = 0.10, size = 12, normalized size = 0.60 \begin {gather*} x \log {\relax (2 )} + \frac {1080 e^{4}}{x + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**2+10*x+25)*ln(2)-1080*exp(4))/(x**2+10*x+25),x)

[Out]

x*log(2) + 1080*exp(4)/(x + 5)

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