3.81.59 \(\int \frac {1925 x-77 e^2 x^2+3850 x^3-77 x^4+e (1925 x^2-154 x^3)+(-154 e x-154 x^2) \log (x)-77 \log ^2(x)}{4 e^2 x^4+8 e x^5+4 x^6+(8 e x^3+8 x^4) \log (x)+4 x^2 \log ^2(x)} \, dx\)

Optimal. Leaf size=22 \[ \frac {77 \left (1-\frac {25 x}{x (e+x)+\log (x)}\right )}{4 x} \]

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Rubi [A]  time = 0.49, antiderivative size = 23, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 4, integrand size = 101, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {6688, 12, 6742, 6686} \begin {gather*} \frac {77}{4 x}-\frac {1925}{4 \left (x^2+e x+\log (x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1925*x - 77*E^2*x^2 + 3850*x^3 - 77*x^4 + E*(1925*x^2 - 154*x^3) + (-154*E*x - 154*x^2)*Log[x] - 77*Log[x
]^2)/(4*E^2*x^4 + 8*E*x^5 + 4*x^6 + (8*E*x^3 + 8*x^4)*Log[x] + 4*x^2*Log[x]^2),x]

[Out]

77/(4*x) - 1925/(4*(E*x + x^2 + Log[x]))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {77 \left (-x \left (-25+e^2 x-50 x^2+x^3+e x (-25+2 x)\right )-2 x (e+x) \log (x)-\log ^2(x)\right )}{4 x^2 (x (e+x)+\log (x))^2} \, dx\\ &=\frac {77}{4} \int \frac {-x \left (-25+e^2 x-50 x^2+x^3+e x (-25+2 x)\right )-2 x (e+x) \log (x)-\log ^2(x)}{x^2 (x (e+x)+\log (x))^2} \, dx\\ &=\frac {77}{4} \int \left (-\frac {1}{x^2}+\frac {25 \left (1+e x+2 x^2\right )}{x \left (e x+x^2+\log (x)\right )^2}\right ) \, dx\\ &=\frac {77}{4 x}+\frac {1925}{4} \int \frac {1+e x+2 x^2}{x \left (e x+x^2+\log (x)\right )^2} \, dx\\ &=\frac {77}{4 x}-\frac {1925}{4 \left (e x+x^2+\log (x)\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.24, size = 23, normalized size = 1.05 \begin {gather*} -\frac {77}{4} \left (-\frac {1}{x}+\frac {25}{e x+x^2+\log (x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1925*x - 77*E^2*x^2 + 3850*x^3 - 77*x^4 + E*(1925*x^2 - 154*x^3) + (-154*E*x - 154*x^2)*Log[x] - 77
*Log[x]^2)/(4*E^2*x^4 + 8*E*x^5 + 4*x^6 + (8*E*x^3 + 8*x^4)*Log[x] + 4*x^2*Log[x]^2),x]

[Out]

(-77*(-x^(-1) + 25/(E*x + x^2 + Log[x])))/4

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fricas [A]  time = 0.57, size = 31, normalized size = 1.41 \begin {gather*} \frac {77 \, {\left (x^{2} + x e - 25 \, x + \log \relax (x)\right )}}{4 \, {\left (x^{3} + x^{2} e + x \log \relax (x)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-77*log(x)^2+(-154*x*exp(1)-154*x^2)*log(x)-77*x^2*exp(1)^2+(-154*x^3+1925*x^2)*exp(1)-77*x^4+3850*
x^3+1925*x)/(4*x^2*log(x)^2+(8*x^3*exp(1)+8*x^4)*log(x)+4*x^4*exp(1)^2+8*x^5*exp(1)+4*x^6),x, algorithm="frica
s")

[Out]

77/4*(x^2 + x*e - 25*x + log(x))/(x^3 + x^2*e + x*log(x))

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-77*log(x)^2+(-154*x*exp(1)-154*x^2)*log(x)-77*x^2*exp(1)^2+(-154*x^3+1925*x^2)*exp(1)-77*x^4+3850*
x^3+1925*x)/(4*x^2*log(x)^2+(8*x^3*exp(1)+8*x^4)*log(x)+4*x^4*exp(1)^2+8*x^5*exp(1)+4*x^6),x, algorithm="giac"
)

[Out]

Timed out

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maple [A]  time = 0.14, size = 21, normalized size = 0.95

method	result	size
risch	\(\frac {77}{4 x}-\frac {1925}{4 \left (x \,{\mathrm e}+x^{2}+\ln \relax (x )\right )}\)	\(21\)
norman	\(\frac {\left (-\frac {1925}{4}+\frac {77 \,{\mathrm e}}{4}\right ) x +\frac {77 x^{2}}{4}+\frac {77 \ln \relax (x )}{4}}{x \left (x \,{\mathrm e}+x^{2}+\ln \relax (x )\right )}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-77*ln(x)^2+(-154*x*exp(1)-154*x^2)*ln(x)-77*x^2*exp(1)^2+(-154*x^3+1925*x^2)*exp(1)-77*x^4+3850*x^3+1925
*x)/(4*x^2*ln(x)^2+(8*x^3*exp(1)+8*x^4)*ln(x)+4*x^4*exp(1)^2+8*x^5*exp(1)+4*x^6),x,method=_RETURNVERBOSE)

[Out]

77/4/x-1925/4/(x*exp(1)+x^2+ln(x))

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maxima [A]  time = 0.39, size = 30, normalized size = 1.36 \begin {gather*} \frac {77 \, {\left (x^{2} + x {\left (e - 25\right )} + \log \relax (x)\right )}}{4 \, {\left (x^{3} + x^{2} e + x \log \relax (x)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-77*log(x)^2+(-154*x*exp(1)-154*x^2)*log(x)-77*x^2*exp(1)^2+(-154*x^3+1925*x^2)*exp(1)-77*x^4+3850*
x^3+1925*x)/(4*x^2*log(x)^2+(8*x^3*exp(1)+8*x^4)*log(x)+4*x^4*exp(1)^2+8*x^5*exp(1)+4*x^6),x, algorithm="maxim
a")

[Out]

77/4*(x^2 + x*(e - 25) + log(x))/(x^3 + x^2*e + x*log(x))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int -\frac {\ln \relax (x)\,\left (154\,x^2+154\,\mathrm {e}\,x\right )-1925\,x+77\,{\ln \relax (x)}^2-\mathrm {e}\,\left (1925\,x^2-154\,x^3\right )+77\,x^2\,{\mathrm {e}}^2-3850\,x^3+77\,x^4}{\ln \relax (x)\,\left (8\,x^4+8\,\mathrm {e}\,x^3\right )+4\,x^2\,{\ln \relax (x)}^2+4\,x^4\,{\mathrm {e}}^2+8\,x^5\,\mathrm {e}+4\,x^6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x)*(154*x*exp(1) + 154*x^2) - 1925*x + 77*log(x)^2 - exp(1)*(1925*x^2 - 154*x^3) + 77*x^2*exp(2) - 3
850*x^3 + 77*x^4)/(log(x)*(8*x^3*exp(1) + 8*x^4) + 4*x^2*log(x)^2 + 4*x^4*exp(2) + 8*x^5*exp(1) + 4*x^6),x)

[Out]

int(-(log(x)*(154*x*exp(1) + 154*x^2) - 1925*x + 77*log(x)^2 - exp(1)*(1925*x^2 - 154*x^3) + 77*x^2*exp(2) - 3
850*x^3 + 77*x^4)/(log(x)*(8*x^3*exp(1) + 8*x^4) + 4*x^2*log(x)^2 + 4*x^4*exp(2) + 8*x^5*exp(1) + 4*x^6), x)

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sympy [A]  time = 0.15, size = 22, normalized size = 1.00 \begin {gather*} - \frac {1925}{4 x^{2} + 4 e x + 4 \log {\relax (x )}} + \frac {77}{4 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-77*ln(x)**2+(-154*x*exp(1)-154*x**2)*ln(x)-77*x**2*exp(1)**2+(-154*x**3+1925*x**2)*exp(1)-77*x**4+
3850*x**3+1925*x)/(4*x**2*ln(x)**2+(8*x**3*exp(1)+8*x**4)*ln(x)+4*x**4*exp(1)**2+8*x**5*exp(1)+4*x**6),x)

[Out]

-1925/(4*x**2 + 4*E*x + 4*log(x)) + 77/(4*x)

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