∫ 150−50x+(150−21x) log(50−7x 2x ) ________________________________________________________________________________________________________________−50x2 +7x3+(300x−142x2+14x3) log(50−7x 2x )+(−450+363x−92x2+7x3) log2(50−7x 2x ) dx

Optimal. Leaf size=30 \[ -2+\frac {x}{x-(3-x) \log \left (-3+\frac {50-x}{2 x}\right )} \]

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Rubi [A] time = 0.23, antiderivative size = 25, normalized size of antiderivative = 0.83, number of steps used = 3, number of rules used = 3, integrand size = 97, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {6688, 6711, 32} \begin {gather*} -\frac {1}{\frac {x}{(x-3) \log \left (\frac {25}{x}-\frac {7}{2}\right )}+1} \end {gather*}

Int[(150 - 50*x + (150 - 21*x)*Log[(50 - 7*x)/(2*x)])/(-50*x^2 + 7*x^3 + (300*x - 142*x^2 + 14*x^3)*Log[(50 -

7*x)/(2*x)] + (-450 + 363*x - 92*x^2 + 7*x^3)*Log[(50 - 7*x)/(2*x)]^2),x]

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N

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

Int[(u_)*((a_.)*(v_)^(p_.) + (b_.)*(w_)^(q_.))^(m_.), x_Symbol] :> With[{c = Simplify[u/(p*w*D[v, x] - q*v*D[w

, x])]}, Dist[c*p, Subst[Int[(b + a*x^p)^m, x], x, v*w^(m*q + 1)], x] /; FreeQ[c, x]] /; FreeQ[{a, b, m, p, q}

, x] && EqQ[p + q*(m*p + 1), 0] && IntegerQ[p] && IntegerQ[m]

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {50 (-3+x)+3 (-50+7 x) \log \left (-\frac {7}{2}+\frac {25}{x}\right )}{(50-7 x) \left (x+(-3+x) \log \left (-\frac {7}{2}+\frac {25}{x}\right )\right )^2} \, dx\\ &=\operatorname {Subst}\left (\int \frac {1}{(1+x)^2} \, dx,x,\frac {x}{(-3+x) \log \left (-\frac {7}{2}+\frac {25}{x}\right )}\right )\\ &=-\frac {1}{1+\frac {x}{(-3+x) \log \left (-\frac {7}{2}+\frac {25}{x}\right )}}\\ \end {aligned} \end {gather*}

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Mathematica [A] time = 5.04, size = 20, normalized size = 0.67 \begin {gather*} \frac {x}{x+(-3+x) \log \left (-\frac {7}{2}+\frac {25}{x}\right )} \end {gather*}

Integrate[(150 - 50*x + (150 - 21*x)*Log[(50 - 7*x)/(2*x)])/(-50*x^2 + 7*x^3 + (300*x - 142*x^2 + 14*x^3)*Log[

(50 - 7*x)/(2*x)] + (-450 + 363*x - 92*x^2 + 7*x^3)*Log[(50 - 7*x)/(2*x)]^2),x]

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fricas [A] time = 0.67, size = 21, normalized size = 0.70 \begin {gather*} \frac {x}{{\left (x - 3\right )} \log \left (-\frac {7 \, x - 50}{2 \, x}\right ) + x} \end {gather*}

integrate(((-21*x+150)*log(1/2*(-7*x+50)/x)-50*x+150)/((7*x^3-92*x^2+363*x-450)*log(1/2*(-7*x+50)/x)^2+(14*x^3

-142*x^2+300*x)*log(1/2*(-7*x+50)/x)+7*x^3-50*x^2),x, algorithm="fricas")

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giac [A] time = 0.20, size = 40, normalized size = 1.33 \begin {gather*} \frac {50}{\frac {3 \, {\left (7 \, x - 50\right )} \log \left (-\frac {7 \, x - 50}{2 \, x}\right )}{x} + 29 \, \log \left (-\frac {7 \, x - 50}{2 \, x}\right ) + 50} \end {gather*}

integrate(((-21*x+150)*log(1/2*(-7*x+50)/x)-50*x+150)/((7*x^3-92*x^2+363*x-450)*log(1/2*(-7*x+50)/x)^2+(14*x^3

-142*x^2+300*x)*log(1/2*(-7*x+50)/x)+7*x^3-50*x^2),x, algorithm="giac")

50/(3*(7*x - 50)*log(-1/2*(7*x - 50)/x)/x + 29*log(-1/2*(7*x - 50)/x) + 50)

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method	result	size

norman	\(\frac {x}{\ln \left (\frac {-7 x +50}{2 x}\right ) x +x -3 \ln \left (\frac {-7 x +50}{2 x}\right )}\)	\(33\)
risch	\(\frac {x}{\ln \left (\frac {-7 x +50}{2 x}\right ) x +x -3 \ln \left (\frac {-7 x +50}{2 x}\right )}\)	\(33\)

int(((-21*x+150)*ln(1/2*(-7*x+50)/x)-50*x+150)/((7*x^3-92*x^2+363*x-450)*ln(1/2*(-7*x+50)/x)^2+(14*x^3-142*x^2

+300*x)*ln(1/2*(-7*x+50)/x)+7*x^3-50*x^2),x,method=_RETURNVERBOSE)

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maxima [A] time = 0.47, size = 33, normalized size = 1.10 \begin {gather*} -\frac {x}{x {\left (\log \relax (2) - 1\right )} + {\left (x - 3\right )} \log \relax (x) - {\left (x - 3\right )} \log \left (-7 \, x + 50\right ) - 3 \, \log \relax (2)} \end {gather*}

integrate(((-21*x+150)*log(1/2*(-7*x+50)/x)-50*x+150)/((7*x^3-92*x^2+363*x-450)*log(1/2*(-7*x+50)/x)^2+(14*x^3

-142*x^2+300*x)*log(1/2*(-7*x+50)/x)+7*x^3-50*x^2),x, algorithm="maxima")

-x/(x*(log(2) - 1) + (x - 3)*log(x) - (x - 3)*log(-7*x + 50) - 3*log(2))

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mupad [B] time = 3.92, size = 21, normalized size = 0.70 \begin {gather*} \frac {x}{x+\ln \left (-\frac {7\,x-50}{2\,x}\right )\,\left (x-3\right )} \end {gather*}

int(-(50*x + log(-((7*x)/2 - 25)/x)*(21*x - 150) - 150)/(log(-((7*x)/2 - 25)/x)^2*(363*x - 92*x^2 + 7*x^3 - 45

0) - 50*x^2 + 7*x^3 + log(-((7*x)/2 - 25)/x)*(300*x - 142*x^2 + 14*x^3)),x)

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sympy [A] time = 0.18, size = 15, normalized size = 0.50 \begin {gather*} \frac {x}{x + \left (x - 3\right ) \log {\left (\frac {25 - \frac {7 x}{2}}{x} \right )}} \end {gather*}

integrate(((-21*x+150)*ln(1/2*(-7*x+50)/x)-50*x+150)/((7*x**3-92*x**2+363*x-450)*ln(1/2*(-7*x+50)/x)**2+(14*x*

*3-142*x**2+300*x)*ln(1/2*(-7*x+50)/x)+7*x**3-50*x**2),x)

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3.51.50 \(\int \frac {150-50 x+(150-21 x) \log (\frac {50-7 x}{2 x})}{-50 x^2+7 x^3+(300 x-142 x^2+14 x^3) \log (\frac {50-7 x}{2 x})+(-450+363 x-92 x^2+7 x^3) \log ^2(\frac {50-7 x}{2 x})} \, dx\)