3.65.35 \(\int \frac {37375-24500 x+4802 x^2+(74875-49500 x+9800 x^2) \log (4)+(37500-25000 x+5000 x^2) \log ^2(4)}{15625-12250 x+2401 x^2+(31250-24750 x+4900 x^2) \log (4)+(15625-12500 x+2500 x^2) \log ^2(4)} \, dx\)

Optimal. Leaf size=29 \[ 2 x-\frac {x}{x+\frac {5}{-2+\frac {4}{25 (4+4 \log (4))}}} \]

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Rubi [A]  time = 0.10, antiderivative size = 28, normalized size of antiderivative = 0.97, number of steps used = 5, number of rules used = 4, integrand size = 79, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {1984, 27, 6, 683} \begin {gather*} 2 x+\frac {125 (1+\log (4))}{125 (1+\log (4))-x (49+50 \log (4))} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(37375 - 24500*x + 4802*x^2 + (74875 - 49500*x + 9800*x^2)*Log[4] + (37500 - 25000*x + 5000*x^2)*Log[4]^2)
/(15625 - 12250*x + 2401*x^2 + (31250 - 24750*x + 4900*x^2)*Log[4] + (15625 - 12500*x + 2500*x^2)*Log[4]^2),x]

[Out]

2*x + (125*(1 + Log[4]))/(125*(1 + Log[4]) - x*(49 + 50*Log[4]))

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

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 683

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rule 1984

Int[(u_)^(p_.)*(v_)^(q_.), x_Symbol] :> Int[ExpandToSum[u, x]^p*ExpandToSum[v, x]^q, x] /; FreeQ[{p, q}, x] &&
QuadraticQ[{u, v}, x] &&  !QuadraticMatchQ[{u, v}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-500 x (1+\log (4)) (49+50 \log (4))+2 x^2 (49+50 \log (4))^2+125 (1+\log (4)) (299+300 \log (4))}{15625 (1+\log (4))^2-250 x (1+\log (4)) (49+50 \log (4))+x^2 (49+50 \log (4))^2} \, dx\\ &=\int \frac {-500 x (1+\log (4)) (49+50 \log (4))+2 x^2 (49+50 \log (4))^2+125 (1+\log (4)) (299+300 \log (4))}{(-125+49 x-125 \log (4)+50 x \log (4))^2} \, dx\\ &=\int \frac {-500 x (1+\log (4)) (49+50 \log (4))+2 x^2 (49+50 \log (4))^2+125 (1+\log (4)) (299+300 \log (4))}{(-125-125 \log (4)+x (49+50 \log (4)))^2} \, dx\\ &=\int \left (2+\frac {125 (1+\log (4)) (49+50 \log (4))}{(125 (1+\log (4))-x (49+50 \log (4)))^2}\right ) \, dx\\ &=2 x+\frac {125 (1+\log (4))}{125 (1+\log (4))-x (49+50 \log (4))}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.03, size = 60, normalized size = 2.07 \begin {gather*} \frac {-125 \left (49+99 \log (4)+50 \log ^2(4)\right )+2 (-125 (1+\log (4))+x (49+50 \log (4)))^2}{(49+50 \log (4)) (-125 (1+\log (4))+x (49+50 \log (4)))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(37375 - 24500*x + 4802*x^2 + (74875 - 49500*x + 9800*x^2)*Log[4] + (37500 - 25000*x + 5000*x^2)*Log
[4]^2)/(15625 - 12250*x + 2401*x^2 + (31250 - 24750*x + 4900*x^2)*Log[4] + (15625 - 12500*x + 2500*x^2)*Log[4]
^2),x]

[Out]

(-125*(49 + 99*Log[4] + 50*Log[4]^2) + 2*(-125*(1 + Log[4]) + x*(49 + 50*Log[4]))^2)/((49 + 50*Log[4])*(-125*(
1 + Log[4]) + x*(49 + 50*Log[4])))

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fricas [A]  time = 0.73, size = 41, normalized size = 1.41 \begin {gather*} \frac {98 \, x^{2} + 50 \, {\left (4 \, x^{2} - 10 \, x - 5\right )} \log \relax (2) - 250 \, x - 125}{50 \, {\left (2 \, x - 5\right )} \log \relax (2) + 49 \, x - 125} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*(5000*x^2-25000*x+37500)*log(2)^2+2*(9800*x^2-49500*x+74875)*log(2)+4802*x^2-24500*x+37375)/(4*(2
500*x^2-12500*x+15625)*log(2)^2+2*(4900*x^2-24750*x+31250)*log(2)+2401*x^2-12250*x+15625),x, algorithm="fricas
")

[Out]

(98*x^2 + 50*(4*x^2 - 10*x - 5)*log(2) - 250*x - 125)/(50*(2*x - 5)*log(2) + 49*x - 125)

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giac [B]  time = 0.21, size = 57, normalized size = 1.97 \begin {gather*} \frac {2 \, {\left (10000 \, x \log \relax (2)^{2} + 9800 \, x \log \relax (2) + 2401 \, x\right )}}{10000 \, \log \relax (2)^{2} + 9800 \, \log \relax (2) + 2401} - \frac {125 \, {\left (2 \, \log \relax (2) + 1\right )}}{100 \, x \log \relax (2) + 49 \, x - 250 \, \log \relax (2) - 125} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*(5000*x^2-25000*x+37500)*log(2)^2+2*(9800*x^2-49500*x+74875)*log(2)+4802*x^2-24500*x+37375)/(4*(2
500*x^2-12500*x+15625)*log(2)^2+2*(4900*x^2-24750*x+31250)*log(2)+2401*x^2-12250*x+15625),x, algorithm="giac")

[Out]

2*(10000*x*log(2)^2 + 9800*x*log(2) + 2401*x)/(10000*log(2)^2 + 9800*log(2) + 2401) - 125*(2*log(2) + 1)/(100*
x*log(2) + 49*x - 250*log(2) - 125)

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maple [A]  time = 0.57, size = 41, normalized size = 1.41

method	result	size
risch	\(2 x -\frac {5 \ln \relax (2)}{2 \left (x \ln \relax (2)-\frac {5 \ln \relax (2)}{2}+\frac {49 x}{100}-\frac {5}{4}\right )}-\frac {5}{4 \left (x \ln \relax (2)-\frac {5 \ln \relax (2)}{2}+\frac {49 x}{100}-\frac {5}{4}\right )}\)	\(41\)
default	\(2 x -\frac {25000 \ln \relax (2)^{2}+24750 \ln \relax (2)+6125}{\left (100 \ln \relax (2)+49\right ) \left (100 x \ln \relax (2)-250 \ln \relax (2)+49 x -125\right )}\)	\(43\)
norman	\(\frac {\left (200 \ln \relax (2)+98\right ) x^{2}-\frac {125 \left (1200 \ln \relax (2)^{2}+1198 \ln \relax (2)+299\right )}{100 \ln \relax (2)+49}}{100 x \ln \relax (2)-250 \ln \relax (2)+49 x -125}\)	\(51\)
gosper	\(\frac {20000 x^{2} \ln \relax (2)^{2}+19600 x^{2} \ln \relax (2)-150000 \ln \relax (2)^{2}+4802 x^{2}-149750 \ln \relax (2)-37375}{\left (100 x \ln \relax (2)-250 \ln \relax (2)+49 x -125\right ) \left (100 \ln \relax (2)+49\right )}\)	\(59\)
meijerg	\(\frac {299 \left (100 \ln \relax (2)+49\right )^{2} x}{125 \left (10000 \ln \relax (2)^{2}+9800 \ln \relax (2)+2401\right ) \left (1+2 \ln \relax (2)\right )^{2} \left (1-\frac {x \left (100 \ln \relax (2)+49\right )}{125 \left (1+2 \ln \relax (2)\right )}\right )}-\frac {1953125 \left (\frac {32 \ln \relax (2)^{2}}{25}+\frac {784 \ln \relax (2)}{625}+\frac {4802}{15625}\right ) \left (1+2 \ln \relax (2)\right ) \left (-\frac {x \left (100 \ln \relax (2)+49\right ) \left (-\frac {3 x \left (100 \ln \relax (2)+49\right )}{125 \left (1+2 \ln \relax (2)\right )}+6\right )}{375 \left (1+2 \ln \relax (2)\right ) \left (1-\frac {x \left (100 \ln \relax (2)+49\right )}{125 \left (1+2 \ln \relax (2)\right )}\right )}-2 \ln \left (1-\frac {x \left (100 \ln \relax (2)+49\right )}{125 \left (1+2 \ln \relax (2)\right )}\right )\right )}{\left (10000 \ln \relax (2)^{2}+9800 \ln \relax (2)+2401\right ) \left (100 \ln \relax (2)+49\right )}+\frac {15625 \left (-\frac {32 \ln \relax (2)^{2}}{5}-\frac {792 \ln \relax (2)}{125}-\frac {196}{125}\right ) \left (\frac {x \left (100 \ln \relax (2)+49\right )}{125 \left (1+2 \ln \relax (2)\right ) \left (1-\frac {x \left (100 \ln \relax (2)+49\right )}{125 \left (1+2 \ln \relax (2)\right )}\right )}+\ln \left (1-\frac {x \left (100 \ln \relax (2)+49\right )}{125 \left (1+2 \ln \relax (2)\right )}\right )\right )}{10000 \ln \relax (2)^{2}+9800 \ln \relax (2)+2401}+\frac {48 \ln \relax (2)^{2} \left (100 \ln \relax (2)+49\right )^{2} x}{5 \left (10000 \ln \relax (2)^{2}+9800 \ln \relax (2)+2401\right ) \left (1+2 \ln \relax (2)\right )^{2} \left (1-\frac {x \left (100 \ln \relax (2)+49\right )}{125 \left (1+2 \ln \relax (2)\right )}\right )}+\frac {1198 \ln \relax (2) \left (100 \ln \relax (2)+49\right )^{2} x}{125 \left (10000 \ln \relax (2)^{2}+9800 \ln \relax (2)+2401\right ) \left (1+2 \ln \relax (2)\right )^{2} \left (1-\frac {x \left (100 \ln \relax (2)+49\right )}{125 \left (1+2 \ln \relax (2)\right )}\right )}\)	\(379\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*(5000*x^2-25000*x+37500)*ln(2)^2+2*(9800*x^2-49500*x+74875)*ln(2)+4802*x^2-24500*x+37375)/(4*(2500*x^2-
12500*x+15625)*ln(2)^2+2*(4900*x^2-24750*x+31250)*ln(2)+2401*x^2-12250*x+15625),x,method=_RETURNVERBOSE)

[Out]

2*x-5/2/(x*ln(2)-5/2*ln(2)+49/100*x-5/4)*ln(2)-5/4/(x*ln(2)-5/2*ln(2)+49/100*x-5/4)

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maxima [A]  time = 0.37, size = 28, normalized size = 0.97 \begin {gather*} 2 \, x - \frac {125 \, {\left (2 \, \log \relax (2) + 1\right )}}{x {\left (100 \, \log \relax (2) + 49\right )} - 250 \, \log \relax (2) - 125} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*(5000*x^2-25000*x+37500)*log(2)^2+2*(9800*x^2-49500*x+74875)*log(2)+4802*x^2-24500*x+37375)/(4*(2
500*x^2-12500*x+15625)*log(2)^2+2*(4900*x^2-24750*x+31250)*log(2)+2401*x^2-12250*x+15625),x, algorithm="maxima
")

[Out]

2*x - 125*(2*log(2) + 1)/(x*(100*log(2) + 49) - 250*log(2) - 125)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {2\,\ln \relax (2)\,\left (9800\,x^2-49500\,x+74875\right )-24500\,x+4\,{\ln \relax (2)}^2\,\left (5000\,x^2-25000\,x+37500\right )+4802\,x^2+37375}{2\,\ln \relax (2)\,\left (4900\,x^2-24750\,x+31250\right )-12250\,x+4\,{\ln \relax (2)}^2\,\left (2500\,x^2-12500\,x+15625\right )+2401\,x^2+15625} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*log(2)*(9800*x^2 - 49500*x + 74875) - 24500*x + 4*log(2)^2*(5000*x^2 - 25000*x + 37500) + 4802*x^2 + 37
375)/(2*log(2)*(4900*x^2 - 24750*x + 31250) - 12250*x + 4*log(2)^2*(2500*x^2 - 12500*x + 15625) + 2401*x^2 + 1
5625),x)

[Out]

int((2*log(2)*(9800*x^2 - 49500*x + 74875) - 24500*x + 4*log(2)^2*(5000*x^2 - 25000*x + 37500) + 4802*x^2 + 37
375)/(2*log(2)*(4900*x^2 - 24750*x + 31250) - 12250*x + 4*log(2)^2*(2500*x^2 - 12500*x + 15625) + 2401*x^2 + 1
5625), x)

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sympy [A]  time = 0.47, size = 26, normalized size = 0.90 \begin {gather*} 2 x + \frac {- 250 \log {\relax (2 )} - 125}{x \left (49 + 100 \log {\relax (2 )}\right ) - 250 \log {\relax (2 )} - 125} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*(5000*x**2-25000*x+37500)*ln(2)**2+2*(9800*x**2-49500*x+74875)*ln(2)+4802*x**2-24500*x+37375)/(4*
(2500*x**2-12500*x+15625)*ln(2)**2+2*(4900*x**2-24750*x+31250)*ln(2)+2401*x**2-12250*x+15625),x)

[Out]

2*x + (-250*log(2) - 125)/(x*(49 + 100*log(2)) - 250*log(2) - 125)

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