3.18.12 \(\int \frac {54+84 x+25 x^2+2 x^3+(-15-26 x-4 x^2) \log (5)+(1+2 x) \log ^2(5)}{36+12 x+x^2+(-12-2 x) \log (5)+\log ^2(5)} \, dx\)

Optimal. Leaf size=24 \[ 3+x+x^2+\log (4)+\frac {x}{2+\frac {1}{3} (x-\log (5))} \]

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Rubi [A]  time = 0.08, antiderivative size = 22, normalized size of antiderivative = 0.92, number of steps used = 4, number of rules used = 3, integrand size = 61, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {1986, 27, 1850} \begin {gather*} x^2+x-\frac {3 (6-\log (5))}{x+6-\log (5)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(54 + 84*x + 25*x^2 + 2*x^3 + (-15 - 26*x - 4*x^2)*Log[5] + (1 + 2*x)*Log[5]^2)/(36 + 12*x + x^2 + (-12 -
2*x)*Log[5] + Log[5]^2),x]

[Out]

x + x^2 - (3*(6 - Log[5]))/(6 + x - Log[5])

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])

Rule 1986

Int[(Pq_)*(u_)^(p_.), x_Symbol] :> Int[Pq*ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && PolyQ[Pq, x] && QuadraticQ
[u, x] &&  !QuadraticMatchQ[u, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {54+84 x+25 x^2+2 x^3+\left (-15-26 x-4 x^2\right ) \log (5)+(1+2 x) \log ^2(5)}{x^2+2 x (6-\log (5))+(-6+\log (5))^2} \, dx\\ &=\int \frac {54+84 x+25 x^2+2 x^3+\left (-15-26 x-4 x^2\right ) \log (5)+(1+2 x) \log ^2(5)}{(6+x-\log (5))^2} \, dx\\ &=\int \left (1+2 x-\frac {3 (-6+\log (5))}{(6+x-\log (5))^2}\right ) \, dx\\ &=x+x^2-\frac {3 (6-\log (5))}{6+x-\log (5)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 39, normalized size = 1.62 \begin {gather*} (6+x-\log (5))^2+\frac {3 (-6+\log (5))}{6+x-\log (5)}+(6+x-\log (5)) (-11+2 \log (5)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(54 + 84*x + 25*x^2 + 2*x^3 + (-15 - 26*x - 4*x^2)*Log[5] + (1 + 2*x)*Log[5]^2)/(36 + 12*x + x^2 + (
-12 - 2*x)*Log[5] + Log[5]^2),x]

[Out]

(6 + x - Log[5])^2 + (3*(-6 + Log[5]))/(6 + x - Log[5]) + (6 + x - Log[5])*(-11 + 2*Log[5])

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fricas [A]  time = 0.66, size = 33, normalized size = 1.38 \begin {gather*} \frac {x^{3} + 7 \, x^{2} - {\left (x^{2} + x - 3\right )} \log \relax (5) + 6 \, x - 18}{x - \log \relax (5) + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x+1)*log(5)^2+(-4*x^2-26*x-15)*log(5)+2*x^3+25*x^2+84*x+54)/(log(5)^2+(-2*x-12)*log(5)+x^2+12*x+
36),x, algorithm="fricas")

[Out]

(x^3 + 7*x^2 - (x^2 + x - 3)*log(5) + 6*x - 18)/(x - log(5) + 6)

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giac [A]  time = 0.21, size = 20, normalized size = 0.83 \begin {gather*} x^{2} + x + \frac {3 \, {\left (\log \relax (5) - 6\right )}}{x - \log \relax (5) + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x+1)*log(5)^2+(-4*x^2-26*x-15)*log(5)+2*x^3+25*x^2+84*x+54)/(log(5)^2+(-2*x-12)*log(5)+x^2+12*x+
36),x, algorithm="giac")

[Out]

x^2 + x + 3*(log(5) - 6)/(x - log(5) + 6)

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maple [A]  time = 0.33, size = 23, normalized size = 0.96

method	result	size
default	\(x^{2}+x -\frac {-3 \ln \relax (5)+18}{-\ln \relax (5)+x +6}\)	\(23\)
risch	\(x^{2}+x -\frac {3 \ln \relax (5)}{\ln \relax (5)-x -6}+\frac {18}{\ln \relax (5)-x -6}\)	\(30\)
norman	\(\frac {-x^{3}+\left (-7+\ln \relax (5)\right ) x^{2}+54+\ln \relax (5)^{2}-15 \ln \relax (5)}{\ln \relax (5)-x -6}\)	\(34\)
gosper	\(\frac {x^{2} \ln \relax (5)-x^{3}+\ln \relax (5)^{2}-7 x^{2}-15 \ln \relax (5)+54}{\ln \relax (5)-x -6}\)	\(37\)
meijerg	\(\left (2 \ln \relax (5)^{2}-26 \ln \relax (5)+84\right ) \left (-\frac {x}{\left (-\ln \relax (5)+6\right ) \left (1+\frac {x}{-\ln \relax (5)+6}\right )}+\ln \left (1+\frac {x}{-\ln \relax (5)+6}\right )\right )+\left (-2 \ln \relax (5)+12\right ) \left (-\ln \relax (5)+6\right ) \left (-\frac {x \left (-\frac {2 x^{2}}{\left (-\ln \relax (5)+6\right )^{2}}+\frac {6 x}{-\ln \relax (5)+6}+12\right )}{4 \left (-\ln \relax (5)+6\right ) \left (1+\frac {x}{-\ln \relax (5)+6}\right )}+3 \ln \left (1+\frac {x}{-\ln \relax (5)+6}\right )\right )+\left (-4 \ln \relax (5)+25\right ) \left (-\ln \relax (5)+6\right ) \left (\frac {x \left (\frac {3 x}{-\ln \relax (5)+6}+6\right )}{3 \left (-\ln \relax (5)+6\right ) \left (1+\frac {x}{-\ln \relax (5)+6}\right )}-2 \ln \left (1+\frac {x}{-\ln \relax (5)+6}\right )\right )+\frac {\ln \relax (5)^{2} x}{\left (-\ln \relax (5)+6\right )^{2} \left (1+\frac {x}{-\ln \relax (5)+6}\right )}-\frac {15 \ln \relax (5) x}{\left (-\ln \relax (5)+6\right )^{2} \left (1+\frac {x}{-\ln \relax (5)+6}\right )}+\frac {54 x}{\left (-\ln \relax (5)+6\right )^{2} \left (1+\frac {x}{-\ln \relax (5)+6}\right )}\)	\(281\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x+1)*ln(5)^2+(-4*x^2-26*x-15)*ln(5)+2*x^3+25*x^2+84*x+54)/(ln(5)^2+(-2*x-12)*ln(5)+x^2+12*x+36),x,meth
od=_RETURNVERBOSE)

[Out]

x^2+x-(-3*ln(5)+18)/(-ln(5)+x+6)

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maxima [A]  time = 0.41, size = 20, normalized size = 0.83 \begin {gather*} x^{2} + x + \frac {3 \, {\left (\log \relax (5) - 6\right )}}{x - \log \relax (5) + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x+1)*log(5)^2+(-4*x^2-26*x-15)*log(5)+2*x^3+25*x^2+84*x+54)/(log(5)^2+(-2*x-12)*log(5)+x^2+12*x+
36),x, algorithm="maxima")

[Out]

x^2 + x + 3*(log(5) - 6)/(x - log(5) + 6)

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mupad [B]  time = 1.21, size = 75, normalized size = 3.12 \begin {gather*} x\,\left (2\,\ln \left (25\right )-\ln \left (625\right )+1\right )+x^2-\frac {\mathrm {atan}\left (\frac {x\,2{}\mathrm {i}-\ln \left (25\right )\,1{}\mathrm {i}+12{}\mathrm {i}}{\sqrt {\ln \left (25\right )-2\,\ln \relax (5)}\,\sqrt {2\,\ln \relax (5)+\ln \left (25\right )-24}}\right )\,\left (\ln \left (125\right )-18\right )\,2{}\mathrm {i}}{\sqrt {\ln \left (25\right )-2\,\ln \relax (5)}\,\sqrt {2\,\ln \relax (5)+\ln \left (25\right )-24}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((84*x - log(5)*(26*x + 4*x^2 + 15) + log(5)^2*(2*x + 1) + 25*x^2 + 2*x^3 + 54)/(12*x - log(5)*(2*x + 12) +
log(5)^2 + x^2 + 36),x)

[Out]

x*(2*log(25) - log(625) + 1) + x^2 - (atan((x*2i - log(25)*1i + 12i)/((log(25) - 2*log(5))^(1/2)*(2*log(5) + l
og(25) - 24)^(1/2)))*(log(125) - 18)*2i)/((log(25) - 2*log(5))^(1/2)*(2*log(5) + log(25) - 24)^(1/2))

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sympy [A]  time = 0.20, size = 17, normalized size = 0.71 \begin {gather*} x^{2} + x + \frac {-18 + 3 \log {\relax (5 )}}{x - \log {\relax (5 )} + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x+1)*ln(5)**2+(-4*x**2-26*x-15)*ln(5)+2*x**3+25*x**2+84*x+54)/(ln(5)**2+(-2*x-12)*ln(5)+x**2+12*
x+36),x)

[Out]

x**2 + x + (-18 + 3*log(5))/(x - log(5) + 6)

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