3.18.50 \(\int \frac {-2880 x^2+1008 x^3-252 x^4+261 x^5-90 x^6+9 x^7+(-320+240 x-64 x^2+48 x^3) \log (5)}{-2880 x^2+2160 x^3-1116 x^4+477 x^5-108 x^6+9 x^7} \, dx\)

Optimal. Leaf size=24 \[ x-\frac {16 \log (5)}{9 (4-x)^2 x}+\log \left (5+x^2\right ) \]

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Rubi [A]  time = 0.15, antiderivative size = 43, normalized size of antiderivative = 1.79, number of steps used = 3, number of rules used = 2, integrand size = 83, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {2074, 260} \begin {gather*} \log \left (x^2+5\right )+x-\frac {\log (5)}{9 (4-x)}-\frac {4 \log (5)}{9 (4-x)^2}-\frac {\log (5)}{9 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2880*x^2 + 1008*x^3 - 252*x^4 + 261*x^5 - 90*x^6 + 9*x^7 + (-320 + 240*x - 64*x^2 + 48*x^3)*Log[5])/(-28
80*x^2 + 2160*x^3 - 1116*x^4 + 477*x^5 - 108*x^6 + 9*x^7),x]

[Out]

x - (4*Log[5])/(9*(4 - x)^2) - Log[5]/(9*(4 - x)) - Log[5]/(9*x) + Log[5 + x^2]

Rule 260

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

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1+\frac {2 x}{5+x^2}+\frac {8 \log (5)}{9 (-4+x)^3}-\frac {\log (5)}{9 (-4+x)^2}+\frac {\log (5)}{9 x^2}\right ) \, dx\\ &=x-\frac {4 \log (5)}{9 (4-x)^2}-\frac {\log (5)}{9 (4-x)}-\frac {\log (5)}{9 x}+2 \int \frac {x}{5+x^2} \, dx\\ &=x-\frac {4 \log (5)}{9 (4-x)^2}-\frac {\log (5)}{9 (4-x)}-\frac {\log (5)}{9 x}+\log \left (5+x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 34, normalized size = 1.42 \begin {gather*} \frac {1}{9} \left (9 x+\frac {(-8+x) \log (5)}{(-4+x)^2}-\frac {\log (5)}{x}+9 \log \left (5+x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2880*x^2 + 1008*x^3 - 252*x^4 + 261*x^5 - 90*x^6 + 9*x^7 + (-320 + 240*x - 64*x^2 + 48*x^3)*Log[5]
)/(-2880*x^2 + 2160*x^3 - 1116*x^4 + 477*x^5 - 108*x^6 + 9*x^7),x]

[Out]

(9*x + ((-8 + x)*Log[5])/(-4 + x)^2 - Log[5]/x + 9*Log[5 + x^2])/9

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fricas [B]  time = 0.64, size = 56, normalized size = 2.33 \begin {gather*} \frac {9 \, x^{4} - 72 \, x^{3} + 144 \, x^{2} + 9 \, {\left (x^{3} - 8 \, x^{2} + 16 \, x\right )} \log \left (x^{2} + 5\right ) - 16 \, \log \relax (5)}{9 \, {\left (x^{3} - 8 \, x^{2} + 16 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((48*x^3-64*x^2+240*x-320)*log(5)+9*x^7-90*x^6+261*x^5-252*x^4+1008*x^3-2880*x^2)/(9*x^7-108*x^6+477
*x^5-1116*x^4+2160*x^3-2880*x^2),x, algorithm="fricas")

[Out]

1/9*(9*x^4 - 72*x^3 + 144*x^2 + 9*(x^3 - 8*x^2 + 16*x)*log(x^2 + 5) - 16*log(5))/(x^3 - 8*x^2 + 16*x)

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giac [A]  time = 0.21, size = 31, normalized size = 1.29 \begin {gather*} x - \frac {\log \relax (5)}{9 \, x} + \frac {x \log \relax (5) - 8 \, \log \relax (5)}{9 \, {\left (x - 4\right )}^{2}} + \log \left (x^{2} + 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((48*x^3-64*x^2+240*x-320)*log(5)+9*x^7-90*x^6+261*x^5-252*x^4+1008*x^3-2880*x^2)/(9*x^7-108*x^6+477
*x^5-1116*x^4+2160*x^3-2880*x^2),x, algorithm="giac")

[Out]

x - 1/9*log(5)/x + 1/9*(x*log(5) - 8*log(5))/(x - 4)^2 + log(x^2 + 5)

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maple [A]  time = 0.08, size = 26, normalized size = 1.08

method	result	size
risch	\(x -\frac {16 \ln \relax (5)}{9 x \left (x^{2}-8 x +16\right )}+\ln \left (x^{2}+5\right )\)	\(26\)
norman	\(\frac {x^{4}-48 x^{2}+128 x -\frac {16 \ln \relax (5)}{9}}{x \left (x -4\right )^{2}}+\ln \left (x^{2}+5\right )\)	\(33\)
default	\(x +\ln \left (x^{2}+5\right )-\frac {4 \ln \relax (5)}{9 \left (x -4\right )^{2}}+\frac {\ln \relax (5)}{9 x -36}-\frac {\ln \relax (5)}{9 x}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((48*x^3-64*x^2+240*x-320)*ln(5)+9*x^7-90*x^6+261*x^5-252*x^4+1008*x^3-2880*x^2)/(9*x^7-108*x^6+477*x^5-11
16*x^4+2160*x^3-2880*x^2),x,method=_RETURNVERBOSE)

[Out]

x-16/9*ln(5)/x/(x^2-8*x+16)+ln(x^2+5)

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maxima [A]  time = 0.69, size = 26, normalized size = 1.08 \begin {gather*} x - \frac {16 \, \log \relax (5)}{9 \, {\left (x^{3} - 8 \, x^{2} + 16 \, x\right )}} + \log \left (x^{2} + 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((48*x^3-64*x^2+240*x-320)*log(5)+9*x^7-90*x^6+261*x^5-252*x^4+1008*x^3-2880*x^2)/(9*x^7-108*x^6+477
*x^5-1116*x^4+2160*x^3-2880*x^2),x, algorithm="maxima")

[Out]

x - 16/9*log(5)/(x^3 - 8*x^2 + 16*x) + log(x^2 + 5)

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mupad [B]  time = 0.13, size = 28, normalized size = 1.17 \begin {gather*} x+\ln \left (x^2+5\right )-\frac {16\,\ln \relax (5)}{9\,x^3-72\,x^2+144\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(5)*(240*x - 64*x^2 + 48*x^3 - 320) - 2880*x^2 + 1008*x^3 - 252*x^4 + 261*x^5 - 90*x^6 + 9*x^7)/(2880
*x^2 - 2160*x^3 + 1116*x^4 - 477*x^5 + 108*x^6 - 9*x^7),x)

[Out]

x + log(x^2 + 5) - (16*log(5))/(144*x - 72*x^2 + 9*x^3)

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sympy [A]  time = 0.93, size = 26, normalized size = 1.08 \begin {gather*} x + \log {\left (x^{2} + 5 \right )} - \frac {16 \log {\relax (5 )}}{9 x^{3} - 72 x^{2} + 144 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((48*x**3-64*x**2+240*x-320)*ln(5)+9*x**7-90*x**6+261*x**5-252*x**4+1008*x**3-2880*x**2)/(9*x**7-108
*x**6+477*x**5-1116*x**4+2160*x**3-2880*x**2),x)

[Out]

x + log(x**2 + 5) - 16*log(5)/(9*x**3 - 72*x**2 + 144*x)

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