3.35.52 \(\int \frac {4-49 x-52 x^2+111 x^3-2 x^4-15 x^5+2 x^6+(-1-2 x) \log (1+2 x)}{-3-7 x-29 x^2-45 x^3+21 x^4+5 x^5-2 x^6+(1+2 x) \log (1+2 x)} \, dx\)

Optimal. Leaf size=30 \[ -x+\log \left (-\left ((3+x) \left (1+\left (3 x-x^2\right )^2\right )\right )+\log (1+2 x)\right ) \]

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Rubi [A]  time = 0.77, antiderivative size = 34, normalized size of antiderivative = 1.13, number of steps used = 4, number of rules used = 3, integrand size = 87, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {6741, 6742, 6684} \begin {gather*} \log \left (x^5-3 x^4-9 x^3+27 x^2+x-\log (2 x+1)+3\right )-x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4 - 49*x - 52*x^2 + 111*x^3 - 2*x^4 - 15*x^5 + 2*x^6 + (-1 - 2*x)*Log[1 + 2*x])/(-3 - 7*x - 29*x^2 - 45*x
^3 + 21*x^4 + 5*x^5 - 2*x^6 + (1 + 2*x)*Log[1 + 2*x]),x]

[Out]

-x + Log[3 + x + 27*x^2 - 9*x^3 - 3*x^4 + x^5 - Log[1 + 2*x]]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rule 6741

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

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 {-4+49 x+52 x^2-111 x^3+2 x^4+15 x^5-2 x^6-(-1-2 x) \log (1+2 x)}{(1+2 x) \left (3+x+27 x^2-9 x^3-3 x^4+x^5-\log (1+2 x)\right )} \, dx\\ &=\int \left (-1+\frac {-1+56 x+81 x^2-66 x^3-19 x^4+10 x^5}{(1+2 x) \left (3+x+27 x^2-9 x^3-3 x^4+x^5-\log (1+2 x)\right )}\right ) \, dx\\ &=-x+\int \frac {-1+56 x+81 x^2-66 x^3-19 x^4+10 x^5}{(1+2 x) \left (3+x+27 x^2-9 x^3-3 x^4+x^5-\log (1+2 x)\right )} \, dx\\ &=-x+\log \left (3+x+27 x^2-9 x^3-3 x^4+x^5-\log (1+2 x)\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 56, normalized size = 1.87 \begin {gather*} -x+\log \left (325-495 (1+2 x)+278 (1+2 x)^2-2 (1+2 x)^3-11 (1+2 x)^4+(1+2 x)^5-32 \log (1+2 x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 - 49*x - 52*x^2 + 111*x^3 - 2*x^4 - 15*x^5 + 2*x^6 + (-1 - 2*x)*Log[1 + 2*x])/(-3 - 7*x - 29*x^2
- 45*x^3 + 21*x^4 + 5*x^5 - 2*x^6 + (1 + 2*x)*Log[1 + 2*x]),x]

[Out]

-x + Log[325 - 495*(1 + 2*x) + 278*(1 + 2*x)^2 - 2*(1 + 2*x)^3 - 11*(1 + 2*x)^4 + (1 + 2*x)^5 - 32*Log[1 + 2*x
]]

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fricas [A]  time = 0.59, size = 36, normalized size = 1.20 \begin {gather*} -x + \log \left (-x^{5} + 3 \, x^{4} + 9 \, x^{3} - 27 \, x^{2} - x + \log \left (2 \, x + 1\right ) - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x-1)*log(2*x+1)+2*x^6-15*x^5-2*x^4+111*x^3-52*x^2-49*x+4)/((2*x+1)*log(2*x+1)-2*x^6+5*x^5+21*x^
4-45*x^3-29*x^2-7*x-3),x, algorithm="fricas")

[Out]

-x + log(-x^5 + 3*x^4 + 9*x^3 - 27*x^2 - x + log(2*x + 1) - 3)

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giac [A]  time = 0.22, size = 34, normalized size = 1.13 \begin {gather*} -x + \log \left (x^{5} - 3 \, x^{4} - 9 \, x^{3} + 27 \, x^{2} + x - \log \left (2 \, x + 1\right ) + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x-1)*log(2*x+1)+2*x^6-15*x^5-2*x^4+111*x^3-52*x^2-49*x+4)/((2*x+1)*log(2*x+1)-2*x^6+5*x^5+21*x^
4-45*x^3-29*x^2-7*x-3),x, algorithm="giac")

[Out]

-x + log(x^5 - 3*x^4 - 9*x^3 + 27*x^2 + x - log(2*x + 1) + 3)

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maple [A]  time = 0.06, size = 35, normalized size = 1.17

method	result	size
norman	\(-x +\ln \left (x^{5}-3 x^{4}-9 x^{3}+27 x^{2}+x -\ln \left (2 x +1\right )+3\right )\)	\(35\)
risch	\(-x +\ln \left (-x^{5}+3 x^{4}+9 x^{3}-27 x^{2}-x +\ln \left (2 x +1\right )-3\right )\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x-1)*ln(2*x+1)+2*x^6-15*x^5-2*x^4+111*x^3-52*x^2-49*x+4)/((2*x+1)*ln(2*x+1)-2*x^6+5*x^5+21*x^4-45*x^3
-29*x^2-7*x-3),x,method=_RETURNVERBOSE)

[Out]

-x+ln(x^5-3*x^4-9*x^3+27*x^2+x-ln(2*x+1)+3)

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maxima [A]  time = 0.41, size = 36, normalized size = 1.20 \begin {gather*} -x + \log \left (-x^{5} + 3 \, x^{4} + 9 \, x^{3} - 27 \, x^{2} - x + \log \left (2 \, x + 1\right ) - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x-1)*log(2*x+1)+2*x^6-15*x^5-2*x^4+111*x^3-52*x^2-49*x+4)/((2*x+1)*log(2*x+1)-2*x^6+5*x^5+21*x^
4-45*x^3-29*x^2-7*x-3),x, algorithm="maxima")

[Out]

-x + log(-x^5 + 3*x^4 + 9*x^3 - 27*x^2 - x + log(2*x + 1) - 3)

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mupad [B]  time = 2.14, size = 34, normalized size = 1.13 \begin {gather*} \ln \left (x-\ln \left (2\,x+1\right )+27\,x^2-9\,x^3-3\,x^4+x^5+3\right )-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((49*x + log(2*x + 1)*(2*x + 1) + 52*x^2 - 111*x^3 + 2*x^4 + 15*x^5 - 2*x^6 - 4)/(7*x - log(2*x + 1)*(2*x +
1) + 29*x^2 + 45*x^3 - 21*x^4 - 5*x^5 + 2*x^6 + 3),x)

[Out]

log(x - log(2*x + 1) + 27*x^2 - 9*x^3 - 3*x^4 + x^5 + 3) - x

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sympy [A]  time = 0.30, size = 31, normalized size = 1.03 \begin {gather*} - x + \log {\left (- x^{5} + 3 x^{4} + 9 x^{3} - 27 x^{2} - x + \log {\left (2 x + 1 \right )} - 3 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x-1)*ln(2*x+1)+2*x**6-15*x**5-2*x**4+111*x**3-52*x**2-49*x+4)/((2*x+1)*ln(2*x+1)-2*x**6+5*x**5+
21*x**4-45*x**3-29*x**2-7*x-3),x)

[Out]

-x + log(-x**5 + 3*x**4 + 9*x**3 - 27*x**2 - x + log(2*x + 1) - 3)

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