3.18.69 \(\int \frac {-16-56 x+55 x^2+146 x^3-28 x^4+x^5+e^3 (8 x^2+9 x^3)+(8 x^2+9 x^3) \log (x)}{112 x+488 x^2+439 x^3-199 x^4+25 x^5-x^6+e^3 (16 x+72 x^2+73 x^3-18 x^4+x^5)+(16 x+72 x^2+73 x^3-18 x^4+x^5) \log (x)} \, dx\)

Optimal. Leaf size=30 \[ \frac {x}{9+\frac {4}{x}-x}-\log \left (-7-e^3+x-\log (x)\right ) \]

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Rubi [A]  time = 0.91, antiderivative size = 32, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 4, integrand size = 135, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {6688, 6742, 775, 6684} \begin {gather*} \frac {9 x+4}{-x^2+9 x+4}-\log \left (-x+\log (x)+e^3+7\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-16 - 56*x + 55*x^2 + 146*x^3 - 28*x^4 + x^5 + E^3*(8*x^2 + 9*x^3) + (8*x^2 + 9*x^3)*Log[x])/(112*x + 488
*x^2 + 439*x^3 - 199*x^4 + 25*x^5 - x^6 + E^3*(16*x + 72*x^2 + 73*x^3 - 18*x^4 + x^5) + (16*x + 72*x^2 + 73*x^
3 - 18*x^4 + x^5)*Log[x]),x]

[Out]

(4 + 9*x)/(4 + 9*x - x^2) - Log[7 + E^3 - x + Log[x]]

Rule 775

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

Rule 6684

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

Rule 6688

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

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 {-16-56 x+\left (55+8 e^3\right ) x^2+\left (146+9 e^3\right ) x^3-28 x^4+x^5+x^2 (8+9 x) \log (x)}{x \left (4+9 x-x^2\right )^2 \left (7 \left (1+\frac {e^3}{7}\right )-x+\log (x)\right )} \, dx\\ &=\int \left (\frac {x (8+9 x)}{\left (-4-9 x+x^2\right )^2}+\frac {-1+x}{x \left (7 \left (1+\frac {e^3}{7}\right )-x+\log (x)\right )}\right ) \, dx\\ &=\int \frac {x (8+9 x)}{\left (-4-9 x+x^2\right )^2} \, dx+\int \frac {-1+x}{x \left (7 \left (1+\frac {e^3}{7}\right )-x+\log (x)\right )} \, dx\\ &=\frac {4+9 x}{4+9 x-x^2}-\log \left (7+e^3-x+\log (x)\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.36, size = 31, normalized size = 1.03 \begin {gather*} -\frac {4+9 x}{-4-9 x+x^2}-\log \left (7+e^3-x+\log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-16 - 56*x + 55*x^2 + 146*x^3 - 28*x^4 + x^5 + E^3*(8*x^2 + 9*x^3) + (8*x^2 + 9*x^3)*Log[x])/(112*x
+ 488*x^2 + 439*x^3 - 199*x^4 + 25*x^5 - x^6 + E^3*(16*x + 72*x^2 + 73*x^3 - 18*x^4 + x^5) + (16*x + 72*x^2 +
73*x^3 - 18*x^4 + x^5)*Log[x]),x]

[Out]

-((4 + 9*x)/(-4 - 9*x + x^2)) - Log[7 + E^3 - x + Log[x]]

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fricas [A]  time = 1.06, size = 36, normalized size = 1.20 \begin {gather*} -\frac {{\left (x^{2} - 9 \, x - 4\right )} \log \left (-x + e^{3} + \log \relax (x) + 7\right ) + 9 \, x + 4}{x^{2} - 9 \, x - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9*x^3+8*x^2)*log(x)+(9*x^3+8*x^2)*exp(3)+x^5-28*x^4+146*x^3+55*x^2-56*x-16)/((x^5-18*x^4+73*x^3+72
*x^2+16*x)*log(x)+(x^5-18*x^4+73*x^3+72*x^2+16*x)*exp(3)-x^6+25*x^5-199*x^4+439*x^3+488*x^2+112*x),x, algorith
m="fricas")

[Out]

-((x^2 - 9*x - 4)*log(-x + e^3 + log(x) + 7) + 9*x + 4)/(x^2 - 9*x - 4)

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giac [A]  time = 0.24, size = 56, normalized size = 1.87 \begin {gather*} -\frac {x^{2} \log \left (-x + e^{3} + \log \relax (x) + 7\right ) - 9 \, x \log \left (-x + e^{3} + \log \relax (x) + 7\right ) + 9 \, x - 4 \, \log \left (-x + e^{3} + \log \relax (x) + 7\right ) + 4}{x^{2} - 9 \, x - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9*x^3+8*x^2)*log(x)+(9*x^3+8*x^2)*exp(3)+x^5-28*x^4+146*x^3+55*x^2-56*x-16)/((x^5-18*x^4+73*x^3+72
*x^2+16*x)*log(x)+(x^5-18*x^4+73*x^3+72*x^2+16*x)*exp(3)-x^6+25*x^5-199*x^4+439*x^3+488*x^2+112*x),x, algorith
m="giac")

[Out]

-(x^2*log(-x + e^3 + log(x) + 7) - 9*x*log(-x + e^3 + log(x) + 7) + 9*x - 4*log(-x + e^3 + log(x) + 7) + 4)/(x
^2 - 9*x - 4)

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maple [A]  time = 0.22, size = 29, normalized size = 0.97

method	result	size
norman	\(-\frac {x^{2}}{x^{2}-9 x -4}-\ln \left ({\mathrm e}^{3}-x +\ln \relax (x )+7\right )\)	\(29\)
risch	\(-\frac {9 x +4}{x^{2}-9 x -4}-\ln \left ({\mathrm e}^{3}-x +\ln \relax (x )+7\right )\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((9*x^3+8*x^2)*ln(x)+(9*x^3+8*x^2)*exp(3)+x^5-28*x^4+146*x^3+55*x^2-56*x-16)/((x^5-18*x^4+73*x^3+72*x^2+16
*x)*ln(x)+(x^5-18*x^4+73*x^3+72*x^2+16*x)*exp(3)-x^6+25*x^5-199*x^4+439*x^3+488*x^2+112*x),x,method=_RETURNVER
BOSE)

[Out]

-x^2/(x^2-9*x-4)-ln(exp(3)-x+ln(x)+7)

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maxima [A]  time = 0.45, size = 30, normalized size = 1.00 \begin {gather*} -\frac {9 \, x + 4}{x^{2} - 9 \, x - 4} - \log \left (-x + e^{3} + \log \relax (x) + 7\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9*x^3+8*x^2)*log(x)+(9*x^3+8*x^2)*exp(3)+x^5-28*x^4+146*x^3+55*x^2-56*x-16)/((x^5-18*x^4+73*x^3+72
*x^2+16*x)*log(x)+(x^5-18*x^4+73*x^3+72*x^2+16*x)*exp(3)-x^6+25*x^5-199*x^4+439*x^3+488*x^2+112*x),x, algorith
m="maxima")

[Out]

-(9*x + 4)/(x^2 - 9*x - 4) - log(-x + e^3 + log(x) + 7)

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mupad [B]  time = 1.32, size = 31, normalized size = 1.03 \begin {gather*} \frac {9\,x+4}{-x^2+9\,x+4}-\ln \left ({\mathrm {e}}^3-x+\ln \relax (x)+7\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)*(8*x^2 + 9*x^3) - 56*x + exp(3)*(8*x^2 + 9*x^3) + 55*x^2 + 146*x^3 - 28*x^4 + x^5 - 16)/(112*x + e
xp(3)*(16*x + 72*x^2 + 73*x^3 - 18*x^4 + x^5) + 488*x^2 + 439*x^3 - 199*x^4 + 25*x^5 - x^6 + log(x)*(16*x + 72
*x^2 + 73*x^3 - 18*x^4 + x^5)),x)

[Out]

(9*x + 4)/(9*x - x^2 + 4) - log(exp(3) - x + log(x) + 7)

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sympy [A]  time = 0.20, size = 26, normalized size = 0.87 \begin {gather*} \frac {- 9 x - 4}{x^{2} - 9 x - 4} - \log {\left (- x + \log {\relax (x )} + 7 + e^{3} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9*x**3+8*x**2)*ln(x)+(9*x**3+8*x**2)*exp(3)+x**5-28*x**4+146*x**3+55*x**2-56*x-16)/((x**5-18*x**4+
73*x**3+72*x**2+16*x)*ln(x)+(x**5-18*x**4+73*x**3+72*x**2+16*x)*exp(3)-x**6+25*x**5-199*x**4+439*x**3+488*x**2
+112*x),x)

[Out]

(-9*x - 4)/(x**2 - 9*x - 4) - log(-x + log(x) + 7 + exp(3))

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