3.102.27 \(\int \frac {135 x-7 x^2-4 x^3+e^3 (10 x-2 x^2)+(-135 x+44 x^2+4 x^3+e^3 (-150+50 x+4 x^2)) \log (27 x+2 x^2+e^3 (30+2 x))}{675 x^3-220 x^4+7 x^5+2 x^6+e^3 (750 x^2-250 x^3+10 x^4+2 x^5)+(270 x^2-34 x^3-4 x^4+e^3 (300 x-40 x^2-4 x^3)) \log (27 x+2 x^2+e^3 (30+2 x))+(27 x+2 x^2+e^3 (30+2 x)) \log ^2(27 x+2 x^2+e^3 (30+2 x))} \, dx\)

Optimal. Leaf size=32 \[ \frac {x}{-x+\frac {\log \left (x \left (-3+\frac {2 (15+x) \left (e^3+x\right )}{x}\right )\right )}{-5+x}} \]

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Rubi [A]  time = 0.55, antiderivative size = 29, normalized size of antiderivative = 0.91, number of steps used = 3, number of rules used = 3, integrand size = 218, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {6688, 6711, 32} \begin {gather*} \frac {1}{1-\frac {(x-5) x}{\log \left (2 e^3 (x+15)+x (2 x+27)\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(135*x - 7*x^2 - 4*x^3 + E^3*(10*x - 2*x^2) + (-135*x + 44*x^2 + 4*x^3 + E^3*(-150 + 50*x + 4*x^2))*Log[27
*x + 2*x^2 + E^3*(30 + 2*x)])/(675*x^3 - 220*x^4 + 7*x^5 + 2*x^6 + E^3*(750*x^2 - 250*x^3 + 10*x^4 + 2*x^5) +
(270*x^2 - 34*x^3 - 4*x^4 + E^3*(300*x - 40*x^2 - 4*x^3))*Log[27*x + 2*x^2 + E^3*(30 + 2*x)] + (27*x + 2*x^2 +
E^3*(30 + 2*x))*Log[27*x + 2*x^2 + E^3*(30 + 2*x)]^2),x]

[Out]

(1 - ((-5 + x)*x)/Log[2*E^3*(15 + x) + x*(27 + 2*x)])^(-1)

Rule 32

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

Rule 6688

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

Rule 6711

Int[(u_)*((a_.)*(v_)^(p_.) + (b_.)*(w_)^(q_.))^(m_.), x_Symbol] :> With[{c = Simplify[u/(p*w*D[v, x] - q*v*D[w
, x])]}, Dist[c*p, Subst[Int[(b + a*x^p)^m, x], x, v*w^(m*q + 1)], x] /; FreeQ[c, x]] /; FreeQ[{a, b, m, p, q}
, x] && EqQ[p + q*(m*p + 1), 0] && IntegerQ[p] && IntegerQ[m]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x \left (135-2 e^3 (-5+x)-7 x-4 x^2\right )+(-5+2 x) \left (2 e^3 (15+x)+x (27+2 x)\right ) \log \left (2 e^3 (15+x)+x (27+2 x)\right )}{\left (30 e^3+\left (27+2 e^3\right ) x+2 x^2\right ) \left ((-5+x) x-\log \left (2 e^3 (15+x)+x (27+2 x)\right )\right )^2} \, dx\\ &=\operatorname {Subst}\left (\int \frac {1}{(-1+x)^2} \, dx,x,\frac {(-5+x) x}{\log \left (2 e^3 (15+x)+x (27+2 x)\right )}\right )\\ &=\frac {1}{1-\frac {(-5+x) x}{\log \left (2 e^3 (15+x)+x (27+2 x)\right )}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 33, normalized size = 1.03 \begin {gather*} \frac {(-5+x) x}{5 x-x^2+\log \left (2 e^3 (15+x)+x (27+2 x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(135*x - 7*x^2 - 4*x^3 + E^3*(10*x - 2*x^2) + (-135*x + 44*x^2 + 4*x^3 + E^3*(-150 + 50*x + 4*x^2))*
Log[27*x + 2*x^2 + E^3*(30 + 2*x)])/(675*x^3 - 220*x^4 + 7*x^5 + 2*x^6 + E^3*(750*x^2 - 250*x^3 + 10*x^4 + 2*x
^5) + (270*x^2 - 34*x^3 - 4*x^4 + E^3*(300*x - 40*x^2 - 4*x^3))*Log[27*x + 2*x^2 + E^3*(30 + 2*x)] + (27*x + 2
*x^2 + E^3*(30 + 2*x))*Log[27*x + 2*x^2 + E^3*(30 + 2*x)]^2),x]

[Out]

((-5 + x)*x)/(5*x - x^2 + Log[2*E^3*(15 + x) + x*(27 + 2*x)])

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fricas [A]  time = 0.62, size = 37, normalized size = 1.16 \begin {gather*} -\frac {x^{2} - 5 \, x}{x^{2} - 5 \, x - \log \left (2 \, x^{2} + 2 \, {\left (x + 15\right )} e^{3} + 27 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x^2+50*x-150)*exp(3)+4*x^3+44*x^2-135*x)*log((2*x+30)*exp(3)+2*x^2+27*x)+(-2*x^2+10*x)*exp(3)-4
*x^3-7*x^2+135*x)/(((2*x+30)*exp(3)+2*x^2+27*x)*log((2*x+30)*exp(3)+2*x^2+27*x)^2+((-4*x^3-40*x^2+300*x)*exp(3
)-4*x^4-34*x^3+270*x^2)*log((2*x+30)*exp(3)+2*x^2+27*x)+(2*x^5+10*x^4-250*x^3+750*x^2)*exp(3)+2*x^6+7*x^5-220*
x^4+675*x^3),x, algorithm="fricas")

[Out]

-(x^2 - 5*x)/(x^2 - 5*x - log(2*x^2 + 2*(x + 15)*e^3 + 27*x))

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x^2+50*x-150)*exp(3)+4*x^3+44*x^2-135*x)*log((2*x+30)*exp(3)+2*x^2+27*x)+(-2*x^2+10*x)*exp(3)-4
*x^3-7*x^2+135*x)/(((2*x+30)*exp(3)+2*x^2+27*x)*log((2*x+30)*exp(3)+2*x^2+27*x)^2+((-4*x^3-40*x^2+300*x)*exp(3
)-4*x^4-34*x^3+270*x^2)*log((2*x+30)*exp(3)+2*x^2+27*x)+(2*x^5+10*x^4-250*x^3+750*x^2)*exp(3)+2*x^6+7*x^5-220*
x^4+675*x^3),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.31, size = 36, normalized size = 1.12

method	result	size
risch	\(-\frac {\left (x -5\right ) x}{x^{2}-5 x -\ln \left (\left (2 x +30\right ) {\mathrm e}^{3}+2 x^{2}+27 x \right )}\)	\(36\)
norman	\(-\frac {\ln \left (\left (2 x +30\right ) {\mathrm e}^{3}+2 x^{2}+27 x \right )}{x^{2}-5 x -\ln \left (\left (2 x +30\right ) {\mathrm e}^{3}+2 x^{2}+27 x \right )}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((4*x^2+50*x-150)*exp(3)+4*x^3+44*x^2-135*x)*ln((2*x+30)*exp(3)+2*x^2+27*x)+(-2*x^2+10*x)*exp(3)-4*x^3-7*
x^2+135*x)/(((2*x+30)*exp(3)+2*x^2+27*x)*ln((2*x+30)*exp(3)+2*x^2+27*x)^2+((-4*x^3-40*x^2+300*x)*exp(3)-4*x^4-
34*x^3+270*x^2)*ln((2*x+30)*exp(3)+2*x^2+27*x)+(2*x^5+10*x^4-250*x^3+750*x^2)*exp(3)+2*x^6+7*x^5-220*x^4+675*x
^3),x,method=_RETURNVERBOSE)

[Out]

-(x-5)*x/(x^2-5*x-ln((2*x+30)*exp(3)+2*x^2+27*x))

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maxima [A]  time = 0.41, size = 39, normalized size = 1.22 \begin {gather*} -\frac {x^{2} - 5 \, x}{x^{2} - 5 \, x - \log \left (2 \, x^{2} + x {\left (2 \, e^{3} + 27\right )} + 30 \, e^{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x^2+50*x-150)*exp(3)+4*x^3+44*x^2-135*x)*log((2*x+30)*exp(3)+2*x^2+27*x)+(-2*x^2+10*x)*exp(3)-4
*x^3-7*x^2+135*x)/(((2*x+30)*exp(3)+2*x^2+27*x)*log((2*x+30)*exp(3)+2*x^2+27*x)^2+((-4*x^3-40*x^2+300*x)*exp(3
)-4*x^4-34*x^3+270*x^2)*log((2*x+30)*exp(3)+2*x^2+27*x)+(2*x^5+10*x^4-250*x^3+750*x^2)*exp(3)+2*x^6+7*x^5-220*
x^4+675*x^3),x, algorithm="maxima")

[Out]

-(x^2 - 5*x)/(x^2 - 5*x - log(2*x^2 + x*(2*e^3 + 27) + 30*e^3))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {135\,x+\ln \left (27\,x+2\,x^2+{\mathrm {e}}^3\,\left (2\,x+30\right )\right )\,\left ({\mathrm {e}}^3\,\left (4\,x^2+50\,x-150\right )-135\,x+44\,x^2+4\,x^3\right )+{\mathrm {e}}^3\,\left (10\,x-2\,x^2\right )-7\,x^2-4\,x^3}{675\,x^3-\ln \left (27\,x+2\,x^2+{\mathrm {e}}^3\,\left (2\,x+30\right )\right )\,\left ({\mathrm {e}}^3\,\left (4\,x^3+40\,x^2-300\,x\right )-270\,x^2+34\,x^3+4\,x^4\right )-220\,x^4+7\,x^5+2\,x^6+{\ln \left (27\,x+2\,x^2+{\mathrm {e}}^3\,\left (2\,x+30\right )\right )}^2\,\left (27\,x+2\,x^2+{\mathrm {e}}^3\,\left (2\,x+30\right )\right )+{\mathrm {e}}^3\,\left (2\,x^5+10\,x^4-250\,x^3+750\,x^2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((135*x + log(27*x + 2*x^2 + exp(3)*(2*x + 30))*(exp(3)*(50*x + 4*x^2 - 150) - 135*x + 44*x^2 + 4*x^3) + ex
p(3)*(10*x - 2*x^2) - 7*x^2 - 4*x^3)/(675*x^3 - log(27*x + 2*x^2 + exp(3)*(2*x + 30))*(exp(3)*(40*x^2 - 300*x
+ 4*x^3) - 270*x^2 + 34*x^3 + 4*x^4) - 220*x^4 + 7*x^5 + 2*x^6 + log(27*x + 2*x^2 + exp(3)*(2*x + 30))^2*(27*x
+ 2*x^2 + exp(3)*(2*x + 30)) + exp(3)*(750*x^2 - 250*x^3 + 10*x^4 + 2*x^5)),x)

[Out]

int((135*x + log(27*x + 2*x^2 + exp(3)*(2*x + 30))*(exp(3)*(50*x + 4*x^2 - 150) - 135*x + 44*x^2 + 4*x^3) + ex
p(3)*(10*x - 2*x^2) - 7*x^2 - 4*x^3)/(675*x^3 - log(27*x + 2*x^2 + exp(3)*(2*x + 30))*(exp(3)*(40*x^2 - 300*x
+ 4*x^3) - 270*x^2 + 34*x^3 + 4*x^4) - 220*x^4 + 7*x^5 + 2*x^6 + log(27*x + 2*x^2 + exp(3)*(2*x + 30))^2*(27*x
+ 2*x^2 + exp(3)*(2*x + 30)) + exp(3)*(750*x^2 - 250*x^3 + 10*x^4 + 2*x^5)), x)

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sympy [A]  time = 0.28, size = 31, normalized size = 0.97 \begin {gather*} \frac {x^{2} - 5 x}{- x^{2} + 5 x + \log {\left (2 x^{2} + 27 x + \left (2 x + 30\right ) e^{3} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x**2+50*x-150)*exp(3)+4*x**3+44*x**2-135*x)*ln((2*x+30)*exp(3)+2*x**2+27*x)+(-2*x**2+10*x)*exp(
3)-4*x**3-7*x**2+135*x)/(((2*x+30)*exp(3)+2*x**2+27*x)*ln((2*x+30)*exp(3)+2*x**2+27*x)**2+((-4*x**3-40*x**2+30
0*x)*exp(3)-4*x**4-34*x**3+270*x**2)*ln((2*x+30)*exp(3)+2*x**2+27*x)+(2*x**5+10*x**4-250*x**3+750*x**2)*exp(3)
+2*x**6+7*x**5-220*x**4+675*x**3),x)

[Out]

(x**2 - 5*x)/(-x**2 + 5*x + log(2*x**2 + 27*x + (2*x + 30)*exp(3)))

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