3.35.66 \(\int \frac {-10-30 x+20 x^2-10 x \log (2)}{375 x+675 x^2+180 x^3-189 x^4-36 x^5+27 x^6-3 x^7+(225 x^2+270 x^3-9 x^4-54 x^5+9 x^6) \log (2)+(45 x^3+27 x^4-9 x^5) \log ^2(2)+3 x^4 \log ^3(2)+(225 x+270 x^2-9 x^3-54 x^4+9 x^5+(90 x^2+54 x^3-18 x^4) \log (2)+9 x^3 \log ^2(2)) \log (x)+(45 x+27 x^2-9 x^3+9 x^2 \log (2)) \log ^2(x)+3 x \log ^3(x)} \, dx\)

Optimal. Leaf size=19 \[ \frac {5}{3 (5+x (3-x+\log (2))+\log (x))^2} \]

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Rubi [A]  time = 0.50, antiderivative size = 21, normalized size of antiderivative = 1.11, number of steps used = 5, number of rules used = 4, integrand size = 199, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {6, 6688, 12, 6686} \begin {gather*} \frac {5}{3 \left (-x^2+x (3+\log (2))+\log (x)+5\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-10 - 30*x + 20*x^2 - 10*x*Log[2])/(375*x + 675*x^2 + 180*x^3 - 189*x^4 - 36*x^5 + 27*x^6 - 3*x^7 + (225*
x^2 + 270*x^3 - 9*x^4 - 54*x^5 + 9*x^6)*Log[2] + (45*x^3 + 27*x^4 - 9*x^5)*Log[2]^2 + 3*x^4*Log[2]^3 + (225*x
+ 270*x^2 - 9*x^3 - 54*x^4 + 9*x^5 + (90*x^2 + 54*x^3 - 18*x^4)*Log[2] + 9*x^3*Log[2]^2)*Log[x] + (45*x + 27*x
^2 - 9*x^3 + 9*x^2*Log[2])*Log[x]^2 + 3*x*Log[x]^3),x]

[Out]

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

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-10+20 x^2+x (-30-10 \log (2))}{375 x+675 x^2+180 x^3-189 x^4-36 x^5+27 x^6-3 x^7+\left (225 x^2+270 x^3-9 x^4-54 x^5+9 x^6\right ) \log (2)+\left (45 x^3+27 x^4-9 x^5\right ) \log ^2(2)+3 x^4 \log ^3(2)+\left (225 x+270 x^2-9 x^3-54 x^4+9 x^5+\left (90 x^2+54 x^3-18 x^4\right ) \log (2)+9 x^3 \log ^2(2)\right ) \log (x)+\left (45 x+27 x^2-9 x^3+9 x^2 \log (2)\right ) \log ^2(x)+3 x \log ^3(x)} \, dx\\ &=\int \frac {-10+20 x^2+x (-30-10 \log (2))}{375 x+675 x^2+180 x^3-36 x^5+27 x^6-3 x^7+\left (225 x^2+270 x^3-9 x^4-54 x^5+9 x^6\right ) \log (2)+\left (45 x^3+27 x^4-9 x^5\right ) \log ^2(2)+x^4 \left (-189+3 \log ^3(2)\right )+\left (225 x+270 x^2-9 x^3-54 x^4+9 x^5+\left (90 x^2+54 x^3-18 x^4\right ) \log (2)+9 x^3 \log ^2(2)\right ) \log (x)+\left (45 x+27 x^2-9 x^3+9 x^2 \log (2)\right ) \log ^2(x)+3 x \log ^3(x)} \, dx\\ &=\int \frac {10 \left (-1+2 x^2-x (3+\log (2))\right )}{3 x \left (5-x^2+x (3+\log (2))+\log (x)\right )^3} \, dx\\ &=\frac {10}{3} \int \frac {-1+2 x^2-x (3+\log (2))}{x \left (5-x^2+x (3+\log (2))+\log (x)\right )^3} \, dx\\ &=\frac {5}{3 \left (5-x^2+x (3+\log (2))+\log (x)\right )^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 21, normalized size = 1.11 \begin {gather*} \frac {5}{3 \left (5-x^2+x (3+\log (2))+\log (x)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-10 - 30*x + 20*x^2 - 10*x*Log[2])/(375*x + 675*x^2 + 180*x^3 - 189*x^4 - 36*x^5 + 27*x^6 - 3*x^7 +
(225*x^2 + 270*x^3 - 9*x^4 - 54*x^5 + 9*x^6)*Log[2] + (45*x^3 + 27*x^4 - 9*x^5)*Log[2]^2 + 3*x^4*Log[2]^3 + (
225*x + 270*x^2 - 9*x^3 - 54*x^4 + 9*x^5 + (90*x^2 + 54*x^3 - 18*x^4)*Log[2] + 9*x^3*Log[2]^2)*Log[x] + (45*x
+ 27*x^2 - 9*x^3 + 9*x^2*Log[2])*Log[x]^2 + 3*x*Log[x]^3),x]

[Out]

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

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fricas [B]  time = 0.65, size = 67, normalized size = 3.53 \begin {gather*} \frac {5}{3 \, {\left (x^{4} + x^{2} \log \relax (2)^{2} - 6 \, x^{3} - x^{2} - 2 \, {\left (x^{3} - 3 \, x^{2} - 5 \, x\right )} \log \relax (2) - 2 \, {\left (x^{2} - x \log \relax (2) - 3 \, x - 5\right )} \log \relax (x) + \log \relax (x)^{2} + 30 \, x + 25\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-10*x*log(2)+20*x^2-30*x-10)/(3*x*log(x)^3+(9*x^2*log(2)-9*x^3+27*x^2+45*x)*log(x)^2+(9*x^3*log(2)^
2+(-18*x^4+54*x^3+90*x^2)*log(2)+9*x^5-54*x^4-9*x^3+270*x^2+225*x)*log(x)+3*x^4*log(2)^3+(-9*x^5+27*x^4+45*x^3
)*log(2)^2+(9*x^6-54*x^5-9*x^4+270*x^3+225*x^2)*log(2)-3*x^7+27*x^6-36*x^5-189*x^4+180*x^3+675*x^2+375*x),x, a
lgorithm="fricas")

[Out]

5/3/(x^4 + x^2*log(2)^2 - 6*x^3 - x^2 - 2*(x^3 - 3*x^2 - 5*x)*log(2) - 2*(x^2 - x*log(2) - 3*x - 5)*log(x) + l
og(x)^2 + 30*x + 25)

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giac [B]  time = 0.22, size = 215, normalized size = 11.32 \begin {gather*} \frac {5 \, {\left (2 \, x^{2} - x \log \relax (2) - 3 \, x - 1\right )}}{3 \, {\left (2 \, x^{6} - 5 \, x^{5} \log \relax (2) + 4 \, x^{4} \log \relax (2)^{2} - x^{3} \log \relax (2)^{3} - 15 \, x^{5} + 24 \, x^{4} \log \relax (2) - 9 \, x^{3} \log \relax (2)^{2} - 4 \, x^{4} \log \relax (x) + 6 \, x^{3} \log \relax (2) \log \relax (x) - 2 \, x^{2} \log \relax (2)^{2} \log \relax (x) + 15 \, x^{4} + 5 \, x^{3} \log \relax (2) - 11 \, x^{2} \log \relax (2)^{2} + 18 \, x^{3} \log \relax (x) - 12 \, x^{2} \log \relax (2) \log \relax (x) + 2 \, x^{2} \log \relax (x)^{2} - x \log \relax (2) \log \relax (x)^{2} + 69 \, x^{3} - 66 \, x^{2} \log \relax (2) + 4 \, x^{2} \log \relax (x) - 12 \, x \log \relax (2) \log \relax (x) - 3 \, x \log \relax (x)^{2} - 39 \, x^{2} - 35 \, x \log \relax (2) - 36 \, x \log \relax (x) - \log \relax (x)^{2} - 105 \, x - 10 \, \log \relax (x) - 25\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-10*x*log(2)+20*x^2-30*x-10)/(3*x*log(x)^3+(9*x^2*log(2)-9*x^3+27*x^2+45*x)*log(x)^2+(9*x^3*log(2)^
2+(-18*x^4+54*x^3+90*x^2)*log(2)+9*x^5-54*x^4-9*x^3+270*x^2+225*x)*log(x)+3*x^4*log(2)^3+(-9*x^5+27*x^4+45*x^3
)*log(2)^2+(9*x^6-54*x^5-9*x^4+270*x^3+225*x^2)*log(2)-3*x^7+27*x^6-36*x^5-189*x^4+180*x^3+675*x^2+375*x),x, a
lgorithm="giac")

[Out]

5/3*(2*x^2 - x*log(2) - 3*x - 1)/(2*x^6 - 5*x^5*log(2) + 4*x^4*log(2)^2 - x^3*log(2)^3 - 15*x^5 + 24*x^4*log(2
) - 9*x^3*log(2)^2 - 4*x^4*log(x) + 6*x^3*log(2)*log(x) - 2*x^2*log(2)^2*log(x) + 15*x^4 + 5*x^3*log(2) - 11*x
^2*log(2)^2 + 18*x^3*log(x) - 12*x^2*log(2)*log(x) + 2*x^2*log(x)^2 - x*log(2)*log(x)^2 + 69*x^3 - 66*x^2*log(
2) + 4*x^2*log(x) - 12*x*log(2)*log(x) - 3*x*log(x)^2 - 39*x^2 - 35*x*log(2) - 36*x*log(x) - log(x)^2 - 105*x
- 10*log(x) - 25)

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maple [A]  time = 0.03, size = 21, normalized size = 1.11

method	result	size
risch	\(\frac {5}{3 \left (x \ln \relax (2)-x^{2}+\ln \relax (x )+3 x +5\right )^{2}}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-10*x*ln(2)+20*x^2-30*x-10)/(3*x*ln(x)^3+(9*x^2*ln(2)-9*x^3+27*x^2+45*x)*ln(x)^2+(9*x^3*ln(2)^2+(-18*x^4+
54*x^3+90*x^2)*ln(2)+9*x^5-54*x^4-9*x^3+270*x^2+225*x)*ln(x)+3*x^4*ln(2)^3+(-9*x^5+27*x^4+45*x^3)*ln(2)^2+(9*x
^6-54*x^5-9*x^4+270*x^3+225*x^2)*ln(2)-3*x^7+27*x^6-36*x^5-189*x^4+180*x^3+675*x^2+375*x),x,method=_RETURNVERB
OSE)

[Out]

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

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maxima [B]  time = 0.58, size = 59, normalized size = 3.11 \begin {gather*} \frac {5}{3 \, {\left (x^{4} - 2 \, x^{3} {\left (\log \relax (2) + 3\right )} + {\left (\log \relax (2)^{2} + 6 \, \log \relax (2) - 1\right )} x^{2} + 10 \, x {\left (\log \relax (2) + 3\right )} - 2 \, {\left (x^{2} - x {\left (\log \relax (2) + 3\right )} - 5\right )} \log \relax (x) + \log \relax (x)^{2} + 25\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-10*x*log(2)+20*x^2-30*x-10)/(3*x*log(x)^3+(9*x^2*log(2)-9*x^3+27*x^2+45*x)*log(x)^2+(9*x^3*log(2)^
2+(-18*x^4+54*x^3+90*x^2)*log(2)+9*x^5-54*x^4-9*x^3+270*x^2+225*x)*log(x)+3*x^4*log(2)^3+(-9*x^5+27*x^4+45*x^3
)*log(2)^2+(9*x^6-54*x^5-9*x^4+270*x^3+225*x^2)*log(2)-3*x^7+27*x^6-36*x^5-189*x^4+180*x^3+675*x^2+375*x),x, a
lgorithm="maxima")

[Out]

5/3/(x^4 - 2*x^3*(log(2) + 3) + (log(2)^2 + 6*log(2) - 1)*x^2 + 10*x*(log(2) + 3) - 2*(x^2 - x*(log(2) + 3) -
5)*log(x) + log(x)^2 + 25)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int -\frac {30\,x+10\,x\,\ln \relax (2)-20\,x^2+10}{375\,x+3\,x^4\,{\ln \relax (2)}^3+3\,x\,{\ln \relax (x)}^3+\ln \relax (x)\,\left (225\,x+9\,x^3\,{\ln \relax (2)}^2+\ln \relax (2)\,\left (-18\,x^4+54\,x^3+90\,x^2\right )+270\,x^2-9\,x^3-54\,x^4+9\,x^5\right )+{\ln \relax (2)}^2\,\left (-9\,x^5+27\,x^4+45\,x^3\right )+\ln \relax (2)\,\left (9\,x^6-54\,x^5-9\,x^4+270\,x^3+225\,x^2\right )+675\,x^2+180\,x^3-189\,x^4-36\,x^5+27\,x^6-3\,x^7+{\ln \relax (x)}^2\,\left (45\,x+9\,x^2\,\ln \relax (2)+27\,x^2-9\,x^3\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(30*x + 10*x*log(2) - 20*x^2 + 10)/(375*x + 3*x^4*log(2)^3 + 3*x*log(x)^3 + log(x)*(225*x + 9*x^3*log(2)^
2 + log(2)*(90*x^2 + 54*x^3 - 18*x^4) + 270*x^2 - 9*x^3 - 54*x^4 + 9*x^5) + log(2)^2*(45*x^3 + 27*x^4 - 9*x^5)
+ log(2)*(225*x^2 + 270*x^3 - 9*x^4 - 54*x^5 + 9*x^6) + 675*x^2 + 180*x^3 - 189*x^4 - 36*x^5 + 27*x^6 - 3*x^7
+ log(x)^2*(45*x + 9*x^2*log(2) + 27*x^2 - 9*x^3)),x)

[Out]

int(-(30*x + 10*x*log(2) - 20*x^2 + 10)/(375*x + 3*x^4*log(2)^3 + 3*x*log(x)^3 + log(x)*(225*x + 9*x^3*log(2)^
2 + log(2)*(90*x^2 + 54*x^3 - 18*x^4) + 270*x^2 - 9*x^3 - 54*x^4 + 9*x^5) + log(2)^2*(45*x^3 + 27*x^4 - 9*x^5)
+ log(2)*(225*x^2 + 270*x^3 - 9*x^4 - 54*x^5 + 9*x^6) + 675*x^2 + 180*x^3 - 189*x^4 - 36*x^5 + 27*x^6 - 3*x^7
+ log(x)^2*(45*x + 9*x^2*log(2) + 27*x^2 - 9*x^3)), x)

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sympy [B]  time = 0.27, size = 82, normalized size = 4.32 \begin {gather*} \frac {5}{3 x^{4} - 18 x^{3} - 6 x^{3} \log {\relax (2 )} - 3 x^{2} + 3 x^{2} \log {\relax (2 )}^{2} + 18 x^{2} \log {\relax (2 )} + 30 x \log {\relax (2 )} + 90 x + \left (- 6 x^{2} + 6 x \log {\relax (2 )} + 18 x + 30\right ) \log {\relax (x )} + 3 \log {\relax (x )}^{2} + 75} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-10*x*ln(2)+20*x**2-30*x-10)/(3*x*ln(x)**3+(9*x**2*ln(2)-9*x**3+27*x**2+45*x)*ln(x)**2+(9*x**3*ln(2
)**2+(-18*x**4+54*x**3+90*x**2)*ln(2)+9*x**5-54*x**4-9*x**3+270*x**2+225*x)*ln(x)+3*x**4*ln(2)**3+(-9*x**5+27*
x**4+45*x**3)*ln(2)**2+(9*x**6-54*x**5-9*x**4+270*x**3+225*x**2)*ln(2)-3*x**7+27*x**6-36*x**5-189*x**4+180*x**
3+675*x**2+375*x),x)

[Out]

5/(3*x**4 - 18*x**3 - 6*x**3*log(2) - 3*x**2 + 3*x**2*log(2)**2 + 18*x**2*log(2) + 30*x*log(2) + 90*x + (-6*x*
*2 + 6*x*log(2) + 18*x + 30)*log(x) + 3*log(x)**2 + 75)

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