Integrand size = 97, antiderivative size = 22 \[ \int \frac {-40 x^3+8 x^4+\left (-60 x^2-120 x^3+12 x^4+e \left (-20 x^6+4 x^7\right )\right ) \log \left (5+10 x-x^2\right )+e \left (-30 x^5-60 x^6+6 x^7\right ) \log ^2\left (5+10 x-x^2\right )}{e \left (-5-10 x+x^2\right )} \, dx=\left (\frac {2}{e}+x^3 \log (5+(10-x) x)\right )^2 \] Output:
(2/exp(1)+x^3*ln((10-x)*x+5))^2
Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {-40 x^3+8 x^4+\left (-60 x^2-120 x^3+12 x^4+e \left (-20 x^6+4 x^7\right )\right ) \log \left (5+10 x-x^2\right )+e \left (-30 x^5-60 x^6+6 x^7\right ) \log ^2\left (5+10 x-x^2\right )}{e \left (-5-10 x+x^2\right )} \, dx=\frac {\left (2+e x^3 \log \left (5+10 x-x^2\right )\right )^2}{e^2} \] Input:
Integrate[(-40*x^3 + 8*x^4 + (-60*x^2 - 120*x^3 + 12*x^4 + E*(-20*x^6 + 4* x^7))*Log[5 + 10*x - x^2] + E*(-30*x^5 - 60*x^6 + 6*x^7)*Log[5 + 10*x - x^ 2]^2)/(E*(-5 - 10*x + x^2)),x]
Output:
(2 + E*x^3*Log[5 + 10*x - x^2])^2/E^2
Time = 0.85 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.041, Rules used = {27, 27, 7239, 7237}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {8 x^4-40 x^3+e \left (6 x^7-60 x^6-30 x^5\right ) \log ^2\left (-x^2+10 x+5\right )+\left (12 x^4-120 x^3-60 x^2+e \left (4 x^7-20 x^6\right )\right ) \log \left (-x^2+10 x+5\right )}{e \left (x^2-10 x-5\right )} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {2 \left (-4 x^4+20 x^3+3 e \left (-x^7+10 x^6+5 x^5\right ) \log ^2\left (-x^2+10 x+5\right )+2 \left (-3 x^4+30 x^3+15 x^2+e \left (5 x^6-x^7\right )\right ) \log \left (-x^2+10 x+5\right )\right )}{-x^2+10 x+5}dx}{e}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2 \int \frac {-4 x^4+20 x^3+3 e \left (-x^7+10 x^6+5 x^5\right ) \log ^2\left (-x^2+10 x+5\right )+2 \left (-3 x^4+30 x^3+15 x^2+e \left (5 x^6-x^7\right )\right ) \log \left (-x^2+10 x+5\right )}{-x^2+10 x+5}dx}{e}\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \frac {2 \int \frac {x^2 \left (e \log \left (-x^2+10 x+5\right ) x^3+2\right ) \left (-2 (x-5) x-3 \left (x^2-10 x-5\right ) \log \left (-x^2+10 x+5\right )\right )}{-x^2+10 x+5}dx}{e}\) |
\(\Big \downarrow \) 7237 |
\(\displaystyle \frac {\left (e x^3 \log \left (-x^2+10 x+5\right )+2\right )^2}{e^2}\) |
Input:
Int[(-40*x^3 + 8*x^4 + (-60*x^2 - 120*x^3 + 12*x^4 + E*(-20*x^6 + 4*x^7))* Log[5 + 10*x - x^2] + E*(-30*x^5 - 60*x^6 + 6*x^7)*Log[5 + 10*x - x^2]^2)/ (E*(-5 - 10*x + x^2)),x]
Output:
(2 + E*x^3*Log[5 + 10*x - x^2])^2/E^2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si mp[q*(y^(m + 1)/(m + 1)), x] /; !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 8.40 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.68
method | result | size |
risch | \(x^{6} \ln \left (-x^{2}+10 x +5\right )^{2}+4 \,{\mathrm e}^{-1} x^{3} \ln \left (-x^{2}+10 x +5\right )\) | \(37\) |
norman | \(x^{6} \ln \left (-x^{2}+10 x +5\right )^{2}+4 \,{\mathrm e}^{-1} x^{3} \ln \left (-x^{2}+10 x +5\right )\) | \(39\) |
parallelrisch | \({\mathrm e}^{-1} \left ({\mathrm e} \ln \left (-x^{2}+10 x +5\right )^{2} x^{6}+4 \ln \left (-x^{2}+10 x +5\right ) x^{3}\right )\) | \(42\) |
Input:
int(((6*x^7-60*x^6-30*x^5)*exp(1)*ln(-x^2+10*x+5)^2+((4*x^7-20*x^6)*exp(1) +12*x^4-120*x^3-60*x^2)*ln(-x^2+10*x+5)+8*x^4-40*x^3)/(x^2-10*x-5)/exp(1), x,method=_RETURNVERBOSE)
Output:
x^6*ln(-x^2+10*x+5)^2+4*exp(-1)*x^3*ln(-x^2+10*x+5)
Time = 0.07 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.77 \[ \int \frac {-40 x^3+8 x^4+\left (-60 x^2-120 x^3+12 x^4+e \left (-20 x^6+4 x^7\right )\right ) \log \left (5+10 x-x^2\right )+e \left (-30 x^5-60 x^6+6 x^7\right ) \log ^2\left (5+10 x-x^2\right )}{e \left (-5-10 x+x^2\right )} \, dx={\left (x^{6} e \log \left (-x^{2} + 10 \, x + 5\right )^{2} + 4 \, x^{3} \log \left (-x^{2} + 10 \, x + 5\right )\right )} e^{\left (-1\right )} \] Input:
integrate(((6*x^7-60*x^6-30*x^5)*exp(1)*log(-x^2+10*x+5)^2+((4*x^7-20*x^6) *exp(1)+12*x^4-120*x^3-60*x^2)*log(-x^2+10*x+5)+8*x^4-40*x^3)/(x^2-10*x-5) /exp(1),x, algorithm="fricas")
Output:
(x^6*e*log(-x^2 + 10*x + 5)^2 + 4*x^3*log(-x^2 + 10*x + 5))*e^(-1)
Time = 0.13 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.45 \[ \int \frac {-40 x^3+8 x^4+\left (-60 x^2-120 x^3+12 x^4+e \left (-20 x^6+4 x^7\right )\right ) \log \left (5+10 x-x^2\right )+e \left (-30 x^5-60 x^6+6 x^7\right ) \log ^2\left (5+10 x-x^2\right )}{e \left (-5-10 x+x^2\right )} \, dx=x^{6} \log {\left (- x^{2} + 10 x + 5 \right )}^{2} + \frac {4 x^{3} \log {\left (- x^{2} + 10 x + 5 \right )}}{e} \] Input:
integrate(((6*x**7-60*x**6-30*x**5)*exp(1)*ln(-x**2+10*x+5)**2+((4*x**7-20 *x**6)*exp(1)+12*x**4-120*x**3-60*x**2)*ln(-x**2+10*x+5)+8*x**4-40*x**3)/( x**2-10*x-5)/exp(1),x)
Output:
x**6*log(-x**2 + 10*x + 5)**2 + 4*x**3*exp(-1)*log(-x**2 + 10*x + 5)
Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (20) = 40\).
Time = 0.17 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.36 \[ \int \frac {-40 x^3+8 x^4+\left (-60 x^2-120 x^3+12 x^4+e \left (-20 x^6+4 x^7\right )\right ) \log \left (5+10 x-x^2\right )+e \left (-30 x^5-60 x^6+6 x^7\right ) \log ^2\left (5+10 x-x^2\right )}{e \left (-5-10 x+x^2\right )} \, dx={\left (x^{6} e \log \left (-x^{2} + 10 \, x + 5\right )^{2} + 4 \, {\left (x^{3} - 575\right )} \log \left (-x^{2} + 10 \, x + 5\right ) + 2300 \, \log \left (x^{2} - 10 \, x - 5\right )\right )} e^{\left (-1\right )} \] Input:
integrate(((6*x^7-60*x^6-30*x^5)*exp(1)*log(-x^2+10*x+5)^2+((4*x^7-20*x^6) *exp(1)+12*x^4-120*x^3-60*x^2)*log(-x^2+10*x+5)+8*x^4-40*x^3)/(x^2-10*x-5) /exp(1),x, algorithm="maxima")
Output:
(x^6*e*log(-x^2 + 10*x + 5)^2 + 4*(x^3 - 575)*log(-x^2 + 10*x + 5) + 2300* log(x^2 - 10*x - 5))*e^(-1)
Time = 0.17 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.77 \[ \int \frac {-40 x^3+8 x^4+\left (-60 x^2-120 x^3+12 x^4+e \left (-20 x^6+4 x^7\right )\right ) \log \left (5+10 x-x^2\right )+e \left (-30 x^5-60 x^6+6 x^7\right ) \log ^2\left (5+10 x-x^2\right )}{e \left (-5-10 x+x^2\right )} \, dx={\left (x^{6} e \log \left (-x^{2} + 10 \, x + 5\right )^{2} + 4 \, x^{3} \log \left (-x^{2} + 10 \, x + 5\right )\right )} e^{\left (-1\right )} \] Input:
integrate(((6*x^7-60*x^6-30*x^5)*exp(1)*log(-x^2+10*x+5)^2+((4*x^7-20*x^6) *exp(1)+12*x^4-120*x^3-60*x^2)*log(-x^2+10*x+5)+8*x^4-40*x^3)/(x^2-10*x-5) /exp(1),x, algorithm="giac")
Output:
(x^6*e*log(-x^2 + 10*x + 5)^2 + 4*x^3*log(-x^2 + 10*x + 5))*e^(-1)
Time = 0.57 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.64 \[ \int \frac {-40 x^3+8 x^4+\left (-60 x^2-120 x^3+12 x^4+e \left (-20 x^6+4 x^7\right )\right ) \log \left (5+10 x-x^2\right )+e \left (-30 x^5-60 x^6+6 x^7\right ) \log ^2\left (5+10 x-x^2\right )}{e \left (-5-10 x+x^2\right )} \, dx=x^6\,{\ln \left (-x^2+10\,x+5\right )}^2+4\,{\mathrm {e}}^{-1}\,x^3\,\ln \left (-x^2+10\,x+5\right ) \] Input:
int((exp(-1)*(log(10*x - x^2 + 5)*(exp(1)*(20*x^6 - 4*x^7) + 60*x^2 + 120* x^3 - 12*x^4) + 40*x^3 - 8*x^4 + exp(1)*log(10*x - x^2 + 5)^2*(30*x^5 + 60 *x^6 - 6*x^7)))/(10*x - x^2 + 5),x)
Output:
x^6*log(10*x - x^2 + 5)^2 + 4*x^3*exp(-1)*log(10*x - x^2 + 5)
Time = 0.30 (sec) , antiderivative size = 107, normalized size of antiderivative = 4.86 \[ \int \frac {-40 x^3+8 x^4+\left (-60 x^2-120 x^3+12 x^4+e \left (-20 x^6+4 x^7\right )\right ) \log \left (5+10 x-x^2\right )+e \left (-30 x^5-60 x^6+6 x^7\right ) \log ^2\left (5+10 x-x^2\right )}{e \left (-5-10 x+x^2\right )} \, dx=\frac {420 \sqrt {30}\, \mathrm {log}\left (-x^{2}+10 x +5\right )-420 \sqrt {30}\, \mathrm {log}\left (-\sqrt {30}+x -5\right )-420 \sqrt {30}\, \mathrm {log}\left (\sqrt {30}+x -5\right )+\mathrm {log}\left (-x^{2}+10 x +5\right )^{2} e \,x^{6}+4 \,\mathrm {log}\left (-x^{2}+10 x +5\right ) x^{3}-2300 \,\mathrm {log}\left (-x^{2}+10 x +5\right )+2300 \,\mathrm {log}\left (-\sqrt {30}+x -5\right )+2300 \,\mathrm {log}\left (\sqrt {30}+x -5\right )}{e} \] Input:
int(((6*x^7-60*x^6-30*x^5)*exp(1)*log(-x^2+10*x+5)^2+((4*x^7-20*x^6)*exp(1 )+12*x^4-120*x^3-60*x^2)*log(-x^2+10*x+5)+8*x^4-40*x^3)/(x^2-10*x-5)/exp(1 ),x)
Output:
(420*sqrt(30)*log( - x**2 + 10*x + 5) - 420*sqrt(30)*log( - sqrt(30) + x - 5) - 420*sqrt(30)*log(sqrt(30) + x - 5) + log( - x**2 + 10*x + 5)**2*e*x* *6 + 4*log( - x**2 + 10*x + 5)*x**3 - 2300*log( - x**2 + 10*x + 5) + 2300* log( - sqrt(30) + x - 5) + 2300*log(sqrt(30) + x - 5))/e