Integrand size = 69, antiderivative size = 29 \[ \int \frac {-320 x+112 x^2-112 x^4+80 x^5-84 x^6+e^{x/4} \left (-480+264 x-188 x^2+40 x^3+12 x^4-46 x^5-7 x^6\right )}{8 x^4} \, dx=\frac {\left (5-\frac {7 x}{2}\right ) \left (e^{x/4}+x\right ) \left (\frac {2}{x}+x\right )^2}{x} \] Output:
(5-7/2*x)*(x+2/x)^2*(exp(1/4*x)+x)/x
Time = 0.70 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.72 \[ \int \frac {-320 x+112 x^2-112 x^4+80 x^5-84 x^6+e^{x/4} \left (-480+264 x-188 x^2+40 x^3+12 x^4-46 x^5-7 x^6\right )}{8 x^4} \, dx=-\frac {e^{x/4} (-10+7 x) \left (2+x^2\right )^2+x \left (-40+28 x+28 x^3-10 x^4+7 x^5\right )}{2 x^3} \] Input:
Integrate[(-320*x + 112*x^2 - 112*x^4 + 80*x^5 - 84*x^6 + E^(x/4)*(-480 + 264*x - 188*x^2 + 40*x^3 + 12*x^4 - 46*x^5 - 7*x^6))/(8*x^4),x]
Output:
-1/2*(E^(x/4)*(-10 + 7*x)*(2 + x^2)^2 + x*(-40 + 28*x + 28*x^3 - 10*x^4 + 7*x^5))/x^3
Leaf count is larger than twice the leaf count of optimal. \(95\) vs. \(2(29)=58\).
Time = 0.49 (sec) , antiderivative size = 95, normalized size of antiderivative = 3.28, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.058, Rules used = {27, 25, 2010, 2009}
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.
\(\displaystyle \int \frac {-84 x^6+80 x^5-112 x^4+112 x^2+e^{x/4} \left (-7 x^6-46 x^5+12 x^4+40 x^3-188 x^2+264 x-480\right )-320 x}{8 x^4} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{8} \int -\frac {84 x^6-80 x^5+112 x^4-112 x^2+320 x+e^{x/4} \left (7 x^6+46 x^5-12 x^4-40 x^3+188 x^2-264 x+480\right )}{x^4}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{8} \int \frac {84 x^6-80 x^5+112 x^4-112 x^2+320 x+e^{x/4} \left (7 x^6+46 x^5-12 x^4-40 x^3+188 x^2-264 x+480\right )}{x^4}dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle -\frac {1}{8} \int \left (\frac {e^{x/4} \left (x^2+2\right ) \left (7 x^4+46 x^3-26 x^2-132 x+240\right )}{x^4}+\frac {4 \left (21 x^5-20 x^4+28 x^3-28 x+80\right )}{x^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{8} \left (-28 x^3+\frac {160 e^{x/4}}{x^3}-28 e^{x/4} x^2+40 x^2-\frac {112 e^{x/4}}{x^2}+\frac {160}{x^2}+40 e^{x/4} x-112 x-112 e^{x/4}+\frac {160 e^{x/4}}{x}-\frac {112}{x}\right )\) |
Input:
Int[(-320*x + 112*x^2 - 112*x^4 + 80*x^5 - 84*x^6 + E^(x/4)*(-480 + 264*x - 188*x^2 + 40*x^3 + 12*x^4 - 46*x^5 - 7*x^6))/(8*x^4),x]
Output:
(-112*E^(x/4) + (160*E^(x/4))/x^3 + 160/x^2 - (112*E^(x/4))/x^2 - 112/x + (160*E^(x/4))/x - 112*x + 40*E^(x/4)*x + 40*x^2 - 28*E^(x/4)*x^2 - 28*x^3) /8
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Leaf count of result is larger than twice the leaf count of optimal. \(58\) vs. \(2(24)=48\).
Time = 0.64 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.03
method | result | size |
risch | \(-\frac {7 x^{3}}{2}+5 x^{2}-14 x +\frac {-112 x +160}{8 x^{2}}-\frac {\left (7 x^{5}-10 x^{4}+28 x^{3}-40 x^{2}+28 x -40\right ) {\mathrm e}^{\frac {x}{4}}}{2 x^{3}}\) | \(59\) |
derivativedivides | \(5 x^{2}-14 x +\frac {20}{x^{2}}-\frac {14}{x}-\frac {7 x^{3}}{2}+\frac {20 \,{\mathrm e}^{\frac {x}{4}}}{x^{3}}-\frac {14 \,{\mathrm e}^{\frac {x}{4}}}{x^{2}}+\frac {20 \,{\mathrm e}^{\frac {x}{4}}}{x}+5 \,{\mathrm e}^{\frac {x}{4}} x -14 \,{\mathrm e}^{\frac {x}{4}}-\frac {7 \,{\mathrm e}^{\frac {x}{4}} x^{2}}{2}\) | \(74\) |
default | \(5 x^{2}-14 x +\frac {20}{x^{2}}-\frac {14}{x}-\frac {7 x^{3}}{2}+\frac {20 \,{\mathrm e}^{\frac {x}{4}}}{x^{3}}-\frac {14 \,{\mathrm e}^{\frac {x}{4}}}{x^{2}}+\frac {20 \,{\mathrm e}^{\frac {x}{4}}}{x}+5 \,{\mathrm e}^{\frac {x}{4}} x -14 \,{\mathrm e}^{\frac {x}{4}}-\frac {7 \,{\mathrm e}^{\frac {x}{4}} x^{2}}{2}\) | \(74\) |
parts | \(5 x^{2}-14 x +\frac {20}{x^{2}}-\frac {14}{x}-\frac {7 x^{3}}{2}+\frac {20 \,{\mathrm e}^{\frac {x}{4}}}{x^{3}}-\frac {14 \,{\mathrm e}^{\frac {x}{4}}}{x^{2}}+\frac {20 \,{\mathrm e}^{\frac {x}{4}}}{x}+5 \,{\mathrm e}^{\frac {x}{4}} x -14 \,{\mathrm e}^{\frac {x}{4}}-\frac {7 \,{\mathrm e}^{\frac {x}{4}} x^{2}}{2}\) | \(74\) |
norman | \(\frac {20 x -14 x^{2}-14 x^{4}+5 x^{5}-\frac {7 x^{6}}{2}-14 \,{\mathrm e}^{\frac {x}{4}} x +20 \,{\mathrm e}^{\frac {x}{4}} x^{2}-14 \,{\mathrm e}^{\frac {x}{4}} x^{3}+5 \,{\mathrm e}^{\frac {x}{4}} x^{4}-\frac {7 \,{\mathrm e}^{\frac {x}{4}} x^{5}}{2}+20 \,{\mathrm e}^{\frac {x}{4}}}{x^{3}}\) | \(78\) |
parallelrisch | \(-\frac {28 x^{6}+28 \,{\mathrm e}^{\frac {x}{4}} x^{5}-40 x^{5}-40 \,{\mathrm e}^{\frac {x}{4}} x^{4}+112 x^{4}+112 \,{\mathrm e}^{\frac {x}{4}} x^{3}-160 \,{\mathrm e}^{\frac {x}{4}} x^{2}+112 x^{2}+112 \,{\mathrm e}^{\frac {x}{4}} x -160 x -160 \,{\mathrm e}^{\frac {x}{4}}}{8 x^{3}}\) | \(79\) |
orering | \(\frac {\left (343 x^{12}+4508 x^{11}-30268 x^{10}+83664 x^{9}+118696 x^{8}+517472 x^{7}+25216 x^{6}+210880 x^{5}-2978448 x^{4}+6846400 x^{3}-16445248 x^{2}-1523200 x +1075200\right ) \left (\left (-7 x^{6}-46 x^{5}+12 x^{4}+40 x^{3}-188 x^{2}+264 x -480\right ) {\mathrm e}^{\frac {x}{4}}-84 x^{6}+80 x^{5}-112 x^{4}+112 x^{2}-320 x \right )}{56 x^{3} \left (147 x^{8}+826 x^{7}-5988 x^{6}+968 x^{5}+41852 x^{4}-91032 x^{3}+37408 x^{2}+36480 x -38400\right ) \left (x^{2}+2\right )^{2}}-\frac {4 \left (343 x^{9}-2352 x^{8}+6678 x^{7}+142972 x^{5}-107840 x^{4}-260728 x^{3}+527424 x^{2}+36960 x -22400\right ) x^{2} \left (\frac {\left (-42 x^{5}-230 x^{4}+48 x^{3}+120 x^{2}-376 x +264\right ) {\mathrm e}^{\frac {x}{4}}+\frac {\left (-7 x^{6}-46 x^{5}+12 x^{4}+40 x^{3}-188 x^{2}+264 x -480\right ) {\mathrm e}^{\frac {x}{4}}}{4}-504 x^{5}+400 x^{4}-448 x^{3}+224 x -320}{8 x^{4}}-\frac {\left (-7 x^{6}-46 x^{5}+12 x^{4}+40 x^{3}-188 x^{2}+264 x -480\right ) {\mathrm e}^{\frac {x}{4}}-84 x^{6}+80 x^{5}-112 x^{4}+112 x^{2}-320 x}{2 x^{5}}\right )}{7 \left (147 x^{8}+826 x^{7}-5988 x^{6}+968 x^{5}+41852 x^{4}-91032 x^{3}+37408 x^{2}+36480 x -38400\right ) \left (x^{2}+2\right )}\) | \(425\) |
Input:
int(1/8*((-7*x^6-46*x^5+12*x^4+40*x^3-188*x^2+264*x-480)*exp(1/4*x)-84*x^6 +80*x^5-112*x^4+112*x^2-320*x)/x^4,x,method=_RETURNVERBOSE)
Output:
-7/2*x^3+5*x^2-14*x+1/8*(-112*x+160)/x^2-1/2*(7*x^5-10*x^4+28*x^3-40*x^2+2 8*x-40)/x^3*exp(1/4*x)
Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (25) = 50\).
Time = 0.10 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.03 \[ \int \frac {-320 x+112 x^2-112 x^4+80 x^5-84 x^6+e^{x/4} \left (-480+264 x-188 x^2+40 x^3+12 x^4-46 x^5-7 x^6\right )}{8 x^4} \, dx=-\frac {7 \, x^{6} - 10 \, x^{5} + 28 \, x^{4} + 28 \, x^{2} + {\left (7 \, x^{5} - 10 \, x^{4} + 28 \, x^{3} - 40 \, x^{2} + 28 \, x - 40\right )} e^{\left (\frac {1}{4} \, x\right )} - 40 \, x}{2 \, x^{3}} \] Input:
integrate(1/8*((-7*x^6-46*x^5+12*x^4+40*x^3-188*x^2+264*x-480)*exp(1/4*x)- 84*x^6+80*x^5-112*x^4+112*x^2-320*x)/x^4,x, algorithm="fricas")
Output:
-1/2*(7*x^6 - 10*x^5 + 28*x^4 + 28*x^2 + (7*x^5 - 10*x^4 + 28*x^3 - 40*x^2 + 28*x - 40)*e^(1/4*x) - 40*x)/x^3
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (20) = 40\).
Time = 0.10 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.07 \[ \int \frac {-320 x+112 x^2-112 x^4+80 x^5-84 x^6+e^{x/4} \left (-480+264 x-188 x^2+40 x^3+12 x^4-46 x^5-7 x^6\right )}{8 x^4} \, dx=- \frac {7 x^{3}}{2} + 5 x^{2} - 14 x - \frac {28 x - 40}{2 x^{2}} + \frac {\left (- 7 x^{5} + 10 x^{4} - 28 x^{3} + 40 x^{2} - 28 x + 40\right ) e^{\frac {x}{4}}}{2 x^{3}} \] Input:
integrate(1/8*((-7*x**6-46*x**5+12*x**4+40*x**3-188*x**2+264*x-480)*exp(1/ 4*x)-84*x**6+80*x**5-112*x**4+112*x**2-320*x)/x**4,x)
Output:
-7*x**3/2 + 5*x**2 - 14*x - (28*x - 40)/(2*x**2) + (-7*x**5 + 10*x**4 - 28 *x**3 + 40*x**2 - 28*x + 40)*exp(x/4)/(2*x**3)
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.07 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.76 \[ \int \frac {-320 x+112 x^2-112 x^4+80 x^5-84 x^6+e^{x/4} \left (-480+264 x-188 x^2+40 x^3+12 x^4-46 x^5-7 x^6\right )}{8 x^4} \, dx=-\frac {7}{2} \, x^{3} + 5 \, x^{2} - \frac {7}{2} \, {\left (x^{2} - 8 \, x + 32\right )} e^{\left (\frac {1}{4} \, x\right )} - 23 \, {\left (x - 4\right )} e^{\left (\frac {1}{4} \, x\right )} - 14 \, x - \frac {14}{x} + \frac {20}{x^{2}} + 5 \, {\rm Ei}\left (\frac {1}{4} \, x\right ) + 6 \, e^{\left (\frac {1}{4} \, x\right )} - \frac {47}{8} \, \Gamma \left (-1, -\frac {1}{4} \, x\right ) - \frac {33}{16} \, \Gamma \left (-2, -\frac {1}{4} \, x\right ) - \frac {15}{16} \, \Gamma \left (-3, -\frac {1}{4} \, x\right ) \] Input:
integrate(1/8*((-7*x^6-46*x^5+12*x^4+40*x^3-188*x^2+264*x-480)*exp(1/4*x)- 84*x^6+80*x^5-112*x^4+112*x^2-320*x)/x^4,x, algorithm="maxima")
Output:
-7/2*x^3 + 5*x^2 - 7/2*(x^2 - 8*x + 32)*e^(1/4*x) - 23*(x - 4)*e^(1/4*x) - 14*x - 14/x + 20/x^2 + 5*Ei(1/4*x) + 6*e^(1/4*x) - 47/8*gamma(-1, -1/4*x) - 33/16*gamma(-2, -1/4*x) - 15/16*gamma(-3, -1/4*x)
Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (25) = 50\).
Time = 0.12 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.69 \[ \int \frac {-320 x+112 x^2-112 x^4+80 x^5-84 x^6+e^{x/4} \left (-480+264 x-188 x^2+40 x^3+12 x^4-46 x^5-7 x^6\right )}{8 x^4} \, dx=-\frac {7 \, x^{6} + 7 \, x^{5} e^{\left (\frac {1}{4} \, x\right )} - 10 \, x^{5} - 10 \, x^{4} e^{\left (\frac {1}{4} \, x\right )} + 28 \, x^{4} + 28 \, x^{3} e^{\left (\frac {1}{4} \, x\right )} - 40 \, x^{2} e^{\left (\frac {1}{4} \, x\right )} + 28 \, x^{2} + 28 \, x e^{\left (\frac {1}{4} \, x\right )} - 40 \, x - 40 \, e^{\left (\frac {1}{4} \, x\right )}}{2 \, x^{3}} \] Input:
integrate(1/8*((-7*x^6-46*x^5+12*x^4+40*x^3-188*x^2+264*x-480)*exp(1/4*x)- 84*x^6+80*x^5-112*x^4+112*x^2-320*x)/x^4,x, algorithm="giac")
Output:
-1/2*(7*x^6 + 7*x^5*e^(1/4*x) - 10*x^5 - 10*x^4*e^(1/4*x) + 28*x^4 + 28*x^ 3*e^(1/4*x) - 40*x^2*e^(1/4*x) + 28*x^2 + 28*x*e^(1/4*x) - 40*x - 40*e^(1/ 4*x))/x^3
Time = 0.09 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.38 \[ \int \frac {-320 x+112 x^2-112 x^4+80 x^5-84 x^6+e^{x/4} \left (-480+264 x-188 x^2+40 x^3+12 x^4-46 x^5-7 x^6\right )}{8 x^4} \, dx=\frac {20\,{\mathrm {e}}^{x/4}+x^2\,\left (20\,{\mathrm {e}}^{x/4}-14\right )-x\,\left (14\,{\mathrm {e}}^{x/4}-20\right )}{x^3}-x^2\,\left (\frac {7\,{\mathrm {e}}^{x/4}}{2}-5\right )-14\,{\mathrm {e}}^{x/4}+x\,\left (5\,{\mathrm {e}}^{x/4}-14\right )-\frac {7\,x^3}{2} \] Input:
int(-(40*x - 14*x^2 + 14*x^4 - 10*x^5 + (21*x^6)/2 + (exp(x/4)*(188*x^2 - 264*x - 40*x^3 - 12*x^4 + 46*x^5 + 7*x^6 + 480))/8)/x^4,x)
Output:
(20*exp(x/4) + x^2*(20*exp(x/4) - 14) - x*(14*exp(x/4) - 20))/x^3 - x^2*(( 7*exp(x/4))/2 - 5) - 14*exp(x/4) + x*(5*exp(x/4) - 14) - (7*x^3)/2
Time = 0.23 (sec) , antiderivative size = 84, normalized size of antiderivative = 2.90 \[ \int \frac {-320 x+112 x^2-112 x^4+80 x^5-84 x^6+e^{x/4} \left (-480+264 x-188 x^2+40 x^3+12 x^4-46 x^5-7 x^6\right )}{8 x^4} \, dx=\frac {-7 e^{\frac {x}{4}} x^{5}+10 e^{\frac {x}{4}} x^{4}-28 e^{\frac {x}{4}} x^{3}+40 e^{\frac {x}{4}} x^{2}-28 e^{\frac {x}{4}} x +40 e^{\frac {x}{4}}-7 x^{6}+10 x^{5}-28 x^{4}-28 x^{2}+40 x}{2 x^{3}} \] Input:
int(1/8*((-7*x^6-46*x^5+12*x^4+40*x^3-188*x^2+264*x-480)*exp(1/4*x)-84*x^6 +80*x^5-112*x^4+112*x^2-320*x)/x^4,x)
Output:
( - 7*e**(x/4)*x**5 + 10*e**(x/4)*x**4 - 28*e**(x/4)*x**3 + 40*e**(x/4)*x* *2 - 28*e**(x/4)*x + 40*e**(x/4) - 7*x**6 + 10*x**5 - 28*x**4 - 28*x**2 + 40*x)/(2*x**3)