Integrand size = 78, antiderivative size = 27 \[ \int \frac {-10+150 x+56 x^2-114 x^3-24 x^4+24 x^5+e^{x^2} \left (150 x+60 x^2-114 x^3-24 x^4+24 x^5\right )}{75+30 x-57 x^2-12 x^3+12 x^4} \, dx=-1+e^{x^2}+x^2+\frac {2 x}{3 \left (-5-x+2 x^2\right )} \] Output:
2*x/(6*x^2-3*x-15)+x^2-1+exp(x^2)
Time = 0.83 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.30 \[ \int \frac {-10+150 x+56 x^2-114 x^3-24 x^4+24 x^5+e^{x^2} \left (150 x+60 x^2-114 x^3-24 x^4+24 x^5\right )}{75+30 x-57 x^2-12 x^3+12 x^4} \, dx=\frac {2}{3} \left (\frac {3 e^{x^2}}{2}+\frac {3 x^2}{2}+\frac {x}{-5-x+2 x^2}\right ) \] Input:
Integrate[(-10 + 150*x + 56*x^2 - 114*x^3 - 24*x^4 + 24*x^5 + E^x^2*(150*x + 60*x^2 - 114*x^3 - 24*x^4 + 24*x^5))/(75 + 30*x - 57*x^2 - 12*x^3 + 12* x^4),x]
Output:
(2*((3*E^x^2)/2 + (3*x^2)/2 + x/(-5 - x + 2*x^2)))/3
Leaf count is larger than twice the leaf count of optimal. \(339\) vs. \(2(27)=54\).
Time = 1.34 (sec) , antiderivative size = 339, normalized size of antiderivative = 12.56, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.064, Rules used = {2463, 27, 27, 7293, 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.
\(\displaystyle \int \frac {24 x^5-24 x^4-114 x^3+56 x^2+e^{x^2} \left (24 x^5-24 x^4-114 x^3+60 x^2+150 x\right )+150 x-10}{12 x^4-12 x^3-57 x^2+30 x+75} \, dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \frac {24 x^5-24 x^4-114 x^3+56 x^2+e^{x^2} \left (24 x^5-24 x^4-114 x^3+60 x^2+150 x\right )+150 x-10}{3 \left (2 x^2-x-5\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \int -\frac {2 \left (-12 x^5+12 x^4+57 x^3-28 x^2-75 x-3 e^{x^2} \left (4 x^5-4 x^4-19 x^3+10 x^2+25 x\right )+5\right )}{\left (-2 x^2+x+5\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {2}{3} \int \frac {-12 x^5+12 x^4+57 x^3-28 x^2-75 x-3 e^{x^2} \left (4 x^5-4 x^4-19 x^3+10 x^2+25 x\right )+5}{\left (-2 x^2+x+5\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {2}{3} \int \left (-\frac {12 x^5}{\left (2 x^2-x-5\right )^2}+\frac {12 x^4}{\left (2 x^2-x-5\right )^2}+\frac {57 x^3}{\left (2 x^2-x-5\right )^2}-\frac {28 x^2}{\left (2 x^2-x-5\right )^2}-3 e^{x^2} x-\frac {75 x}{\left (2 x^2-x-5\right )^2}+\frac {5}{\left (2 x^2-x-5\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2}{3} \left (-\frac {6 x^3}{41}+\frac {57 (x+10) x^2}{41 \left (-2 x^2+x+5\right )}-\frac {237 x^2}{82}-\frac {28 (x+10) x}{41 \left (-2 x^2+x+5\right )}-\frac {3 e^{x^2}}{2}-\frac {5 (1-4 x)}{41 \left (-2 x^2+x+5\right )}-\frac {75 (x+10)}{41 \left (-2 x^2+x+5\right )}-\frac {12 (x+10) x^4}{41 \left (-2 x^2+x+5\right )}+\frac {12 (x+10) x^3}{41 \left (-2 x^2+x+5\right )}+\frac {45 x}{41}+\frac {57 \left (1681-61 \sqrt {41}\right ) \log \left (-4 x-\sqrt {41}+1\right )}{13448}+\frac {3 \left (1681-661 \sqrt {41}\right ) \log \left (-4 x-\sqrt {41}+1\right )}{3362}-\frac {3 \left (38663-3203 \sqrt {41}\right ) \log \left (-4 x-\sqrt {41}+1\right )}{13448}+\frac {225 \log \left (-4 x-\sqrt {41}+1\right )}{41 \sqrt {41}}-\frac {3 \left (38663+3203 \sqrt {41}\right ) \log \left (-4 x+\sqrt {41}+1\right )}{13448}+\frac {3 \left (1681+661 \sqrt {41}\right ) \log \left (-4 x+\sqrt {41}+1\right )}{3362}+\frac {57 \left (1681+61 \sqrt {41}\right ) \log \left (-4 x+\sqrt {41}+1\right )}{13448}-\frac {225 \log \left (-4 x+\sqrt {41}+1\right )}{41 \sqrt {41}}\right )\) |
Input:
Int[(-10 + 150*x + 56*x^2 - 114*x^3 - 24*x^4 + 24*x^5 + E^x^2*(150*x + 60* x^2 - 114*x^3 - 24*x^4 + 24*x^5))/(75 + 30*x - 57*x^2 - 12*x^3 + 12*x^4),x ]
Output:
(-2*((-3*E^x^2)/2 + (45*x)/41 - (237*x^2)/82 - (6*x^3)/41 - (5*(1 - 4*x))/ (41*(5 + x - 2*x^2)) - (75*(10 + x))/(41*(5 + x - 2*x^2)) - (28*x*(10 + x) )/(41*(5 + x - 2*x^2)) + (57*x^2*(10 + x))/(41*(5 + x - 2*x^2)) + (12*x^3* (10 + x))/(41*(5 + x - 2*x^2)) - (12*x^4*(10 + x))/(41*(5 + x - 2*x^2)) + (225*Log[1 - Sqrt[41] - 4*x])/(41*Sqrt[41]) - (3*(38663 - 3203*Sqrt[41])*L og[1 - Sqrt[41] - 4*x])/13448 + (3*(1681 - 661*Sqrt[41])*Log[1 - Sqrt[41] - 4*x])/3362 + (57*(1681 - 61*Sqrt[41])*Log[1 - Sqrt[41] - 4*x])/13448 - ( 225*Log[1 + Sqrt[41] - 4*x])/(41*Sqrt[41]) + (57*(1681 + 61*Sqrt[41])*Log[ 1 + Sqrt[41] - 4*x])/13448 + (3*(1681 + 661*Sqrt[41])*Log[1 + Sqrt[41] - 4 *x])/3362 - (3*(38663 + 3203*Sqrt[41])*Log[1 + Sqrt[41] - 4*x])/13448))/3
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Time = 0.90 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81
method | result | size |
risch | \(x^{2}+\frac {x}{3 x^{2}-\frac {3}{2} x -\frac {15}{2}}+{\mathrm e}^{x^{2}}\) | \(22\) |
parts | \(x^{2}+\frac {x}{3 x^{2}-\frac {3}{2} x -\frac {15}{2}}+{\mathrm e}^{x^{2}}\) | \(22\) |
norman | \(\frac {-\frac {11 x}{6}-x^{3}+2 x^{4}+2 x^{2} {\mathrm e}^{x^{2}}-{\mathrm e}^{x^{2}} x -5 \,{\mathrm e}^{x^{2}}-\frac {25}{2}}{2 x^{2}-x -5}\) | \(51\) |
parallelrisch | \(\frac {12 x^{4}-6 x^{3}+12 x^{2} {\mathrm e}^{x^{2}}-75-6 \,{\mathrm e}^{x^{2}} x -11 x -30 \,{\mathrm e}^{x^{2}}}{12 x^{2}-6 x -30}\) | \(52\) |
orering | \(\frac {\left (192 x^{10}-288 x^{9}-96 x^{8}-320 x^{7}-4468 x^{6}+6094 x^{5}+14300 x^{4}-7671 x^{3}-10975 x^{2}-4315 x -7085\right ) \left (\left (24 x^{5}-24 x^{4}-114 x^{3}+60 x^{2}+150 x \right ) {\mathrm e}^{x^{2}}+24 x^{5}-24 x^{4}-114 x^{3}+56 x^{2}+150 x -10\right )}{8 \left (48 x^{9}-72 x^{8}-324 x^{7}+346 x^{6}+814 x^{5}-462 x^{4}-738 x^{3}-10 x^{2}+15 x +25\right ) \left (12 x^{4}-12 x^{3}-57 x^{2}+30 x +75\right )}-\frac {\left (2 x^{2}-x -5\right ) \left (48 x^{7}-48 x^{6}-92 x^{4}-791 x^{3}+538 x^{2}+1425 x +20\right ) \left (\frac {\left (120 x^{4}-96 x^{3}-342 x^{2}+120 x +150\right ) {\mathrm e}^{x^{2}}+2 \left (24 x^{5}-24 x^{4}-114 x^{3}+60 x^{2}+150 x \right ) {\mathrm e}^{x^{2}} x +120 x^{4}-96 x^{3}-342 x^{2}+112 x +150}{12 x^{4}-12 x^{3}-57 x^{2}+30 x +75}-\frac {\left (\left (24 x^{5}-24 x^{4}-114 x^{3}+60 x^{2}+150 x \right ) {\mathrm e}^{x^{2}}+24 x^{5}-24 x^{4}-114 x^{3}+56 x^{2}+150 x -10\right ) \left (48 x^{3}-36 x^{2}-114 x +30\right )}{\left (12 x^{4}-12 x^{3}-57 x^{2}+30 x +75\right )^{2}}\right )}{8 \left (48 x^{9}-72 x^{8}-324 x^{7}+346 x^{6}+814 x^{5}-462 x^{4}-738 x^{3}-10 x^{2}+15 x +25\right )}\) | \(459\) |
Input:
int(((24*x^5-24*x^4-114*x^3+60*x^2+150*x)*exp(x^2)+24*x^5-24*x^4-114*x^3+5 6*x^2+150*x-10)/(12*x^4-12*x^3-57*x^2+30*x+75),x,method=_RETURNVERBOSE)
Output:
x^2+1/3*x/(x^2-1/2*x-5/2)+exp(x^2)
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (24) = 48\).
Time = 0.10 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.81 \[ \int \frac {-10+150 x+56 x^2-114 x^3-24 x^4+24 x^5+e^{x^2} \left (150 x+60 x^2-114 x^3-24 x^4+24 x^5\right )}{75+30 x-57 x^2-12 x^3+12 x^4} \, dx=\frac {6 \, x^{4} - 3 \, x^{3} - 15 \, x^{2} + 3 \, {\left (2 \, x^{2} - x - 5\right )} e^{\left (x^{2}\right )} + 2 \, x}{3 \, {\left (2 \, x^{2} - x - 5\right )}} \] Input:
integrate(((24*x^5-24*x^4-114*x^3+60*x^2+150*x)*exp(x^2)+24*x^5-24*x^4-114 *x^3+56*x^2+150*x-10)/(12*x^4-12*x^3-57*x^2+30*x+75),x, algorithm="fricas" )
Output:
1/3*(6*x^4 - 3*x^3 - 15*x^2 + 3*(2*x^2 - x - 5)*e^(x^2) + 2*x)/(2*x^2 - x - 5)
Time = 0.09 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {-10+150 x+56 x^2-114 x^3-24 x^4+24 x^5+e^{x^2} \left (150 x+60 x^2-114 x^3-24 x^4+24 x^5\right )}{75+30 x-57 x^2-12 x^3+12 x^4} \, dx=x^{2} + \frac {2 x}{6 x^{2} - 3 x - 15} + e^{x^{2}} \] Input:
integrate(((24*x**5-24*x**4-114*x**3+60*x**2+150*x)*exp(x**2)+24*x**5-24*x **4-114*x**3+56*x**2+150*x-10)/(12*x**4-12*x**3-57*x**2+30*x+75),x)
Output:
x**2 + 2*x/(6*x**2 - 3*x - 15) + exp(x**2)
Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (24) = 48\).
Time = 0.15 (sec) , antiderivative size = 120, normalized size of antiderivative = 4.44 \[ \int \frac {-10+150 x+56 x^2-114 x^3-24 x^4+24 x^5+e^{x^2} \left (150 x+60 x^2-114 x^3-24 x^4+24 x^5\right )}{75+30 x-57 x^2-12 x^3+12 x^4} \, dx=x^{2} - \frac {551 \, x + 1205}{82 \, {\left (2 \, x^{2} - x - 5\right )}} + \frac {241 \, x + 155}{41 \, {\left (2 \, x^{2} - x - 5\right )}} + \frac {19 \, {\left (31 \, x + 105\right )}}{82 \, {\left (2 \, x^{2} - x - 5\right )}} - \frac {28 \, {\left (21 \, x + 5\right )}}{123 \, {\left (2 \, x^{2} - x - 5\right )}} + \frac {10 \, {\left (4 \, x - 1\right )}}{123 \, {\left (2 \, x^{2} - x - 5\right )}} - \frac {50 \, {\left (x + 10\right )}}{41 \, {\left (2 \, x^{2} - x - 5\right )}} + e^{\left (x^{2}\right )} \] Input:
integrate(((24*x^5-24*x^4-114*x^3+60*x^2+150*x)*exp(x^2)+24*x^5-24*x^4-114 *x^3+56*x^2+150*x-10)/(12*x^4-12*x^3-57*x^2+30*x+75),x, algorithm="maxima" )
Output:
x^2 - 1/82*(551*x + 1205)/(2*x^2 - x - 5) + 1/41*(241*x + 155)/(2*x^2 - x - 5) + 19/82*(31*x + 105)/(2*x^2 - x - 5) - 28/123*(21*x + 5)/(2*x^2 - x - 5) + 10/123*(4*x - 1)/(2*x^2 - x - 5) - 50/41*(x + 10)/(2*x^2 - x - 5) + e^(x^2)
Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (24) = 48\).
Time = 0.12 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.04 \[ \int \frac {-10+150 x+56 x^2-114 x^3-24 x^4+24 x^5+e^{x^2} \left (150 x+60 x^2-114 x^3-24 x^4+24 x^5\right )}{75+30 x-57 x^2-12 x^3+12 x^4} \, dx=\frac {6 \, x^{4} - 3 \, x^{3} + 6 \, x^{2} e^{\left (x^{2}\right )} - 15 \, x^{2} - 3 \, x e^{\left (x^{2}\right )} + 2 \, x - 15 \, e^{\left (x^{2}\right )}}{3 \, {\left (2 \, x^{2} - x - 5\right )}} \] Input:
integrate(((24*x^5-24*x^4-114*x^3+60*x^2+150*x)*exp(x^2)+24*x^5-24*x^4-114 *x^3+56*x^2+150*x-10)/(12*x^4-12*x^3-57*x^2+30*x+75),x, algorithm="giac")
Output:
1/3*(6*x^4 - 3*x^3 + 6*x^2*e^(x^2) - 15*x^2 - 3*x*e^(x^2) + 2*x - 15*e^(x^ 2))/(2*x^2 - x - 5)
Time = 3.69 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {-10+150 x+56 x^2-114 x^3-24 x^4+24 x^5+e^{x^2} \left (150 x+60 x^2-114 x^3-24 x^4+24 x^5\right )}{75+30 x-57 x^2-12 x^3+12 x^4} \, dx={\mathrm {e}}^{x^2}-\frac {2\,x}{3\,\left (-2\,x^2+x+5\right )}+x^2 \] Input:
int((150*x + exp(x^2)*(150*x + 60*x^2 - 114*x^3 - 24*x^4 + 24*x^5) + 56*x^ 2 - 114*x^3 - 24*x^4 + 24*x^5 - 10)/(30*x - 57*x^2 - 12*x^3 + 12*x^4 + 75) ,x)
Output:
exp(x^2) - (2*x)/(3*(x - 2*x^2 + 5)) + x^2
Time = 0.16 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.04 \[ \int \frac {-10+150 x+56 x^2-114 x^3-24 x^4+24 x^5+e^{x^2} \left (150 x+60 x^2-114 x^3-24 x^4+24 x^5\right )}{75+30 x-57 x^2-12 x^3+12 x^4} \, dx=\frac {6 e^{x^{2}} x^{2}-3 e^{x^{2}} x -15 e^{x^{2}}+6 x^{4}-3 x^{3}-11 x^{2}-10}{6 x^{2}-3 x -15} \] Input:
int(((24*x^5-24*x^4-114*x^3+60*x^2+150*x)*exp(x^2)+24*x^5-24*x^4-114*x^3+5 6*x^2+150*x-10)/(12*x^4-12*x^3-57*x^2+30*x+75),x)
Output:
(6*e**(x**2)*x**2 - 3*e**(x**2)*x - 15*e**(x**2) + 6*x**4 - 3*x**3 - 11*x* *2 - 10)/(3*(2*x**2 - x - 5))