Integrand size = 12, antiderivative size = 64 \[ \int \frac {1}{\left (6-5 x+x^2\right )^{7/2}} \, dx=\frac {2 (5-2 x)}{5 \left (6-5 x+x^2\right )^{5/2}}-\frac {32 (5-2 x)}{15 \left (6-5 x+x^2\right )^{3/2}}+\frac {256 (5-2 x)}{15 \sqrt {6-5 x+x^2}} \] Output:
2/5*(5-2*x)/(x^2-5*x+6)^(5/2)-32/15*(5-2*x)/(x^2-5*x+6)^(3/2)+256/15*(5-2* x)/(x^2-5*x+6)^(1/2)
Time = 0.23 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\left (6-5 x+x^2\right )^{7/2}} \, dx=-\frac {2 \sqrt {6-5 x+x^2} \left (-22575+47030 x-38800 x^2+15840 x^3-3200 x^4+256 x^5\right )}{15 (-3+x)^3 (-2+x)^3} \] Input:
Integrate[(6 - 5*x + x^2)^(-7/2),x]
Output:
(-2*Sqrt[6 - 5*x + x^2]*(-22575 + 47030*x - 38800*x^2 + 15840*x^3 - 3200*x ^4 + 256*x^5))/(15*(-3 + x)^3*(-2 + x)^3)
Time = 0.31 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.08, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1089, 1089, 1088}
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.
\(\displaystyle \int \frac {1}{\left (x^2-5 x+6\right )^{7/2}} \, dx\) |
\(\Big \downarrow \) 1089 |
\(\displaystyle \frac {2 (5-2 x)}{5 \left (x^2-5 x+6\right )^{5/2}}-\frac {16}{5} \int \frac {1}{\left (x^2-5 x+6\right )^{5/2}}dx\) |
\(\Big \downarrow \) 1089 |
\(\displaystyle \frac {2 (5-2 x)}{5 \left (x^2-5 x+6\right )^{5/2}}-\frac {16}{5} \left (\frac {2 (5-2 x)}{3 \left (x^2-5 x+6\right )^{3/2}}-\frac {8}{3} \int \frac {1}{\left (x^2-5 x+6\right )^{3/2}}dx\right )\) |
\(\Big \downarrow \) 1088 |
\(\displaystyle \frac {2 (5-2 x)}{5 \left (x^2-5 x+6\right )^{5/2}}-\frac {16}{5} \left (\frac {2 (5-2 x)}{3 \left (x^2-5 x+6\right )^{3/2}}-\frac {16 (5-2 x)}{3 \sqrt {x^2-5 x+6}}\right )\) |
Input:
Int[(6 - 5*x + x^2)^(-7/2),x]
Output:
(2*(5 - 2*x))/(5*(6 - 5*x + x^2)^(5/2)) - (16*((2*(5 - 2*x))/(3*(6 - 5*x + x^2)^(3/2)) - (16*(5 - 2*x))/(3*Sqrt[6 - 5*x + x^2])))/5
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[-2*((b + 2*c*x)/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] - Simp[2*c*((2*p + 3)/((p + 1)*(b^2 - 4*a*c))) Int[(a + b*x + c*x^2)^(p + 1), x], x] /; Fre eQ[{a, b, c}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[3*p])
Time = 0.61 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.59
method | result | size |
trager | \(-\frac {2 \left (256 x^{5}-3200 x^{4}+15840 x^{3}-38800 x^{2}+47030 x -22575\right )}{15 \left (x^{2}-5 x +6\right )^{\frac {5}{2}}}\) | \(38\) |
risch | \(-\frac {2 \left (256 x^{5}-3200 x^{4}+15840 x^{3}-38800 x^{2}+47030 x -22575\right )}{15 \left (x^{2}-5 x +6\right )^{\frac {5}{2}}}\) | \(38\) |
gosper | \(-\frac {2 \left (x -2\right ) \left (-3+x \right ) \left (256 x^{5}-3200 x^{4}+15840 x^{3}-38800 x^{2}+47030 x -22575\right )}{15 \left (x^{2}-5 x +6\right )^{\frac {7}{2}}}\) | \(44\) |
orering | \(-\frac {2 \left (x -2\right ) \left (-3+x \right ) \left (256 x^{5}-3200 x^{4}+15840 x^{3}-38800 x^{2}+47030 x -22575\right )}{15 \left (x^{2}-5 x +6\right )^{\frac {7}{2}}}\) | \(44\) |
default | \(-\frac {2 \left (-5+2 x \right )}{5 \left (x^{2}-5 x +6\right )^{\frac {5}{2}}}+\frac {-\frac {32}{3}+\frac {64 x}{15}}{\left (x^{2}-5 x +6\right )^{\frac {3}{2}}}-\frac {256 \left (-5+2 x \right )}{15 \sqrt {x^{2}-5 x +6}}\) | \(53\) |
Input:
int(1/(x^2-5*x+6)^(7/2),x,method=_RETURNVERBOSE)
Output:
-2/15*(256*x^5-3200*x^4+15840*x^3-38800*x^2+47030*x-22575)/(x^2-5*x+6)^(5/ 2)
Time = 0.09 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.53 \[ \int \frac {1}{\left (6-5 x+x^2\right )^{7/2}} \, dx=-\frac {2 \, {\left (256 \, x^{6} - 3840 \, x^{5} + 23808 \, x^{4} - 78080 \, x^{3} + 142848 \, x^{2} + {\left (256 \, x^{5} - 3200 \, x^{4} + 15840 \, x^{3} - 38800 \, x^{2} + 47030 \, x - 22575\right )} \sqrt {x^{2} - 5 \, x + 6} - 138240 \, x + 55296\right )}}{15 \, {\left (x^{6} - 15 \, x^{5} + 93 \, x^{4} - 305 \, x^{3} + 558 \, x^{2} - 540 \, x + 216\right )}} \] Input:
integrate(1/(x^2-5*x+6)^(7/2),x, algorithm="fricas")
Output:
-2/15*(256*x^6 - 3840*x^5 + 23808*x^4 - 78080*x^3 + 142848*x^2 + (256*x^5 - 3200*x^4 + 15840*x^3 - 38800*x^2 + 47030*x - 22575)*sqrt(x^2 - 5*x + 6) - 138240*x + 55296)/(x^6 - 15*x^5 + 93*x^4 - 305*x^3 + 558*x^2 - 540*x + 2 16)
\[ \int \frac {1}{\left (6-5 x+x^2\right )^{7/2}} \, dx=\int \frac {1}{\left (x^{2} - 5 x + 6\right )^{\frac {7}{2}}}\, dx \] Input:
integrate(1/(x**2-5*x+6)**(7/2),x)
Output:
Integral((x**2 - 5*x + 6)**(-7/2), x)
Time = 0.03 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.19 \[ \int \frac {1}{\left (6-5 x+x^2\right )^{7/2}} \, dx=-\frac {512 \, x}{15 \, \sqrt {x^{2} - 5 \, x + 6}} + \frac {256}{3 \, \sqrt {x^{2} - 5 \, x + 6}} + \frac {64 \, x}{15 \, {\left (x^{2} - 5 \, x + 6\right )}^{\frac {3}{2}}} - \frac {32}{3 \, {\left (x^{2} - 5 \, x + 6\right )}^{\frac {3}{2}}} - \frac {4 \, x}{5 \, {\left (x^{2} - 5 \, x + 6\right )}^{\frac {5}{2}}} + \frac {2}{{\left (x^{2} - 5 \, x + 6\right )}^{\frac {5}{2}}} \] Input:
integrate(1/(x^2-5*x+6)^(7/2),x, algorithm="maxima")
Output:
-512/15*x/sqrt(x^2 - 5*x + 6) + 256/3/sqrt(x^2 - 5*x + 6) + 64/15*x/(x^2 - 5*x + 6)^(3/2) - 32/3/(x^2 - 5*x + 6)^(3/2) - 4/5*x/(x^2 - 5*x + 6)^(5/2) + 2/(x^2 - 5*x + 6)^(5/2)
Time = 0.18 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.58 \[ \int \frac {1}{\left (6-5 x+x^2\right )^{7/2}} \, dx=-\frac {2 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (4 \, {\left (2 \, x - 25\right )} x + 495\right )} x - 2425\right )} x + 23515\right )} x - 22575\right )}}{15 \, {\left (x^{2} - 5 \, x + 6\right )}^{\frac {5}{2}}} \] Input:
integrate(1/(x^2-5*x+6)^(7/2),x, algorithm="giac")
Output:
-2/15*(2*(8*(2*(4*(2*x - 25)*x + 495)*x - 2425)*x + 23515)*x - 22575)/(x^2 - 5*x + 6)^(5/2)
Time = 9.88 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\left (6-5 x+x^2\right )^{7/2}} \, dx=\frac {788\,x-512\,x\,{\left (x^2-5\,x+6\right )}^2+1280\,{\left (x^2-5\,x+6\right )}^2-160\,x^2+64\,x\,\left (x^2-5\,x+6\right )-930}{{\left (x^2-5\,x+6\right )}^{3/2}\,\left (15\,x^2-75\,x+90\right )} \] Input:
int(1/(x^2 - 5*x + 6)^(7/2),x)
Output:
(788*x - 512*x*(x^2 - 5*x + 6)^2 + 1280*(x^2 - 5*x + 6)^2 - 160*x^2 + 64*x *(x^2 - 5*x + 6) - 930)/((x^2 - 5*x + 6)^(3/2)*(15*x^2 - 75*x + 90))
Time = 0.22 (sec) , antiderivative size = 143, normalized size of antiderivative = 2.23 \[ \int \frac {1}{\left (6-5 x+x^2\right )^{7/2}} \, dx=\frac {-512 \sqrt {x^{2}-5 x +6}\, x^{5}+6400 \sqrt {x^{2}-5 x +6}\, x^{4}-31680 \sqrt {x^{2}-5 x +6}\, x^{3}+77600 \sqrt {x^{2}-5 x +6}\, x^{2}-94060 \sqrt {x^{2}-5 x +6}\, x +45150 \sqrt {x^{2}-5 x +6}+512 x^{6}-7680 x^{5}+47616 x^{4}-156160 x^{3}+285696 x^{2}-276480 x +110592}{15 x^{6}-225 x^{5}+1395 x^{4}-4575 x^{3}+8370 x^{2}-8100 x +3240} \] Input:
int(1/(x^2-5*x+6)^(7/2),x)
Output:
(2*( - 256*sqrt(x**2 - 5*x + 6)*x**5 + 3200*sqrt(x**2 - 5*x + 6)*x**4 - 15 840*sqrt(x**2 - 5*x + 6)*x**3 + 38800*sqrt(x**2 - 5*x + 6)*x**2 - 47030*sq rt(x**2 - 5*x + 6)*x + 22575*sqrt(x**2 - 5*x + 6) + 256*x**6 - 3840*x**5 + 23808*x**4 - 78080*x**3 + 142848*x**2 - 138240*x + 55296))/(15*(x**6 - 15 *x**5 + 93*x**4 - 305*x**3 + 558*x**2 - 540*x + 216))