Integrand size = 27, antiderivative size = 105 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^3}{(3+2 x)^{7/2}} \, dx=-\frac {325}{128 (3+2 x)^{5/2}}+\frac {7925}{384 (3+2 x)^{3/2}}-\frac {16005}{128 \sqrt {3+2 x}}-\frac {17201}{128} \sqrt {3+2 x}+\frac {10475}{384} (3+2 x)^{3/2}-\frac {3519}{640} (3+2 x)^{5/2}+\frac {81}{128} (3+2 x)^{7/2}-\frac {3}{128} (3+2 x)^{9/2} \]
-325/128/(3+2*x)^(5/2)+7925/384/(3+2*x)^(3/2)+10475/384*(3+2*x)^(3/2)-3519 /640*(3+2*x)^(5/2)+81/128*(3+2*x)^(7/2)-3/128*(3+2*x)^(9/2)-16005/128/(3+2 *x)^(1/2)-17201/128*(3+2*x)^(1/2)
Time = 0.04 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.46 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^3}{(3+2 x)^{7/2}} \, dx=-\frac {51162+85070 x+41805 x^2+3195 x^3-1940 x^4-702 x^5-135 x^6+45 x^7}{15 (3+2 x)^{5/2}} \]
-1/15*(51162 + 85070*x + 41805*x^2 + 3195*x^3 - 1940*x^4 - 702*x^5 - 135*x ^6 + 45*x^7)/(3 + 2*x)^(5/2)
Time = 0.22 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {1195, 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.
\(\displaystyle \int \frac {(5-x) \left (3 x^2+5 x+2\right )^3}{(2 x+3)^{7/2}} \, dx\) |
\(\Big \downarrow \) 1195 |
\(\displaystyle \int \left (-\frac {27}{128} (2 x+3)^{7/2}+\frac {567}{128} (2 x+3)^{5/2}-\frac {3519}{128} (2 x+3)^{3/2}+\frac {10475}{128} \sqrt {2 x+3}-\frac {17201}{128 \sqrt {2 x+3}}+\frac {16005}{128 (2 x+3)^{3/2}}-\frac {7925}{128 (2 x+3)^{5/2}}+\frac {1625}{128 (2 x+3)^{7/2}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {3}{128} (2 x+3)^{9/2}+\frac {81}{128} (2 x+3)^{7/2}-\frac {3519}{640} (2 x+3)^{5/2}+\frac {10475}{384} (2 x+3)^{3/2}-\frac {17201}{128} \sqrt {2 x+3}-\frac {16005}{128 \sqrt {2 x+3}}+\frac {7925}{384 (2 x+3)^{3/2}}-\frac {325}{128 (2 x+3)^{5/2}}\) |
-325/(128*(3 + 2*x)^(5/2)) + 7925/(384*(3 + 2*x)^(3/2)) - 16005/(128*Sqrt[ 3 + 2*x]) - (17201*Sqrt[3 + 2*x])/128 + (10475*(3 + 2*x)^(3/2))/384 - (351 9*(3 + 2*x)^(5/2))/640 + (81*(3 + 2*x)^(7/2))/128 - (3*(3 + 2*x)^(9/2))/12 8
3.26.49.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Time = 0.38 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.41
method | result | size |
pseudoelliptic | \(-\frac {3 \left (x^{7}-3 x^{6}-\frac {78}{5} x^{5}-\frac {388}{9} x^{4}+71 x^{3}+929 x^{2}+\frac {17014}{9} x +\frac {17054}{15}\right )}{\left (3+2 x \right )^{\frac {5}{2}}}\) | \(43\) |
gosper | \(-\frac {45 x^{7}-135 x^{6}-702 x^{5}-1940 x^{4}+3195 x^{3}+41805 x^{2}+85070 x +51162}{15 \left (3+2 x \right )^{\frac {5}{2}}}\) | \(45\) |
trager | \(-\frac {45 x^{7}-135 x^{6}-702 x^{5}-1940 x^{4}+3195 x^{3}+41805 x^{2}+85070 x +51162}{15 \left (3+2 x \right )^{\frac {5}{2}}}\) | \(45\) |
risch | \(-\frac {45 x^{7}-135 x^{6}-702 x^{5}-1940 x^{4}+3195 x^{3}+41805 x^{2}+85070 x +51162}{15 \left (3+2 x \right )^{\frac {5}{2}}}\) | \(45\) |
derivativedivides | \(-\frac {325}{128 \left (3+2 x \right )^{\frac {5}{2}}}+\frac {7925}{384 \left (3+2 x \right )^{\frac {3}{2}}}+\frac {10475 \left (3+2 x \right )^{\frac {3}{2}}}{384}-\frac {3519 \left (3+2 x \right )^{\frac {5}{2}}}{640}+\frac {81 \left (3+2 x \right )^{\frac {7}{2}}}{128}-\frac {3 \left (3+2 x \right )^{\frac {9}{2}}}{128}-\frac {16005}{128 \sqrt {3+2 x}}-\frac {17201 \sqrt {3+2 x}}{128}\) | \(74\) |
default | \(-\frac {325}{128 \left (3+2 x \right )^{\frac {5}{2}}}+\frac {7925}{384 \left (3+2 x \right )^{\frac {3}{2}}}+\frac {10475 \left (3+2 x \right )^{\frac {3}{2}}}{384}-\frac {3519 \left (3+2 x \right )^{\frac {5}{2}}}{640}+\frac {81 \left (3+2 x \right )^{\frac {7}{2}}}{128}-\frac {3 \left (3+2 x \right )^{\frac {9}{2}}}{128}-\frac {16005}{128 \sqrt {3+2 x}}-\frac {17201 \sqrt {3+2 x}}{128}\) | \(74\) |
meijerg | \(\frac {32 \sqrt {3}\, \left (\frac {3 \sqrt {\pi }}{4}-\frac {3 \sqrt {\pi }}{4 \left (1+\frac {2 x}{3}\right )^{\frac {5}{2}}}\right )}{81 \sqrt {\pi }}+\frac {584 \sqrt {3}\, \left (\frac {\sqrt {\pi }}{2}-\frac {\sqrt {\pi }\, \left (\frac {40 x}{3}+8\right )}{16 \left (1+\frac {2 x}{3}\right )^{\frac {5}{2}}}\right )}{135 \sqrt {\pi }}+\frac {58 \sqrt {3}\, \left (2 \sqrt {\pi }-\frac {\sqrt {\pi }\, \left (\frac {40}{3} x^{2}+\frac {80}{3} x +16\right )}{8 \left (1+\frac {2 x}{3}\right )^{\frac {5}{2}}}\right )}{3 \sqrt {\pi }}+\frac {1339 \sqrt {3}\, \left (-12 \sqrt {\pi }+\frac {3 \sqrt {\pi }\, \left (\frac {320}{27} x^{3}+\frac {320}{3} x^{2}+\frac {640}{3} x +128\right )}{32 \left (1+\frac {2 x}{3}\right )^{\frac {5}{2}}}\right )}{30 \sqrt {\pi }}+\frac {109 \sqrt {3}\, \left (32 \sqrt {\pi }-\frac {\sqrt {\pi }\, \left (-\frac {160}{81} x^{4}+\frac {640}{27} x^{3}+\frac {640}{3} x^{2}+\frac {1280}{3} x +256\right )}{8 \left (1+\frac {2 x}{3}\right )^{\frac {5}{2}}}\right )}{2 \sqrt {\pi }}+\frac {297 \sqrt {3}\, \left (-64 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (\frac {128}{81} x^{5}-\frac {640}{81} x^{4}+\frac {2560}{27} x^{3}+\frac {2560}{3} x^{2}+\frac {5120}{3} x +1024\right )}{16 \left (1+\frac {2 x}{3}\right )^{\frac {5}{2}}}\right )}{10 \sqrt {\pi }}-\frac {729 \sqrt {3}\, \left (-\frac {512 \sqrt {\pi }}{3}+\frac {\sqrt {\pi }\, \left (\frac {10240}{2187} x^{7}-\frac {10240}{729} x^{6}+\frac {4096}{81} x^{5}-\frac {20480}{81} x^{4}+\frac {81920}{27} x^{3}+\frac {81920}{3} x^{2}+\frac {163840}{3} x +32768\right )}{192 \left (1+\frac {2 x}{3}\right )^{\frac {5}{2}}}\right )}{160 \sqrt {\pi }}\) | \(294\) |
Time = 0.26 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.58 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^3}{(3+2 x)^{7/2}} \, dx=-\frac {{\left (45 \, x^{7} - 135 \, x^{6} - 702 \, x^{5} - 1940 \, x^{4} + 3195 \, x^{3} + 41805 \, x^{2} + 85070 \, x + 51162\right )} \sqrt {2 \, x + 3}}{15 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} \]
-1/15*(45*x^7 - 135*x^6 - 702*x^5 - 1940*x^4 + 3195*x^3 + 41805*x^2 + 8507 0*x + 51162)*sqrt(2*x + 3)/(8*x^3 + 36*x^2 + 54*x + 27)
Time = 1.42 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^3}{(3+2 x)^{7/2}} \, dx=- \frac {3 \left (2 x + 3\right )^{\frac {9}{2}}}{128} + \frac {81 \left (2 x + 3\right )^{\frac {7}{2}}}{128} - \frac {3519 \left (2 x + 3\right )^{\frac {5}{2}}}{640} + \frac {10475 \left (2 x + 3\right )^{\frac {3}{2}}}{384} - \frac {17201 \sqrt {2 x + 3}}{128} - \frac {16005}{128 \sqrt {2 x + 3}} + \frac {7925}{384 \left (2 x + 3\right )^{\frac {3}{2}}} - \frac {325}{128 \left (2 x + 3\right )^{\frac {5}{2}}} \]
-3*(2*x + 3)**(9/2)/128 + 81*(2*x + 3)**(7/2)/128 - 3519*(2*x + 3)**(5/2)/ 640 + 10475*(2*x + 3)**(3/2)/384 - 17201*sqrt(2*x + 3)/128 - 16005/(128*sq rt(2*x + 3)) + 7925/(384*(2*x + 3)**(3/2)) - 325/(128*(2*x + 3)**(5/2))
Time = 0.19 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.66 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^3}{(3+2 x)^{7/2}} \, dx=-\frac {3}{128} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} + \frac {81}{128} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - \frac {3519}{640} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + \frac {10475}{384} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - \frac {17201}{128} \, \sqrt {2 \, x + 3} - \frac {5 \, {\left (9603 \, {\left (2 \, x + 3\right )}^{2} - 3170 \, x - 4560\right )}}{384 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}}} \]
-3/128*(2*x + 3)^(9/2) + 81/128*(2*x + 3)^(7/2) - 3519/640*(2*x + 3)^(5/2) + 10475/384*(2*x + 3)^(3/2) - 17201/128*sqrt(2*x + 3) - 5/384*(9603*(2*x + 3)^2 - 3170*x - 4560)/(2*x + 3)^(5/2)
Time = 0.29 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.66 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^3}{(3+2 x)^{7/2}} \, dx=-\frac {3}{128} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} + \frac {81}{128} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - \frac {3519}{640} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + \frac {10475}{384} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - \frac {17201}{128} \, \sqrt {2 \, x + 3} - \frac {5 \, {\left (9603 \, {\left (2 \, x + 3\right )}^{2} - 3170 \, x - 4560\right )}}{384 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}}} \]
-3/128*(2*x + 3)^(9/2) + 81/128*(2*x + 3)^(7/2) - 3519/640*(2*x + 3)^(5/2) + 10475/384*(2*x + 3)^(3/2) - 17201/128*sqrt(2*x + 3) - 5/384*(9603*(2*x + 3)^2 - 3170*x - 4560)/(2*x + 3)^(5/2)
Time = 0.03 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.65 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^3}{(3+2 x)^{7/2}} \, dx=\frac {\frac {7925\,x}{192}-\frac {16005\,{\left (2\,x+3\right )}^2}{128}+\frac {475}{8}}{{\left (2\,x+3\right )}^{5/2}}-\frac {17201\,\sqrt {2\,x+3}}{128}+\frac {10475\,{\left (2\,x+3\right )}^{3/2}}{384}-\frac {3519\,{\left (2\,x+3\right )}^{5/2}}{640}+\frac {81\,{\left (2\,x+3\right )}^{7/2}}{128}-\frac {3\,{\left (2\,x+3\right )}^{9/2}}{128} \]