Integrand size = 27, antiderivative size = 105 \[ \int \frac {(5-2 x)^3 \left (2+3 x+x^2\right )^2}{(4+3 x)^{3/2}} \, dx=-\frac {97336}{6561 \sqrt {4+3 x}}-\frac {122728 \sqrt {4+3 x}}{6561}-\frac {15134 (4+3 x)^{3/2}}{6561}+\frac {13088 (4+3 x)^{5/2}}{6561}+\frac {10046 (4+3 x)^{7/2}}{45927}-\frac {1732 (4+3 x)^{9/2}}{19683}+\frac {520 (4+3 x)^{11/2}}{72171}-\frac {16 (4+3 x)^{13/2}}{85293} \] Output:
-97336/6561/(4+3*x)^(1/2)-122728/6561*(4+3*x)^(1/2)-15134/6561*(4+3*x)^(3/ 2)+13088/6561*(4+3*x)^(5/2)+10046/45927*(4+3*x)^(7/2)-1732/19683*(4+3*x)^( 9/2)+520/72171*(4+3*x)^(11/2)-16/85293*(4+3*x)^(13/2)
Time = 0.06 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.46 \[ \int \frac {(5-2 x)^3 \left (2+3 x+x^2\right )^2}{(4+3 x)^{3/2}} \, dx=-\frac {2 \left (64697524-1207156116 x-881934147 x^2+277363872 x^3+185228613 x^4-52421418 x^5-14023044 x^6+4041576 x^7\right )}{19702683 \sqrt {4+3 x}} \] Input:
Integrate[((5 - 2*x)^3*(2 + 3*x + x^2)^2)/(4 + 3*x)^(3/2),x]
Output:
(-2*(64697524 - 1207156116*x - 881934147*x^2 + 277363872*x^3 + 185228613*x ^4 - 52421418*x^5 - 14023044*x^6 + 4041576*x^7))/(19702683*Sqrt[4 + 3*x])
Time = 0.36 (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.
\(\displaystyle \int \frac {(5-2 x)^3 \left (x^2+3 x+2\right )^2}{(3 x+4)^{3/2}} \, dx\) |
\(\Big \downarrow \) 1195 |
\(\displaystyle \int \left (-\frac {8 (3 x+4)^{11/2}}{2187}+\frac {260 (3 x+4)^{9/2}}{2187}-\frac {866}{729} (3 x+4)^{7/2}+\frac {5023 (3 x+4)^{5/2}}{2187}+\frac {32720 (3 x+4)^{3/2}}{2187}-\frac {7567}{729} \sqrt {3 x+4}-\frac {61364}{2187 \sqrt {3 x+4}}+\frac {48668}{2187 (3 x+4)^{3/2}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {16 (3 x+4)^{13/2}}{85293}+\frac {520 (3 x+4)^{11/2}}{72171}-\frac {1732 (3 x+4)^{9/2}}{19683}+\frac {10046 (3 x+4)^{7/2}}{45927}+\frac {13088 (3 x+4)^{5/2}}{6561}-\frac {15134 (3 x+4)^{3/2}}{6561}-\frac {122728 \sqrt {3 x+4}}{6561}-\frac {97336}{6561 \sqrt {3 x+4}}\) |
Input:
Int[((5 - 2*x)^3*(2 + 3*x + x^2)^2)/(4 + 3*x)^(3/2),x]
Output:
-97336/(6561*Sqrt[4 + 3*x]) - (122728*Sqrt[4 + 3*x])/6561 - (15134*(4 + 3* x)^(3/2))/6561 + (13088*(4 + 3*x)^(5/2))/6561 + (10046*(4 + 3*x)^(7/2))/45 927 - (1732*(4 + 3*x)^(9/2))/19683 + (520*(4 + 3*x)^(11/2))/72171 - (16*(4 + 3*x)^(13/2))/85293
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 = 1.69 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.43
method | result | size |
gosper | \(-\frac {2 \left (4041576 x^{7}-14023044 x^{6}-52421418 x^{5}+185228613 x^{4}+277363872 x^{3}-881934147 x^{2}-1207156116 x +64697524\right )}{19702683 \sqrt {3 x +4}}\) | \(45\) |
trager | \(-\frac {2 \left (4041576 x^{7}-14023044 x^{6}-52421418 x^{5}+185228613 x^{4}+277363872 x^{3}-881934147 x^{2}-1207156116 x +64697524\right )}{19702683 \sqrt {3 x +4}}\) | \(45\) |
risch | \(-\frac {2 \left (4041576 x^{7}-14023044 x^{6}-52421418 x^{5}+185228613 x^{4}+277363872 x^{3}-881934147 x^{2}-1207156116 x +64697524\right )}{19702683 \sqrt {3 x +4}}\) | \(45\) |
pseudoelliptic | \(\frac {-\frac {16}{39} x^{7}+\frac {1832}{1287} x^{6}+\frac {61636}{11583} x^{5}-\frac {4573546}{243243} x^{4}-\frac {20545472}{729729} x^{3}+\frac {195985366}{2189187} x^{2}+\frac {804770744}{6567561} x -\frac {129395048}{19702683}}{\sqrt {3 x +4}}\) | \(45\) |
derivativedivides | \(-\frac {97336}{6561 \sqrt {3 x +4}}-\frac {122728 \sqrt {3 x +4}}{6561}-\frac {15134 \left (3 x +4\right )^{\frac {3}{2}}}{6561}+\frac {13088 \left (3 x +4\right )^{\frac {5}{2}}}{6561}+\frac {10046 \left (3 x +4\right )^{\frac {7}{2}}}{45927}-\frac {1732 \left (3 x +4\right )^{\frac {9}{2}}}{19683}+\frac {520 \left (3 x +4\right )^{\frac {11}{2}}}{72171}-\frac {16 \left (3 x +4\right )^{\frac {13}{2}}}{85293}\) | \(74\) |
default | \(-\frac {97336}{6561 \sqrt {3 x +4}}-\frac {122728 \sqrt {3 x +4}}{6561}-\frac {15134 \left (3 x +4\right )^{\frac {3}{2}}}{6561}+\frac {13088 \left (3 x +4\right )^{\frac {5}{2}}}{6561}+\frac {10046 \left (3 x +4\right )^{\frac {7}{2}}}{45927}-\frac {1732 \left (3 x +4\right )^{\frac {9}{2}}}{19683}+\frac {520 \left (3 x +4\right )^{\frac {11}{2}}}{72171}-\frac {16 \left (3 x +4\right )^{\frac {13}{2}}}{85293}\) | \(74\) |
orering | \(\frac {2 \left (4041576 x^{7}-14023044 x^{6}-52421418 x^{5}+185228613 x^{4}+277363872 x^{3}-881934147 x^{2}-1207156116 x +64697524\right ) \left (5-2 x \right )^{3} \left (x^{2}+3 x +2\right )^{2}}{19702683 \sqrt {3 x +4}\, \left (2+x \right )^{2} \left (x +1\right )^{2} \left (-5+2 x \right )^{3}}\) | \(79\) |
meijerg | \(\frac {\frac {500 \sqrt {\pi }}{3}-\frac {500 \sqrt {\pi }}{3 \sqrt {1+\frac {3 x}{4}}}}{\sqrt {\pi }}+\frac {\frac {8320 \sqrt {\pi }}{81}-\frac {520 \sqrt {\pi }\, \left (-\frac {9}{8} x^{2}+6 x +16\right )}{81 \sqrt {1+\frac {3 x}{4}}}}{\sqrt {\pi }}+\frac {-800 \sqrt {\pi }+\frac {100 \sqrt {\pi }\, \left (8+3 x \right )}{\sqrt {1+\frac {3 x}{4}}}}{\sqrt {\pi }}-\frac {32768 \left (-\frac {16 \sqrt {\pi }}{5}+\frac {\sqrt {\pi }\, \left (\frac {27}{8} x^{3}-9 x^{2}+48 x +128\right )}{40 \sqrt {1+\frac {3 x}{4}}}\right )}{81 \sqrt {\pi }}-\frac {23296 \left (\frac {128 \sqrt {\pi }}{35}-\frac {\sqrt {\pi }\, \left (-\frac {405}{128} x^{4}+\frac {27}{4} x^{3}-18 x^{2}+96 x +256\right )}{70 \sqrt {1+\frac {3 x}{4}}}\right )}{243 \sqrt {\pi }}+\frac {-\frac {27787264 \sqrt {\pi }}{45927}+\frac {27136 \sqrt {\pi }\, \left (\frac {1701}{256} x^{5}-\frac {405}{32} x^{4}+27 x^{3}-72 x^{2}+384 x +1024\right )}{45927 \sqrt {1+\frac {3 x}{4}}}}{\sqrt {\pi }}+\frac {\frac {16777216 \sqrt {\pi }}{168399}-\frac {8192 \sqrt {\pi }\, \left (-\frac {15309}{2048} x^{6}+\frac {1701}{128} x^{5}-\frac {405}{16} x^{4}+54 x^{3}-144 x^{2}+768 x +2048\right )}{168399 \sqrt {1+\frac {3 x}{4}}}}{\sqrt {\pi }}-\frac {131072 \left (-\frac {2048 \sqrt {\pi }}{429}+\frac {\sqrt {\pi }\, \left (\frac {72171}{1024} x^{7}-\frac {15309}{128} x^{6}+\frac {1701}{8} x^{5}-405 x^{4}+864 x^{3}-2304 x^{2}+12288 x +32768\right )}{6864 \sqrt {1+\frac {3 x}{4}}}\right )}{6561 \sqrt {\pi }}\) | \(324\) |
Input:
int((5-2*x)^3*(x^2+3*x+2)^2/(3*x+4)^(3/2),x,method=_RETURNVERBOSE)
Output:
-2/19702683*(4041576*x^7-14023044*x^6-52421418*x^5+185228613*x^4+277363872 *x^3-881934147*x^2-1207156116*x+64697524)/(3*x+4)^(1/2)
Time = 0.07 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.42 \[ \int \frac {(5-2 x)^3 \left (2+3 x+x^2\right )^2}{(4+3 x)^{3/2}} \, dx=-\frac {2 \, {\left (4041576 \, x^{7} - 14023044 \, x^{6} - 52421418 \, x^{5} + 185228613 \, x^{4} + 277363872 \, x^{3} - 881934147 \, x^{2} - 1207156116 \, x + 64697524\right )}}{19702683 \, \sqrt {3 \, x + 4}} \] Input:
integrate((5-2*x)^3*(x^2+3*x+2)^2/(4+3*x)^(3/2),x, algorithm="fricas")
Output:
-2/19702683*(4041576*x^7 - 14023044*x^6 - 52421418*x^5 + 185228613*x^4 + 2 77363872*x^3 - 881934147*x^2 - 1207156116*x + 64697524)/sqrt(3*x + 4)
Time = 1.12 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int \frac {(5-2 x)^3 \left (2+3 x+x^2\right )^2}{(4+3 x)^{3/2}} \, dx=- \frac {16 \left (3 x + 4\right )^{\frac {13}{2}}}{85293} + \frac {520 \left (3 x + 4\right )^{\frac {11}{2}}}{72171} - \frac {1732 \left (3 x + 4\right )^{\frac {9}{2}}}{19683} + \frac {10046 \left (3 x + 4\right )^{\frac {7}{2}}}{45927} + \frac {13088 \left (3 x + 4\right )^{\frac {5}{2}}}{6561} - \frac {15134 \left (3 x + 4\right )^{\frac {3}{2}}}{6561} - \frac {122728 \sqrt {3 x + 4}}{6561} - \frac {97336}{6561 \sqrt {3 x + 4}} \] Input:
integrate((5-2*x)**3*(x**2+3*x+2)**2/(4+3*x)**(3/2),x)
Output:
-16*(3*x + 4)**(13/2)/85293 + 520*(3*x + 4)**(11/2)/72171 - 1732*(3*x + 4) **(9/2)/19683 + 10046*(3*x + 4)**(7/2)/45927 + 13088*(3*x + 4)**(5/2)/6561 - 15134*(3*x + 4)**(3/2)/6561 - 122728*sqrt(3*x + 4)/6561 - 97336/(6561*s qrt(3*x + 4))
Time = 0.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.70 \[ \int \frac {(5-2 x)^3 \left (2+3 x+x^2\right )^2}{(4+3 x)^{3/2}} \, dx=-\frac {16}{85293} \, {\left (3 \, x + 4\right )}^{\frac {13}{2}} + \frac {520}{72171} \, {\left (3 \, x + 4\right )}^{\frac {11}{2}} - \frac {1732}{19683} \, {\left (3 \, x + 4\right )}^{\frac {9}{2}} + \frac {10046}{45927} \, {\left (3 \, x + 4\right )}^{\frac {7}{2}} + \frac {13088}{6561} \, {\left (3 \, x + 4\right )}^{\frac {5}{2}} - \frac {15134}{6561} \, {\left (3 \, x + 4\right )}^{\frac {3}{2}} - \frac {122728}{6561} \, \sqrt {3 \, x + 4} - \frac {97336}{6561 \, \sqrt {3 \, x + 4}} \] Input:
integrate((5-2*x)^3*(x^2+3*x+2)^2/(4+3*x)^(3/2),x, algorithm="maxima")
Output:
-16/85293*(3*x + 4)^(13/2) + 520/72171*(3*x + 4)^(11/2) - 1732/19683*(3*x + 4)^(9/2) + 10046/45927*(3*x + 4)^(7/2) + 13088/6561*(3*x + 4)^(5/2) - 15 134/6561*(3*x + 4)^(3/2) - 122728/6561*sqrt(3*x + 4) - 97336/6561/sqrt(3*x + 4)
Time = 0.21 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.70 \[ \int \frac {(5-2 x)^3 \left (2+3 x+x^2\right )^2}{(4+3 x)^{3/2}} \, dx=-\frac {16}{85293} \, {\left (3 \, x + 4\right )}^{\frac {13}{2}} + \frac {520}{72171} \, {\left (3 \, x + 4\right )}^{\frac {11}{2}} - \frac {1732}{19683} \, {\left (3 \, x + 4\right )}^{\frac {9}{2}} + \frac {10046}{45927} \, {\left (3 \, x + 4\right )}^{\frac {7}{2}} + \frac {13088}{6561} \, {\left (3 \, x + 4\right )}^{\frac {5}{2}} - \frac {15134}{6561} \, {\left (3 \, x + 4\right )}^{\frac {3}{2}} - \frac {122728}{6561} \, \sqrt {3 \, x + 4} - \frac {97336}{6561 \, \sqrt {3 \, x + 4}} \] Input:
integrate((5-2*x)^3*(x^2+3*x+2)^2/(4+3*x)^(3/2),x, algorithm="giac")
Output:
-16/85293*(3*x + 4)^(13/2) + 520/72171*(3*x + 4)^(11/2) - 1732/19683*(3*x + 4)^(9/2) + 10046/45927*(3*x + 4)^(7/2) + 13088/6561*(3*x + 4)^(5/2) - 15 134/6561*(3*x + 4)^(3/2) - 122728/6561*sqrt(3*x + 4) - 97336/6561/sqrt(3*x + 4)
Time = 0.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.70 \[ \int \frac {(5-2 x)^3 \left (2+3 x+x^2\right )^2}{(4+3 x)^{3/2}} \, dx=\frac {13088\,{\left (3\,x+4\right )}^{5/2}}{6561}-\frac {122728\,\sqrt {3\,x+4}}{6561}-\frac {15134\,{\left (3\,x+4\right )}^{3/2}}{6561}-\frac {97336}{6561\,\sqrt {3\,x+4}}+\frac {10046\,{\left (3\,x+4\right )}^{7/2}}{45927}-\frac {1732\,{\left (3\,x+4\right )}^{9/2}}{19683}+\frac {520\,{\left (3\,x+4\right )}^{11/2}}{72171}-\frac {16\,{\left (3\,x+4\right )}^{13/2}}{85293} \] Input:
int(-((2*x - 5)^3*(3*x + x^2 + 2)^2)/(3*x + 4)^(3/2),x)
Output:
(13088*(3*x + 4)^(5/2))/6561 - (122728*(3*x + 4)^(1/2))/6561 - (15134*(3*x + 4)^(3/2))/6561 - 97336/(6561*(3*x + 4)^(1/2)) + (10046*(3*x + 4)^(7/2)) /45927 - (1732*(3*x + 4)^(9/2))/19683 + (520*(3*x + 4)^(11/2))/72171 - (16 *(3*x + 4)^(13/2))/85293
Time = 0.17 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.43 \[ \int \frac {(5-2 x)^3 \left (2+3 x+x^2\right )^2}{(4+3 x)^{3/2}} \, dx=\frac {-\frac {16}{39} x^{7}+\frac {1832}{1287} x^{6}+\frac {61636}{11583} x^{5}-\frac {4573546}{243243} x^{4}-\frac {20545472}{729729} x^{3}+\frac {195985366}{2189187} x^{2}+\frac {804770744}{6567561} x -\frac {129395048}{19702683}}{\sqrt {3 x +4}} \] Input:
int((5-2*x)^3*(x^2+3*x+2)^2/(4+3*x)^(3/2),x)
Output:
(2*( - 4041576*x**7 + 14023044*x**6 + 52421418*x**5 - 185228613*x**4 - 277 363872*x**3 + 881934147*x**2 + 1207156116*x - 64697524))/(19702683*sqrt(3* x + 4))