Integrand size = 27, antiderivative size = 92 \[ \int (5-2 x)^2 (4+3 x)^{3/2} \left (2+3 x+x^2\right )^2 \, dx=\frac {4232 (4+3 x)^{5/2}}{10935}-\frac {184}{567} (4+3 x)^{7/2}-\frac {802 (4+3 x)^{9/2}}{6561}+\frac {2636 (4+3 x)^{11/2}}{24057}+\frac {74 (4+3 x)^{13/2}}{3159}-\frac {56 (4+3 x)^{15/2}}{10935}+\frac {8 (4+3 x)^{17/2}}{37179} \] Output:
4232/10935*(4+3*x)^(5/2)-184/567*(4+3*x)^(7/2)-802/6561*(4+3*x)^(9/2)+2636 /24057*(4+3*x)^(11/2)+74/3159*(4+3*x)^(13/2)-56/10935*(4+3*x)^(15/2)+8/371 79*(4+3*x)^(17/2)
Time = 0.05 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.47 \[ \int (5-2 x)^2 (4+3 x)^{3/2} \left (2+3 x+x^2\right )^2 \, dx=\frac {2 (4+3 x)^{5/2} \left (1613306444+3953083980 x+2487414195 x^2-449069670 x^3-618492105 x^4+2918916 x^5+43783740 x^6\right )}{558242685} \] Input:
Integrate[(5 - 2*x)^2*(4 + 3*x)^(3/2)*(2 + 3*x + x^2)^2,x]
Output:
(2*(4 + 3*x)^(5/2)*(1613306444 + 3953083980*x + 2487414195*x^2 - 449069670 *x^3 - 618492105*x^4 + 2918916*x^5 + 43783740*x^6))/558242685
Time = 0.33 (sec) , antiderivative size = 92, 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.
\(\displaystyle \int (5-2 x)^2 (3 x+4)^{3/2} \left (x^2+3 x+2\right )^2 \, dx\) |
\(\Big \downarrow \) 1195 |
\(\displaystyle \int \left (\frac {4}{729} (3 x+4)^{15/2}-\frac {28}{243} (3 x+4)^{13/2}+\frac {37}{81} (3 x+4)^{11/2}+\frac {1318}{729} (3 x+4)^{9/2}-\frac {401}{243} (3 x+4)^{7/2}-\frac {92}{27} (3 x+4)^{5/2}+\frac {2116}{729} (3 x+4)^{3/2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {8 (3 x+4)^{17/2}}{37179}-\frac {56 (3 x+4)^{15/2}}{10935}+\frac {74 (3 x+4)^{13/2}}{3159}+\frac {2636 (3 x+4)^{11/2}}{24057}-\frac {802 (3 x+4)^{9/2}}{6561}-\frac {184}{567} (3 x+4)^{7/2}+\frac {4232 (3 x+4)^{5/2}}{10935}\) |
Input:
Int[(5 - 2*x)^2*(4 + 3*x)^(3/2)*(2 + 3*x + x^2)^2,x]
Output:
(4232*(4 + 3*x)^(5/2))/10935 - (184*(4 + 3*x)^(7/2))/567 - (802*(4 + 3*x)^ (9/2))/6561 + (2636*(4 + 3*x)^(11/2))/24057 + (74*(4 + 3*x)^(13/2))/3159 - (56*(4 + 3*x)^(15/2))/10935 + (8*(4 + 3*x)^(17/2))/37179
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 = 2.12 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.43
method | result | size |
gosper | \(\frac {2 \left (3 x +4\right )^{\frac {5}{2}} \left (43783740 x^{6}+2918916 x^{5}-618492105 x^{4}-449069670 x^{3}+2487414195 x^{2}+3953083980 x +1613306444\right )}{558242685}\) | \(40\) |
pseudoelliptic | \(\frac {2 \left (3 x +4\right )^{\frac {5}{2}} \left (43783740 x^{6}+2918916 x^{5}-618492105 x^{4}-449069670 x^{3}+2487414195 x^{2}+3953083980 x +1613306444\right )}{558242685}\) | \(40\) |
trager | \(\left (\frac {24}{17} x^{8}+\frac {328}{85} x^{7}-\frac {18986}{1105} x^{6}-\frac {2461132}{36465} x^{5}+\frac {1208594}{196911} x^{4}+\frac {1305045656}{4135131} x^{3}+\frac {11051288936}{20675655} x^{2}+\frac {67979132224}{186080895} x +\frac {51625806208}{558242685}\right ) \sqrt {3 x +4}\) | \(49\) |
risch | \(\frac {2 \left (394053660 x^{8}+1077080004 x^{7}-4795835121 x^{6}-18838734894 x^{5}+1713181995 x^{4}+88090581780 x^{3}+149192400636 x^{2}+101968698336 x +25812903104\right ) \sqrt {3 x +4}}{558242685}\) | \(50\) |
derivativedivides | \(\frac {4232 \left (3 x +4\right )^{\frac {5}{2}}}{10935}-\frac {184 \left (3 x +4\right )^{\frac {7}{2}}}{567}-\frac {802 \left (3 x +4\right )^{\frac {9}{2}}}{6561}+\frac {2636 \left (3 x +4\right )^{\frac {11}{2}}}{24057}+\frac {74 \left (3 x +4\right )^{\frac {13}{2}}}{3159}-\frac {56 \left (3 x +4\right )^{\frac {15}{2}}}{10935}+\frac {8 \left (3 x +4\right )^{\frac {17}{2}}}{37179}\) | \(65\) |
default | \(\frac {4232 \left (3 x +4\right )^{\frac {5}{2}}}{10935}-\frac {184 \left (3 x +4\right )^{\frac {7}{2}}}{567}-\frac {802 \left (3 x +4\right )^{\frac {9}{2}}}{6561}+\frac {2636 \left (3 x +4\right )^{\frac {11}{2}}}{24057}+\frac {74 \left (3 x +4\right )^{\frac {13}{2}}}{3159}-\frac {56 \left (3 x +4\right )^{\frac {15}{2}}}{10935}+\frac {8 \left (3 x +4\right )^{\frac {17}{2}}}{37179}\) | \(65\) |
orering | \(\frac {2 \left (43783740 x^{6}+2918916 x^{5}-618492105 x^{4}-449069670 x^{3}+2487414195 x^{2}+3953083980 x +1613306444\right ) \left (3 x +4\right )^{\frac {5}{2}} \left (5-2 x \right )^{2} \left (x^{2}+3 x +2\right )^{2}}{558242685 \left (x +1\right )^{2} \left (2+x \right )^{2} \left (-5+2 x \right )^{2}}\) | \(74\) |
meijerg | \(\frac {-\frac {827392 \sqrt {\pi }}{8505}+\frac {51712 \sqrt {\pi }\, \left (\frac {2835}{128} x^{4}+\frac {675}{16} x^{3}+\frac {27}{8} x^{2}-6 x +16\right ) \sqrt {1+\frac {3 x}{4}}}{8505}}{\sqrt {\pi }}+\frac {\frac {22528 \sqrt {\pi }}{63}-\frac {2816 \sqrt {\pi }\, \left (-\frac {135}{16} x^{3}-18 x^{2}-3 x +8\right ) \sqrt {1+\frac {3 x}{4}}}{63}}{\sqrt {\pi }}+\frac {-\frac {1280 \sqrt {\pi }}{3}+\frac {640 \sqrt {\pi }\, \left (\frac {9}{8} x^{2}+3 x +2\right ) \sqrt {1+\frac {3 x}{4}}}{3}}{\sqrt {\pi }}-\frac {31744 \left (\frac {128 \sqrt {\pi }}{3465}-\frac {\sqrt {\pi }\, \left (-\frac {25515}{128} x^{5}-\frac {2835}{8} x^{4}-\frac {135}{8} x^{3}+27 x^{2}-48 x +128\right ) \sqrt {1+\frac {3 x}{4}}}{3465}\right )}{27 \sqrt {\pi }}-\frac {88064 \left (-\frac {1024 \sqrt {\pi }}{45045}+\frac {4 \sqrt {\pi }\, \left (\frac {841995}{2048} x^{6}+\frac {178605}{256} x^{5}+\frac {2835}{128} x^{4}-\frac {135}{4} x^{3}+54 x^{2}-96 x +256\right ) \sqrt {1+\frac {3 x}{4}}}{45045}\right )}{81 \sqrt {\pi }}+\frac {\frac {67108864 \sqrt {\pi }}{32837805}-\frac {65536 \sqrt {\pi }\, \left (-\frac {6567561}{4096} x^{7}-\frac {168399}{64} x^{6}-\frac {15309}{256} x^{5}+\frac {2835}{32} x^{4}-135 x^{3}+216 x^{2}-384 x +1024\right ) \sqrt {1+\frac {3 x}{4}}}{32837805}}{\sqrt {\pi }}+\frac {-\frac {1073741824 \sqrt {\pi }}{558242685}+\frac {524288 \sqrt {\pi }\, \left (\frac {98513415}{32768} x^{8}+\frac {19702683}{4096} x^{7}+\frac {168399}{2048} x^{6}-\frac {15309}{128} x^{5}+\frac {2835}{16} x^{4}-270 x^{3}+432 x^{2}-768 x +2048\right ) \sqrt {1+\frac {3 x}{4}}}{558242685}}{\sqrt {\pi }}\) | \(338\) |
Input:
int((5-2*x)^2*(3*x+4)^(3/2)*(x^2+3*x+2)^2,x,method=_RETURNVERBOSE)
Output:
2/558242685*(3*x+4)^(5/2)*(43783740*x^6+2918916*x^5-618492105*x^4-44906967 0*x^3+2487414195*x^2+3953083980*x+1613306444)
Time = 0.08 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.53 \[ \int (5-2 x)^2 (4+3 x)^{3/2} \left (2+3 x+x^2\right )^2 \, dx=\frac {2}{558242685} \, {\left (394053660 \, x^{8} + 1077080004 \, x^{7} - 4795835121 \, x^{6} - 18838734894 \, x^{5} + 1713181995 \, x^{4} + 88090581780 \, x^{3} + 149192400636 \, x^{2} + 101968698336 \, x + 25812903104\right )} \sqrt {3 \, x + 4} \] Input:
integrate((5-2*x)^2*(4+3*x)^(3/2)*(x^2+3*x+2)^2,x, algorithm="fricas")
Output:
2/558242685*(394053660*x^8 + 1077080004*x^7 - 4795835121*x^6 - 18838734894 *x^5 + 1713181995*x^4 + 88090581780*x^3 + 149192400636*x^2 + 101968698336* x + 25812903104)*sqrt(3*x + 4)
Time = 0.84 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.89 \[ \int (5-2 x)^2 (4+3 x)^{3/2} \left (2+3 x+x^2\right )^2 \, dx=\frac {8 \left (3 x + 4\right )^{\frac {17}{2}}}{37179} - \frac {56 \left (3 x + 4\right )^{\frac {15}{2}}}{10935} + \frac {74 \left (3 x + 4\right )^{\frac {13}{2}}}{3159} + \frac {2636 \left (3 x + 4\right )^{\frac {11}{2}}}{24057} - \frac {802 \left (3 x + 4\right )^{\frac {9}{2}}}{6561} - \frac {184 \left (3 x + 4\right )^{\frac {7}{2}}}{567} + \frac {4232 \left (3 x + 4\right )^{\frac {5}{2}}}{10935} \] Input:
integrate((5-2*x)**2*(4+3*x)**(3/2)*(x**2+3*x+2)**2,x)
Output:
8*(3*x + 4)**(17/2)/37179 - 56*(3*x + 4)**(15/2)/10935 + 74*(3*x + 4)**(13 /2)/3159 + 2636*(3*x + 4)**(11/2)/24057 - 802*(3*x + 4)**(9/2)/6561 - 184* (3*x + 4)**(7/2)/567 + 4232*(3*x + 4)**(5/2)/10935
Time = 0.03 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.70 \[ \int (5-2 x)^2 (4+3 x)^{3/2} \left (2+3 x+x^2\right )^2 \, dx=\frac {8}{37179} \, {\left (3 \, x + 4\right )}^{\frac {17}{2}} - \frac {56}{10935} \, {\left (3 \, x + 4\right )}^{\frac {15}{2}} + \frac {74}{3159} \, {\left (3 \, x + 4\right )}^{\frac {13}{2}} + \frac {2636}{24057} \, {\left (3 \, x + 4\right )}^{\frac {11}{2}} - \frac {802}{6561} \, {\left (3 \, x + 4\right )}^{\frac {9}{2}} - \frac {184}{567} \, {\left (3 \, x + 4\right )}^{\frac {7}{2}} + \frac {4232}{10935} \, {\left (3 \, x + 4\right )}^{\frac {5}{2}} \] Input:
integrate((5-2*x)^2*(4+3*x)^(3/2)*(x^2+3*x+2)^2,x, algorithm="maxima")
Output:
8/37179*(3*x + 4)^(17/2) - 56/10935*(3*x + 4)^(15/2) + 74/3159*(3*x + 4)^( 13/2) + 2636/24057*(3*x + 4)^(11/2) - 802/6561*(3*x + 4)^(9/2) - 184/567*( 3*x + 4)^(7/2) + 4232/10935*(3*x + 4)^(5/2)
Time = 0.23 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.70 \[ \int (5-2 x)^2 (4+3 x)^{3/2} \left (2+3 x+x^2\right )^2 \, dx=\frac {8}{37179} \, {\left (3 \, x + 4\right )}^{\frac {17}{2}} - \frac {56}{10935} \, {\left (3 \, x + 4\right )}^{\frac {15}{2}} + \frac {74}{3159} \, {\left (3 \, x + 4\right )}^{\frac {13}{2}} + \frac {2636}{24057} \, {\left (3 \, x + 4\right )}^{\frac {11}{2}} - \frac {802}{6561} \, {\left (3 \, x + 4\right )}^{\frac {9}{2}} - \frac {184}{567} \, {\left (3 \, x + 4\right )}^{\frac {7}{2}} + \frac {4232}{10935} \, {\left (3 \, x + 4\right )}^{\frac {5}{2}} \] Input:
integrate((5-2*x)^2*(4+3*x)^(3/2)*(x^2+3*x+2)^2,x, algorithm="giac")
Output:
8/37179*(3*x + 4)^(17/2) - 56/10935*(3*x + 4)^(15/2) + 74/3159*(3*x + 4)^( 13/2) + 2636/24057*(3*x + 4)^(11/2) - 802/6561*(3*x + 4)^(9/2) - 184/567*( 3*x + 4)^(7/2) + 4232/10935*(3*x + 4)^(5/2)
Time = 0.04 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.70 \[ \int (5-2 x)^2 (4+3 x)^{3/2} \left (2+3 x+x^2\right )^2 \, dx=\frac {4232\,{\left (3\,x+4\right )}^{5/2}}{10935}-\frac {184\,{\left (3\,x+4\right )}^{7/2}}{567}-\frac {802\,{\left (3\,x+4\right )}^{9/2}}{6561}+\frac {2636\,{\left (3\,x+4\right )}^{11/2}}{24057}+\frac {74\,{\left (3\,x+4\right )}^{13/2}}{3159}-\frac {56\,{\left (3\,x+4\right )}^{15/2}}{10935}+\frac {8\,{\left (3\,x+4\right )}^{17/2}}{37179} \] Input:
int((2*x - 5)^2*(3*x + 4)^(3/2)*(3*x + x^2 + 2)^2,x)
Output:
(4232*(3*x + 4)^(5/2))/10935 - (184*(3*x + 4)^(7/2))/567 - (802*(3*x + 4)^ (9/2))/6561 + (2636*(3*x + 4)^(11/2))/24057 + (74*(3*x + 4)^(13/2))/3159 - (56*(3*x + 4)^(15/2))/10935 + (8*(3*x + 4)^(17/2))/37179
Time = 0.16 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.52 \[ \int (5-2 x)^2 (4+3 x)^{3/2} \left (2+3 x+x^2\right )^2 \, dx=\frac {2 \sqrt {3 x +4}\, \left (394053660 x^{8}+1077080004 x^{7}-4795835121 x^{6}-18838734894 x^{5}+1713181995 x^{4}+88090581780 x^{3}+149192400636 x^{2}+101968698336 x +25812903104\right )}{558242685} \] Input:
int((5-2*x)^2*(4+3*x)^(3/2)*(x^2+3*x+2)^2,x)
Output:
(2*sqrt(3*x + 4)*(394053660*x**8 + 1077080004*x**7 - 4795835121*x**6 - 188 38734894*x**5 + 1713181995*x**4 + 88090581780*x**3 + 149192400636*x**2 + 1 01968698336*x + 25812903104))/558242685