Integrand size = 25, antiderivative size = 79 \[ \int \frac {(5-2 x) \left (2+3 x+x^2\right )^2}{(4+3 x)^{3/2}} \, dx=-\frac {184}{729 \sqrt {4+3 x}}-\frac {200}{729} \sqrt {4+3 x}-\frac {122 (4+3 x)^{3/2}}{2187}+\frac {104 (4+3 x)^{5/2}}{3645}+\frac {38 (4+3 x)^{7/2}}{5103}-\frac {4 (4+3 x)^{9/2}}{6561} \] Output:
-184/729/(4+3*x)^(1/2)-200/729*(4+3*x)^(1/2)-122/2187*(4+3*x)^(3/2)+104/36 45*(4+3*x)^(5/2)+38/5103*(4+3*x)^(7/2)-4/6561*(4+3*x)^(9/2)
Time = 0.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.48 \[ \int \frac {(5-2 x) \left (2+3 x+x^2\right )^2}{(4+3 x)^{3/2}} \, dx=\frac {2 \left (99404+611364 x+631683 x^2+155412 x^3-44145 x^4-17010 x^5\right )}{229635 \sqrt {4+3 x}} \] Input:
Integrate[((5 - 2*x)*(2 + 3*x + x^2)^2)/(4 + 3*x)^(3/2),x]
Output:
(2*(99404 + 611364*x + 631683*x^2 + 155412*x^3 - 44145*x^4 - 17010*x^5))/( 229635*Sqrt[4 + 3*x])
Time = 0.34 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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.
\(\displaystyle \int \frac {(5-2 x) \left (x^2+3 x+2\right )^2}{(3 x+4)^{3/2}} \, dx\) |
\(\Big \downarrow \) 1195 |
\(\displaystyle \int \left (-\frac {2}{243} (3 x+4)^{7/2}+\frac {19}{243} (3 x+4)^{5/2}+\frac {52}{243} (3 x+4)^{3/2}-\frac {61}{243} \sqrt {3 x+4}-\frac {100}{243 \sqrt {3 x+4}}+\frac {92}{243 (3 x+4)^{3/2}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {4 (3 x+4)^{9/2}}{6561}+\frac {38 (3 x+4)^{7/2}}{5103}+\frac {104 (3 x+4)^{5/2}}{3645}-\frac {122 (3 x+4)^{3/2}}{2187}-\frac {200}{729} \sqrt {3 x+4}-\frac {184}{729 \sqrt {3 x+4}}\) |
Input:
Int[((5 - 2*x)*(2 + 3*x + x^2)^2)/(4 + 3*x)^(3/2),x]
Output:
-184/(729*Sqrt[4 + 3*x]) - (200*Sqrt[4 + 3*x])/729 - (122*(4 + 3*x)^(3/2)) /2187 + (104*(4 + 3*x)^(5/2))/3645 + (38*(4 + 3*x)^(7/2))/5103 - (4*(4 + 3 *x)^(9/2))/6561
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 = 1.45 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.44
method | result | size |
gosper | \(-\frac {2 \left (17010 x^{5}+44145 x^{4}-155412 x^{3}-631683 x^{2}-611364 x -99404\right )}{229635 \sqrt {3 x +4}}\) | \(35\) |
trager | \(-\frac {2 \left (17010 x^{5}+44145 x^{4}-155412 x^{3}-631683 x^{2}-611364 x -99404\right )}{229635 \sqrt {3 x +4}}\) | \(35\) |
risch | \(-\frac {2 \left (17010 x^{5}+44145 x^{4}-155412 x^{3}-631683 x^{2}-611364 x -99404\right )}{229635 \sqrt {3 x +4}}\) | \(35\) |
pseudoelliptic | \(\frac {-\frac {4}{27} x^{5}-\frac {218}{567} x^{4}+\frac {11512}{8505} x^{3}+\frac {140374}{25515} x^{2}+\frac {407576}{76545} x +\frac {198808}{229635}}{\sqrt {3 x +4}}\) | \(35\) |
derivativedivides | \(-\frac {184}{729 \sqrt {3 x +4}}-\frac {200 \sqrt {3 x +4}}{729}-\frac {122 \left (3 x +4\right )^{\frac {3}{2}}}{2187}+\frac {104 \left (3 x +4\right )^{\frac {5}{2}}}{3645}+\frac {38 \left (3 x +4\right )^{\frac {7}{2}}}{5103}-\frac {4 \left (3 x +4\right )^{\frac {9}{2}}}{6561}\) | \(56\) |
default | \(-\frac {184}{729 \sqrt {3 x +4}}-\frac {200 \sqrt {3 x +4}}{729}-\frac {122 \left (3 x +4\right )^{\frac {3}{2}}}{2187}+\frac {104 \left (3 x +4\right )^{\frac {5}{2}}}{3645}+\frac {38 \left (3 x +4\right )^{\frac {7}{2}}}{5103}-\frac {4 \left (3 x +4\right )^{\frac {9}{2}}}{6561}\) | \(56\) |
orering | \(\frac {2 \left (17010 x^{5}+44145 x^{4}-155412 x^{3}-631683 x^{2}-611364 x -99404\right ) \left (5-2 x \right ) \left (x^{2}+3 x +2\right )^{2}}{229635 \sqrt {3 x +4}\, \left (-5+2 x \right ) \left (2+x \right )^{2} \left (x +1\right )^{2}}\) | \(67\) |
meijerg | \(\frac {\frac {20 \sqrt {\pi }}{3}-\frac {20 \sqrt {\pi }}{3 \sqrt {1+\frac {3 x}{4}}}}{\sqrt {\pi }}+\frac {\frac {5248 \sqrt {\pi }}{81}-\frac {328 \sqrt {\pi }\, \left (-\frac {9}{8} x^{2}+6 x +16\right )}{81 \sqrt {1+\frac {3 x}{4}}}}{\sqrt {\pi }}+\frac {-\frac {416 \sqrt {\pi }}{9}+\frac {52 \sqrt {\pi }\, \left (8+3 x \right )}{9 \sqrt {1+\frac {3 x}{4}}}}{\sqrt {\pi }}+\frac {-\frac {4096 \sqrt {\pi }}{405}+\frac {32 \sqrt {\pi }\, \left (\frac {27}{8} x^{3}-9 x^{2}+48 x +128\right )}{405 \sqrt {1+\frac {3 x}{4}}}}{\sqrt {\pi }}-\frac {1792 \left (\frac {128 \sqrt {\pi }}{35}-\frac {\sqrt {\pi }\, \left (-\frac {405}{128} x^{4}+\frac {27}{4} x^{3}-18 x^{2}+96 x +256\right )}{70 \sqrt {1+\frac {3 x}{4}}}\right )}{243 \sqrt {\pi }}-\frac {2048 \left (-\frac {256 \sqrt {\pi }}{63}+\frac {\sqrt {\pi }\, \left (\frac {1701}{256} x^{5}-\frac {405}{32} x^{4}+27 x^{3}-72 x^{2}+384 x +1024\right )}{252 \sqrt {1+\frac {3 x}{4}}}\right )}{729 \sqrt {\pi }}\) | \(213\) |
Input:
int((5-2*x)*(x^2+3*x+2)^2/(3*x+4)^(3/2),x,method=_RETURNVERBOSE)
Output:
-2/229635*(17010*x^5+44145*x^4-155412*x^3-631683*x^2-611364*x-99404)/(3*x+ 4)^(1/2)
Time = 0.07 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.43 \[ \int \frac {(5-2 x) \left (2+3 x+x^2\right )^2}{(4+3 x)^{3/2}} \, dx=-\frac {2 \, {\left (17010 \, x^{5} + 44145 \, x^{4} - 155412 \, x^{3} - 631683 \, x^{2} - 611364 \, x - 99404\right )}}{229635 \, \sqrt {3 \, x + 4}} \] Input:
integrate((5-2*x)*(x^2+3*x+2)^2/(4+3*x)^(3/2),x, algorithm="fricas")
Output:
-2/229635*(17010*x^5 + 44145*x^4 - 155412*x^3 - 631683*x^2 - 611364*x - 99 404)/sqrt(3*x + 4)
Time = 1.06 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.89 \[ \int \frac {(5-2 x) \left (2+3 x+x^2\right )^2}{(4+3 x)^{3/2}} \, dx=- \frac {4 \left (3 x + 4\right )^{\frac {9}{2}}}{6561} + \frac {38 \left (3 x + 4\right )^{\frac {7}{2}}}{5103} + \frac {104 \left (3 x + 4\right )^{\frac {5}{2}}}{3645} - \frac {122 \left (3 x + 4\right )^{\frac {3}{2}}}{2187} - \frac {200 \sqrt {3 x + 4}}{729} - \frac {184}{729 \sqrt {3 x + 4}} \] Input:
integrate((5-2*x)*(x**2+3*x+2)**2/(4+3*x)**(3/2),x)
Output:
-4*(3*x + 4)**(9/2)/6561 + 38*(3*x + 4)**(7/2)/5103 + 104*(3*x + 4)**(5/2) /3645 - 122*(3*x + 4)**(3/2)/2187 - 200*sqrt(3*x + 4)/729 - 184/(729*sqrt( 3*x + 4))
Time = 0.04 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.70 \[ \int \frac {(5-2 x) \left (2+3 x+x^2\right )^2}{(4+3 x)^{3/2}} \, dx=-\frac {4}{6561} \, {\left (3 \, x + 4\right )}^{\frac {9}{2}} + \frac {38}{5103} \, {\left (3 \, x + 4\right )}^{\frac {7}{2}} + \frac {104}{3645} \, {\left (3 \, x + 4\right )}^{\frac {5}{2}} - \frac {122}{2187} \, {\left (3 \, x + 4\right )}^{\frac {3}{2}} - \frac {200}{729} \, \sqrt {3 \, x + 4} - \frac {184}{729 \, \sqrt {3 \, x + 4}} \] Input:
integrate((5-2*x)*(x^2+3*x+2)^2/(4+3*x)^(3/2),x, algorithm="maxima")
Output:
-4/6561*(3*x + 4)^(9/2) + 38/5103*(3*x + 4)^(7/2) + 104/3645*(3*x + 4)^(5/ 2) - 122/2187*(3*x + 4)^(3/2) - 200/729*sqrt(3*x + 4) - 184/729/sqrt(3*x + 4)
Time = 0.23 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.70 \[ \int \frac {(5-2 x) \left (2+3 x+x^2\right )^2}{(4+3 x)^{3/2}} \, dx=-\frac {4}{6561} \, {\left (3 \, x + 4\right )}^{\frac {9}{2}} + \frac {38}{5103} \, {\left (3 \, x + 4\right )}^{\frac {7}{2}} + \frac {104}{3645} \, {\left (3 \, x + 4\right )}^{\frac {5}{2}} - \frac {122}{2187} \, {\left (3 \, x + 4\right )}^{\frac {3}{2}} - \frac {200}{729} \, \sqrt {3 \, x + 4} - \frac {184}{729 \, \sqrt {3 \, x + 4}} \] Input:
integrate((5-2*x)*(x^2+3*x+2)^2/(4+3*x)^(3/2),x, algorithm="giac")
Output:
-4/6561*(3*x + 4)^(9/2) + 38/5103*(3*x + 4)^(7/2) + 104/3645*(3*x + 4)^(5/ 2) - 122/2187*(3*x + 4)^(3/2) - 200/729*sqrt(3*x + 4) - 184/729/sqrt(3*x + 4)
Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.70 \[ \int \frac {(5-2 x) \left (2+3 x+x^2\right )^2}{(4+3 x)^{3/2}} \, dx=\frac {104\,{\left (3\,x+4\right )}^{5/2}}{3645}-\frac {200\,\sqrt {3\,x+4}}{729}-\frac {122\,{\left (3\,x+4\right )}^{3/2}}{2187}-\frac {184}{729\,\sqrt {3\,x+4}}+\frac {38\,{\left (3\,x+4\right )}^{7/2}}{5103}-\frac {4\,{\left (3\,x+4\right )}^{9/2}}{6561} \] Input:
int(-((2*x - 5)*(3*x + x^2 + 2)^2)/(3*x + 4)^(3/2),x)
Output:
(104*(3*x + 4)^(5/2))/3645 - (200*(3*x + 4)^(1/2))/729 - (122*(3*x + 4)^(3 /2))/2187 - 184/(729*(3*x + 4)^(1/2)) + (38*(3*x + 4)^(7/2))/5103 - (4*(3* x + 4)^(9/2))/6561
Time = 0.18 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.44 \[ \int \frac {(5-2 x) \left (2+3 x+x^2\right )^2}{(4+3 x)^{3/2}} \, dx=\frac {-\frac {4}{27} x^{5}-\frac {218}{567} x^{4}+\frac {11512}{8505} x^{3}+\frac {140374}{25515} x^{2}+\frac {407576}{76545} x +\frac {198808}{229635}}{\sqrt {3 x +4}} \] Input:
int((5-2*x)*(x^2+3*x+2)^2/(4+3*x)^(3/2),x)
Output:
(2*( - 17010*x**5 - 44145*x**4 + 155412*x**3 + 631683*x**2 + 611364*x + 99 404))/(229635*sqrt(3*x + 4))