Integrand size = 24, antiderivative size = 92 \[ \int \frac {(2+3 x)^4 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx=\frac {290521}{64 \sqrt {1-2 x}}+\frac {381073}{32} \sqrt {1-2 x}-\frac {832951}{192} (1-2 x)^{3/2}+\frac {121359}{80} (1-2 x)^{5/2}-\frac {159111}{448} (1-2 x)^{7/2}+\frac {1545}{32} (1-2 x)^{9/2}-\frac {2025}{704} (1-2 x)^{11/2} \] Output:
290521/64/(1-2*x)^(1/2)+381073/32*(1-2*x)^(1/2)-832951/192*(1-2*x)^(3/2)+1 21359/80*(1-2*x)^(5/2)-159111/448*(1-2*x)^(7/2)+1545/32*(1-2*x)^(9/2)-2025 /704*(1-2*x)^(11/2)
Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.47 \[ \int \frac {(2+3 x)^4 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx=\frac {15380984-15214664 x-6831172 x^2-4819932 x^3-2899485 x^4-1146600 x^5-212625 x^6}{1155 \sqrt {1-2 x}} \] Input:
Integrate[((2 + 3*x)^4*(3 + 5*x)^2)/(1 - 2*x)^(3/2),x]
Output:
(15380984 - 15214664*x - 6831172*x^2 - 4819932*x^3 - 2899485*x^4 - 1146600 *x^5 - 212625*x^6)/(1155*Sqrt[1 - 2*x])
Time = 0.22 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {99, 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.
\(\displaystyle \int \frac {(3 x+2)^4 (5 x+3)^2}{(1-2 x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (\frac {2025}{64} (1-2 x)^{9/2}-\frac {13905}{32} (1-2 x)^{7/2}+\frac {159111}{64} (1-2 x)^{5/2}-\frac {121359}{16} (1-2 x)^{3/2}+\frac {832951}{64} \sqrt {1-2 x}-\frac {381073}{32 \sqrt {1-2 x}}+\frac {290521}{64 (1-2 x)^{3/2}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2025}{704} (1-2 x)^{11/2}+\frac {1545}{32} (1-2 x)^{9/2}-\frac {159111}{448} (1-2 x)^{7/2}+\frac {121359}{80} (1-2 x)^{5/2}-\frac {832951}{192} (1-2 x)^{3/2}+\frac {381073}{32} \sqrt {1-2 x}+\frac {290521}{64 \sqrt {1-2 x}}\) |
Input:
Int[((2 + 3*x)^4*(3 + 5*x)^2)/(1 - 2*x)^(3/2),x]
Output:
290521/(64*Sqrt[1 - 2*x]) + (381073*Sqrt[1 - 2*x])/32 - (832951*(1 - 2*x)^ (3/2))/192 + (121359*(1 - 2*x)^(5/2))/80 - (159111*(1 - 2*x)^(7/2))/448 + (1545*(1 - 2*x)^(9/2))/32 - (2025*(1 - 2*x)^(11/2))/704
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Time = 0.21 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.43
method | result | size |
gosper | \(-\frac {212625 x^{6}+1146600 x^{5}+2899485 x^{4}+4819932 x^{3}+6831172 x^{2}+15214664 x -15380984}{1155 \sqrt {1-2 x}}\) | \(40\) |
risch | \(-\frac {212625 x^{6}+1146600 x^{5}+2899485 x^{4}+4819932 x^{3}+6831172 x^{2}+15214664 x -15380984}{1155 \sqrt {1-2 x}}\) | \(40\) |
pseudoelliptic | \(\frac {-212625 x^{6}-1146600 x^{5}-2899485 x^{4}-4819932 x^{3}-6831172 x^{2}-15214664 x +15380984}{1155 \sqrt {1-2 x}}\) | \(40\) |
orering | \(\frac {\left (-1+2 x \right ) \left (212625 x^{6}+1146600 x^{5}+2899485 x^{4}+4819932 x^{3}+6831172 x^{2}+15214664 x -15380984\right )}{1155 \left (1-2 x \right )^{\frac {3}{2}}}\) | \(45\) |
trager | \(\frac {\left (212625 x^{6}+1146600 x^{5}+2899485 x^{4}+4819932 x^{3}+6831172 x^{2}+15214664 x -15380984\right ) \sqrt {1-2 x}}{-1155+2310 x}\) | \(47\) |
derivativedivides | \(\frac {290521}{64 \sqrt {1-2 x}}+\frac {381073 \sqrt {1-2 x}}{32}-\frac {832951 \left (1-2 x \right )^{\frac {3}{2}}}{192}+\frac {121359 \left (1-2 x \right )^{\frac {5}{2}}}{80}-\frac {159111 \left (1-2 x \right )^{\frac {7}{2}}}{448}+\frac {1545 \left (1-2 x \right )^{\frac {9}{2}}}{32}-\frac {2025 \left (1-2 x \right )^{\frac {11}{2}}}{704}\) | \(65\) |
default | \(\frac {290521}{64 \sqrt {1-2 x}}+\frac {381073 \sqrt {1-2 x}}{32}-\frac {832951 \left (1-2 x \right )^{\frac {3}{2}}}{192}+\frac {121359 \left (1-2 x \right )^{\frac {5}{2}}}{80}-\frac {159111 \left (1-2 x \right )^{\frac {7}{2}}}{448}+\frac {1545 \left (1-2 x \right )^{\frac {9}{2}}}{32}-\frac {2025 \left (1-2 x \right )^{\frac {11}{2}}}{704}\) | \(65\) |
meijerg | \(-\frac {144 \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {1-2 x}}\right )}{\sqrt {\pi }}+\frac {-1344 \sqrt {\pi }+\frac {168 \sqrt {\pi }\, \left (-8 x +8\right )}{\sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {1306 \left (\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-8 x^{2}-16 x +16\right )}{6 \sqrt {1-2 x}}\right )}{\sqrt {\pi }}+\frac {-\frac {21648 \sqrt {\pi }}{5}+\frac {1353 \sqrt {\pi }\, \left (-64 x^{3}-64 x^{2}-128 x +128\right )}{40 \sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {12609 \left (\frac {128 \sqrt {\pi }}{35}-\frac {\sqrt {\pi }\, \left (-160 x^{4}-128 x^{3}-128 x^{2}-256 x +256\right )}{70 \sqrt {1-2 x}}\right )}{16 \sqrt {\pi }}+\frac {-\frac {6960 \sqrt {\pi }}{7}+\frac {435 \sqrt {\pi }\, \left (-896 x^{5}-640 x^{4}-512 x^{3}-512 x^{2}-1024 x +1024\right )}{448 \sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {2025 \left (\frac {1024 \sqrt {\pi }}{231}-\frac {\sqrt {\pi }\, \left (-2688 x^{6}-1792 x^{5}-1280 x^{4}-1024 x^{3}-1024 x^{2}-2048 x +2048\right )}{462 \sqrt {1-2 x}}\right )}{64 \sqrt {\pi }}\) | \(266\) |
Input:
int((2+3*x)^4*(3+5*x)^2/(1-2*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/1155/(1-2*x)^(1/2)*(212625*x^6+1146600*x^5+2899485*x^4+4819932*x^3+6831 172*x^2+15214664*x-15380984)
Time = 0.06 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.50 \[ \int \frac {(2+3 x)^4 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx=\frac {{\left (212625 \, x^{6} + 1146600 \, x^{5} + 2899485 \, x^{4} + 4819932 \, x^{3} + 6831172 \, x^{2} + 15214664 \, x - 15380984\right )} \sqrt {-2 \, x + 1}}{1155 \, {\left (2 \, x - 1\right )}} \] Input:
integrate((2+3*x)^4*(3+5*x)^2/(1-2*x)^(3/2),x, algorithm="fricas")
Output:
1/1155*(212625*x^6 + 1146600*x^5 + 2899485*x^4 + 4819932*x^3 + 6831172*x^2 + 15214664*x - 15380984)*sqrt(-2*x + 1)/(2*x - 1)
Time = 0.93 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.89 \[ \int \frac {(2+3 x)^4 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx=- \frac {2025 \left (1 - 2 x\right )^{\frac {11}{2}}}{704} + \frac {1545 \left (1 - 2 x\right )^{\frac {9}{2}}}{32} - \frac {159111 \left (1 - 2 x\right )^{\frac {7}{2}}}{448} + \frac {121359 \left (1 - 2 x\right )^{\frac {5}{2}}}{80} - \frac {832951 \left (1 - 2 x\right )^{\frac {3}{2}}}{192} + \frac {381073 \sqrt {1 - 2 x}}{32} + \frac {290521}{64 \sqrt {1 - 2 x}} \] Input:
integrate((2+3*x)**4*(3+5*x)**2/(1-2*x)**(3/2),x)
Output:
-2025*(1 - 2*x)**(11/2)/704 + 1545*(1 - 2*x)**(9/2)/32 - 159111*(1 - 2*x)* *(7/2)/448 + 121359*(1 - 2*x)**(5/2)/80 - 832951*(1 - 2*x)**(3/2)/192 + 38 1073*sqrt(1 - 2*x)/32 + 290521/(64*sqrt(1 - 2*x))
Time = 0.04 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.70 \[ \int \frac {(2+3 x)^4 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx=-\frac {2025}{704} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {1545}{32} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {159111}{448} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {121359}{80} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {832951}{192} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {381073}{32} \, \sqrt {-2 \, x + 1} + \frac {290521}{64 \, \sqrt {-2 \, x + 1}} \] Input:
integrate((2+3*x)^4*(3+5*x)^2/(1-2*x)^(3/2),x, algorithm="maxima")
Output:
-2025/704*(-2*x + 1)^(11/2) + 1545/32*(-2*x + 1)^(9/2) - 159111/448*(-2*x + 1)^(7/2) + 121359/80*(-2*x + 1)^(5/2) - 832951/192*(-2*x + 1)^(3/2) + 38 1073/32*sqrt(-2*x + 1) + 290521/64/sqrt(-2*x + 1)
Time = 0.12 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00 \[ \int \frac {(2+3 x)^4 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx=\frac {2025}{704} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {1545}{32} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {159111}{448} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {121359}{80} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {832951}{192} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {381073}{32} \, \sqrt {-2 \, x + 1} + \frac {290521}{64 \, \sqrt {-2 \, x + 1}} \] Input:
integrate((2+3*x)^4*(3+5*x)^2/(1-2*x)^(3/2),x, algorithm="giac")
Output:
2025/704*(2*x - 1)^5*sqrt(-2*x + 1) + 1545/32*(2*x - 1)^4*sqrt(-2*x + 1) + 159111/448*(2*x - 1)^3*sqrt(-2*x + 1) + 121359/80*(2*x - 1)^2*sqrt(-2*x + 1) - 832951/192*(-2*x + 1)^(3/2) + 381073/32*sqrt(-2*x + 1) + 290521/64/s qrt(-2*x + 1)
Time = 0.02 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.70 \[ \int \frac {(2+3 x)^4 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx=\frac {290521}{64\,\sqrt {1-2\,x}}+\frac {381073\,\sqrt {1-2\,x}}{32}-\frac {832951\,{\left (1-2\,x\right )}^{3/2}}{192}+\frac {121359\,{\left (1-2\,x\right )}^{5/2}}{80}-\frac {159111\,{\left (1-2\,x\right )}^{7/2}}{448}+\frac {1545\,{\left (1-2\,x\right )}^{9/2}}{32}-\frac {2025\,{\left (1-2\,x\right )}^{11/2}}{704} \] Input:
int(((3*x + 2)^4*(5*x + 3)^2)/(1 - 2*x)^(3/2),x)
Output:
290521/(64*(1 - 2*x)^(1/2)) + (381073*(1 - 2*x)^(1/2))/32 - (832951*(1 - 2 *x)^(3/2))/192 + (121359*(1 - 2*x)^(5/2))/80 - (159111*(1 - 2*x)^(7/2))/44 8 + (1545*(1 - 2*x)^(9/2))/32 - (2025*(1 - 2*x)^(11/2))/704
Time = 0.17 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.43 \[ \int \frac {(2+3 x)^4 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx=\frac {-212625 x^{6}-1146600 x^{5}-2899485 x^{4}-4819932 x^{3}-6831172 x^{2}-15214664 x +15380984}{1155 \sqrt {-2 x +1}} \] Input:
int((2+3*x)^4*(3+5*x)^2/(1-2*x)^(3/2),x)
Output:
( - 212625*x**6 - 1146600*x**5 - 2899485*x**4 - 4819932*x**3 - 6831172*x** 2 - 15214664*x + 15380984)/(1155*sqrt( - 2*x + 1))