Integrand size = 24, antiderivative size = 92 \[ \int \frac {(2+3 x)^4 (3+5 x)^2}{\sqrt {1-2 x}} \, dx=-\frac {290521}{64} \sqrt {1-2 x}+\frac {381073}{96} (1-2 x)^{3/2}-\frac {832951}{320} (1-2 x)^{5/2}+\frac {17337}{16} (1-2 x)^{7/2}-\frac {17679}{64} (1-2 x)^{9/2}+\frac {13905}{352} (1-2 x)^{11/2}-\frac {2025}{832} (1-2 x)^{13/2} \]
381073/96*(1-2*x)^(3/2)-832951/320*(1-2*x)^(5/2)+17337/16*(1-2*x)^(7/2)-17 679/64*(1-2*x)^(9/2)+13905/352*(1-2*x)^(11/2)-2025/832*(1-2*x)^(13/2)-2905 21/64*(1-2*x)^(1/2)
Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.47 \[ \int \frac {(2+3 x)^4 (3+5 x)^2}{\sqrt {1-2 x}} \, dx=-\frac {\sqrt {1-2 x} \left (4994536+4685656 x+5587044 x^2+5576580 x^3+3954645 x^4+1709100 x^5+334125 x^6\right )}{2145} \]
-1/2145*(Sqrt[1 - 2*x]*(4994536 + 4685656*x + 5587044*x^2 + 5576580*x^3 + 3954645*x^4 + 1709100*x^5 + 334125*x^6))
Time = 0.19 (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.
\(\displaystyle \int \frac {(3 x+2)^4 (5 x+3)^2}{\sqrt {1-2 x}} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (\frac {2025}{64} (1-2 x)^{11/2}-\frac {13905}{32} (1-2 x)^{9/2}+\frac {159111}{64} (1-2 x)^{7/2}-\frac {121359}{16} (1-2 x)^{5/2}+\frac {832951}{64} (1-2 x)^{3/2}-\frac {381073}{32} \sqrt {1-2 x}+\frac {290521}{64 \sqrt {1-2 x}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2025}{832} (1-2 x)^{13/2}+\frac {13905}{352} (1-2 x)^{11/2}-\frac {17679}{64} (1-2 x)^{9/2}+\frac {17337}{16} (1-2 x)^{7/2}-\frac {832951}{320} (1-2 x)^{5/2}+\frac {381073}{96} (1-2 x)^{3/2}-\frac {290521}{64} \sqrt {1-2 x}\) |
(-290521*Sqrt[1 - 2*x])/64 + (381073*(1 - 2*x)^(3/2))/96 - (832951*(1 - 2* x)^(5/2))/320 + (17337*(1 - 2*x)^(7/2))/16 - (17679*(1 - 2*x)^(9/2))/64 + (13905*(1 - 2*x)^(11/2))/352 - (2025*(1 - 2*x)^(13/2))/832
3.21.13.3.1 Defintions of rubi rules used
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 = 1.03 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.42
method | result | size |
trager | \(\left (-\frac {2025}{13} x^{6}-\frac {113940}{143} x^{5}-\frac {263643}{143} x^{4}-\frac {371772}{143} x^{3}-\frac {1862348}{715} x^{2}-\frac {4685656}{2145} x -\frac {4994536}{2145}\right ) \sqrt {1-2 x}\) | \(39\) |
gosper | \(-\frac {\sqrt {1-2 x}\, \left (334125 x^{6}+1709100 x^{5}+3954645 x^{4}+5576580 x^{3}+5587044 x^{2}+4685656 x +4994536\right )}{2145}\) | \(40\) |
pseudoelliptic | \(-\frac {\sqrt {1-2 x}\, \left (334125 x^{6}+1709100 x^{5}+3954645 x^{4}+5576580 x^{3}+5587044 x^{2}+4685656 x +4994536\right )}{2145}\) | \(40\) |
risch | \(\frac {\left (-1+2 x \right ) \left (334125 x^{6}+1709100 x^{5}+3954645 x^{4}+5576580 x^{3}+5587044 x^{2}+4685656 x +4994536\right )}{2145 \sqrt {1-2 x}}\) | \(45\) |
derivativedivides | \(\frac {381073 \left (1-2 x \right )^{\frac {3}{2}}}{96}-\frac {832951 \left (1-2 x \right )^{\frac {5}{2}}}{320}+\frac {17337 \left (1-2 x \right )^{\frac {7}{2}}}{16}-\frac {17679 \left (1-2 x \right )^{\frac {9}{2}}}{64}+\frac {13905 \left (1-2 x \right )^{\frac {11}{2}}}{352}-\frac {2025 \left (1-2 x \right )^{\frac {13}{2}}}{832}-\frac {290521 \sqrt {1-2 x}}{64}\) | \(65\) |
default | \(\frac {381073 \left (1-2 x \right )^{\frac {3}{2}}}{96}-\frac {832951 \left (1-2 x \right )^{\frac {5}{2}}}{320}+\frac {17337 \left (1-2 x \right )^{\frac {7}{2}}}{16}-\frac {17679 \left (1-2 x \right )^{\frac {9}{2}}}{64}+\frac {13905 \left (1-2 x \right )^{\frac {11}{2}}}{352}-\frac {2025 \left (1-2 x \right )^{\frac {13}{2}}}{832}-\frac {290521 \sqrt {1-2 x}}{64}\) | \(65\) |
meijerg | \(-\frac {72 \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {1-2 x}\right )}{\sqrt {\pi }}+\frac {448 \sqrt {\pi }-56 \sqrt {\pi }\, \left (8 x +8\right ) \sqrt {1-2 x}}{\sqrt {\pi }}-\frac {653 \left (-\frac {16 \sqrt {\pi }}{15}+\frac {\sqrt {\pi }\, \left (24 x^{2}+16 x +16\right ) \sqrt {1-2 x}}{15}\right )}{\sqrt {\pi }}+\frac {\frac {21648 \sqrt {\pi }}{35}-\frac {1353 \sqrt {\pi }\, \left (320 x^{3}+192 x^{2}+128 x +128\right ) \sqrt {1-2 x}}{280}}{\sqrt {\pi }}-\frac {12609 \left (-\frac {256 \sqrt {\pi }}{315}+\frac {\sqrt {\pi }\, \left (1120 x^{4}+640 x^{3}+384 x^{2}+256 x +256\right ) \sqrt {1-2 x}}{315}\right )}{32 \sqrt {\pi }}+\frac {\frac {6960 \sqrt {\pi }}{77}-\frac {435 \sqrt {\pi }\, \left (8064 x^{5}+4480 x^{4}+2560 x^{3}+1536 x^{2}+1024 x +1024\right ) \sqrt {1-2 x}}{4928}}{\sqrt {\pi }}-\frac {2025 \left (-\frac {2048 \sqrt {\pi }}{3003}+\frac {\sqrt {\pi }\, \left (29568 x^{6}+16128 x^{5}+8960 x^{4}+5120 x^{3}+3072 x^{2}+2048 x +2048\right ) \sqrt {1-2 x}}{3003}\right )}{128 \sqrt {\pi }}\) | \(268\) |
(-2025/13*x^6-113940/143*x^5-263643/143*x^4-371772/143*x^3-1862348/715*x^2 -4685656/2145*x-4994536/2145)*(1-2*x)^(1/2)
Time = 0.22 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.42 \[ \int \frac {(2+3 x)^4 (3+5 x)^2}{\sqrt {1-2 x}} \, dx=-\frac {1}{2145} \, {\left (334125 \, x^{6} + 1709100 \, x^{5} + 3954645 \, x^{4} + 5576580 \, x^{3} + 5587044 \, x^{2} + 4685656 \, x + 4994536\right )} \sqrt {-2 \, x + 1} \]
-1/2145*(334125*x^6 + 1709100*x^5 + 3954645*x^4 + 5576580*x^3 + 5587044*x^ 2 + 4685656*x + 4994536)*sqrt(-2*x + 1)
Time = 0.74 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.89 \[ \int \frac {(2+3 x)^4 (3+5 x)^2}{\sqrt {1-2 x}} \, dx=- \frac {2025 \left (1 - 2 x\right )^{\frac {13}{2}}}{832} + \frac {13905 \left (1 - 2 x\right )^{\frac {11}{2}}}{352} - \frac {17679 \left (1 - 2 x\right )^{\frac {9}{2}}}{64} + \frac {17337 \left (1 - 2 x\right )^{\frac {7}{2}}}{16} - \frac {832951 \left (1 - 2 x\right )^{\frac {5}{2}}}{320} + \frac {381073 \left (1 - 2 x\right )^{\frac {3}{2}}}{96} - \frac {290521 \sqrt {1 - 2 x}}{64} \]
-2025*(1 - 2*x)**(13/2)/832 + 13905*(1 - 2*x)**(11/2)/352 - 17679*(1 - 2*x )**(9/2)/64 + 17337*(1 - 2*x)**(7/2)/16 - 832951*(1 - 2*x)**(5/2)/320 + 38 1073*(1 - 2*x)**(3/2)/96 - 290521*sqrt(1 - 2*x)/64
Time = 0.21 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.70 \[ \int \frac {(2+3 x)^4 (3+5 x)^2}{\sqrt {1-2 x}} \, dx=-\frac {2025}{832} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} + \frac {13905}{352} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {17679}{64} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {17337}{16} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {832951}{320} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {381073}{96} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {290521}{64} \, \sqrt {-2 \, x + 1} \]
-2025/832*(-2*x + 1)^(13/2) + 13905/352*(-2*x + 1)^(11/2) - 17679/64*(-2*x + 1)^(9/2) + 17337/16*(-2*x + 1)^(7/2) - 832951/320*(-2*x + 1)^(5/2) + 38 1073/96*(-2*x + 1)^(3/2) - 290521/64*sqrt(-2*x + 1)
Time = 0.28 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.08 \[ \int \frac {(2+3 x)^4 (3+5 x)^2}{\sqrt {1-2 x}} \, dx=-\frac {2025}{832} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} - \frac {13905}{352} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {17679}{64} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {17337}{16} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {832951}{320} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {381073}{96} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {290521}{64} \, \sqrt {-2 \, x + 1} \]
-2025/832*(2*x - 1)^6*sqrt(-2*x + 1) - 13905/352*(2*x - 1)^5*sqrt(-2*x + 1 ) - 17679/64*(2*x - 1)^4*sqrt(-2*x + 1) - 17337/16*(2*x - 1)^3*sqrt(-2*x + 1) - 832951/320*(2*x - 1)^2*sqrt(-2*x + 1) + 381073/96*(-2*x + 1)^(3/2) - 290521/64*sqrt(-2*x + 1)
Time = 0.03 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.70 \[ \int \frac {(2+3 x)^4 (3+5 x)^2}{\sqrt {1-2 x}} \, dx=\frac {381073\,{\left (1-2\,x\right )}^{3/2}}{96}-\frac {290521\,\sqrt {1-2\,x}}{64}-\frac {832951\,{\left (1-2\,x\right )}^{5/2}}{320}+\frac {17337\,{\left (1-2\,x\right )}^{7/2}}{16}-\frac {17679\,{\left (1-2\,x\right )}^{9/2}}{64}+\frac {13905\,{\left (1-2\,x\right )}^{11/2}}{352}-\frac {2025\,{\left (1-2\,x\right )}^{13/2}}{832} \]