Integrand size = 24, antiderivative size = 79 \[ \int \frac {(2+3 x)^2 (3+5 x)^3}{\sqrt {1-2 x}} \, dx=-\frac {65219}{32} \sqrt {1-2 x}+\frac {48279}{32} (1-2 x)^{3/2}-\frac {64317}{80} (1-2 x)^{5/2}+\frac {28555}{112} (1-2 x)^{7/2}-\frac {4225}{96} (1-2 x)^{9/2}+\frac {1125}{352} (1-2 x)^{11/2} \] Output:
-65219/32*(1-2*x)^(1/2)+48279/32*(1-2*x)^(3/2)-64317/80*(1-2*x)^(5/2)+2855 5/112*(1-2*x)^(7/2)-4225/96*(1-2*x)^(9/2)+1125/352*(1-2*x)^(11/2)
Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.48 \[ \int \frac {(2+3 x)^2 (3+5 x)^3}{\sqrt {1-2 x}} \, dx=-\frac {\sqrt {1-2 x} \left (1292672+1167932 x+1252938 x^2+1024475 x^3+518000 x^4+118125 x^5\right )}{1155} \] Input:
Integrate[((2 + 3*x)^2*(3 + 5*x)^3)/Sqrt[1 - 2*x],x]
Output:
-1/1155*(Sqrt[1 - 2*x]*(1292672 + 1167932*x + 1252938*x^2 + 1024475*x^3 + 518000*x^4 + 118125*x^5))
Time = 0.18 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, 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.
\(\displaystyle \int \frac {(3 x+2)^2 (5 x+3)^3}{\sqrt {1-2 x}} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (-\frac {1125}{32} (1-2 x)^{9/2}+\frac {12675}{32} (1-2 x)^{7/2}-\frac {28555}{16} (1-2 x)^{5/2}+\frac {64317}{16} (1-2 x)^{3/2}-\frac {144837}{32} \sqrt {1-2 x}+\frac {65219}{32 \sqrt {1-2 x}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1125}{352} (1-2 x)^{11/2}-\frac {4225}{96} (1-2 x)^{9/2}+\frac {28555}{112} (1-2 x)^{7/2}-\frac {64317}{80} (1-2 x)^{5/2}+\frac {48279}{32} (1-2 x)^{3/2}-\frac {65219}{32} \sqrt {1-2 x}\) |
Input:
Int[((2 + 3*x)^2*(3 + 5*x)^3)/Sqrt[1 - 2*x],x]
Output:
(-65219*Sqrt[1 - 2*x])/32 + (48279*(1 - 2*x)^(3/2))/32 - (64317*(1 - 2*x)^ (5/2))/80 + (28555*(1 - 2*x)^(7/2))/112 - (4225*(1 - 2*x)^(9/2))/96 + (112 5*(1 - 2*x)^(11/2))/352
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 = 34, normalized size of antiderivative = 0.43
method | result | size |
trager | \(\left (-\frac {1125}{11} x^{5}-\frac {14800}{33} x^{4}-\frac {204895}{231} x^{3}-\frac {417646}{385} x^{2}-\frac {1167932}{1155} x -\frac {1292672}{1155}\right ) \sqrt {1-2 x}\) | \(34\) |
gosper | \(-\frac {\sqrt {1-2 x}\, \left (118125 x^{5}+518000 x^{4}+1024475 x^{3}+1252938 x^{2}+1167932 x +1292672\right )}{1155}\) | \(35\) |
pseudoelliptic | \(-\frac {\sqrt {1-2 x}\, \left (118125 x^{5}+518000 x^{4}+1024475 x^{3}+1252938 x^{2}+1167932 x +1292672\right )}{1155}\) | \(35\) |
risch | \(\frac {\left (-1+2 x \right ) \left (118125 x^{5}+518000 x^{4}+1024475 x^{3}+1252938 x^{2}+1167932 x +1292672\right )}{1155 \sqrt {1-2 x}}\) | \(40\) |
orering | \(\frac {\left (-1+2 x \right ) \left (118125 x^{5}+518000 x^{4}+1024475 x^{3}+1252938 x^{2}+1167932 x +1292672\right )}{1155 \sqrt {1-2 x}}\) | \(40\) |
derivativedivides | \(-\frac {65219 \sqrt {1-2 x}}{32}+\frac {48279 \left (1-2 x \right )^{\frac {3}{2}}}{32}-\frac {64317 \left (1-2 x \right )^{\frac {5}{2}}}{80}+\frac {28555 \left (1-2 x \right )^{\frac {7}{2}}}{112}-\frac {4225 \left (1-2 x \right )^{\frac {9}{2}}}{96}+\frac {1125 \left (1-2 x \right )^{\frac {11}{2}}}{352}\) | \(56\) |
default | \(-\frac {65219 \sqrt {1-2 x}}{32}+\frac {48279 \left (1-2 x \right )^{\frac {3}{2}}}{32}-\frac {64317 \left (1-2 x \right )^{\frac {5}{2}}}{80}+\frac {28555 \left (1-2 x \right )^{\frac {7}{2}}}{112}-\frac {4225 \left (1-2 x \right )^{\frac {9}{2}}}{96}+\frac {1125 \left (1-2 x \right )^{\frac {11}{2}}}{352}\) | \(56\) |
meijerg | \(-\frac {54 \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {1-2 x}\right )}{\sqrt {\pi }}+\frac {288 \sqrt {\pi }-36 \sqrt {\pi }\, \left (8 x +8\right ) \sqrt {1-2 x}}{\sqrt {\pi }}-\frac {2763 \left (-\frac {16 \sqrt {\pi }}{15}+\frac {\sqrt {\pi }\, \left (24 x^{2}+16 x +16\right ) \sqrt {1-2 x}}{15}\right )}{8 \sqrt {\pi }}+\frac {\frac {1766 \sqrt {\pi }}{7}-\frac {883 \sqrt {\pi }\, \left (320 x^{3}+192 x^{2}+128 x +128\right ) \sqrt {1-2 x}}{448}}{\sqrt {\pi }}-\frac {3525 \left (-\frac {256 \sqrt {\pi }}{315}+\frac {\sqrt {\pi }\, \left (1120 x^{4}+640 x^{3}+384 x^{2}+256 x +256\right ) \sqrt {1-2 x}}{315}\right )}{32 \sqrt {\pi }}+\frac {\frac {1000 \sqrt {\pi }}{77}-\frac {125 \sqrt {\pi }\, \left (8064 x^{5}+4480 x^{4}+2560 x^{3}+1536 x^{2}+1024 x +1024\right ) \sqrt {1-2 x}}{9856}}{\sqrt {\pi }}\) | \(215\) |
Input:
int((2+3*x)^2*(3+5*x)^3/(1-2*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
(-1125/11*x^5-14800/33*x^4-204895/231*x^3-417646/385*x^2-1167932/1155*x-12 92672/1155)*(1-2*x)^(1/2)
Time = 0.07 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.43 \[ \int \frac {(2+3 x)^2 (3+5 x)^3}{\sqrt {1-2 x}} \, dx=-\frac {1}{1155} \, {\left (118125 \, x^{5} + 518000 \, x^{4} + 1024475 \, x^{3} + 1252938 \, x^{2} + 1167932 \, x + 1292672\right )} \sqrt {-2 \, x + 1} \] Input:
integrate((2+3*x)^2*(3+5*x)^3/(1-2*x)^(1/2),x, algorithm="fricas")
Output:
-1/1155*(118125*x^5 + 518000*x^4 + 1024475*x^3 + 1252938*x^2 + 1167932*x + 1292672)*sqrt(-2*x + 1)
Time = 0.66 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.89 \[ \int \frac {(2+3 x)^2 (3+5 x)^3}{\sqrt {1-2 x}} \, dx=\frac {1125 \left (1 - 2 x\right )^{\frac {11}{2}}}{352} - \frac {4225 \left (1 - 2 x\right )^{\frac {9}{2}}}{96} + \frac {28555 \left (1 - 2 x\right )^{\frac {7}{2}}}{112} - \frac {64317 \left (1 - 2 x\right )^{\frac {5}{2}}}{80} + \frac {48279 \left (1 - 2 x\right )^{\frac {3}{2}}}{32} - \frac {65219 \sqrt {1 - 2 x}}{32} \] Input:
integrate((2+3*x)**2*(3+5*x)**3/(1-2*x)**(1/2),x)
Output:
1125*(1 - 2*x)**(11/2)/352 - 4225*(1 - 2*x)**(9/2)/96 + 28555*(1 - 2*x)**( 7/2)/112 - 64317*(1 - 2*x)**(5/2)/80 + 48279*(1 - 2*x)**(3/2)/32 - 65219*s qrt(1 - 2*x)/32
Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.70 \[ \int \frac {(2+3 x)^2 (3+5 x)^3}{\sqrt {1-2 x}} \, dx=\frac {1125}{352} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {4225}{96} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {28555}{112} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {64317}{80} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {48279}{32} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {65219}{32} \, \sqrt {-2 \, x + 1} \] Input:
integrate((2+3*x)^2*(3+5*x)^3/(1-2*x)^(1/2),x, algorithm="maxima")
Output:
1125/352*(-2*x + 1)^(11/2) - 4225/96*(-2*x + 1)^(9/2) + 28555/112*(-2*x + 1)^(7/2) - 64317/80*(-2*x + 1)^(5/2) + 48279/32*(-2*x + 1)^(3/2) - 65219/3 2*sqrt(-2*x + 1)
Time = 0.13 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.05 \[ \int \frac {(2+3 x)^2 (3+5 x)^3}{\sqrt {1-2 x}} \, dx=-\frac {1125}{352} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {4225}{96} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {28555}{112} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {64317}{80} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {48279}{32} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {65219}{32} \, \sqrt {-2 \, x + 1} \] Input:
integrate((2+3*x)^2*(3+5*x)^3/(1-2*x)^(1/2),x, algorithm="giac")
Output:
-1125/352*(2*x - 1)^5*sqrt(-2*x + 1) - 4225/96*(2*x - 1)^4*sqrt(-2*x + 1) - 28555/112*(2*x - 1)^3*sqrt(-2*x + 1) - 64317/80*(2*x - 1)^2*sqrt(-2*x + 1) + 48279/32*(-2*x + 1)^(3/2) - 65219/32*sqrt(-2*x + 1)
Time = 0.02 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.70 \[ \int \frac {(2+3 x)^2 (3+5 x)^3}{\sqrt {1-2 x}} \, dx=\frac {48279\,{\left (1-2\,x\right )}^{3/2}}{32}-\frac {65219\,\sqrt {1-2\,x}}{32}-\frac {64317\,{\left (1-2\,x\right )}^{5/2}}{80}+\frac {28555\,{\left (1-2\,x\right )}^{7/2}}{112}-\frac {4225\,{\left (1-2\,x\right )}^{9/2}}{96}+\frac {1125\,{\left (1-2\,x\right )}^{11/2}}{352} \] Input:
int(((3*x + 2)^2*(5*x + 3)^3)/(1 - 2*x)^(1/2),x)
Output:
(48279*(1 - 2*x)^(3/2))/32 - (65219*(1 - 2*x)^(1/2))/32 - (64317*(1 - 2*x) ^(5/2))/80 + (28555*(1 - 2*x)^(7/2))/112 - (4225*(1 - 2*x)^(9/2))/96 + (11 25*(1 - 2*x)^(11/2))/352
Time = 0.16 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.42 \[ \int \frac {(2+3 x)^2 (3+5 x)^3}{\sqrt {1-2 x}} \, dx=\frac {\sqrt {-2 x +1}\, \left (-118125 x^{5}-518000 x^{4}-1024475 x^{3}-1252938 x^{2}-1167932 x -1292672\right )}{1155} \] Input:
int((2+3*x)^2*(3+5*x)^3/(1-2*x)^(1/2),x)
Output:
(sqrt( - 2*x + 1)*( - 118125*x**5 - 518000*x**4 - 1024475*x**3 - 1252938*x **2 - 1167932*x - 1292672))/1155