\(\int \sqrt {1-2 x} (2+3 x)^4 (3+5 x)^3 \, dx\) [1815]

Optimal result

Integrand size = 24, antiderivative size = 105 \[ \int \sqrt {1-2 x} (2+3 x)^4 (3+5 x)^3 \, dx=-\frac {3195731}{384} (1-2 x)^{3/2}+\frac {9836211}{640} (1-2 x)^{5/2}-\frac {1853313}{128} (1-2 x)^{7/2}+\frac {9504551 (1-2 x)^{9/2}}{1152}-\frac {4177401 (1-2 x)^{11/2}}{1408}+\frac {1101465 (1-2 x)^{13/2}}{1664}-\frac {10755}{128} (1-2 x)^{15/2}+\frac {10125 (1-2 x)^{17/2}}{2176} \]

[Out]

-3195731/384*(1-2*x)^(3/2)+9836211/640*(1-2*x)^(5/2)-1853313/128*(1-2*x)^(7/2)+9504551/1152*(1-2*x)^(9/2)-4177
401/1408*(1-2*x)^(11/2)+1101465/1664*(1-2*x)^(13/2)-10755/128*(1-2*x)^(15/2)+10125/2176*(1-2*x)^(17/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {90} \[ \int \sqrt {1-2 x} (2+3 x)^4 (3+5 x)^3 \, dx=\frac {10125 (1-2 x)^{17/2}}{2176}-\frac {10755}{128} (1-2 x)^{15/2}+\frac {1101465 (1-2 x)^{13/2}}{1664}-\frac {4177401 (1-2 x)^{11/2}}{1408}+\frac {9504551 (1-2 x)^{9/2}}{1152}-\frac {1853313}{128} (1-2 x)^{7/2}+\frac {9836211}{640} (1-2 x)^{5/2}-\frac {3195731}{384} (1-2 x)^{3/2} \]

[In]

Int[Sqrt[1 - 2*x]*(2 + 3*x)^4*(3 + 5*x)^3,x]

[Out]

(-3195731*(1 - 2*x)^(3/2))/384 + (9836211*(1 - 2*x)^(5/2))/640 - (1853313*(1 - 2*x)^(7/2))/128 + (9504551*(1 -
2*x)^(9/2))/1152 - (4177401*(1 - 2*x)^(11/2))/1408 + (1101465*(1 - 2*x)^(13/2))/1664 - (10755*(1 - 2*x)^(15/2
))/128 + (10125*(1 - 2*x)^(17/2))/2176

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(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]))

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3195731}{128} \sqrt {1-2 x}-\frac {9836211}{128} (1-2 x)^{3/2}+\frac {12973191}{128} (1-2 x)^{5/2}-\frac {9504551}{128} (1-2 x)^{7/2}+\frac {4177401}{128} (1-2 x)^{9/2}-\frac {1101465}{128} (1-2 x)^{11/2}+\frac {161325}{128} (1-2 x)^{13/2}-\frac {10125}{128} (1-2 x)^{15/2}\right ) \, dx \\ & = -\frac {3195731}{384} (1-2 x)^{3/2}+\frac {9836211}{640} (1-2 x)^{5/2}-\frac {1853313}{128} (1-2 x)^{7/2}+\frac {9504551 (1-2 x)^{9/2}}{1152}-\frac {4177401 (1-2 x)^{11/2}}{1408}+\frac {1101465 (1-2 x)^{13/2}}{1664}-\frac {10755}{128} (1-2 x)^{15/2}+\frac {10125 (1-2 x)^{17/2}}{2176} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.46 \[ \int \sqrt {1-2 x} (2+3 x)^4 (3+5 x)^3 \, dx=-\frac {(1-2 x)^{3/2} \left (171312832+466679856 x+906777120 x^2+1299289000 x^3+1320982290 x^4+894452625 x^5+360231300 x^6+65154375 x^7\right )}{109395} \]

[In]

Integrate[Sqrt[1 - 2*x]*(2 + 3*x)^4*(3 + 5*x)^3,x]

[Out]

-1/109395*((1 - 2*x)^(3/2)*(171312832 + 466679856*x + 906777120*x^2 + 1299289000*x^3 + 1320982290*x^4 + 894452
625*x^5 + 360231300*x^6 + 65154375*x^7))

Maple [A] (verified)

Time = 0.97 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.43

method	result	size
gosper	\(-\frac {\left (1-2 x \right )^{\frac {3}{2}} \left (65154375 x^{7}[phone] x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right )}{109395}\)	\(45\)
trager	\(\left (\frac {20250}{17} x^{8}+\frac {101835}{17} x^{7}+\frac {2886210}{221} x^{6}+\frac {38833599}{2431} x^{5}+\frac {255519142}{21879} x^{4}+\frac {102853048}{21879} x^{3}+\frac {8860864}{36465} x^{2}-\frac {124054192}{109395} x -\frac {171312832}{109395}\right ) \sqrt {1-2 x}\)	\(49\)
pseudoelliptic	\(\frac {\sqrt {1-2 x}\, \left (130308750 x^{8}[phone] x^{7}[phone] x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}-124054192 x -171312832\right )}{109395}\)	\(50\)
risch	\(-\frac {\left (130308750 x^{8}[phone] x^{7}[phone] x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}-124054192 x -171312832\right ) \left (-1+2 x \right )}{109395 \sqrt {1-2 x}}\)	\(55\)
derivativedivides	\(-\frac {3195731 \left (1-2 x \right )^{\frac {3}{2}}}{384}+\frac {9836211 \left (1-2 x \right )^{\frac {5}{2}}}{640}-\frac {1853313 \left (1-2 x \right )^{\frac {7}{2}}}{128}+\frac {9504551 \left (1-2 x \right )^{\frac {9}{2}}}{1152}-\frac {4177401 \left (1-2 x \right )^{\frac {11}{2}}}{1408}+\frac {1101465 \left (1-2 x \right )^{\frac {13}{2}}}{1664}-\frac {10755 \left (1-2 x \right )^{\frac {15}{2}}}{128}+\frac {10125 \left (1-2 x \right )^{\frac {17}{2}}}{2176}\)	\(74\)
default	\(-\frac {3195731 \left (1-2 x \right )^{\frac {3}{2}}}{384}+\frac {9836211 \left (1-2 x \right )^{\frac {5}{2}}}{640}-\frac {1853313 \left (1-2 x \right )^{\frac {7}{2}}}{128}+\frac {9504551 \left (1-2 x \right )^{\frac {9}{2}}}{1152}-\frac {4177401 \left (1-2 x \right )^{\frac {11}{2}}}{1408}+\frac {1101465 \left (1-2 x \right )^{\frac {13}{2}}}{1664}-\frac {10755 \left (1-2 x \right )^{\frac {15}{2}}}{128}+\frac {10125 \left (1-2 x \right )^{\frac {17}{2}}}{2176}\)	\(74\)
meijerg	\(\frac {144 \sqrt {\pi }-72 \sqrt {\pi }\, \left (2-4 x \right ) \sqrt {1-2 x}}{\sqrt {\pi }}-\frac {594 \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (6 x +2\right )}{15}\right )}{\sqrt {\pi }}+\frac {\frac {14928 \sqrt {\pi }}{35}-\frac {1866 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (60 x^{2}+24 x +8\right )}{35}}{\sqrt {\pi }}-\frac {1831 \left (-\frac {64 \sqrt {\pi }}{315}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (280 x^{3}+120 x^{2}+48 x +16\right )}{315}\right )}{\sqrt {\pi }}+\frac {\frac {245192 \sqrt {\pi }}{1155}-\frac {30649 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (5040 x^{4}+2240 x^{3}+960 x^{2}+384 x +128\right )}{18480}}{\sqrt {\pi }}-\frac {86535 \left (-\frac {1024 \sqrt {\pi }}{9009}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (22176 x^{5}+10080 x^{4}+4480 x^{3}+1920 x^{2}+768 x +256\right )}{9009}\right )}{128 \sqrt {\pi }}+\frac {\frac {16080 \sqrt {\pi }}{1001}-\frac {1005 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (192192 x^{6}+88704 x^{5}+40320 x^{4}+17920 x^{3}+7680 x^{2}+3072 x +1024\right )}{64064}}{\sqrt {\pi }}-\frac {10125 \left (-\frac {8192 \sqrt {\pi }}{109395}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (823680 x^{7}+384384 x^{6}+177408 x^{5}+80640 x^{4}+35840 x^{3}+15360 x^{2}+6144 x +2048\right )}{109395}\right )}{512 \sqrt {\pi }}\)	\(331\)

[In]

int((2+3*x)^4*(3+5*x)^3*(1-2*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/109395*(1-2*x)^(3/2)*(65154375*x^7+360231300*x^6+894452625*x^5+1320982290*x^4+1299289000*x^3+906777120*x^2+
466679856*x+171312832)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.47 \[ \int \sqrt {1-2 x} (2+3 x)^4 (3+5 x)^3 \, dx=\frac {1}{109395} \, {\left (130308750 \, x^{8} + 655308225 \, x^{7} + 1428673950 \, x^{6} + 1747511955 \, x^{5} + 1277595710 \, x^{4} + 514265240 \, x^{3} + 26582592 \, x^{2} - 124054192 \, x - 171312832\right )} \sqrt {-2 \, x + 1} \]

[In]

integrate((2+3*x)^4*(3+5*x)^3*(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

1/109395*(130308750*x^8 + 655308225*x^7 + 1428673950*x^6 + 1747511955*x^5 + 1277595710*x^4 + 514265240*x^3 + 2
6582592*x^2 - 124054192*x - 171312832)*sqrt(-2*x + 1)

Sympy [A] (verification not implemented)

Time = 0.76 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int \sqrt {1-2 x} (2+3 x)^4 (3+5 x)^3 \, dx=\frac {10125 \left (1 - 2 x\right )^{\frac {17}{2}}}{2176} - \frac {10755 \left (1 - 2 x\right )^{\frac {15}{2}}}{128} + \frac {1101465 \left (1 - 2 x\right )^{\frac {13}{2}}}{1664} - \frac {4177401 \left (1 - 2 x\right )^{\frac {11}{2}}}{1408} + \frac {9504551 \left (1 - 2 x\right )^{\frac {9}{2}}}{1152} - \frac {1853313 \left (1 - 2 x\right )^{\frac {7}{2}}}{128} + \frac {9836211 \left (1 - 2 x\right )^{\frac {5}{2}}}{640} - \frac {3195731 \left (1 - 2 x\right )^{\frac {3}{2}}}{384} \]

[In]

integrate((2+3*x)**4*(3+5*x)**3*(1-2*x)**(1/2),x)

[Out]

10125*(1 - 2*x)**(17/2)/2176 - 10755*(1 - 2*x)**(15/2)/128 + 1101465*(1 - 2*x)**(13/2)/1664 - 4177401*(1 - 2*x
)**(11/2)/1408 + 9504551*(1 - 2*x)**(9/2)/1152 - 1853313*(1 - 2*x)**(7/2)/128 + 9836211*(1 - 2*x)**(5/2)/640 -
3195731*(1 - 2*x)**(3/2)/384

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.70 \[ \int \sqrt {1-2 x} (2+3 x)^4 (3+5 x)^3 \, dx=\frac {10125}{2176} \, {\left (-2 \, x + 1\right )}^{\frac {17}{2}} - \frac {10755}{128} \, {\left (-2 \, x + 1\right )}^{\frac {15}{2}} + \frac {1101465}{1664} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - \frac {4177401}{1408} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {9504551}{1152} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {1853313}{128} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {9836211}{640} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {3195731}{384} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \]

[In]

integrate((2+3*x)^4*(3+5*x)^3*(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

10125/2176*(-2*x + 1)^(17/2) - 10755/128*(-2*x + 1)^(15/2) + 1101465/1664*(-2*x + 1)^(13/2) - 4177401/1408*(-2
*x + 1)^(11/2) + 9504551/1152*(-2*x + 1)^(9/2) - 1853313/128*(-2*x + 1)^(7/2) + 9836211/640*(-2*x + 1)^(5/2) -
3195731/384*(-2*x + 1)^(3/2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.16 \[ \int \sqrt {1-2 x} (2+3 x)^4 (3+5 x)^3 \, dx=\frac {10125}{2176} \, {\left (2 \, x - 1\right )}^{8} \sqrt {-2 \, x + 1} + \frac {10755}{128} \, {\left (2 \, x - 1\right )}^{7} \sqrt {-2 \, x + 1} + \frac {1101465}{1664} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + \frac {4177401}{1408} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {9504551}{1152} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {1853313}{128} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {9836211}{640} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {3195731}{384} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \]

[In]

integrate((2+3*x)^4*(3+5*x)^3*(1-2*x)^(1/2),x, algorithm="giac")

[Out]

10125/2176*(2*x - 1)^8*sqrt(-2*x + 1) + 10755/128*(2*x - 1)^7*sqrt(-2*x + 1) + 1101465/1664*(2*x - 1)^6*sqrt(-
2*x + 1) + 4177401/1408*(2*x - 1)^5*sqrt(-2*x + 1) + 9504551/1152*(2*x - 1)^4*sqrt(-2*x + 1) + 1853313/128*(2*
x - 1)^3*sqrt(-2*x + 1) + 9836211/640*(2*x - 1)^2*sqrt(-2*x + 1) - 3195731/384*(-2*x + 1)^(3/2)

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.70 \[ \int \sqrt {1-2 x} (2+3 x)^4 (3+5 x)^3 \, dx=\frac {9836211\,{\left (1-2\,x\right )}^{5/2}}{640}-\frac {3195731\,{\left (1-2\,x\right )}^{3/2}}{384}-\frac {1853313\,{\left (1-2\,x\right )}^{7/2}}{128}+\frac {9504551\,{\left (1-2\,x\right )}^{9/2}}{1152}-\frac {4177401\,{\left (1-2\,x\right )}^{11/2}}{1408}+\frac {1101465\,{\left (1-2\,x\right )}^{13/2}}{1664}-\frac {10755\,{\left (1-2\,x\right )}^{15/2}}{128}+\frac {10125\,{\left (1-2\,x\right )}^{17/2}}{2176} \]

[In]

int((1 - 2*x)^(1/2)*(3*x + 2)^4*(5*x + 3)^3,x)

[Out]

(9836211*(1 - 2*x)^(5/2))/640 - (3195731*(1 - 2*x)^(3/2))/384 - (1853313*(1 - 2*x)^(7/2))/128 + (9504551*(1 -
2*x)^(9/2))/1152 - (4177401*(1 - 2*x)^(11/2))/1408 + (1101465*(1 - 2*x)^(13/2))/1664 - (10755*(1 - 2*x)^(15/2)
)/128 + (10125*(1 - 2*x)^(17/2))/2176