3.20.56 \(\int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^3 \, dx\) [1956]

Optimal. Leaf size=92 \[ -\frac {65219}{64} (1-2 x)^{7/2}+\frac {100793}{48} (1-2 x)^{9/2}-\frac {121359}{64} (1-2 x)^{11/2}+\frac {98209}{104} (1-2 x)^{13/2}-\frac {17337}{64} (1-2 x)^{15/2}+\frac {675}{16} (1-2 x)^{17/2}-\frac {3375 (1-2 x)^{19/2}}{1216} \]

[Out]

-65219/64*(1-2*x)^(7/2)+100793/48*(1-2*x)^(9/2)-121359/64*(1-2*x)^(11/2)+98209/104*(1-2*x)^(13/2)-17337/64*(1-
2*x)^(15/2)+675/16*(1-2*x)^(17/2)-3375/1216*(1-2*x)^(19/2)

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Rubi [A]
time = 0.01, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {90} \begin {gather*} -\frac {3375 (1-2 x)^{19/2}}{1216}+\frac {675}{16} (1-2 x)^{17/2}-\frac {17337}{64} (1-2 x)^{15/2}+\frac {98209}{104} (1-2 x)^{13/2}-\frac {121359}{64} (1-2 x)^{11/2}+\frac {100793}{48} (1-2 x)^{9/2}-\frac {65219}{64} (1-2 x)^{7/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-65219*(1 - 2*x)^(7/2))/64 + (100793*(1 - 2*x)^(9/2))/48 - (121359*(1 - 2*x)^(11/2))/64 + (98209*(1 - 2*x)^(1
3/2))/104 - (17337*(1 - 2*x)^(15/2))/64 + (675*(1 - 2*x)^(17/2))/16 - (3375*(1 - 2*x)^(19/2))/1216

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*} \int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^3 \, dx &=\int \left (\frac {456533}{64} (1-2 x)^{5/2}-\frac {302379}{16} (1-2 x)^{7/2}+\frac {1334949}{64} (1-2 x)^{9/2}-\frac {98209}{8} (1-2 x)^{11/2}+\frac {260055}{64} (1-2 x)^{13/2}-\frac {11475}{16} (1-2 x)^{15/2}+\frac {3375}{64} (1-2 x)^{17/2}\right ) \, dx\\ &=-\frac {65219}{64} (1-2 x)^{7/2}+\frac {100793}{48} (1-2 x)^{9/2}-\frac {121359}{64} (1-2 x)^{11/2}+\frac {98209}{104} (1-2 x)^{13/2}-\frac {17337}{64} (1-2 x)^{15/2}+\frac {675}{16} (1-2 x)^{17/2}-\frac {3375 (1-2 x)^{19/2}}{1216}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 43, normalized size = 0.47 \begin {gather*} -\frac {1}{741} (1-2 x)^{7/2} \left (76018+372070 x+914049 x^2+1346367 x^3+1204398 x^4+605475 x^5+131625 x^6\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/741*((1 - 2*x)^(7/2)*(76018 + 372070*x + 914049*x^2 + 1346367*x^3 + 1204398*x^4 + 605475*x^5 + 131625*x^6))

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Maple [A]
time = 0.11, size = 65, normalized size = 0.71

method	result	size
gosper	\(-\frac {\left (131625 x^{6}+605475 x^{5}+1204398 x^{4}+1346367 x^{3}+914049 x^{2}+372070 x +76018\right ) \left (1-2 x \right )^{\frac {7}{2}}}{741}\)	\(40\)
trager	\(\left (\frac {27000}{19} x^{9}+\frac {83700}{19} x^{8}+\frac {81006}{19} x^{7}-\frac {60205}{247} x^{6}-\frac {741033}{247} x^{5}-\frac {1118224}{741} x^{4}+\frac {281231}{741} x^{3}+\frac {135385}{247} x^{2}+\frac {84038}{741} x -\frac {76018}{741}\right ) \sqrt {1-2 x}\)	\(54\)
risch	\(-\frac {\left (1053000 x^{9}+3264300 x^{8}+3159234 x^{7}-180615 x^{6}-2223099 x^{5}-1118224 x^{4}+281231 x^{3}+406155 x^{2}+84038 x -76018\right ) \left (-1+2 x \right )}{741 \sqrt {1-2 x}}\)	\(60\)
derivativedivides	\(-\frac {65219 \left (1-2 x \right )^{\frac {7}{2}}}{64}+\frac {100793 \left (1-2 x \right )^{\frac {9}{2}}}{48}-\frac {121359 \left (1-2 x \right )^{\frac {11}{2}}}{64}+\frac {98209 \left (1-2 x \right )^{\frac {13}{2}}}{104}-\frac {17337 \left (1-2 x \right )^{\frac {15}{2}}}{64}+\frac {675 \left (1-2 x \right )^{\frac {17}{2}}}{16}-\frac {3375 \left (1-2 x \right )^{\frac {19}{2}}}{1216}\)	\(65\)
default	\(-\frac {65219 \left (1-2 x \right )^{\frac {7}{2}}}{64}+\frac {100793 \left (1-2 x \right )^{\frac {9}{2}}}{48}-\frac {121359 \left (1-2 x \right )^{\frac {11}{2}}}{64}+\frac {98209 \left (1-2 x \right )^{\frac {13}{2}}}{104}-\frac {17337 \left (1-2 x \right )^{\frac {15}{2}}}{64}+\frac {675 \left (1-2 x \right )^{\frac {17}{2}}}{16}-\frac {3375 \left (1-2 x \right )^{\frac {19}{2}}}{1216}\)	\(65\)
meijerg	\(\frac {\frac {216 \sqrt {\pi }}{7}-\frac {108 \sqrt {\pi }\, \left (-16 x^{3}+24 x^{2}-12 x +2\right ) \sqrt {1-2 x}}{7}}{\sqrt {\pi }}-\frac {7695 \left (-\frac {32 \sqrt {\pi }}{945}+\frac {4 \sqrt {\pi }\, \left (-448 x^{4}+608 x^{3}-240 x^{2}+8 x +8\right ) \sqrt {1-2 x}}{945}\right )}{8 \sqrt {\pi }}+\frac {\frac {164 \sqrt {\pi }}{7}-\frac {41 \sqrt {\pi }\, \left (-4032 x^{5}+5152 x^{4}-1808 x^{3}+24 x^{2}+16 x +16\right ) \sqrt {1-2 x}}{28}}{\sqrt {\pi }}-\frac {256785 \left (-\frac {256 \sqrt {\pi }}{45045}+\frac {2 \sqrt {\pi }\, \left (-118272 x^{6}+145152 x^{5}-47488 x^{4}+320 x^{3}+192 x^{2}+128 x +128\right ) \sqrt {1-2 x}}{45045}\right )}{128 \sqrt {\pi }}+\frac {\frac {328 \sqrt {\pi }}{91}-\frac {41 \sqrt {\pi }\, \left (-768768 x^{7}+916608 x^{6}-286272 x^{5}+1120 x^{4}+640 x^{3}+384 x^{2}+256 x +256\right ) \sqrt {1-2 x}}{2912}}{\sqrt {\pi }}-\frac {192375 \left (-\frac {4096 \sqrt {\pi }}{2297295}+\frac {4 \sqrt {\pi }\, \left (-9225216 x^{8}[phone] x^{7}-3252480 x^{6}+8064 x^{5}+4480 x^{4}+2560 x^{3}+1536 x^{2}+1024 x +1024\right ) \sqrt {1-2 x}}{2297295}\right )}{512 \sqrt {\pi }}+\frac {\frac {18000 \sqrt {\pi }}{323323}-\frac {1125 \sqrt {\pi }\, \left (-52276224 x^{9}[phone] x^{8}-17681664 x^{7}+29568 x^{6}+16128 x^{5}+8960 x^{4}+5120 x^{3}+3072 x^{2}+2048 x +2048\right ) \sqrt {1-2 x}}{41385344}}{\sqrt {\pi }}\)	\(373\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-65219/64*(1-2*x)^(7/2)+100793/48*(1-2*x)^(9/2)-121359/64*(1-2*x)^(11/2)+98209/104*(1-2*x)^(13/2)-17337/64*(1-
2*x)^(15/2)+675/16*(1-2*x)^(17/2)-3375/1216*(1-2*x)^(19/2)

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Maxima [A]
time = 0.30, size = 64, normalized size = 0.70 \begin {gather*} -\frac {3375}{1216} \, {\left (-2 \, x + 1\right )}^{\frac {19}{2}} + \frac {675}{16} \, {\left (-2 \, x + 1\right )}^{\frac {17}{2}} - \frac {17337}{64} \, {\left (-2 \, x + 1\right )}^{\frac {15}{2}} + \frac {98209}{104} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - \frac {121359}{64} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {100793}{48} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {65219}{64} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3375/1216*(-2*x + 1)^(19/2) + 675/16*(-2*x + 1)^(17/2) - 17337/64*(-2*x + 1)^(15/2) + 98209/104*(-2*x + 1)^(1
3/2) - 121359/64*(-2*x + 1)^(11/2) + 100793/48*(-2*x + 1)^(9/2) - 65219/64*(-2*x + 1)^(7/2)

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Fricas [A]
time = 1.74, size = 54, normalized size = 0.59 \begin {gather*} \frac {1}{741} \, {\left (1053000 \, x^{9} + 3264300 \, x^{8} + 3159234 \, x^{7} - 180615 \, x^{6} - 2223099 \, x^{5} - 1118224 \, x^{4} + 281231 \, x^{3} + 406155 \, x^{2} + 84038 \, x - 76018\right )} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/741*(1053000*x^9 + 3264300*x^8 + 3159234*x^7 - 180615*x^6 - 2223099*x^5 - 1118224*x^4 + 281231*x^3 + 406155*
x^2 + 84038*x - 76018)*sqrt(-2*x + 1)

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Sympy [A]
time = 15.11, size = 82, normalized size = 0.89 \begin {gather*} - \frac {3375 \left (1 - 2 x\right )^{\frac {19}{2}}}{1216} + \frac {675 \left (1 - 2 x\right )^{\frac {17}{2}}}{16} - \frac {17337 \left (1 - 2 x\right )^{\frac {15}{2}}}{64} + \frac {98209 \left (1 - 2 x\right )^{\frac {13}{2}}}{104} - \frac {121359 \left (1 - 2 x\right )^{\frac {11}{2}}}{64} + \frac {100793 \left (1 - 2 x\right )^{\frac {9}{2}}}{48} - \frac {65219 \left (1 - 2 x\right )^{\frac {7}{2}}}{64} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3375*(1 - 2*x)**(19/2)/1216 + 675*(1 - 2*x)**(17/2)/16 - 17337*(1 - 2*x)**(15/2)/64 + 98209*(1 - 2*x)**(13/2)
/104 - 121359*(1 - 2*x)**(11/2)/64 + 100793*(1 - 2*x)**(9/2)/48 - 65219*(1 - 2*x)**(7/2)/64

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Giac [A]
time = 0.67, size = 113, normalized size = 1.23 \begin {gather*} \frac {3375}{1216} \, {\left (2 \, x - 1\right )}^{9} \sqrt {-2 \, x + 1} + \frac {675}{16} \, {\left (2 \, x - 1\right )}^{8} \sqrt {-2 \, x + 1} + \frac {17337}{64} \, {\left (2 \, x - 1\right )}^{7} \sqrt {-2 \, x + 1} + \frac {98209}{104} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + \frac {121359}{64} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {100793}{48} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {65219}{64} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3375/1216*(2*x - 1)^9*sqrt(-2*x + 1) + 675/16*(2*x - 1)^8*sqrt(-2*x + 1) + 17337/64*(2*x - 1)^7*sqrt(-2*x + 1)
+ 98209/104*(2*x - 1)^6*sqrt(-2*x + 1) + 121359/64*(2*x - 1)^5*sqrt(-2*x + 1) + 100793/48*(2*x - 1)^4*sqrt(-2
*x + 1) + 65219/64*(2*x - 1)^3*sqrt(-2*x + 1)

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Mupad [B]
time = 0.03, size = 64, normalized size = 0.70 \begin {gather*} \frac {100793\,{\left (1-2\,x\right )}^{9/2}}{48}-\frac {65219\,{\left (1-2\,x\right )}^{7/2}}{64}-\frac {121359\,{\left (1-2\,x\right )}^{11/2}}{64}+\frac {98209\,{\left (1-2\,x\right )}^{13/2}}{104}-\frac {17337\,{\left (1-2\,x\right )}^{15/2}}{64}+\frac {675\,{\left (1-2\,x\right )}^{17/2}}{16}-\frac {3375\,{\left (1-2\,x\right )}^{19/2}}{1216} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(100793*(1 - 2*x)^(9/2))/48 - (65219*(1 - 2*x)^(7/2))/64 - (121359*(1 - 2*x)^(11/2))/64 + (98209*(1 - 2*x)^(13
/2))/104 - (17337*(1 - 2*x)^(15/2))/64 + (675*(1 - 2*x)^(17/2))/16 - (3375*(1 - 2*x)^(19/2))/1216

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