3.20.55 \(\int (1-2 x)^{5/2} (2+3 x)^4 (3+5 x)^3 \, dx\) [1955]

Optimal. Leaf size=105 \[ -\frac {456533}{128} (1-2 x)^{7/2}+\frac {3278737}{384} (1-2 x)^{9/2}-\frac {1179381}{128} (1-2 x)^{11/2}+\frac {9504551 (1-2 x)^{13/2}}{1664}-\frac {1392467}{640} (1-2 x)^{15/2}+\frac {1101465 (1-2 x)^{17/2}}{2176}-\frac {161325 (1-2 x)^{19/2}}{2432}+\frac {3375}{896} (1-2 x)^{21/2} \]

[Out]

-456533/128*(1-2*x)^(7/2)+3278737/384*(1-2*x)^(9/2)-1179381/128*(1-2*x)^(11/2)+9504551/1664*(1-2*x)^(13/2)-139
2467/640*(1-2*x)^(15/2)+1101465/2176*(1-2*x)^(17/2)-161325/2432*(1-2*x)^(19/2)+3375/896*(1-2*x)^(21/2)

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Rubi [A]
time = 0.01, antiderivative size = 105, 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}{896} (1-2 x)^{21/2}-\frac {161325 (1-2 x)^{19/2}}{2432}+\frac {1101465 (1-2 x)^{17/2}}{2176}-\frac {1392467}{640} (1-2 x)^{15/2}+\frac {9504551 (1-2 x)^{13/2}}{1664}-\frac {1179381}{128} (1-2 x)^{11/2}+\frac {3278737}{384} (1-2 x)^{9/2}-\frac {456533}{128} (1-2 x)^{7/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-456533*(1 - 2*x)^(7/2))/128 + (3278737*(1 - 2*x)^(9/2))/384 - (1179381*(1 - 2*x)^(11/2))/128 + (9504551*(1 -
2*x)^(13/2))/1664 - (1392467*(1 - 2*x)^(15/2))/640 + (1101465*(1 - 2*x)^(17/2))/2176 - (161325*(1 - 2*x)^(19/
2))/2432 + (3375*(1 - 2*x)^(21/2))/896

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)^4 (3+5 x)^3 \, dx &=\int \left (\frac {3195731}{128} (1-2 x)^{5/2}-\frac {9836211}{128} (1-2 x)^{7/2}+\frac {12973191}{128} (1-2 x)^{9/2}-\frac {9504551}{128} (1-2 x)^{11/2}+\frac {4177401}{128} (1-2 x)^{13/2}-\frac {1101465}{128} (1-2 x)^{15/2}+\frac {161325}{128} (1-2 x)^{17/2}-\frac {10125}{128} (1-2 x)^{19/2}\right ) \, dx\\ &=-\frac {456533}{128} (1-2 x)^{7/2}+\frac {3278737}{384} (1-2 x)^{9/2}-\frac {1179381}{128} (1-2 x)^{11/2}+\frac {9504551 (1-2 x)^{13/2}}{1664}-\frac {1392467}{640} (1-2 x)^{15/2}+\frac {1101465 (1-2 x)^{17/2}}{2176}-\frac {161325 (1-2 x)^{19/2}}{2432}+\frac {3375}{896} (1-2 x)^{21/2}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 48, normalized size = 0.46 \begin {gather*} -\frac {(1-2 x)^{7/2} \left (115708576+619493392 x+1740153744 x^2+3089723448 x^3+3583371246 x^4+2642319225 x^5+1127763000 x^6+212574375 x^7\right )}{440895} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/440895*((1 - 2*x)^(7/2)*(115708576 + 619493392*x + 1740153744*x^2 + 3089723448*x^3 + 3583371246*x^4 + 26423
19225*x^5 + 1127763000*x^6 + 212574375*x^7))

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Maple [A]
time = 0.11, size = 74, normalized size = 0.70

method	result	size
gosper	\(-\frac {\left (212574375 x^{7}[phone] x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right ) \left (1-2 x \right )^{\frac {7}{2}}}{440895}\)	\(45\)
trager	\(\left (\frac {27000}{7} x^{10}+\frac {1952100}{133} x^{9}+\frac {45542790}{2261} x^{8}+\frac {90080587}{11305} x^{7}-\frac {169357858}{20995} x^{6}-\frac {204644913}{20995} x^{5}-\frac {27740810}{12597} x^{4}+\frac {168589384}{88179} x^{3}+\frac {196101232}{146965} x^{2}+\frac {74758064}{440895} x -\frac {115708576}{440895}\right ) \sqrt {1-2 x}\)	\(59\)
risch	\(-\frac {\left (1700595000 x^{10}[phone] x^{9}[phone] x^{8}[phone] x^{7}-3556515018 x^{6}-4297543173 x^{5}-970928350 x^{4}[phone] x^{3}[phone] x^{2}[phone] x -115708576\right ) \left (-1+2 x \right )}{440895 \sqrt {1-2 x}}\)	\(65\)
derivativedivides	\(-\frac {456533 \left (1-2 x \right )^{\frac {7}{2}}}{128}+\frac {3278737 \left (1-2 x \right )^{\frac {9}{2}}}{384}-\frac {1179381 \left (1-2 x \right )^{\frac {11}{2}}}{128}+\frac {9504551 \left (1-2 x \right )^{\frac {13}{2}}}{1664}-\frac {1392467 \left (1-2 x \right )^{\frac {15}{2}}}{640}+\frac {1101465 \left (1-2 x \right )^{\frac {17}{2}}}{2176}-\frac {161325 \left (1-2 x \right )^{\frac {19}{2}}}{2432}+\frac {3375 \left (1-2 x \right )^{\frac {21}{2}}}{896}\)	\(74\)
default	\(-\frac {456533 \left (1-2 x \right )^{\frac {7}{2}}}{128}+\frac {3278737 \left (1-2 x \right )^{\frac {9}{2}}}{384}-\frac {1179381 \left (1-2 x \right )^{\frac {11}{2}}}{128}+\frac {9504551 \left (1-2 x \right )^{\frac {13}{2}}}{1664}-\frac {1392467 \left (1-2 x \right )^{\frac {15}{2}}}{640}+\frac {1101465 \left (1-2 x \right )^{\frac {17}{2}}}{2176}-\frac {161325 \left (1-2 x \right )^{\frac {19}{2}}}{2432}+\frac {3375 \left (1-2 x \right )^{\frac {21}{2}}}{896}\)	\(74\)
meijerg	\(\frac {\frac {432 \sqrt {\pi }}{7}-\frac {216 \sqrt {\pi }\, \left (-16 x^{3}+24 x^{2}-12 x +2\right ) \sqrt {1-2 x}}{7}}{\sqrt {\pi }}-\frac {4455 \left (-\frac {32 \sqrt {\pi }}{945}+\frac {4 \sqrt {\pi }\, \left (-448 x^{4}+608 x^{3}-240 x^{2}+8 x +8\right ) \sqrt {1-2 x}}{945}\right )}{2 \sqrt {\pi }}+\frac {\frac {4976 \sqrt {\pi }}{77}-\frac {311 \sqrt {\pi }\, \left (-4032 x^{5}+5152 x^{4}-1808 x^{3}+24 x^{2}+16 x +16\right ) \sqrt {1-2 x}}{77}}{\sqrt {\pi }}-\frac {27465 \left (-\frac {256 \sqrt {\pi }}{45045}+\frac {2 \sqrt {\pi }\, \left (-118272 x^{6}+145152 x^{5}-47488 x^{4}+320 x^{3}+192 x^{2}+128 x +128\right ) \sqrt {1-2 x}}{45045}\right )}{4 \sqrt {\pi }}+\frac {\frac {245192 \sqrt {\pi }}{15015}-\frac {30649 \sqrt {\pi }\, \left (-768768 x^{7}+916608 x^{6}-286272 x^{5}+1120 x^{4}+640 x^{3}+384 x^{2}+256 x +256\right ) \sqrt {1-2 x}}{480480}}{\sqrt {\pi }}-\frac {1298025 \left (-\frac {4096 \sqrt {\pi }}{2297295}+\frac {4 \sqrt {\pi }\, \left (-9225216 x^{8}[phone] x^{7}-3252480 x^{6}+8064 x^{5}+4480 x^{4}+2560 x^{3}+1536 x^{2}+1024 x +1024\right ) \sqrt {1-2 x}}{2297295}\right )}{512 \sqrt {\pi }}+\frac {\frac {241200 \sqrt {\pi }}{323323}-\frac {15075 \sqrt {\pi }\, \left (-52276224 x^{9}[phone] x^{8}-17681664 x^{7}+29568 x^{6}+16128 x^{5}+8960 x^{4}+5120 x^{3}+3072 x^{2}+2048 x +2048\right ) \sqrt {1-2 x}}{41385344}}{\sqrt {\pi }}-\frac {151875 \left (-\frac {32768 \sqrt {\pi }}{43648605}+\frac {\sqrt {\pi }\, \left (-2270281728 x^{10}[phone] x^{9}-743288832 x^{8}+878592 x^{7}+473088 x^{6}+258048 x^{5}+143360 x^{4}+81920 x^{3}+49152 x^{2}+32768 x +32768\right ) \sqrt {1-2 x}}{43648605}\right )}{2048 \sqrt {\pi }}\)	\(446\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-456533/128*(1-2*x)^(7/2)+3278737/384*(1-2*x)^(9/2)-1179381/128*(1-2*x)^(11/2)+9504551/1664*(1-2*x)^(13/2)-139
2467/640*(1-2*x)^(15/2)+1101465/2176*(1-2*x)^(17/2)-161325/2432*(1-2*x)^(19/2)+3375/896*(1-2*x)^(21/2)

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Maxima [A]
time = 0.27, size = 73, normalized size = 0.70 \begin {gather*} \frac {3375}{896} \, {\left (-2 \, x + 1\right )}^{\frac {21}{2}} - \frac {161325}{2432} \, {\left (-2 \, x + 1\right )}^{\frac {19}{2}} + \frac {1101465}{2176} \, {\left (-2 \, x + 1\right )}^{\frac {17}{2}} - \frac {1392467}{640} \, {\left (-2 \, x + 1\right )}^{\frac {15}{2}} + \frac {9504551}{1664} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - \frac {1179381}{128} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {3278737}{384} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {456533}{128} \, {\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)^4*(3+5*x)^3,x, algorithm="maxima")

[Out]

3375/896*(-2*x + 1)^(21/2) - 161325/2432*(-2*x + 1)^(19/2) + 1101465/2176*(-2*x + 1)^(17/2) - 1392467/640*(-2*
x + 1)^(15/2) + 9504551/1664*(-2*x + 1)^(13/2) - 1179381/128*(-2*x + 1)^(11/2) + 3278737/384*(-2*x + 1)^(9/2)
- 456533/128*(-2*x + 1)^(7/2)

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Fricas [A]
time = 1.26, size = 59, normalized size = 0.56 \begin {gather*} \frac {1}{440895} \, {\left (1700595000 \, x^{10} + 6471211500 \, x^{9} + 8880844050 \, x^{8} + 3513142893 \, x^{7} - 3556515018 \, x^{6} - 4297543173 \, x^{5} - 970928350 \, x^{4} + 842946920 \, x^{3} + 588303696 \, x^{2} + 74758064 \, x - 115708576\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)^4*(3+5*x)^3,x, algorithm="fricas")

[Out]

1/440895*(1700595000*x^10 + 6471211500*x^9 + 8880844050*x^8 + 3513142893*x^7 - 3556515018*x^6 - 4297543173*x^5
- 970928350*x^4 + 842946920*x^3 + 588303696*x^2 + 74758064*x - 115708576)*sqrt(-2*x + 1)

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Sympy [A]
time = 17.42, size = 94, normalized size = 0.90 \begin {gather*} \frac {3375 \left (1 - 2 x\right )^{\frac {21}{2}}}{896} - \frac {161325 \left (1 - 2 x\right )^{\frac {19}{2}}}{2432} + \frac {1101465 \left (1 - 2 x\right )^{\frac {17}{2}}}{2176} - \frac {1392467 \left (1 - 2 x\right )^{\frac {15}{2}}}{640} + \frac {9504551 \left (1 - 2 x\right )^{\frac {13}{2}}}{1664} - \frac {1179381 \left (1 - 2 x\right )^{\frac {11}{2}}}{128} + \frac {3278737 \left (1 - 2 x\right )^{\frac {9}{2}}}{384} - \frac {456533 \left (1 - 2 x\right )^{\frac {7}{2}}}{128} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3375*(1 - 2*x)**(21/2)/896 - 161325*(1 - 2*x)**(19/2)/2432 + 1101465*(1 - 2*x)**(17/2)/2176 - 1392467*(1 - 2*x
)**(15/2)/640 + 9504551*(1 - 2*x)**(13/2)/1664 - 1179381*(1 - 2*x)**(11/2)/128 + 3278737*(1 - 2*x)**(9/2)/384
- 456533*(1 - 2*x)**(7/2)/128

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Giac [A]
time = 0.63, size = 129, normalized size = 1.23 \begin {gather*} \frac {3375}{896} \, {\left (2 \, x - 1\right )}^{10} \sqrt {-2 \, x + 1} + \frac {161325}{2432} \, {\left (2 \, x - 1\right )}^{9} \sqrt {-2 \, x + 1} + \frac {1101465}{2176} \, {\left (2 \, x - 1\right )}^{8} \sqrt {-2 \, x + 1} + \frac {1392467}{640} \, {\left (2 \, x - 1\right )}^{7} \sqrt {-2 \, x + 1} + \frac {9504551}{1664} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + \frac {1179381}{128} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {3278737}{384} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {456533}{128} \, {\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)^4*(3+5*x)^3,x, algorithm="giac")

[Out]

3375/896*(2*x - 1)^10*sqrt(-2*x + 1) + 161325/2432*(2*x - 1)^9*sqrt(-2*x + 1) + 1101465/2176*(2*x - 1)^8*sqrt(
-2*x + 1) + 1392467/640*(2*x - 1)^7*sqrt(-2*x + 1) + 9504551/1664*(2*x - 1)^6*sqrt(-2*x + 1) + 1179381/128*(2*
x - 1)^5*sqrt(-2*x + 1) + 3278737/384*(2*x - 1)^4*sqrt(-2*x + 1) + 456533/128*(2*x - 1)^3*sqrt(-2*x + 1)

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Mupad [B]
time = 0.03, size = 73, normalized size = 0.70 \begin {gather*} \frac {3278737\,{\left (1-2\,x\right )}^{9/2}}{384}-\frac {456533\,{\left (1-2\,x\right )}^{7/2}}{128}-\frac {1179381\,{\left (1-2\,x\right )}^{11/2}}{128}+\frac {9504551\,{\left (1-2\,x\right )}^{13/2}}{1664}-\frac {1392467\,{\left (1-2\,x\right )}^{15/2}}{640}+\frac {1101465\,{\left (1-2\,x\right )}^{17/2}}{2176}-\frac {161325\,{\left (1-2\,x\right )}^{19/2}}{2432}+\frac {3375\,{\left (1-2\,x\right )}^{21/2}}{896} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3278737*(1 - 2*x)^(9/2))/384 - (456533*(1 - 2*x)^(7/2))/128 - (1179381*(1 - 2*x)^(11/2))/128 + (9504551*(1 -
2*x)^(13/2))/1664 - (1392467*(1 - 2*x)^(15/2))/640 + (1101465*(1 - 2*x)^(17/2))/2176 - (161325*(1 - 2*x)^(19/2
))/2432 + (3375*(1 - 2*x)^(21/2))/896

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