3.20.42 \(\int (1-2 x)^{5/2} (2+3 x)^4 (3+5 x)^2 \, dx\) [1942]

Optimal. Leaf size=92 \[ -\frac {41503}{64} (1-2 x)^{7/2}+\frac {381073}{288} (1-2 x)^{9/2}-\frac {832951}{704} (1-2 x)^{11/2}+\frac {121359}{208} (1-2 x)^{13/2}-\frac {53037}{320} (1-2 x)^{15/2}+\frac {13905}{544} (1-2 x)^{17/2}-\frac {2025 (1-2 x)^{19/2}}{1216} \]

[Out]

-41503/64*(1-2*x)^(7/2)+381073/288*(1-2*x)^(9/2)-832951/704*(1-2*x)^(11/2)+121359/208*(1-2*x)^(13/2)-53037/320
*(1-2*x)^(15/2)+13905/544*(1-2*x)^(17/2)-2025/1216*(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 {2025 (1-2 x)^{19/2}}{1216}+\frac {13905}{544} (1-2 x)^{17/2}-\frac {53037}{320} (1-2 x)^{15/2}+\frac {121359}{208} (1-2 x)^{13/2}-\frac {832951}{704} (1-2 x)^{11/2}+\frac {381073}{288} (1-2 x)^{9/2}-\frac {41503}{64} (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)^2,x]

[Out]

(-41503*(1 - 2*x)^(7/2))/64 + (381073*(1 - 2*x)^(9/2))/288 - (832951*(1 - 2*x)^(11/2))/704 + (121359*(1 - 2*x)
^(13/2))/208 - (53037*(1 - 2*x)^(15/2))/320 + (13905*(1 - 2*x)^(17/2))/544 - (2025*(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)^4 (3+5 x)^2 \, dx &=\int \left (\frac {290521}{64} (1-2 x)^{5/2}-\frac {381073}{32} (1-2 x)^{7/2}+\frac {832951}{64} (1-2 x)^{9/2}-\frac {121359}{16} (1-2 x)^{11/2}+\frac {159111}{64} (1-2 x)^{13/2}-\frac {13905}{32} (1-2 x)^{15/2}+\frac {2025}{64} (1-2 x)^{17/2}\right ) \, dx\\ &=-\frac {41503}{64} (1-2 x)^{7/2}+\frac {381073}{288} (1-2 x)^{9/2}-\frac {832951}{704} (1-2 x)^{11/2}+\frac {121359}{208} (1-2 x)^{13/2}-\frac {53037}{320} (1-2 x)^{15/2}+\frac {13905}{544} (1-2 x)^{17/2}-\frac {2025 (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 {(1-2 x)^{7/2} \left (138993368+673648856 x+1634664492 x^2+2374399764 x^3+2092364703 x^4+1035520200 x^5+221524875 x^6\right )}{2078505} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2078505*((1 - 2*x)^(7/2)*(138993368 + 673648856*x + 1634664492*x^2 + 2374399764*x^3 + 2092364703*x^4 + 1035
520200*x^5 + 221524875*x^6))

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Maple [A]
time = 0.10, size = 65, normalized size = 0.71

method	result	size
gosper	\(-\frac {\left (221524875 x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right ) \left (1-2 x \right )^{\frac {7}{2}}}{2078505}\)	\(40\)
trager	\(\left (\frac {16200}{19} x^{9}+\frac {874260}{323} x^{8}+\frac {4383702}{1615} x^{7}-\frac {1228101}{20995} x^{6}-\frac {432979246}{230945} x^{5}-\frac {414549835}{415701} x^{4}+\frac {92349572}{415701} x^{3}+\frac {246436076}{692835} x^{2}+\frac {160311352}{2078505} x -\frac {138993368}{2078505}\right ) \sqrt {1-2 x}\)	\(54\)
risch	\(-\frac {\left (1772199000 x^{9}[phone] x^{8}[phone] x^{7}-121581999 x^{6}-3896813214 x^{5}-2072749175 x^{4}[phone] x^{3}[phone] x^{2}[phone] x -138993368\right ) \left (-1+2 x \right )}{2078505 \sqrt {1-2 x}}\)	\(60\)
derivativedivides	\(-\frac {41503 \left (1-2 x \right )^{\frac {7}{2}}}{64}+\frac {381073 \left (1-2 x \right )^{\frac {9}{2}}}{288}-\frac {832951 \left (1-2 x \right )^{\frac {11}{2}}}{704}+\frac {121359 \left (1-2 x \right )^{\frac {13}{2}}}{208}-\frac {53037 \left (1-2 x \right )^{\frac {15}{2}}}{320}+\frac {13905 \left (1-2 x \right )^{\frac {17}{2}}}{544}-\frac {2025 \left (1-2 x \right )^{\frac {19}{2}}}{1216}\)	\(65\)
default	\(-\frac {41503 \left (1-2 x \right )^{\frac {7}{2}}}{64}+\frac {381073 \left (1-2 x \right )^{\frac {9}{2}}}{288}-\frac {832951 \left (1-2 x \right )^{\frac {11}{2}}}{704}+\frac {121359 \left (1-2 x \right )^{\frac {13}{2}}}{208}-\frac {53037 \left (1-2 x \right )^{\frac {15}{2}}}{320}+\frac {13905 \left (1-2 x \right )^{\frac {17}{2}}}{544}-\frac {2025 \left (1-2 x \right )^{\frac {19}{2}}}{1216}\)	\(65\)
meijerg	\(\frac {\frac {144 \sqrt {\pi }}{7}-\frac {72 \sqrt {\pi }\, \left (-16 x^{3}+24 x^{2}-12 x +2\right ) \sqrt {1-2 x}}{7}}{\sqrt {\pi }}-\frac {630 \left (-\frac {32 \sqrt {\pi }}{945}+\frac {4 \sqrt {\pi }\, \left (-448 x^{4}+608 x^{3}-240 x^{2}+8 x +8\right ) \sqrt {1-2 x}}{945}\right )}{\sqrt {\pi }}+\frac {\frac {10448 \sqrt {\pi }}{693}-\frac {653 \sqrt {\pi }\, \left (-4032 x^{5}+5152 x^{4}-1808 x^{3}+24 x^{2}+16 x +16\right ) \sqrt {1-2 x}}{693}}{\sqrt {\pi }}-\frac {20295 \left (-\frac {256 \sqrt {\pi }}{45045}+\frac {2 \sqrt {\pi }\, \left (-118272 x^{6}+145152 x^{5}-47488 x^{4}+320 x^{3}+192 x^{2}+128 x +128\right ) \sqrt {1-2 x}}{45045}\right )}{16 \sqrt {\pi }}+\frac {\frac {11208 \sqrt {\pi }}{5005}-\frac {1401 \sqrt {\pi }\, \left (-768768 x^{7}+916608 x^{6}-286272 x^{5}+1120 x^{4}+640 x^{3}+384 x^{2}+256 x +256\right ) \sqrt {1-2 x}}{160160}}{\sqrt {\pi }}-\frac {58725 \left (-\frac {4096 \sqrt {\pi }}{2297295}+\frac {4 \sqrt {\pi }\, \left (-9225216 x^{8}[phone] x^{7}-3252480 x^{6}+8064 x^{5}+4480 x^{4}+2560 x^{3}+1536 x^{2}+1024 x +1024\right ) \sqrt {1-2 x}}{2297295}\right )}{256 \sqrt {\pi }}+\frac {\frac {10800 \sqrt {\pi }}{323323}-\frac {675 \sqrt {\pi }\, \left (-52276224 x^{9}[phone] x^{8}-17681664 x^{7}+29568 x^{6}+16128 x^{5}+8960 x^{4}+5120 x^{3}+3072 x^{2}+2048 x +2048\right ) \sqrt {1-2 x}}{41385344}}{\sqrt {\pi }}\)	\(373\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-41503/64*(1-2*x)^(7/2)+381073/288*(1-2*x)^(9/2)-832951/704*(1-2*x)^(11/2)+121359/208*(1-2*x)^(13/2)-53037/320
*(1-2*x)^(15/2)+13905/544*(1-2*x)^(17/2)-2025/1216*(1-2*x)^(19/2)

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Maxima [A]
time = 0.26, size = 64, normalized size = 0.70 \begin {gather*} -\frac {2025}{1216} \, {\left (-2 \, x + 1\right )}^{\frac {19}{2}} + \frac {13905}{544} \, {\left (-2 \, x + 1\right )}^{\frac {17}{2}} - \frac {53037}{320} \, {\left (-2 \, x + 1\right )}^{\frac {15}{2}} + \frac {121359}{208} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - \frac {832951}{704} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {381073}{288} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {41503}{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)^4*(3+5*x)^2,x, algorithm="maxima")

[Out]

-2025/1216*(-2*x + 1)^(19/2) + 13905/544*(-2*x + 1)^(17/2) - 53037/320*(-2*x + 1)^(15/2) + 121359/208*(-2*x +
1)^(13/2) - 832951/704*(-2*x + 1)^(11/2) + 381073/288*(-2*x + 1)^(9/2) - 41503/64*(-2*x + 1)^(7/2)

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Fricas [A]
time = 1.33, size = 54, normalized size = 0.59 \begin {gather*} \frac {1}{2078505} \, {\left (1772199000 \, x^{9} + 5625863100 \, x^{8} + 5641824474 \, x^{7} - 121581999 \, x^{6} - 3896813214 \, x^{5} - 2072749175 \, x^{4} + 461747860 \, x^{3} + 739308228 \, x^{2} + 160311352 \, x - 138993368\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)^2,x, algorithm="fricas")

[Out]

1/2078505*(1772199000*x^9 + 5625863100*x^8 + 5641824474*x^7 - 121581999*x^6 - 3896813214*x^5 - 2072749175*x^4
+ 461747860*x^3 + 739308228*x^2 + 160311352*x - 138993368)*sqrt(-2*x + 1)

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Sympy [A]
time = 13.55, size = 82, normalized size = 0.89 \begin {gather*} - \frac {2025 \left (1 - 2 x\right )^{\frac {19}{2}}}{1216} + \frac {13905 \left (1 - 2 x\right )^{\frac {17}{2}}}{544} - \frac {53037 \left (1 - 2 x\right )^{\frac {15}{2}}}{320} + \frac {121359 \left (1 - 2 x\right )^{\frac {13}{2}}}{208} - \frac {832951 \left (1 - 2 x\right )^{\frac {11}{2}}}{704} + \frac {381073 \left (1 - 2 x\right )^{\frac {9}{2}}}{288} - \frac {41503 \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)**4*(3+5*x)**2,x)

[Out]

-2025*(1 - 2*x)**(19/2)/1216 + 13905*(1 - 2*x)**(17/2)/544 - 53037*(1 - 2*x)**(15/2)/320 + 121359*(1 - 2*x)**(
13/2)/208 - 832951*(1 - 2*x)**(11/2)/704 + 381073*(1 - 2*x)**(9/2)/288 - 41503*(1 - 2*x)**(7/2)/64

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Giac [A]
time = 0.54, size = 113, normalized size = 1.23 \begin {gather*} \frac {2025}{1216} \, {\left (2 \, x - 1\right )}^{9} \sqrt {-2 \, x + 1} + \frac {13905}{544} \, {\left (2 \, x - 1\right )}^{8} \sqrt {-2 \, x + 1} + \frac {53037}{320} \, {\left (2 \, x - 1\right )}^{7} \sqrt {-2 \, x + 1} + \frac {121359}{208} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + \frac {832951}{704} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {381073}{288} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {41503}{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)^4*(3+5*x)^2,x, algorithm="giac")

[Out]

2025/1216*(2*x - 1)^9*sqrt(-2*x + 1) + 13905/544*(2*x - 1)^8*sqrt(-2*x + 1) + 53037/320*(2*x - 1)^7*sqrt(-2*x
+ 1) + 121359/208*(2*x - 1)^6*sqrt(-2*x + 1) + 832951/704*(2*x - 1)^5*sqrt(-2*x + 1) + 381073/288*(2*x - 1)^4*
sqrt(-2*x + 1) + 41503/64*(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 {381073\,{\left (1-2\,x\right )}^{9/2}}{288}-\frac {41503\,{\left (1-2\,x\right )}^{7/2}}{64}-\frac {832951\,{\left (1-2\,x\right )}^{11/2}}{704}+\frac {121359\,{\left (1-2\,x\right )}^{13/2}}{208}-\frac {53037\,{\left (1-2\,x\right )}^{15/2}}{320}+\frac {13905\,{\left (1-2\,x\right )}^{17/2}}{544}-\frac {2025\,{\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)^4*(5*x + 3)^2,x)

[Out]

(381073*(1 - 2*x)^(9/2))/288 - (41503*(1 - 2*x)^(7/2))/64 - (832951*(1 - 2*x)^(11/2))/704 + (121359*(1 - 2*x)^
(13/2))/208 - (53037*(1 - 2*x)^(15/2))/320 + (13905*(1 - 2*x)^(17/2))/544 - (2025*(1 - 2*x)^(19/2))/1216

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