3.20.29 \(\int (1-2 x)^{5/2} (2+3 x)^5 (3+5 x) \, dx\) [1929]

Optimal. Leaf size=92 \[ -\frac {26411}{64} (1-2 x)^{7/2}+\frac {60025}{72} (1-2 x)^{9/2}-\frac {519645}{704} (1-2 x)^{11/2}+\frac {37485}{104} (1-2 x)^{13/2}-\frac {6489}{64} (1-2 x)^{15/2}+\frac {1053}{68} (1-2 x)^{17/2}-\frac {1215 (1-2 x)^{19/2}}{1216} \]

[Out]

-26411/64*(1-2*x)^(7/2)+60025/72*(1-2*x)^(9/2)-519645/704*(1-2*x)^(11/2)+37485/104*(1-2*x)^(13/2)-6489/64*(1-2
*x)^(15/2)+1053/68*(1-2*x)^(17/2)-1215/1216*(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 = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {78} \begin {gather*} -\frac {1215 (1-2 x)^{19/2}}{1216}+\frac {1053}{68} (1-2 x)^{17/2}-\frac {6489}{64} (1-2 x)^{15/2}+\frac {37485}{104} (1-2 x)^{13/2}-\frac {519645}{704} (1-2 x)^{11/2}+\frac {60025}{72} (1-2 x)^{9/2}-\frac {26411}{64} (1-2 x)^{7/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-26411*(1 - 2*x)^(7/2))/64 + (60025*(1 - 2*x)^(9/2))/72 - (519645*(1 - 2*x)^(11/2))/704 + (37485*(1 - 2*x)^(1
3/2))/104 - (6489*(1 - 2*x)^(15/2))/64 + (1053*(1 - 2*x)^(17/2))/68 - (1215*(1 - 2*x)^(19/2))/1216

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin {align*} \int (1-2 x)^{5/2} (2+3 x)^5 (3+5 x) \, dx &=\int \left (\frac {184877}{64} (1-2 x)^{5/2}-\frac {60025}{8} (1-2 x)^{7/2}+\frac {519645}{64} (1-2 x)^{9/2}-\frac {37485}{8} (1-2 x)^{11/2}+\frac {97335}{64} (1-2 x)^{13/2}-\frac {1053}{4} (1-2 x)^{15/2}+\frac {1215}{64} (1-2 x)^{17/2}\right ) \, dx\\ &=-\frac {26411}{64} (1-2 x)^{7/2}+\frac {60025}{72} (1-2 x)^{9/2}-\frac {519645}{704} (1-2 x)^{11/2}+\frac {37485}{104} (1-2 x)^{13/2}-\frac {6489}{64} (1-2 x)^{15/2}+\frac {1053}{68} (1-2 x)^{17/2}-\frac {1215 (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 (18122584+86950792 x+208370124 x^2+298438668 x^3+259076961 x^4+126243117 x^5+26582985 x^6\right )}{415701} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/415701*((1 - 2*x)^(7/2)*(18122584 + 86950792*x + 208370124*x^2 + 298438668*x^3 + 259076961*x^4 + 126243117*
x^5 + 26582985*x^6))

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Maple [A]
time = 0.12, size = 65, normalized size = 0.71

method	result	size
gosper	\(-\frac {\left (26582985 x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right ) \left (1-2 x \right )^{\frac {7}{2}}}{415701}\)	\(40\)
trager	\(\left (\frac {9720}{19} x^{9}+\frac {536868}{323} x^{8}+\frac {557262}{323} x^{7}+\frac {95571}{4199} x^{6}-\frac {54009375}{46189} x^{5}-\frac {273280105}{415701} x^{4}+\frac {53353244}{415701} x^{3}+\frac {31954540}{138567} x^{2}+\frac {21784712}{415701} x -\frac {18122584}{415701}\right ) \sqrt {1-2 x}\)	\(54\)
risch	\(-\frac {\left (212663880 x^{9}[phone] x^{8}[phone] x^{7}+9461529 x^{6}-486084375 x^{5}-273280105 x^{4}[phone] x^{3}[phone] x^{2}[phone] x -18122584\right ) \left (-1+2 x \right )}{415701 \sqrt {1-2 x}}\)	\(60\)
derivativedivides	\(-\frac {26411 \left (1-2 x \right )^{\frac {7}{2}}}{64}+\frac {60025 \left (1-2 x \right )^{\frac {9}{2}}}{72}-\frac {519645 \left (1-2 x \right )^{\frac {11}{2}}}{704}+\frac {37485 \left (1-2 x \right )^{\frac {13}{2}}}{104}-\frac {6489 \left (1-2 x \right )^{\frac {15}{2}}}{64}+\frac {1053 \left (1-2 x \right )^{\frac {17}{2}}}{68}-\frac {1215 \left (1-2 x \right )^{\frac {19}{2}}}{1216}\)	\(65\)
default	\(-\frac {26411 \left (1-2 x \right )^{\frac {7}{2}}}{64}+\frac {60025 \left (1-2 x \right )^{\frac {9}{2}}}{72}-\frac {519645 \left (1-2 x \right )^{\frac {11}{2}}}{704}+\frac {37485 \left (1-2 x \right )^{\frac {13}{2}}}{104}-\frac {6489 \left (1-2 x \right )^{\frac {15}{2}}}{64}+\frac {1053 \left (1-2 x \right )^{\frac {17}{2}}}{68}-\frac {1215 \left (1-2 x \right )^{\frac {19}{2}}}{1216}\)	\(65\)
meijerg	\(\frac {\frac {96 \sqrt {\pi }}{7}-\frac {48 \sqrt {\pi }\, \left (-16 x^{3}+24 x^{2}-12 x +2\right ) \sqrt {1-2 x}}{7}}{\sqrt {\pi }}-\frac {825 \left (-\frac {32 \sqrt {\pi }}{945}+\frac {4 \sqrt {\pi }\, \left (-448 x^{4}+608 x^{3}-240 x^{2}+8 x +8\right ) \sqrt {1-2 x}}{945}\right )}{2 \sqrt {\pi }}+\frac {\frac {320 \sqrt {\pi }}{33}-\frac {20 \sqrt {\pi }\, \left (-4032 x^{5}+5152 x^{4}-1808 x^{3}+24 x^{2}+16 x +16\right ) \sqrt {1-2 x}}{33}}{\sqrt {\pi }}-\frac {12825 \left (-\frac {256 \sqrt {\pi }}{45045}+\frac {2 \sqrt {\pi }\, \left (-118272 x^{6}+145152 x^{5}-47488 x^{4}+320 x^{3}+192 x^{2}+128 x +128\right ) \sqrt {1-2 x}}{45045}\right )}{16 \sqrt {\pi }}+\frac {\frac {1392 \sqrt {\pi }}{1001}-\frac {87 \sqrt {\pi }\, \left (-768768 x^{7}+916608 x^{6}-286272 x^{5}+1120 x^{4}+640 x^{3}+384 x^{2}+256 x +256\right ) \sqrt {1-2 x}}{16016}}{\sqrt {\pi }}-\frac {71685 \left (-\frac {4096 \sqrt {\pi }}{2297295}+\frac {4 \sqrt {\pi }\, \left (-9225216 x^{8}[phone] x^{7}-3252480 x^{6}+8064 x^{5}+4480 x^{4}+2560 x^{3}+1536 x^{2}+1024 x +1024\right ) \sqrt {1-2 x}}{2297295}\right )}{512 \sqrt {\pi }}+\frac {\frac {6480 \sqrt {\pi }}{323323}-\frac {405 \sqrt {\pi }\, \left (-52276224 x^{9}[phone] x^{8}-17681664 x^{7}+29568 x^{6}+16128 x^{5}+8960 x^{4}+5120 x^{3}+3072 x^{2}+2048 x +2048\right ) \sqrt {1-2 x}}{41385344}}{\sqrt {\pi }}\)	\(373\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-26411/64*(1-2*x)^(7/2)+60025/72*(1-2*x)^(9/2)-519645/704*(1-2*x)^(11/2)+37485/104*(1-2*x)^(13/2)-6489/64*(1-2
*x)^(15/2)+1053/68*(1-2*x)^(17/2)-1215/1216*(1-2*x)^(19/2)

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Maxima [A]
time = 0.27, size = 64, normalized size = 0.70 \begin {gather*} -\frac {1215}{1216} \, {\left (-2 \, x + 1\right )}^{\frac {19}{2}} + \frac {1053}{68} \, {\left (-2 \, x + 1\right )}^{\frac {17}{2}} - \frac {6489}{64} \, {\left (-2 \, x + 1\right )}^{\frac {15}{2}} + \frac {37485}{104} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - \frac {519645}{704} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {60025}{72} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {26411}{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)^5*(3+5*x),x, algorithm="maxima")

[Out]

-1215/1216*(-2*x + 1)^(19/2) + 1053/68*(-2*x + 1)^(17/2) - 6489/64*(-2*x + 1)^(15/2) + 37485/104*(-2*x + 1)^(1
3/2) - 519645/704*(-2*x + 1)^(11/2) + 60025/72*(-2*x + 1)^(9/2) - 26411/64*(-2*x + 1)^(7/2)

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Fricas [A]
time = 0.99, size = 54, normalized size = 0.59 \begin {gather*} \frac {1}{415701} \, {\left (212663880 \, x^{9} + 690949116 \, x^{8} + 717196194 \, x^{7} + 9461529 \, x^{6} - 486084375 \, x^{5} - 273280105 \, x^{4} + 53353244 \, x^{3} + 95863620 \, x^{2} + 21784712 \, x - 18122584\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)^5*(3+5*x),x, algorithm="fricas")

[Out]

1/415701*(212663880*x^9 + 690949116*x^8 + 717196194*x^7 + 9461529*x^6 - 486084375*x^5 - 273280105*x^4 + 533532
44*x^3 + 95863620*x^2 + 21784712*x - 18122584)*sqrt(-2*x + 1)

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Sympy [A]
time = 15.32, size = 82, normalized size = 0.89 \begin {gather*} - \frac {1215 \left (1 - 2 x\right )^{\frac {19}{2}}}{1216} + \frac {1053 \left (1 - 2 x\right )^{\frac {17}{2}}}{68} - \frac {6489 \left (1 - 2 x\right )^{\frac {15}{2}}}{64} + \frac {37485 \left (1 - 2 x\right )^{\frac {13}{2}}}{104} - \frac {519645 \left (1 - 2 x\right )^{\frac {11}{2}}}{704} + \frac {60025 \left (1 - 2 x\right )^{\frac {9}{2}}}{72} - \frac {26411 \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)**5*(3+5*x),x)

[Out]

-1215*(1 - 2*x)**(19/2)/1216 + 1053*(1 - 2*x)**(17/2)/68 - 6489*(1 - 2*x)**(15/2)/64 + 37485*(1 - 2*x)**(13/2)
/104 - 519645*(1 - 2*x)**(11/2)/704 + 60025*(1 - 2*x)**(9/2)/72 - 26411*(1 - 2*x)**(7/2)/64

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Giac [A]
time = 0.54, size = 113, normalized size = 1.23 \begin {gather*} \frac {1215}{1216} \, {\left (2 \, x - 1\right )}^{9} \sqrt {-2 \, x + 1} + \frac {1053}{68} \, {\left (2 \, x - 1\right )}^{8} \sqrt {-2 \, x + 1} + \frac {6489}{64} \, {\left (2 \, x - 1\right )}^{7} \sqrt {-2 \, x + 1} + \frac {37485}{104} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + \frac {519645}{704} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {60025}{72} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {26411}{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)^5*(3+5*x),x, algorithm="giac")

[Out]

1215/1216*(2*x - 1)^9*sqrt(-2*x + 1) + 1053/68*(2*x - 1)^8*sqrt(-2*x + 1) + 6489/64*(2*x - 1)^7*sqrt(-2*x + 1)
+ 37485/104*(2*x - 1)^6*sqrt(-2*x + 1) + 519645/704*(2*x - 1)^5*sqrt(-2*x + 1) + 60025/72*(2*x - 1)^4*sqrt(-2
*x + 1) + 26411/64*(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 {60025\,{\left (1-2\,x\right )}^{9/2}}{72}-\frac {26411\,{\left (1-2\,x\right )}^{7/2}}{64}-\frac {519645\,{\left (1-2\,x\right )}^{11/2}}{704}+\frac {37485\,{\left (1-2\,x\right )}^{13/2}}{104}-\frac {6489\,{\left (1-2\,x\right )}^{15/2}}{64}+\frac {1053\,{\left (1-2\,x\right )}^{17/2}}{68}-\frac {1215\,{\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)^5*(5*x + 3),x)

[Out]

(60025*(1 - 2*x)^(9/2))/72 - (26411*(1 - 2*x)^(7/2))/64 - (519645*(1 - 2*x)^(11/2))/704 + (37485*(1 - 2*x)^(13
/2))/104 - (6489*(1 - 2*x)^(15/2))/64 + (1053*(1 - 2*x)^(17/2))/68 - (1215*(1 - 2*x)^(19/2))/1216

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