3.20.28 \(\int (1-2 x)^{5/2} (2+3 x)^6 (3+5 x) \, dx\) [1928]

3.20.28.1 Optimal result

Integrand size = 22, antiderivative size = 105 \[ \int (1-2 x)^{5/2} (2+3 x)^6 (3+5 x) \, dx=-\frac {184877}{128} (1-2 x)^{7/2}+\frac {3916031 (1-2 x)^{9/2}}{1152}-\frac {5078115 (1-2 x)^{11/2}}{1408}+\frac {3658095 (1-2 x)^{13/2}}{1664}-\frac {105399}{128} (1-2 x)^{15/2}+\frac {409941 (1-2 x)^{17/2}}{2176}-\frac {59049 (1-2 x)^{19/2}}{2432}+\frac {1215}{896} (1-2 x)^{21/2} \]

output

-184877/128*(1-2*x)^(7/2)+3916031/1152*(1-2*x)^(9/2)-5078115/1408*(1-2*x)^ 
(11/2)+3658095/1664*(1-2*x)^(13/2)-105399/128*(1-2*x)^(15/2)+409941/2176*( 
1-2*x)^(17/2)-59049/2432*(1-2*x)^(19/2)+1215/896*(1-2*x)^(21/2)
 

3.20.28.2 Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.46 \[ \int (1-2 x)^{5/2} (2+3 x)^6 (3+5 x) \, dx=-\frac {(1-2 x)^{7/2} \left (323646080+1706820416 x+4700947104 x^2+8157896208 x^3+9228315096 x^4+6628858236 x^5+2753997246 x^6+505076715 x^7\right )}{2909907} \]

input

Integrate[(1 - 2*x)^(5/2)*(2 + 3*x)^6*(3 + 5*x),x]
 

output

-1/2909907*((1 - 2*x)^(7/2)*(323646080 + 1706820416*x + 4700947104*x^2 + 8 
157896208*x^3 + 9228315096*x^4 + 6628858236*x^5 + 2753997246*x^6 + 5050767 
15*x^7))
 

3.20.28.3 Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {86, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int (1-2 x)^{5/2} (3 x+2)^6 (5 x+3) \, dx\)

\(\Big \downarrow \) 86

\(\displaystyle \int \left (-\frac {3645}{128} (1-2 x)^{19/2}+\frac {59049}{128} (1-2 x)^{17/2}-\frac {409941}{128} (1-2 x)^{15/2}+\frac {1580985}{128} (1-2 x)^{13/2}-\frac {3658095}{128} (1-2 x)^{11/2}+\frac {5078115}{128} (1-2 x)^{9/2}-\frac {3916031}{128} (1-2 x)^{7/2}+\frac {1294139}{128} (1-2 x)^{5/2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1215}{896} (1-2 x)^{21/2}-\frac {59049 (1-2 x)^{19/2}}{2432}+\frac {409941 (1-2 x)^{17/2}}{2176}-\frac {105399}{128} (1-2 x)^{15/2}+\frac {3658095 (1-2 x)^{13/2}}{1664}-\frac {5078115 (1-2 x)^{11/2}}{1408}+\frac {3916031 (1-2 x)^{9/2}}{1152}-\frac {184877}{128} (1-2 x)^{7/2}\)

input

Int[(1 - 2*x)^(5/2)*(2 + 3*x)^6*(3 + 5*x),x]
 

output

(-184877*(1 - 2*x)^(7/2))/128 + (3916031*(1 - 2*x)^(9/2))/1152 - (5078115* 
(1 - 2*x)^(11/2))/1408 + (3658095*(1 - 2*x)^(13/2))/1664 - (105399*(1 - 2* 
x)^(15/2))/128 + (409941*(1 - 2*x)^(17/2))/2176 - (59049*(1 - 2*x)^(19/2)) 
/2432 + (1215*(1 - 2*x)^(21/2))/896
 

3.20.28.3.1 Defintions of rubi rules used

rule 86

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ 
.), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; 
 FreeQ[{a, b, c, d, e, f, n}, x] && ((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]))))
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

3.20.28.4 Maple [A] (verified)

Time = 1.05 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.43

method	result	size
gosper	\(-\frac {\left (1-2 x \right )^{\frac {7}{2}} \left (505076715 x^{7}[phone] x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right )}{2909907}\)	\(45\)
pseudoelliptic	\(\frac {\left (505076715 x^{7}[phone] x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {1-2 x}\, \left (-1+2 x \right )^{3}}{2909907}\)	\(52\)
trager	\(\left (\frac {9720}{7} x^{10}+\frac {729972}{133} x^{9}+\frac {17881398}{2261} x^{8}+\frac {8002431}{2261} x^{7}-\frac {12204126}{4199} x^{6}-\frac {183272148}{46189} x^{5}-\frac {433962824}{415701} x^{4}+\frac {2155110064}{2909907} x^{3}+\frac {552074144}{969969} x^{2}+\frac {235056064}{2909907} x -\frac {323646080}{2909907}\right ) \sqrt {1-2 x}\)	\(59\)
risch	\(-\frac {\left (4040613720 x^{10}[phone] x^{9}[phone] x^{8}[phone] x^{7}-8457459318 x^{6}-11546145324 x^{5}-3037739768 x^{4}[phone] x^{3}[phone] x^{2}[phone] x -323646080\right ) \left (-1+2 x \right )}{2909907 \sqrt {1-2 x}}\)	\(65\)
derivativedivides	\(-\frac {184877 \left (1-2 x \right )^{\frac {7}{2}}}{128}+\frac {3916031 \left (1-2 x \right )^{\frac {9}{2}}}{1152}-\frac {5078115 \left (1-2 x \right )^{\frac {11}{2}}}{1408}+\frac {3658095 \left (1-2 x \right )^{\frac {13}{2}}}{1664}-\frac {105399 \left (1-2 x \right )^{\frac {15}{2}}}{128}+\frac {409941 \left (1-2 x \right )^{\frac {17}{2}}}{2176}-\frac {59049 \left (1-2 x \right )^{\frac {19}{2}}}{2432}+\frac {1215 \left (1-2 x \right )^{\frac {21}{2}}}{896}\)	\(74\)
default	\(-\frac {184877 \left (1-2 x \right )^{\frac {7}{2}}}{128}+\frac {3916031 \left (1-2 x \right )^{\frac {9}{2}}}{1152}-\frac {5078115 \left (1-2 x \right )^{\frac {11}{2}}}{1408}+\frac {3658095 \left (1-2 x \right )^{\frac {13}{2}}}{1664}-\frac {105399 \left (1-2 x \right )^{\frac {15}{2}}}{128}+\frac {409941 \left (1-2 x \right )^{\frac {17}{2}}}{2176}-\frac {59049 \left (1-2 x \right )^{\frac {19}{2}}}{2432}+\frac {1215 \left (1-2 x \right )^{\frac {21}{2}}}{896}\)	\(74\)
meijerg	\(\frac {\frac {192 \sqrt {\pi }}{7}-\frac {96 \sqrt {\pi }\, \left (-16 x^{3}+24 x^{2}-12 x +2\right ) \sqrt {1-2 x}}{7}}{\sqrt {\pi }}-\frac {960 \left (-\frac {32 \sqrt {\pi }}{945}+\frac {4 \sqrt {\pi }\, \left (-448 x^{4}+608 x^{3}-240 x^{2}+8 x +8\right ) \sqrt {1-2 x}}{945}\right )}{\sqrt {\pi }}+\frac {\frac {2080 \sqrt {\pi }}{77}-\frac {130 \sqrt {\pi }\, \left (-4032 x^{5}+5152 x^{4}-1808 x^{3}+24 x^{2}+16 x +16\right ) \sqrt {1-2 x}}{77}}{\sqrt {\pi }}-\frac {22275 \left (-\frac {256 \sqrt {\pi }}{45045}+\frac {2 \sqrt {\pi }\, \left (-118272 x^{6}+145152 x^{5}-47488 x^{4}+320 x^{3}+192 x^{2}+128 x +128\right ) \sqrt {1-2 x}}{45045}\right )}{8 \sqrt {\pi }}+\frac {\frac {6432 \sqrt {\pi }}{1001}-\frac {201 \sqrt {\pi }\, \left (-768768 x^{7}+916608 x^{6}-286272 x^{5}+1120 x^{4}+640 x^{3}+384 x^{2}+256 x +256\right ) \sqrt {1-2 x}}{8008}}{\sqrt {\pi }}-\frac {61965 \left (-\frac {4096 \sqrt {\pi }}{2297295}+\frac {4 \sqrt {\pi }\, \left (-9225216 x^{8}[phone] x^{7}-3252480 x^{6}+8064 x^{5}+4480 x^{4}+2560 x^{3}+1536 x^{2}+1024 x +1024\right ) \sqrt {1-2 x}}{2297295}\right )}{64 \sqrt {\pi }}+\frac {\frac {89424 \sqrt {\pi }}{323323}-\frac {5589 \sqrt {\pi }\, \left (-52276224 x^{9}[phone] x^{8}-17681664 x^{7}+29568 x^{6}+16128 x^{5}+8960 x^{4}+5120 x^{3}+3072 x^{2}+2048 x +2048\right ) \sqrt {1-2 x}}{41385344}}{\sqrt {\pi }}-\frac {54675 \left (-\frac {32768 \sqrt {\pi }}{43648605}+\frac {\sqrt {\pi }\, \left (-2270281728 x^{10}[phone] x^{9}-743288832 x^{8}+878592 x^{7}+473088 x^{6}+258048 x^{5}+143360 x^{4}+81920 x^{3}+49152 x^{2}+32768 x +32768\right ) \sqrt {1-2 x}}{43648605}\right )}{2048 \sqrt {\pi }}\)	\(446\)

input

int((1-2*x)^(5/2)*(2+3*x)^6*(3+5*x),x,method=_RETURNVERBOSE)
 

output

-1/2909907*(1-2*x)^(7/2)*(505076715*x^7+2753997246*x^6+6628858236*x^5+9228 
315096*x^4+8157896208*x^3+4700947104*x^2+1706820416*x[phone])
 

3.20.28.5 Fricas [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.56 \[ \int (1-2 x)^{5/2} (2+3 x)^6 (3+5 x) \, dx=\frac {1}{2909907} \, {\left (4040613720 \, x^{10} + 15971057388 \, x^{9} + 23013359226 \, x^{8} + 10299128697 \, x^{7} - 8457459318 \, x^{6} - 11546145324 \, x^{5} - 3037739768 \, x^{4} + 2155110064 \, x^{3} + 1656222432 \, x^{2} + 235056064 \, x - 323646080\right )} \sqrt {-2 \, x + 1} \]

input

integrate((1-2*x)^(5/2)*(2+3*x)^6*(3+5*x),x, algorithm="fricas")
 

output

1/2909907*(4040613720*x^10 + 15971057388*x^9 + 23013359226*x^8 + 102991286 
97*x^7 - 8457459318*x^6 - 11546145324*x^5 - 3037739768*x^4 + 2155110064*x^ 
3 + 1656222432*x^2 + 235056064*x - 323646080)*sqrt(-2*x + 1)
 

3.20.28.6 Sympy [A] (verification not implemented)

Time = 1.07 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int (1-2 x)^{5/2} (2+3 x)^6 (3+5 x) \, dx=\frac {1215 \left (1 - 2 x\right )^{\frac {21}{2}}}{896} - \frac {59049 \left (1 - 2 x\right )^{\frac {19}{2}}}{2432} + \frac {409941 \left (1 - 2 x\right )^{\frac {17}{2}}}{2176} - \frac {105399 \left (1 - 2 x\right )^{\frac {15}{2}}}{128} + \frac {3658095 \left (1 - 2 x\right )^{\frac {13}{2}}}{1664} - \frac {5078115 \left (1 - 2 x\right )^{\frac {11}{2}}}{1408} + \frac {3916031 \left (1 - 2 x\right )^{\frac {9}{2}}}{1152} - \frac {184877 \left (1 - 2 x\right )^{\frac {7}{2}}}{128} \]

input

integrate((1-2*x)**(5/2)*(2+3*x)**6*(3+5*x),x)
 

output

1215*(1 - 2*x)**(21/2)/896 - 59049*(1 - 2*x)**(19/2)/2432 + 409941*(1 - 2* 
x)**(17/2)/2176 - 105399*(1 - 2*x)**(15/2)/128 + 3658095*(1 - 2*x)**(13/2) 
/1664 - 5078115*(1 - 2*x)**(11/2)/1408 + 3916031*(1 - 2*x)**(9/2)/1152 - 1 
84877*(1 - 2*x)**(7/2)/128
 

3.20.28.7 Maxima [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.70 \[ \int (1-2 x)^{5/2} (2+3 x)^6 (3+5 x) \, dx=\frac {1215}{896} \, {\left (-2 \, x + 1\right )}^{\frac {21}{2}} - \frac {59049}{2432} \, {\left (-2 \, x + 1\right )}^{\frac {19}{2}} + \frac {409941}{2176} \, {\left (-2 \, x + 1\right )}^{\frac {17}{2}} - \frac {105399}{128} \, {\left (-2 \, x + 1\right )}^{\frac {15}{2}} + \frac {3658095}{1664} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - \frac {5078115}{1408} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {3916031}{1152} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {184877}{128} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} \]

input

integrate((1-2*x)^(5/2)*(2+3*x)^6*(3+5*x),x, algorithm="maxima")
 

output

1215/896*(-2*x + 1)^(21/2) - 59049/2432*(-2*x + 1)^(19/2) + 409941/2176*(- 
2*x + 1)^(17/2) - 105399/128*(-2*x + 1)^(15/2) + 3658095/1664*(-2*x + 1)^( 
13/2) - 5078115/1408*(-2*x + 1)^(11/2) + 3916031/1152*(-2*x + 1)^(9/2) - 1 
84877/128*(-2*x + 1)^(7/2)
 

3.20.28.8 Giac [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.23 \[ \int (1-2 x)^{5/2} (2+3 x)^6 (3+5 x) \, dx=\frac {1215}{896} \, {\left (2 \, x - 1\right )}^{10} \sqrt {-2 \, x + 1} + \frac {59049}{2432} \, {\left (2 \, x - 1\right )}^{9} \sqrt {-2 \, x + 1} + \frac {409941}{2176} \, {\left (2 \, x - 1\right )}^{8} \sqrt {-2 \, x + 1} + \frac {105399}{128} \, {\left (2 \, x - 1\right )}^{7} \sqrt {-2 \, x + 1} + \frac {3658095}{1664} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + \frac {5078115}{1408} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {3916031}{1152} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {184877}{128} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} \]

input

integrate((1-2*x)^(5/2)*(2+3*x)^6*(3+5*x),x, algorithm="giac")
 

output

1215/896*(2*x - 1)^10*sqrt(-2*x + 1) + 59049/2432*(2*x - 1)^9*sqrt(-2*x + 
1) + 409941/2176*(2*x - 1)^8*sqrt(-2*x + 1) + 105399/128*(2*x - 1)^7*sqrt( 
-2*x + 1) + 3658095/1664*(2*x - 1)^6*sqrt(-2*x + 1) + 5078115/1408*(2*x - 
1)^5*sqrt(-2*x + 1) + 3916031/1152*(2*x - 1)^4*sqrt(-2*x + 1) + 184877/128 
*(2*x - 1)^3*sqrt(-2*x + 1)
 

3.20.28.9 Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.70 \[ \int (1-2 x)^{5/2} (2+3 x)^6 (3+5 x) \, dx=\frac {3916031\,{\left (1-2\,x\right )}^{9/2}}{1152}-\frac {184877\,{\left (1-2\,x\right )}^{7/2}}{128}-\frac {5078115\,{\left (1-2\,x\right )}^{11/2}}{1408}+\frac {3658095\,{\left (1-2\,x\right )}^{13/2}}{1664}-\frac {105399\,{\left (1-2\,x\right )}^{15/2}}{128}+\frac {409941\,{\left (1-2\,x\right )}^{17/2}}{2176}-\frac {59049\,{\left (1-2\,x\right )}^{19/2}}{2432}+\frac {1215\,{\left (1-2\,x\right )}^{21/2}}{896} \]

input

int((1 - 2*x)^(5/2)*(3*x + 2)^6*(5*x + 3),x)
 

output

(3916031*(1 - 2*x)^(9/2))/1152 - (184877*(1 - 2*x)^(7/2))/128 - (5078115*( 
1 - 2*x)^(11/2))/1408 + (3658095*(1 - 2*x)^(13/2))/1664 - (105399*(1 - 2*x 
)^(15/2))/128 + (409941*(1 - 2*x)^(17/2))/2176 - (59049*(1 - 2*x)^(19/2))/ 
2432 + (1215*(1 - 2*x)^(21/2))/896