\(\int (1-2 x) (2+3 x)^m (3+5 x)^3 \, dx\) [3187]

Optimal result

Integrand size = 20, antiderivative size = 91 \[ \int (1-2 x) (2+3 x)^m (3+5 x)^3 \, dx=-\frac {7 (2+3 x)^{1+m}}{243 (1+m)}+\frac {107 (2+3 x)^{2+m}}{243 (2+m)}-\frac {185 (2+3 x)^{3+m}}{81 (3+m)}+\frac {1025 (2+3 x)^{4+m}}{243 (4+m)}-\frac {250 (2+3 x)^{5+m}}{243 (5+m)} \]

[Out]

-7/243*(2+3*x)^(1+m)/(1+m)+107/243*(2+3*x)^(2+m)/(2+m)-185/81*(2+3*x)^(3+m)/(3+m)+1025/243*(2+3*x)^(4+m)/(4+m)
-250/243*(2+3*x)^(5+m)/(5+m)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78} \[ \int (1-2 x) (2+3 x)^m (3+5 x)^3 \, dx=-\frac {7 (3 x+2)^{m+1}}{243 (m+1)}+\frac {107 (3 x+2)^{m+2}}{243 (m+2)}-\frac {185 (3 x+2)^{m+3}}{81 (m+3)}+\frac {1025 (3 x+2)^{m+4}}{243 (m+4)}-\frac {250 (3 x+2)^{m+5}}{243 (m+5)} \]

[In]

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

[Out]

(-7*(2 + 3*x)^(1 + m))/(243*(1 + m)) + (107*(2 + 3*x)^(2 + m))/(243*(2 + m)) - (185*(2 + 3*x)^(3 + m))/(81*(3
+ m)) + (1025*(2 + 3*x)^(4 + m))/(243*(4 + m)) - (250*(2 + 3*x)^(5 + m))/(243*(5 + m))

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*} \text {integral}& = \int \left (-\frac {7}{81} (2+3 x)^m+\frac {107}{81} (2+3 x)^{1+m}-\frac {185}{27} (2+3 x)^{2+m}+\frac {1025}{81} (2+3 x)^{3+m}-\frac {250}{81} (2+3 x)^{4+m}\right ) \, dx \\ & = -\frac {7 (2+3 x)^{1+m}}{243 (1+m)}+\frac {107 (2+3 x)^{2+m}}{243 (2+m)}-\frac {185 (2+3 x)^{3+m}}{81 (3+m)}+\frac {1025 (2+3 x)^{4+m}}{243 (4+m)}-\frac {250 (2+3 x)^{5+m}}{243 (5+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.82 \[ \int (1-2 x) (2+3 x)^m (3+5 x)^3 \, dx=\frac {1}{243} (2+3 x)^{1+m} \left (-\frac {7}{1+m}+\frac {107 (2+3 x)}{2+m}-\frac {555 (2+3 x)^2}{3+m}+\frac {1025 (2+3 x)^3}{4+m}-\frac {250 (2+3 x)^4}{5+m}\right ) \]

[In]

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

[Out]

((2 + 3*x)^(1 + m)*(-7/(1 + m) + (107*(2 + 3*x))/(2 + m) - (555*(2 + 3*x)^2)/(3 + m) + (1025*(2 + 3*x)^3)/(4 +
m) - (250*(2 + 3*x)^4)/(5 + m)))/243

Maple [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3.

Time = 1.60 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.11

method	result	size
meijerg	\(27 \,2^{m} x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (1,-m ;2;-\frac {3 x}{2}\right )+81 \,2^{-1+m} x^{2} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (2,-m ;3;-\frac {3 x}{2}\right )-15 \,2^{m} x^{3} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (3,-m ;4;-\frac {3 x}{2}\right )-325 \,2^{-2+m} x^{4} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (4,-m ;5;-\frac {3 x}{2}\right )-25 \,2^{1+m} x^{5} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (5,-m ;6;-\frac {3 x}{2}\right )\)	\(101\)
gosper	\(-\frac {\left (2+3 x \right )^{1+m} \left (6750 m^{4} x^{4}+8775 m^{4} x^{3}+67500 m^{3} x^{4}+1215 m^{4} x^{2}+78525 m^{3} x^{3}+236250 m^{2} x^{4}-2187 m^{4} x -2970 m^{3} x^{2}+251775 m^{2} x^{3}+337500 m \,x^{4}-729 m^{4}-30051 m^{3} x -44865 m^{2} x^{2}+337275 m \,x^{3}+162000 x^{4}-8748 m^{3}-121833 m^{2} x -95580 m \,x^{2}+155250 x^{3}-33183 m^{2}-188589 x m -54900 x^{2}-49620 m -94620 x -24400\right )}{81 \left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right )}\)	\(187\)
risch	\(-\frac {\left (20250 m^{4} x^{5}+39825 m^{4} x^{4}+202500 m^{3} x^{5}+21195 m^{4} x^{3}+370575 m^{3} x^{4}+708750 m^{2} x^{5}-4131 m^{4} x^{2}+148140 m^{3} x^{3}+1227825 m^{2} x^{4}+1012500 m \,x^{5}-6561 m^{4} x -96093 m^{3} x^{2}+368955 m^{2} x^{3}+1686825 m \,x^{4}+486000 x^{5}-1458 m^{4}-86346 m^{3} x -455229 m^{2} x^{2}+387810 m \,x^{3}+789750 x^{4}-17496 m^{3}-343215 m^{2} x -756927 m \,x^{2}+145800 x^{3}-66366 m^{2}-526038 x m -393660 x^{2}-99240 m -262440 x -48800\right ) \left (2+3 x \right )^{m}}{81 \left (4+m \right ) \left (5+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\)	\(220\)
norman	\(-\frac {250 x^{5} {\mathrm e}^{m \ln \left (2+3 x \right )}}{5+m}+\frac {2 \left (729 m^{4}+8748 m^{3}+33183 m^{2}+49620 m +24400\right ) {\mathrm e}^{m \ln \left (2+3 x \right )}}{81 \left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right )}-\frac {25 \left (59 m +195\right ) x^{4} {\mathrm e}^{m \ln \left (2+3 x \right )}}{3 \left (m^{2}+9 m +20\right )}-\frac {5 \left (471 m^{2}+1879 m +1620\right ) x^{3} {\mathrm e}^{m \ln \left (2+3 x \right )}}{9 \left (m^{3}+12 m^{2}+47 m +60\right )}+\frac {\left (459 m^{3}+10218 m^{2}+40363 m +43740\right ) x^{2} {\mathrm e}^{m \ln \left (2+3 x \right )}}{9 m^{4}+126 m^{3}+639 m^{2}+1386 m +1080}+\frac {\left (2187 m^{4}+28782 m^{3}+114405 m^{2}+175346 m +87480\right ) x \,{\mathrm e}^{m \ln \left (2+3 x \right )}}{27 m^{5}+405 m^{4}+2295 m^{3}+6075 m^{2}+7398 m +3240}\)	\(251\)
parallelrisch	\(-\frac {-97600 \left (2+3 x \right )^{m}+2455650 x^{4} \left (2+3 x \right )^{m} m^{2}+296280 x^{3} \left (2+3 x \right )^{m} m^{3}-8262 x^{2} \left (2+3 x \right )^{m} m^{4}+3373650 x^{4} \left (2+3 x \right )^{m} m +737910 x^{3} \left (2+3 x \right )^{m} m^{2}-192186 x^{2} \left (2+3 x \right )^{m} m^{3}-13122 x \left (2+3 x \right )^{m} m^{4}+775620 x^{3} \left (2+3 x \right )^{m} m -910458 x^{2} \left (2+3 x \right )^{m} m^{2}-172692 x \left (2+3 x \right )^{m} m^{3}-1513854 x^{2} \left (2+3 x \right )^{m} m -686430 x \left (2+3 x \right )^{m} m^{2}-1052076 x \left (2+3 x \right )^{m} m +405000 x^{5} \left (2+3 x \right )^{m} m^{3}+79650 x^{4} \left (2+3 x \right )^{m} m^{4}+1417500 x^{5} \left (2+3 x \right )^{m} m^{2}+741150 x^{4} \left (2+3 x \right )^{m} m^{3}+42390 x^{3} \left (2+3 x \right )^{m} m^{4}+2025000 x^{5} \left (2+3 x \right )^{m} m +972000 \left (2+3 x \right )^{m} x^{5}+1579500 \left (2+3 x \right )^{m} x^{4}-2916 \left (2+3 x \right )^{m} m^{4}+291600 \left (2+3 x \right )^{m} x^{3}-34992 \left (2+3 x \right )^{m} m^{3}-787320 \left (2+3 x \right )^{m} x^{2}-132732 \left (2+3 x \right )^{m} m^{2}-524880 \left (2+3 x \right )^{m} x -198480 \left (2+3 x \right )^{m} m +40500 x^{5} \left (2+3 x \right )^{m} m^{4}}{162 \left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right )}\)	\(424\)

[In]

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

[Out]

27*2^m*x*hypergeom([1,-m],[2],-3/2*x)+81*2^(-1+m)*x^2*hypergeom([2,-m],[3],-3/2*x)-15*2^m*x^3*hypergeom([3,-m]
,[4],-3/2*x)-325*2^(-2+m)*x^4*hypergeom([4,-m],[5],-3/2*x)-25*2^(1+m)*x^5*hypergeom([5,-m],[6],-3/2*x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 175 vs. \(2 (81) = 162\).

Time = 0.24 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.92 \[ \int (1-2 x) (2+3 x)^m (3+5 x)^3 \, dx=-\frac {{\left (20250 \, {\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} x^{5} + 675 \, {\left (59 \, m^{4} + 549 \, m^{3} + 1819 \, m^{2} + 2499 \, m + 1170\right )} x^{4} - 1458 \, m^{4} + 45 \, {\left (471 \, m^{4} + 3292 \, m^{3} + 8199 \, m^{2} + 8618 \, m + 3240\right )} x^{3} - 17496 \, m^{3} - 9 \, {\left (459 \, m^{4} + 10677 \, m^{3} + 50581 \, m^{2} + 84103 \, m + 43740\right )} x^{2} - 66366 \, m^{2} - 3 \, {\left (2187 \, m^{4} + 28782 \, m^{3} + 114405 \, m^{2} + 175346 \, m + 87480\right )} x - 99240 \, m - 48800\right )} {\left (3 \, x + 2\right )}^{m}}{81 \, {\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )}} \]

[In]

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

[Out]

-1/81*(20250*(m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*x^5 + 675*(59*m^4 + 549*m^3 + 1819*m^2 + 2499*m + 1170)*x^4 -
1458*m^4 + 45*(471*m^4 + 3292*m^3 + 8199*m^2 + 8618*m + 3240)*x^3 - 17496*m^3 - 9*(459*m^4 + 10677*m^3 + 5058
1*m^2 + 84103*m + 43740)*x^2 - 66366*m^2 - 3*(2187*m^4 + 28782*m^3 + 114405*m^2 + 175346*m + 87480)*x - 99240*
m - 48800)*(3*x + 2)^m/(m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1822 vs. \(2 (75) = 150\).

Time = 0.68 (sec) , antiderivative size = 1822, normalized size of antiderivative = 20.02 \[ \int (1-2 x) (2+3 x)^m (3+5 x)^3 \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((-243000*x**4*log(3*x + 2)/(236196*x**4 + 629856*x**3 + 629856*x**2 + 279936*x + 46656) - 648000*x**
3*log(3*x + 2)/(236196*x**4 + 629856*x**3 + 629856*x**2 + 279936*x + 46656) - 332100*x**3/(236196*x**4 + 62985
6*x**3 + 629856*x**2 + 279936*x + 46656) - 648000*x**2*log(3*x + 2)/(236196*x**4 + 629856*x**3 + 629856*x**2 +
279936*x + 46656) - 634230*x**2/(236196*x**4 + 629856*x**3 + 629856*x**2 + 279936*x + 46656) - 288000*x*log(3
*x + 2)/(236196*x**4 + 629856*x**3 + 629856*x**2 + 279936*x + 46656) - 404124*x/(236196*x**4 + 629856*x**3 + 6
29856*x**2 + 279936*x + 46656) - 48000*log(3*x + 2)/(236196*x**4 + 629856*x**3 + 629856*x**2 + 279936*x + 4665
6) - 85915/(236196*x**4 + 629856*x**3 + 629856*x**2 + 279936*x + 46656), Eq(m, -5)), (-121500*x**4/(39366*x**3
+ 78732*x**2 + 52488*x + 11664) + 166050*x**3*log(3*x + 2)/(39366*x**3 + 78732*x**2 + 52488*x + 11664) + 3321
00*x**2*log(3*x + 2)/(39366*x**3 + 78732*x**2 + 52488*x + 11664) + 353970*x**2/(39366*x**3 + 78732*x**2 + 5248
8*x + 11664) + 221400*x*log(3*x + 2)/(39366*x**3 + 78732*x**2 + 52488*x + 11664) + 326997*x/(39366*x**3 + 7873
2*x**2 + 52488*x + 11664) + 49200*log(3*x + 2)/(39366*x**3 + 78732*x**2 + 52488*x + 11664) + 84692/(39366*x**3
+ 78732*x**2 + 52488*x + 11664), Eq(m, -4)), (-6750*x**4/(1458*x**2 + 1944*x + 648) + 450*x**3/(1458*x**2 + 1
944*x + 648) - 3330*x**2*log(3*x + 2)/(1458*x**2 + 1944*x + 648) - 4440*x*log(3*x + 2)/(1458*x**2 + 1944*x + 6
48) - 8814*x/(1458*x**2 + 1944*x + 648) - 1480*log(3*x + 2)/(1458*x**2 + 1944*x + 648) - 4407/(1458*x**2 + 194
4*x + 648), Eq(m, -3)), (-13500*x**4/(1458*x + 972) - 8325*x**3/(1458*x + 972) + 9360*x**2/(1458*x + 972) + 64
2*x*log(3*x + 2)/(1458*x + 972) + 428*log(3*x + 2)/(1458*x + 972) - 3946/(1458*x + 972), Eq(m, -2)), (-125*x**
4/6 - 475*x**3/27 + 545*x**2/54 + 1097*x/81 - 7*log(3*x + 2)/243, Eq(m, -1)), (-20250*m**4*x**5*(3*x + 2)**m/(
81*m**5 + 1215*m**4 + 6885*m**3 + 18225*m**2 + 22194*m + 9720) - 39825*m**4*x**4*(3*x + 2)**m/(81*m**5 + 1215*
m**4 + 6885*m**3 + 18225*m**2 + 22194*m + 9720) - 21195*m**4*x**3*(3*x + 2)**m/(81*m**5 + 1215*m**4 + 6885*m**
3 + 18225*m**2 + 22194*m + 9720) + 4131*m**4*x**2*(3*x + 2)**m/(81*m**5 + 1215*m**4 + 6885*m**3 + 18225*m**2 +
22194*m + 9720) + 6561*m**4*x*(3*x + 2)**m/(81*m**5 + 1215*m**4 + 6885*m**3 + 18225*m**2 + 22194*m + 9720) +
1458*m**4*(3*x + 2)**m/(81*m**5 + 1215*m**4 + 6885*m**3 + 18225*m**2 + 22194*m + 9720) - 202500*m**3*x**5*(3*x
+ 2)**m/(81*m**5 + 1215*m**4 + 6885*m**3 + 18225*m**2 + 22194*m + 9720) - 370575*m**3*x**4*(3*x + 2)**m/(81*m
**5 + 1215*m**4 + 6885*m**3 + 18225*m**2 + 22194*m + 9720) - 148140*m**3*x**3*(3*x + 2)**m/(81*m**5 + 1215*m**
4 + 6885*m**3 + 18225*m**2 + 22194*m + 9720) + 96093*m**3*x**2*(3*x + 2)**m/(81*m**5 + 1215*m**4 + 6885*m**3 +
18225*m**2 + 22194*m + 9720) + 86346*m**3*x*(3*x + 2)**m/(81*m**5 + 1215*m**4 + 6885*m**3 + 18225*m**2 + 2219
4*m + 9720) + 17496*m**3*(3*x + 2)**m/(81*m**5 + 1215*m**4 + 6885*m**3 + 18225*m**2 + 22194*m + 9720) - 708750
*m**2*x**5*(3*x + 2)**m/(81*m**5 + 1215*m**4 + 6885*m**3 + 18225*m**2 + 22194*m + 9720) - 1227825*m**2*x**4*(3
*x + 2)**m/(81*m**5 + 1215*m**4 + 6885*m**3 + 18225*m**2 + 22194*m + 9720) - 368955*m**2*x**3*(3*x + 2)**m/(81
*m**5 + 1215*m**4 + 6885*m**3 + 18225*m**2 + 22194*m + 9720) + 455229*m**2*x**2*(3*x + 2)**m/(81*m**5 + 1215*m
**4 + 6885*m**3 + 18225*m**2 + 22194*m + 9720) + 343215*m**2*x*(3*x + 2)**m/(81*m**5 + 1215*m**4 + 6885*m**3 +
18225*m**2 + 22194*m + 9720) + 66366*m**2*(3*x + 2)**m/(81*m**5 + 1215*m**4 + 6885*m**3 + 18225*m**2 + 22194*
m + 9720) - 1012500*m*x**5*(3*x + 2)**m/(81*m**5 + 1215*m**4 + 6885*m**3 + 18225*m**2 + 22194*m + 9720) - 1686
825*m*x**4*(3*x + 2)**m/(81*m**5 + 1215*m**4 + 6885*m**3 + 18225*m**2 + 22194*m + 9720) - 387810*m*x**3*(3*x +
2)**m/(81*m**5 + 1215*m**4 + 6885*m**3 + 18225*m**2 + 22194*m + 9720) + 756927*m*x**2*(3*x + 2)**m/(81*m**5 +
1215*m**4 + 6885*m**3 + 18225*m**2 + 22194*m + 9720) + 526038*m*x*(3*x + 2)**m/(81*m**5 + 1215*m**4 + 6885*m*
*3 + 18225*m**2 + 22194*m + 9720) + 99240*m*(3*x + 2)**m/(81*m**5 + 1215*m**4 + 6885*m**3 + 18225*m**2 + 22194
*m + 9720) - 486000*x**5*(3*x + 2)**m/(81*m**5 + 1215*m**4 + 6885*m**3 + 18225*m**2 + 22194*m + 9720) - 789750
*x**4*(3*x + 2)**m/(81*m**5 + 1215*m**4 + 6885*m**3 + 18225*m**2 + 22194*m + 9720) - 145800*x**3*(3*x + 2)**m/
(81*m**5 + 1215*m**4 + 6885*m**3 + 18225*m**2 + 22194*m + 9720) + 393660*x**2*(3*x + 2)**m/(81*m**5 + 1215*m**
4 + 6885*m**3 + 18225*m**2 + 22194*m + 9720) + 262440*x*(3*x + 2)**m/(81*m**5 + 1215*m**4 + 6885*m**3 + 18225*
m**2 + 22194*m + 9720) + 48800*(3*x + 2)**m/(81*m**5 + 1215*m**4 + 6885*m**3 + 18225*m**2 + 22194*m + 9720), T
rue))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 295 vs. \(2 (81) = 162\).

Time = 0.21 (sec) , antiderivative size = 295, normalized size of antiderivative = 3.24 \[ \int (1-2 x) (2+3 x)^m (3+5 x)^3 \, dx=-\frac {250 \, {\left (81 \, {\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} x^{5} + 54 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} x^{4} - 144 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} x^{3} + 288 \, {\left (m^{2} + m\right )} x^{2} - 384 \, m x + 256\right )} {\left (3 \, x + 2\right )}^{m}}{81 \, {\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )}} - \frac {325 \, {\left (27 \, {\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} x^{4} + 18 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} x^{3} - 36 \, {\left (m^{2} + m\right )} x^{2} + 48 \, m x - 32\right )} {\left (3 \, x + 2\right )}^{m}}{27 \, {\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )}} - \frac {5 \, {\left (27 \, {\left (m^{2} + 3 \, m + 2\right )} x^{3} + 18 \, {\left (m^{2} + m\right )} x^{2} - 24 \, m x + 16\right )} {\left (3 \, x + 2\right )}^{m}}{3 \, {\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )}} + \frac {9 \, {\left (9 \, {\left (m + 1\right )} x^{2} + 6 \, m x - 4\right )} {\left (3 \, x + 2\right )}^{m}}{m^{2} + 3 \, m + 2} + \frac {9 \, {\left (3 \, x + 2\right )}^{m + 1}}{m + 1} \]

[In]

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

[Out]

-250/81*(81*(m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*x^5 + 54*(m^4 + 6*m^3 + 11*m^2 + 6*m)*x^4 - 144*(m^3 + 3*m^2 +
2*m)*x^3 + 288*(m^2 + m)*x^2 - 384*m*x + 256)*(3*x + 2)^m/(m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120) - 3
25/27*(27*(m^3 + 6*m^2 + 11*m + 6)*x^4 + 18*(m^3 + 3*m^2 + 2*m)*x^3 - 36*(m^2 + m)*x^2 + 48*m*x - 32)*(3*x + 2
)^m/(m^4 + 10*m^3 + 35*m^2 + 50*m + 24) - 5/3*(27*(m^2 + 3*m + 2)*x^3 + 18*(m^2 + m)*x^2 - 24*m*x + 16)*(3*x +
2)^m/(m^3 + 6*m^2 + 11*m + 6) + 9*(9*(m + 1)*x^2 + 6*m*x - 4)*(3*x + 2)^m/(m^2 + 3*m + 2) + 9*(3*x + 2)^(m +
1)/(m + 1)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 423 vs. \(2 (81) = 162\).

Time = 0.28 (sec) , antiderivative size = 423, normalized size of antiderivative = 4.65 \[ \int (1-2 x) (2+3 x)^m (3+5 x)^3 \, dx=-\frac {20250 \, m^{4} {\left (3 \, x + 2\right )}^{m} x^{5} + 39825 \, m^{4} {\left (3 \, x + 2\right )}^{m} x^{4} + 202500 \, m^{3} {\left (3 \, x + 2\right )}^{m} x^{5} + 21195 \, m^{4} {\left (3 \, x + 2\right )}^{m} x^{3} + 370575 \, m^{3} {\left (3 \, x + 2\right )}^{m} x^{4} + 708750 \, m^{2} {\left (3 \, x + 2\right )}^{m} x^{5} - 4131 \, m^{4} {\left (3 \, x + 2\right )}^{m} x^{2} + 148140 \, m^{3} {\left (3 \, x + 2\right )}^{m} x^{3} + 1227825 \, m^{2} {\left (3 \, x + 2\right )}^{m} x^{4} + 1012500 \, m {\left (3 \, x + 2\right )}^{m} x^{5} - 6561 \, m^{4} {\left (3 \, x + 2\right )}^{m} x - 96093 \, m^{3} {\left (3 \, x + 2\right )}^{m} x^{2} + 368955 \, m^{2} {\left (3 \, x + 2\right )}^{m} x^{3} + 1686825 \, m {\left (3 \, x + 2\right )}^{m} x^{4} + 486000 \, {\left (3 \, x + 2\right )}^{m} x^{5} - 1458 \, m^{4} {\left (3 \, x + 2\right )}^{m} - 86346 \, m^{3} {\left (3 \, x + 2\right )}^{m} x - 455229 \, m^{2} {\left (3 \, x + 2\right )}^{m} x^{2} + 387810 \, m {\left (3 \, x + 2\right )}^{m} x^{3} + 789750 \, {\left (3 \, x + 2\right )}^{m} x^{4} - 17496 \, m^{3} {\left (3 \, x + 2\right )}^{m} - 343215 \, m^{2} {\left (3 \, x + 2\right )}^{m} x - 756927 \, m {\left (3 \, x + 2\right )}^{m} x^{2} + 145800 \, {\left (3 \, x + 2\right )}^{m} x^{3} - 66366 \, m^{2} {\left (3 \, x + 2\right )}^{m} - 526038 \, m {\left (3 \, x + 2\right )}^{m} x - 393660 \, {\left (3 \, x + 2\right )}^{m} x^{2} - 99240 \, m {\left (3 \, x + 2\right )}^{m} - 262440 \, {\left (3 \, x + 2\right )}^{m} x - 48800 \, {\left (3 \, x + 2\right )}^{m}}{81 \, {\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )}} \]

[In]

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

[Out]

-1/81*(20250*m^4*(3*x + 2)^m*x^5 + 39825*m^4*(3*x + 2)^m*x^4 + 202500*m^3*(3*x + 2)^m*x^5 + 21195*m^4*(3*x + 2
)^m*x^3 + 370575*m^3*(3*x + 2)^m*x^4 + 708750*m^2*(3*x + 2)^m*x^5 - 4131*m^4*(3*x + 2)^m*x^2 + 148140*m^3*(3*x
+ 2)^m*x^3 + 1227825*m^2*(3*x + 2)^m*x^4 + 1012500*m*(3*x + 2)^m*x^5 - 6561*m^4*(3*x + 2)^m*x - 96093*m^3*(3*
x + 2)^m*x^2 + 368955*m^2*(3*x + 2)^m*x^3 + 1686825*m*(3*x + 2)^m*x^4 + 486000*(3*x + 2)^m*x^5 - 1458*m^4*(3*x
+ 2)^m - 86346*m^3*(3*x + 2)^m*x - 455229*m^2*(3*x + 2)^m*x^2 + 387810*m*(3*x + 2)^m*x^3 + 789750*(3*x + 2)^m
*x^4 - 17496*m^3*(3*x + 2)^m - 343215*m^2*(3*x + 2)^m*x - 756927*m*(3*x + 2)^m*x^2 + 145800*(3*x + 2)^m*x^3 -
66366*m^2*(3*x + 2)^m - 526038*m*(3*x + 2)^m*x - 393660*(3*x + 2)^m*x^2 - 99240*m*(3*x + 2)^m - 262440*(3*x +
2)^m*x - 48800*(3*x + 2)^m)/(m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)

Mupad [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 313, normalized size of antiderivative = 3.44 \[ \int (1-2 x) (2+3 x)^m (3+5 x)^3 \, dx={\left (3\,x+2\right )}^m\,\left (\frac {1458\,m^4+17496\,m^3+66366\,m^2+99240\,m+48800}{81\,m^5+1215\,m^4+6885\,m^3+18225\,m^2+22194\,m+9720}-\frac {x^3\,\left (21195\,m^4+148140\,m^3+368955\,m^2+387810\,m+145800\right )}{81\,m^5+1215\,m^4+6885\,m^3+18225\,m^2+22194\,m+9720}+\frac {x^2\,\left (4131\,m^4+96093\,m^3+455229\,m^2+756927\,m+393660\right )}{81\,m^5+1215\,m^4+6885\,m^3+18225\,m^2+22194\,m+9720}-\frac {x^5\,\left (20250\,m^4+202500\,m^3+708750\,m^2+1012500\,m+486000\right )}{81\,m^5+1215\,m^4+6885\,m^3+18225\,m^2+22194\,m+9720}-\frac {x^4\,\left (39825\,m^4+370575\,m^3+1227825\,m^2+1686825\,m+789750\right )}{81\,m^5+1215\,m^4+6885\,m^3+18225\,m^2+22194\,m+9720}+\frac {x\,\left (6561\,m^4+86346\,m^3+343215\,m^2+526038\,m+262440\right )}{81\,m^5+1215\,m^4+6885\,m^3+18225\,m^2+22194\,m+9720}\right ) \]

[In]

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

[Out]

(3*x + 2)^m*((99240*m + 66366*m^2 + 17496*m^3 + 1458*m^4 + 48800)/(22194*m + 18225*m^2 + 6885*m^3 + 1215*m^4 +
81*m^5 + 9720) - (x^3*(387810*m + 368955*m^2 + 148140*m^3 + 21195*m^4 + 145800))/(22194*m + 18225*m^2 + 6885*
m^3 + 1215*m^4 + 81*m^5 + 9720) + (x^2*(756927*m + 455229*m^2 + 96093*m^3 + 4131*m^4 + 393660))/(22194*m + 182
25*m^2 + 6885*m^3 + 1215*m^4 + 81*m^5 + 9720) - (x^5*(1012500*m + 708750*m^2 + 202500*m^3 + 20250*m^4 + 486000
))/(22194*m + 18225*m^2 + 6885*m^3 + 1215*m^4 + 81*m^5 + 9720) - (x^4*(1686825*m + 1227825*m^2 + 370575*m^3 +
39825*m^4 + 789750))/(22194*m + 18225*m^2 + 6885*m^3 + 1215*m^4 + 81*m^5 + 9720) + (x*(526038*m + 343215*m^2 +
86346*m^3 + 6561*m^4 + 262440))/(22194*m + 18225*m^2 + 6885*m^3 + 1215*m^4 + 81*m^5 + 9720))