\(\int (1-2 x) (2+3 x)^m (3+5 x)^2 \, dx\) [3188]

Optimal result

Integrand size = 20, antiderivative size = 73 \[ \int (1-2 x) (2+3 x)^m (3+5 x)^2 \, dx=\frac {7 (2+3 x)^{1+m}}{81 (1+m)}-\frac {8 (2+3 x)^{2+m}}{9 (2+m)}+\frac {65 (2+3 x)^{3+m}}{27 (3+m)}-\frac {50 (2+3 x)^{4+m}}{81 (4+m)} \]

[Out]

7/81*(2+3*x)^(1+m)/(1+m)-8/9*(2+3*x)^(2+m)/(2+m)+65/27*(2+3*x)^(3+m)/(3+m)-50/81*(2+3*x)^(4+m)/(4+m)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 73, 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} \[ \int (1-2 x) (2+3 x)^m (3+5 x)^2 \, dx=\frac {7 (3 x+2)^{m+1}}{81 (m+1)}-\frac {8 (3 x+2)^{m+2}}{9 (m+2)}+\frac {65 (3 x+2)^{m+3}}{27 (m+3)}-\frac {50 (3 x+2)^{m+4}}{81 (m+4)} \]

[In]

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

[Out]

(7*(2 + 3*x)^(1 + m))/(81*(1 + m)) - (8*(2 + 3*x)^(2 + m))/(9*(2 + m)) + (65*(2 + 3*x)^(3 + m))/(27*(3 + m)) -
(50*(2 + 3*x)^(4 + m))/(81*(4 + 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}{27} (2+3 x)^m-\frac {8}{3} (2+3 x)^{1+m}+\frac {65}{9} (2+3 x)^{2+m}-\frac {50}{27} (2+3 x)^{3+m}\right ) \, dx \\ & = \frac {7 (2+3 x)^{1+m}}{81 (1+m)}-\frac {8 (2+3 x)^{2+m}}{9 (2+m)}+\frac {65 (2+3 x)^{3+m}}{27 (3+m)}-\frac {50 (2+3 x)^{4+m}}{81 (4+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.84 \[ \int (1-2 x) (2+3 x)^m (3+5 x)^2 \, dx=\frac {1}{81} (2+3 x)^{1+m} \left (\frac {7}{1+m}-\frac {72 (2+3 x)}{2+m}+\frac {195 (2+3 x)^2}{3+m}-\frac {50 (2+3 x)^3}{4+m}\right ) \]

[In]

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

[Out]

((2 + 3*x)^(1 + m)*(7/(1 + m) - (72*(2 + 3*x))/(2 + m) + (195*(2 + 3*x)^2)/(3 + m) - (50*(2 + 3*x)^3)/(4 + m))
)/81

Maple [C] (verified)

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

Time = 1.57 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.10

method	result	size
meijerg	\(9 \,2^{m} x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (1,-m ;2;-\frac {3 x}{2}\right )+3 \,2^{1+m} x^{2} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (2,-m ;3;-\frac {3 x}{2}\right )-\frac {35 \,2^{m} x^{3} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (3,-m ;4;-\frac {3 x}{2}\right )}{3}-25 \,2^{-1+m} x^{4} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (4,-m ;5;-\frac {3 x}{2}\right )\)	\(80\)
gosper	\(-\frac {\left (2+3 x \right )^{1+m} \left (450 m^{3} x^{3}+315 m^{3} x^{2}+2700 m^{2} x^{3}-108 m^{3} x +1305 m^{2} x^{2}+4950 m \,x^{3}-81 m^{3}-1284 m^{2} x +1710 m \,x^{2}+2700 x^{3}-657 m^{2}-2952 x m +720 x^{2}-1322 m -1776 x -760\right )}{27 \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}\)	\(120\)
risch	\(-\frac {\left (1350 m^{3} x^{4}+1845 m^{3} x^{3}+8100 m^{2} x^{4}+306 m^{3} x^{2}+9315 m^{2} x^{3}+14850 m \,x^{4}-459 m^{3} x -1242 m^{2} x^{2}+15030 m \,x^{3}+8100 x^{4}-162 m^{3}-4539 m^{2} x -5436 m \,x^{2}+7560 x^{3}-1314 m^{2}-9870 x m -3888 x^{2}-2644 m -5832 x -1520\right ) \left (2+3 x \right )^{m}}{27 \left (3+m \right ) \left (4+m \right ) \left (2+m \right ) \left (1+m \right )}\)	\(145\)
norman	\(-\frac {50 x^{4} {\mathrm e}^{m \ln \left (2+3 x \right )}}{4+m}+\frac {2 \left (81 m^{3}+657 m^{2}+1322 m +760\right ) {\mathrm e}^{m \ln \left (2+3 x \right )}}{27 \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}-\frac {5 \left (41 m +84\right ) x^{3} {\mathrm e}^{m \ln \left (2+3 x \right )}}{3 \left (m^{2}+7 m +12\right )}-\frac {2 \left (17 m^{2}-86 m -216\right ) x^{2} {\mathrm e}^{m \ln \left (2+3 x \right )}}{3 \left (m^{3}+9 m^{2}+26 m +24\right )}+\frac {\left (153 m^{3}+1513 m^{2}+3290 m +1944\right ) x \,{\mathrm e}^{m \ln \left (2+3 x \right )}}{9 m^{4}+90 m^{3}+315 m^{2}+450 m +216}\)	\(182\)
parallelrisch	\(-\frac {-3040 \left (2+3 x \right )^{m}+16200 x^{4} \left (2+3 x \right )^{m} m^{2}+3690 x^{3} \left (2+3 x \right )^{m} m^{3}+29700 x^{4} \left (2+3 x \right )^{m} m +18630 x^{3} \left (2+3 x \right )^{m} m^{2}+612 x^{2} \left (2+3 x \right )^{m} m^{3}+30060 x^{3} \left (2+3 x \right )^{m} m -2484 x^{2} \left (2+3 x \right )^{m} m^{2}-918 x \left (2+3 x \right )^{m} m^{3}-10872 x^{2} \left (2+3 x \right )^{m} m -9078 x \left (2+3 x \right )^{m} m^{2}-19740 x \left (2+3 x \right )^{m} m +2700 x^{4} \left (2+3 x \right )^{m} m^{3}+16200 \left (2+3 x \right )^{m} x^{4}+15120 \left (2+3 x \right )^{m} x^{3}-324 \left (2+3 x \right )^{m} m^{3}-7776 \left (2+3 x \right )^{m} x^{2}-2628 \left (2+3 x \right )^{m} m^{2}-11664 \left (2+3 x \right )^{m} x -5288 \left (2+3 x \right )^{m} m}{54 \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}\)	\(279\)

[In]

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

[Out]

9*2^m*x*hypergeom([1,-m],[2],-3/2*x)+3*2^(1+m)*x^2*hypergeom([2,-m],[3],-3/2*x)-35/3*2^m*x^3*hypergeom([3,-m],
[4],-3/2*x)-25*2^(-1+m)*x^4*hypergeom([4,-m],[5],-3/2*x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.64 \[ \int (1-2 x) (2+3 x)^m (3+5 x)^2 \, dx=-\frac {{\left (1350 \, {\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} x^{4} + 45 \, {\left (41 \, m^{3} + 207 \, m^{2} + 334 \, m + 168\right )} x^{3} - 162 \, m^{3} + 18 \, {\left (17 \, m^{3} - 69 \, m^{2} - 302 \, m - 216\right )} x^{2} - 1314 \, m^{2} - 3 \, {\left (153 \, m^{3} + 1513 \, m^{2} + 3290 \, m + 1944\right )} x - 2644 \, m - 1520\right )} {\left (3 \, x + 2\right )}^{m}}{27 \, {\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )}} \]

[In]

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

[Out]

-1/27*(1350*(m^3 + 6*m^2 + 11*m + 6)*x^4 + 45*(41*m^3 + 207*m^2 + 334*m + 168)*x^3 - 162*m^3 + 18*(17*m^3 - 69
*m^2 - 302*m - 216)*x^2 - 1314*m^2 - 3*(153*m^3 + 1513*m^2 + 3290*m + 1944)*x - 2644*m - 1520)*(3*x + 2)^m/(m^
4 + 10*m^3 + 35*m^2 + 50*m + 24)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1017 vs. \(2 (60) = 120\).

Time = 0.48 (sec) , antiderivative size = 1017, normalized size of antiderivative = 13.93 \[ \int (1-2 x) (2+3 x)^m (3+5 x)^2 \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((-4050*x**3*log(3*x + 2)/(6561*x**3 + 13122*x**2 + 8748*x + 1944) - 8100*x**2*log(3*x + 2)/(6561*x**
3 + 13122*x**2 + 8748*x + 1944) - 5265*x**2/(6561*x**3 + 13122*x**2 + 8748*x + 1944) - 5400*x*log(3*x + 2)/(65
61*x**3 + 13122*x**2 + 8748*x + 1944) - 6696*x/(6561*x**3 + 13122*x**2 + 8748*x + 1944) - 1200*log(3*x + 2)/(6
561*x**3 + 13122*x**2 + 8748*x + 1944) - 2131/(6561*x**3 + 13122*x**2 + 8748*x + 1944), Eq(m, -4)), (-900*x**3
/(486*x**2 + 648*x + 216) + 1170*x**2*log(3*x + 2)/(486*x**2 + 648*x + 216) + 1560*x*log(3*x + 2)/(486*x**2 +
648*x + 216) + 1344*x/(486*x**2 + 648*x + 216) + 520*log(3*x + 2)/(486*x**2 + 648*x + 216) + 627/(486*x**2 + 6
48*x + 216), Eq(m, -3)), (-75*x**3/(27*x + 18) + 45*x**2/(27*x + 18) - 24*x*log(3*x + 2)/(27*x + 18) - 16*log(
3*x + 2)/(27*x + 18) - 43/(27*x + 18), Eq(m, -2)), (-50*x**3/9 - 5*x**2/18 + 118*x/27 + 7*log(3*x + 2)/81, Eq(
m, -1)), (-1350*m**3*x**4*(3*x + 2)**m/(27*m**4 + 270*m**3 + 945*m**2 + 1350*m + 648) - 1845*m**3*x**3*(3*x +
2)**m/(27*m**4 + 270*m**3 + 945*m**2 + 1350*m + 648) - 306*m**3*x**2*(3*x + 2)**m/(27*m**4 + 270*m**3 + 945*m*
*2 + 1350*m + 648) + 459*m**3*x*(3*x + 2)**m/(27*m**4 + 270*m**3 + 945*m**2 + 1350*m + 648) + 162*m**3*(3*x +
2)**m/(27*m**4 + 270*m**3 + 945*m**2 + 1350*m + 648) - 8100*m**2*x**4*(3*x + 2)**m/(27*m**4 + 270*m**3 + 945*m
**2 + 1350*m + 648) - 9315*m**2*x**3*(3*x + 2)**m/(27*m**4 + 270*m**3 + 945*m**2 + 1350*m + 648) + 1242*m**2*x
**2*(3*x + 2)**m/(27*m**4 + 270*m**3 + 945*m**2 + 1350*m + 648) + 4539*m**2*x*(3*x + 2)**m/(27*m**4 + 270*m**3
+ 945*m**2 + 1350*m + 648) + 1314*m**2*(3*x + 2)**m/(27*m**4 + 270*m**3 + 945*m**2 + 1350*m + 648) - 14850*m*
x**4*(3*x + 2)**m/(27*m**4 + 270*m**3 + 945*m**2 + 1350*m + 648) - 15030*m*x**3*(3*x + 2)**m/(27*m**4 + 270*m*
*3 + 945*m**2 + 1350*m + 648) + 5436*m*x**2*(3*x + 2)**m/(27*m**4 + 270*m**3 + 945*m**2 + 1350*m + 648) + 9870
*m*x*(3*x + 2)**m/(27*m**4 + 270*m**3 + 945*m**2 + 1350*m + 648) + 2644*m*(3*x + 2)**m/(27*m**4 + 270*m**3 + 9
45*m**2 + 1350*m + 648) - 8100*x**4*(3*x + 2)**m/(27*m**4 + 270*m**3 + 945*m**2 + 1350*m + 648) - 7560*x**3*(3
*x + 2)**m/(27*m**4 + 270*m**3 + 945*m**2 + 1350*m + 648) + 3888*x**2*(3*x + 2)**m/(27*m**4 + 270*m**3 + 945*m
**2 + 1350*m + 648) + 5832*x*(3*x + 2)**m/(27*m**4 + 270*m**3 + 945*m**2 + 1350*m + 648) + 1520*(3*x + 2)**m/(
27*m**4 + 270*m**3 + 945*m**2 + 1350*m + 648), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 183 vs. \(2 (65) = 130\).

Time = 0.23 (sec) , antiderivative size = 183, normalized size of antiderivative = 2.51 \[ \int (1-2 x) (2+3 x)^m (3+5 x)^2 \, dx=-\frac {50 \, {\left (27 \, {\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} x^{4} + 18 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} x^{3} - 36 \, {\left (m^{2} + m\right )} x^{2} + 48 \, m x - 32\right )} {\left (3 \, x + 2\right )}^{m}}{27 \, {\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )}} - \frac {35 \, {\left (27 \, {\left (m^{2} + 3 \, m + 2\right )} x^{3} + 18 \, {\left (m^{2} + m\right )} x^{2} - 24 \, m x + 16\right )} {\left (3 \, x + 2\right )}^{m}}{27 \, {\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )}} + \frac {4 \, {\left (9 \, {\left (m + 1\right )} x^{2} + 6 \, m x - 4\right )} {\left (3 \, x + 2\right )}^{m}}{3 \, {\left (m^{2} + 3 \, m + 2\right )}} + \frac {3 \, {\left (3 \, x + 2\right )}^{m + 1}}{m + 1} \]

[In]

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

[Out]

-50/27*(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) - 35/27*(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) + 4/3*(9*(m + 1)*x^2 + 6*m*x - 4)*(3*x + 2)^m/(m^2 + 3*m + 2) + 3*(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. 278 vs. \(2 (65) = 130\).

Time = 0.28 (sec) , antiderivative size = 278, normalized size of antiderivative = 3.81 \[ \int (1-2 x) (2+3 x)^m (3+5 x)^2 \, dx=-\frac {1350 \, m^{3} {\left (3 \, x + 2\right )}^{m} x^{4} + 1845 \, m^{3} {\left (3 \, x + 2\right )}^{m} x^{3} + 8100 \, m^{2} {\left (3 \, x + 2\right )}^{m} x^{4} + 306 \, m^{3} {\left (3 \, x + 2\right )}^{m} x^{2} + 9315 \, m^{2} {\left (3 \, x + 2\right )}^{m} x^{3} + 14850 \, m {\left (3 \, x + 2\right )}^{m} x^{4} - 459 \, m^{3} {\left (3 \, x + 2\right )}^{m} x - 1242 \, m^{2} {\left (3 \, x + 2\right )}^{m} x^{2} + 15030 \, m {\left (3 \, x + 2\right )}^{m} x^{3} + 8100 \, {\left (3 \, x + 2\right )}^{m} x^{4} - 162 \, m^{3} {\left (3 \, x + 2\right )}^{m} - 4539 \, m^{2} {\left (3 \, x + 2\right )}^{m} x - 5436 \, m {\left (3 \, x + 2\right )}^{m} x^{2} + 7560 \, {\left (3 \, x + 2\right )}^{m} x^{3} - 1314 \, m^{2} {\left (3 \, x + 2\right )}^{m} - 9870 \, m {\left (3 \, x + 2\right )}^{m} x - 3888 \, {\left (3 \, x + 2\right )}^{m} x^{2} - 2644 \, m {\left (3 \, x + 2\right )}^{m} - 5832 \, {\left (3 \, x + 2\right )}^{m} x - 1520 \, {\left (3 \, x + 2\right )}^{m}}{27 \, {\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )}} \]

[In]

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

[Out]

-1/27*(1350*m^3*(3*x + 2)^m*x^4 + 1845*m^3*(3*x + 2)^m*x^3 + 8100*m^2*(3*x + 2)^m*x^4 + 306*m^3*(3*x + 2)^m*x^
2 + 9315*m^2*(3*x + 2)^m*x^3 + 14850*m*(3*x + 2)^m*x^4 - 459*m^3*(3*x + 2)^m*x - 1242*m^2*(3*x + 2)^m*x^2 + 15
030*m*(3*x + 2)^m*x^3 + 8100*(3*x + 2)^m*x^4 - 162*m^3*(3*x + 2)^m - 4539*m^2*(3*x + 2)^m*x - 5436*m*(3*x + 2)
^m*x^2 + 7560*(3*x + 2)^m*x^3 - 1314*m^2*(3*x + 2)^m - 9870*m*(3*x + 2)^m*x - 3888*(3*x + 2)^m*x^2 - 2644*m*(3
*x + 2)^m - 5832*(3*x + 2)^m*x - 1520*(3*x + 2)^m)/(m^4 + 10*m^3 + 35*m^2 + 50*m + 24)

Mupad [B] (verification not implemented)

Time = 2.91 (sec) , antiderivative size = 211, normalized size of antiderivative = 2.89 \[ \int (1-2 x) (2+3 x)^m (3+5 x)^2 \, dx={\left (3\,x+2\right )}^m\,\left (\frac {162\,m^3+1314\,m^2+2644\,m+1520}{27\,m^4+270\,m^3+945\,m^2+1350\,m+648}+\frac {x\,\left (459\,m^3+4539\,m^2+9870\,m+5832\right )}{27\,m^4+270\,m^3+945\,m^2+1350\,m+648}+\frac {x^2\,\left (-306\,m^3+1242\,m^2+5436\,m+3888\right )}{27\,m^4+270\,m^3+945\,m^2+1350\,m+648}-\frac {x^4\,\left (1350\,m^3+8100\,m^2+14850\,m+8100\right )}{27\,m^4+270\,m^3+945\,m^2+1350\,m+648}-\frac {x^3\,\left (1845\,m^3+9315\,m^2+15030\,m+7560\right )}{27\,m^4+270\,m^3+945\,m^2+1350\,m+648}\right ) \]

[In]

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

[Out]

(3*x + 2)^m*((2644*m + 1314*m^2 + 162*m^3 + 1520)/(1350*m + 945*m^2 + 270*m^3 + 27*m^4 + 648) + (x*(9870*m + 4
539*m^2 + 459*m^3 + 5832))/(1350*m + 945*m^2 + 270*m^3 + 27*m^4 + 648) + (x^2*(5436*m + 1242*m^2 - 306*m^3 + 3
888))/(1350*m + 945*m^2 + 270*m^3 + 27*m^4 + 648) - (x^4*(14850*m + 8100*m^2 + 1350*m^3 + 8100))/(1350*m + 945
*m^2 + 270*m^3 + 27*m^4 + 648) - (x^3*(15030*m + 9315*m^2 + 1845*m^3 + 7560))/(1350*m + 945*m^2 + 270*m^3 + 27
*m^4 + 648))