\(\int (1-2 x) (2+3 x)^m (3+5 x) \, dx\) [3189]

Optimal result

Integrand size = 18, antiderivative size = 55 \[ \int (1-2 x) (2+3 x)^m (3+5 x) \, dx=-\frac {7 (2+3 x)^{1+m}}{27 (1+m)}+\frac {37 (2+3 x)^{2+m}}{27 (2+m)}-\frac {10 (2+3 x)^{3+m}}{27 (3+m)} \]

[Out]

-7/27*(2+3*x)^(1+m)/(1+m)+37/27*(2+3*x)^(2+m)/(2+m)-10/27*(2+3*x)^(3+m)/(3+m)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {78} \[ \int (1-2 x) (2+3 x)^m (3+5 x) \, dx=-\frac {7 (3 x+2)^{m+1}}{27 (m+1)}+\frac {37 (3 x+2)^{m+2}}{27 (m+2)}-\frac {10 (3 x+2)^{m+3}}{27 (m+3)} \]

[In]

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

[Out]

(-7*(2 + 3*x)^(1 + m))/(27*(1 + m)) + (37*(2 + 3*x)^(2 + m))/(27*(2 + m)) - (10*(2 + 3*x)^(3 + m))/(27*(3 + 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}{9} (2+3 x)^m+\frac {37}{9} (2+3 x)^{1+m}-\frac {10}{9} (2+3 x)^{2+m}\right ) \, dx \\ & = -\frac {7 (2+3 x)^{1+m}}{27 (1+m)}+\frac {37 (2+3 x)^{2+m}}{27 (2+m)}-\frac {10 (2+3 x)^{3+m}}{27 (3+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.85 \[ \int (1-2 x) (2+3 x)^m (3+5 x) \, dx=\frac {1}{27} (2+3 x)^{1+m} \left (-\frac {7}{1+m}+\frac {37 (2+3 x)}{2+m}-\frac {10 (2+3 x)^2}{3+m}\right ) \]

[In]

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

[Out]

((2 + 3*x)^(1 + m)*(-7/(1 + m) + (37*(2 + 3*x))/(2 + m) - (10*(2 + 3*x)^2)/(3 + m)))/27

Maple [C] (verified)

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

Time = 1.49 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.11

method	result	size
meijerg	\(3 \,2^{m} x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (1,-m ;2;-\frac {3 x}{2}\right )-2^{-1+m} x^{2} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (2,-m ;3;-\frac {3 x}{2}\right )-\frac {5 \,2^{1+m} x^{3} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (3,-m ;4;-\frac {3 x}{2}\right )}{3}\)	\(61\)
gosper	\(-\frac {\left (2+3 x \right )^{1+m} \left (90 m^{2} x^{2}+9 m^{2} x +270 m \,x^{2}-27 m^{2}-84 x m +180 x^{2}-141 m -93 x -100\right )}{27 \left (m^{3}+6 m^{2}+11 m +6\right )}\)	\(69\)
risch	\(-\frac {\left (270 m^{2} x^{3}+207 m^{2} x^{2}+810 m \,x^{3}-63 m^{2} x +288 m \,x^{2}+540 x^{3}-54 m^{2}-591 x m +81 x^{2}-282 m -486 x -200\right ) \left (2+3 x \right )^{m}}{27 \left (2+m \right ) \left (3+m \right ) \left (1+m \right )}\)	\(86\)
norman	\(-\frac {10 x^{3} {\mathrm e}^{m \ln \left (2+3 x \right )}}{3+m}+\frac {2 \left (27 m^{2}+141 m +100\right ) {\mathrm e}^{m \ln \left (2+3 x \right )}}{27 \left (m^{3}+6 m^{2}+11 m +6\right )}-\frac {\left (23 m +9\right ) x^{2} {\mathrm e}^{m \ln \left (2+3 x \right )}}{3 \left (m^{2}+5 m +6\right )}+\frac {\left (21 m^{2}+197 m +162\right ) x \,{\mathrm e}^{m \ln \left (2+3 x \right )}}{9 m^{3}+54 m^{2}+99 m +54}\)	\(123\)
parallelrisch	\(-\frac {540 x^{3} \left (2+3 x \right )^{m} m^{2}+1620 x^{3} \left (2+3 x \right )^{m} m +414 x^{2} \left (2+3 x \right )^{m} m^{2}+1080 \left (2+3 x \right )^{m} x^{3}+576 x^{2} \left (2+3 x \right )^{m} m -126 x \left (2+3 x \right )^{m} m^{2}+162 \left (2+3 x \right )^{m} x^{2}-1182 x \left (2+3 x \right )^{m} m -108 \left (2+3 x \right )^{m} m^{2}-972 \left (2+3 x \right )^{m} x -564 \left (2+3 x \right )^{m} m -400 \left (2+3 x \right )^{m}}{54 \left (m^{3}+6 m^{2}+11 m +6\right )}\)	\(164\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.36 \[ \int (1-2 x) (2+3 x)^m (3+5 x) \, dx=-\frac {{\left (270 \, {\left (m^{2} + 3 \, m + 2\right )} x^{3} + 9 \, {\left (23 \, m^{2} + 32 \, m + 9\right )} x^{2} - 54 \, m^{2} - 3 \, {\left (21 \, m^{2} + 197 \, m + 162\right )} x - 282 \, m - 200\right )} {\left (3 \, x + 2\right )}^{m}}{27 \, {\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )}} \]

[In]

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

[Out]

-1/27*(270*(m^2 + 3*m + 2)*x^3 + 9*(23*m^2 + 32*m + 9)*x^2 - 54*m^2 - 3*(21*m^2 + 197*m + 162)*x - 282*m - 200
)*(3*x + 2)^m/(m^3 + 6*m^2 + 11*m + 6)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 488 vs. \(2 (44) = 88\).

Time = 0.34 (sec) , antiderivative size = 488, normalized size of antiderivative = 8.87 \[ \int (1-2 x) (2+3 x)^m (3+5 x) \, dx=\begin {cases} - \frac {180 x^{2} \log {\left (3 x + 2 \right )}}{486 x^{2} + 648 x + 216} - \frac {240 x \log {\left (3 x + 2 \right )}}{486 x^{2} + 648 x + 216} - \frac {222 x}{486 x^{2} + 648 x + 216} - \frac {80 \log {\left (3 x + 2 \right )}}{486 x^{2} + 648 x + 216} - \frac {141}{486 x^{2} + 648 x + 216} & \text {for}\: m = -3 \\- \frac {90 x^{2}}{81 x + 54} + \frac {111 x \log {\left (3 x + 2 \right )}}{81 x + 54} + \frac {74 \log {\left (3 x + 2 \right )}}{81 x + 54} + \frac {47}{81 x + 54} & \text {for}\: m = -2 \\- \frac {5 x^{2}}{3} + \frac {17 x}{9} - \frac {7 \log {\left (3 x + 2 \right )}}{27} & \text {for}\: m = -1 \\- \frac {270 m^{2} x^{3} \left (3 x + 2\right )^{m}}{27 m^{3} + 162 m^{2} + 297 m + 162} - \frac {207 m^{2} x^{2} \left (3 x + 2\right )^{m}}{27 m^{3} + 162 m^{2} + 297 m + 162} + \frac {63 m^{2} x \left (3 x + 2\right )^{m}}{27 m^{3} + 162 m^{2} + 297 m + 162} + \frac {54 m^{2} \left (3 x + 2\right )^{m}}{27 m^{3} + 162 m^{2} + 297 m + 162} - \frac {810 m x^{3} \left (3 x + 2\right )^{m}}{27 m^{3} + 162 m^{2} + 297 m + 162} - \frac {288 m x^{2} \left (3 x + 2\right )^{m}}{27 m^{3} + 162 m^{2} + 297 m + 162} + \frac {591 m x \left (3 x + 2\right )^{m}}{27 m^{3} + 162 m^{2} + 297 m + 162} + \frac {282 m \left (3 x + 2\right )^{m}}{27 m^{3} + 162 m^{2} + 297 m + 162} - \frac {540 x^{3} \left (3 x + 2\right )^{m}}{27 m^{3} + 162 m^{2} + 297 m + 162} - \frac {81 x^{2} \left (3 x + 2\right )^{m}}{27 m^{3} + 162 m^{2} + 297 m + 162} + \frac {486 x \left (3 x + 2\right )^{m}}{27 m^{3} + 162 m^{2} + 297 m + 162} + \frac {200 \left (3 x + 2\right )^{m}}{27 m^{3} + 162 m^{2} + 297 m + 162} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-180*x**2*log(3*x + 2)/(486*x**2 + 648*x + 216) - 240*x*log(3*x + 2)/(486*x**2 + 648*x + 216) - 222
*x/(486*x**2 + 648*x + 216) - 80*log(3*x + 2)/(486*x**2 + 648*x + 216) - 141/(486*x**2 + 648*x + 216), Eq(m, -
3)), (-90*x**2/(81*x + 54) + 111*x*log(3*x + 2)/(81*x + 54) + 74*log(3*x + 2)/(81*x + 54) + 47/(81*x + 54), Eq
(m, -2)), (-5*x**2/3 + 17*x/9 - 7*log(3*x + 2)/27, Eq(m, -1)), (-270*m**2*x**3*(3*x + 2)**m/(27*m**3 + 162*m**
2 + 297*m + 162) - 207*m**2*x**2*(3*x + 2)**m/(27*m**3 + 162*m**2 + 297*m + 162) + 63*m**2*x*(3*x + 2)**m/(27*
m**3 + 162*m**2 + 297*m + 162) + 54*m**2*(3*x + 2)**m/(27*m**3 + 162*m**2 + 297*m + 162) - 810*m*x**3*(3*x + 2
)**m/(27*m**3 + 162*m**2 + 297*m + 162) - 288*m*x**2*(3*x + 2)**m/(27*m**3 + 162*m**2 + 297*m + 162) + 591*m*x
*(3*x + 2)**m/(27*m**3 + 162*m**2 + 297*m + 162) + 282*m*(3*x + 2)**m/(27*m**3 + 162*m**2 + 297*m + 162) - 540
*x**3*(3*x + 2)**m/(27*m**3 + 162*m**2 + 297*m + 162) - 81*x**2*(3*x + 2)**m/(27*m**3 + 162*m**2 + 297*m + 162
) + 486*x*(3*x + 2)**m/(27*m**3 + 162*m**2 + 297*m + 162) + 200*(3*x + 2)**m/(27*m**3 + 162*m**2 + 297*m + 162
), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (49) = 98\).

Time = 0.20 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.85 \[ \int (1-2 x) (2+3 x)^m (3+5 x) \, dx=-\frac {10 \, {\left (27 \, {\left (m^{2} + 3 \, m + 2\right )} x^{3} + 18 \, {\left (m^{2} + m\right )} x^{2} - 24 \, m x + 16\right )} {\left (3 \, x + 2\right )}^{m}}{27 \, {\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )}} - \frac {{\left (9 \, {\left (m + 1\right )} x^{2} + 6 \, m x - 4\right )} {\left (3 \, x + 2\right )}^{m}}{9 \, {\left (m^{2} + 3 \, m + 2\right )}} + \frac {{\left (3 \, x + 2\right )}^{m + 1}}{m + 1} \]

[In]

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

[Out]

-10/27*(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) - 1/9*(9
*(m + 1)*x^2 + 6*m*x - 4)*(3*x + 2)^m/(m^2 + 3*m + 2) + (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. 163 vs. \(2 (49) = 98\).

Time = 0.27 (sec) , antiderivative size = 163, normalized size of antiderivative = 2.96 \[ \int (1-2 x) (2+3 x)^m (3+5 x) \, dx=-\frac {270 \, m^{2} {\left (3 \, x + 2\right )}^{m} x^{3} + 207 \, m^{2} {\left (3 \, x + 2\right )}^{m} x^{2} + 810 \, m {\left (3 \, x + 2\right )}^{m} x^{3} - 63 \, m^{2} {\left (3 \, x + 2\right )}^{m} x + 288 \, m {\left (3 \, x + 2\right )}^{m} x^{2} + 540 \, {\left (3 \, x + 2\right )}^{m} x^{3} - 54 \, m^{2} {\left (3 \, x + 2\right )}^{m} - 591 \, m {\left (3 \, x + 2\right )}^{m} x + 81 \, {\left (3 \, x + 2\right )}^{m} x^{2} - 282 \, m {\left (3 \, x + 2\right )}^{m} - 486 \, {\left (3 \, x + 2\right )}^{m} x - 200 \, {\left (3 \, x + 2\right )}^{m}}{27 \, {\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )}} \]

[In]

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

[Out]

-1/27*(270*m^2*(3*x + 2)^m*x^3 + 207*m^2*(3*x + 2)^m*x^2 + 810*m*(3*x + 2)^m*x^3 - 63*m^2*(3*x + 2)^m*x + 288*
m*(3*x + 2)^m*x^2 + 540*(3*x + 2)^m*x^3 - 54*m^2*(3*x + 2)^m - 591*m*(3*x + 2)^m*x + 81*(3*x + 2)^m*x^2 - 282*
m*(3*x + 2)^m - 486*(3*x + 2)^m*x - 200*(3*x + 2)^m)/(m^3 + 6*m^2 + 11*m + 6)

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 130, normalized size of antiderivative = 2.36 \[ \int (1-2 x) (2+3 x)^m (3+5 x) \, dx={\left (3\,x+2\right )}^m\,\left (\frac {54\,m^2+282\,m+200}{27\,m^3+162\,m^2+297\,m+162}+\frac {x\,\left (63\,m^2+591\,m+486\right )}{27\,m^3+162\,m^2+297\,m+162}-\frac {x^2\,\left (207\,m^2+288\,m+81\right )}{27\,m^3+162\,m^2+297\,m+162}-\frac {x^3\,\left (270\,m^2+810\,m+540\right )}{27\,m^3+162\,m^2+297\,m+162}\right ) \]

[In]

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

[Out]

(3*x + 2)^m*((282*m + 54*m^2 + 200)/(297*m + 162*m^2 + 27*m^3 + 162) + (x*(591*m + 63*m^2 + 486))/(297*m + 162
*m^2 + 27*m^3 + 162) - (x^2*(288*m + 207*m^2 + 81))/(297*m + 162*m^2 + 27*m^3 + 162) - (x^3*(810*m + 270*m^2 +
540))/(297*m + 162*m^2 + 27*m^3 + 162))