3.3188 \(\int (1-2 x) (2+3 x)^m (3+5 x)^2 \, dx\)
Optimal. Leaf size=73 \[ \frac{7 (3 x+2)^{m+1}}{81 (m+1)}-\frac{8 (3 x+2)^{m+2}}{9 (m+2)}+\frac{65 (3 x+2)^{m+3}}{27 (m+3)}-\frac{50 (3 x+2)^{m+4}}{81 (m+4)} \]
[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))
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Rubi [A] time = 0.016229, antiderivative size = 73, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used =
{77} \[ \frac{7 (3 x+2)^{m+1}}{81 (m+1)}-\frac{8 (3 x+2)^{m+2}}{9 (m+2)}+\frac{65 (3 x+2)^{m+3}}{27 (m+3)}-\frac{50 (3 x+2)^{m+4}}{81 (m+4)} \]
Antiderivative was successfully verified.
[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 77
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) (2+3 x)^m (3+5 x)^2 \, dx &=\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] time = 0.0209004, size = 61, normalized size = 0.84 \[ \frac{1}{81} (3 x+2)^{m+1} \left (-\frac{50 (3 x+2)^3}{m+4}+\frac{195 (3 x+2)^2}{m+3}-\frac{72 (3 x+2)}{m+2}+\frac{7}{m+1}\right ) \]
Antiderivative was successfully verified.
[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
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Maple [A] time = 0.006, size = 120, normalized size = 1.6 \begin{align*} -{\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\,mx+720\,{x}^{2}-1322\,m-1776\,x-760 \right ) }{27\,{m}^{4}+270\,{m}^{3}+945\,{m}^{2}+1350\,m+648}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((1-2*x)*(2+3*x)^m*(3+5*x)^2,x)
[Out]
-1/27*(2+3*x)^(1+m)*(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*m*x+720*x^2-1322*m-1776*x-760)/(m^4+10*m^3+35*m^2+50*m+24)
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((1-2*x)*(2+3*x)^m*(3+5*x)^2,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 1.30739, size = 343, normalized size = 4.7 \begin{align*} -\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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Sympy [A] time = 1.31413, size = 1018, normalized size = 13.95 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((1-2*x)*(2+3*x)**m*(3+5*x)**2,x)
[Out]
Piecewise((-4050*x**3*log(x + 2/3)/(6561*x**3 + 13122*x**2 + 8748*x + 1944) + 5022*x**3/(6561*x**3 + 13122*x**
2 + 8748*x + 1944) - 8100*x**2*log(x + 2/3)/(6561*x**3 + 13122*x**2 + 8748*x + 1944) + 4779*x**2/(6561*x**3 +
13122*x**2 + 8748*x + 1944) - 5400*x*log(x + 2/3)/(6561*x**3 + 13122*x**2 + 8748*x + 1944) - 1200*log(x + 2/3)
/(6561*x**3 + 13122*x**2 + 8748*x + 1944) - 643/(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(x + 2/3)/(486*x**2 + 648*x + 216) - 1008*x**2/(486*x**2 + 648*x +
216) + 1560*x*log(x + 2/3)/(486*x**2 + 648*x + 216) + 520*log(x + 2/3)/(486*x**2 + 648*x + 216) + 179/(486*x**
2 + 648*x + 216), Eq(m, -3)), (-75*x**3/(27*x + 18) + 45*x**2/(27*x + 18) - 24*x*log(x + 2/3)/(27*x + 18) - 16
*log(x + 2/3)/(27*x + 18) - 43/(27*x + 18), Eq(m, -2)), (-50*x**3/9 - 5*x**2/18 + 118*x/27 + 7*log(x + 2/3)/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 + 9
45*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) - 148
50*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 + 2
70*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 + 945*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))
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Giac [B] time = 2.31557, size = 375, normalized size = 5.14 \begin{align*} -\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)