3.3160 \(\int (1-2 x) (2+3 x)^m (3+5 x)^2 \, dx\)

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 = 0.0644679, 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 \[ \frac{7 (3 x+2)^{m+1}}{81 (m+1)}-\frac{8 (3 x+2)^{m+2}}{9 (m+2)}+\frac{65 (3 x+2)^{m+3}}{27 (m+3)}-\frac{50 (3 x+2)^{m+4}}{81 (m+4)} \]

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))

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Rubi in Sympy [A]  time = 10.0384, size = 60, normalized size = 0.82 \[ - \frac{50 \left (3 x + 2\right )^{m + 4}}{81 \left (m + 4\right )} + \frac{65 \left (3 x + 2\right )^{m + 3}}{27 \left (m + 3\right )} - \frac{8 \left (3 x + 2\right )^{m + 2}}{9 \left (m + 2\right )} + \frac{7 \left (3 x + 2\right )^{m + 1}}{81 \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)*(2+3*x)**m*(3+5*x)**2,x)

[Out]

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

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Mathematica [A]  time = 0.0499164, size = 106, normalized size = 1.45 \[ -\frac{(3 x+2)^{m+1} \left (9 m^3 (2 x-1) (5 x+3)^2+3 m^2 \left (900 x^3+435 x^2-428 x-219\right )+2 m \left (2475 x^3+855 x^2-1476 x-661\right )+4 \left (675 x^3+180 x^2-444 x-190\right )\right )}{27 (m+1) (m+2) (m+3) (m+4)} \]

Antiderivative was successfully verified.

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

[Out]

-((2 + 3*x)^(1 + m)*(9*m^3*(-1 + 2*x)*(3 + 5*x)^2 + 4*(-190 - 444*x + 180*x^2 +
675*x^3) + 3*m^2*(-219 - 428*x + 435*x^2 + 900*x^3) + 2*m*(-661 - 1476*x + 855*x
^2 + 2475*x^3)))/(27*(1 + m)*(2 + m)*(3 + m)*(4 + m))

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Maple [A]  time = 0.01, size = 120, normalized size = 1.6 \[ -{\frac{ \left ( 2+3\,x \right ) ^{1+m} \left ( 450\,{m}^{3}{x}^{3}+315\,{m}^{3}{x}^{2}+2700\,{m}^{2}{x}^{3}-108\,{m}^{3}x+1305\,{m}^{2}{x}^{2}+4950\,m{x}^{3}-81\,{m}^{3}-1284\,{m}^{2}x+1710\,m{x}^{2}+2700\,{x}^{3}-657\,{m}^{2}-2952\,mx+720\,{x}^{2}-1322\,m-1776\,x-760 \right ) }{27\,{m}^{4}+270\,{m}^{3}+945\,{m}^{2}+1350\,m+648}} \]

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]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.236186, size = 162, normalized size = 2.22 \[ -\frac{{\left (1350 \,{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} x^{4} + 45 \,{\left (41 \, m^{3} + 207 \, m^{2} + 334 \, m + 168\right )} x^{3} - 162 \, m^{3} + 18 \,{\left (17 \, m^{3} - 69 \, m^{2} - 302 \, m - 216\right )} x^{2} - 1314 \, m^{2} - 3 \,{\left (153 \, m^{3} + 1513 \, m^{2} + 3290 \, m + 1944\right )} x - 2644 \, m - 1520\right )}{\left (3 \, x + 2\right )}^{m}}{27 \,{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x + 2)^m*(5*x + 3)^2*(2*x - 1),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 = 2.17886, size = 1018, normalized size = 13.95 \[ \text{result too large to display} \]

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) + 50
22*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*lo
g(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) + 156
0*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 + 135
0*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 + 6
48) + 459*m**3*x*(3*x + 2)**m/(27*m**4 + 270*m**3 + 945*m**2 + 1350*m + 648) + 1
62*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/(2
7*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 + 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/XCAS [A]  time = 0.228423, size = 429, normalized size = 5.88 \[ -\frac{1350 \, m^{3} x^{4} e^{\left (m{\rm ln}\left (3 \, x + 2\right )\right )} + 1845 \, m^{3} x^{3} e^{\left (m{\rm ln}\left (3 \, x + 2\right )\right )} + 8100 \, m^{2} x^{4} e^{\left (m{\rm ln}\left (3 \, x + 2\right )\right )} + 306 \, m^{3} x^{2} e^{\left (m{\rm ln}\left (3 \, x + 2\right )\right )} + 9315 \, m^{2} x^{3} e^{\left (m{\rm ln}\left (3 \, x + 2\right )\right )} + 14850 \, m x^{4} e^{\left (m{\rm ln}\left (3 \, x + 2\right )\right )} - 459 \, m^{3} x e^{\left (m{\rm ln}\left (3 \, x + 2\right )\right )} - 1242 \, m^{2} x^{2} e^{\left (m{\rm ln}\left (3 \, x + 2\right )\right )} + 15030 \, m x^{3} e^{\left (m{\rm ln}\left (3 \, x + 2\right )\right )} + 8100 \, x^{4} e^{\left (m{\rm ln}\left (3 \, x + 2\right )\right )} - 162 \, m^{3} e^{\left (m{\rm ln}\left (3 \, x + 2\right )\right )} - 4539 \, m^{2} x e^{\left (m{\rm ln}\left (3 \, x + 2\right )\right )} - 5436 \, m x^{2} e^{\left (m{\rm ln}\left (3 \, x + 2\right )\right )} + 7560 \, x^{3} e^{\left (m{\rm ln}\left (3 \, x + 2\right )\right )} - 1314 \, m^{2} e^{\left (m{\rm ln}\left (3 \, x + 2\right )\right )} - 9870 \, m x e^{\left (m{\rm ln}\left (3 \, x + 2\right )\right )} - 3888 \, x^{2} e^{\left (m{\rm ln}\left (3 \, x + 2\right )\right )} - 2644 \, m e^{\left (m{\rm ln}\left (3 \, x + 2\right )\right )} - 5832 \, x e^{\left (m{\rm ln}\left (3 \, x + 2\right )\right )} - 1520 \, e^{\left (m{\rm ln}\left (3 \, x + 2\right )\right )}}{27 \,{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/27*(1350*m^3*x^4*e^(m*ln(3*x + 2)) + 1845*m^3*x^3*e^(m*ln(3*x + 2)) + 8100*m^
2*x^4*e^(m*ln(3*x + 2)) + 306*m^3*x^2*e^(m*ln(3*x + 2)) + 9315*m^2*x^3*e^(m*ln(3
*x + 2)) + 14850*m*x^4*e^(m*ln(3*x + 2)) - 459*m^3*x*e^(m*ln(3*x + 2)) - 1242*m^
2*x^2*e^(m*ln(3*x + 2)) + 15030*m*x^3*e^(m*ln(3*x + 2)) + 8100*x^4*e^(m*ln(3*x +
2)) - 162*m^3*e^(m*ln(3*x + 2)) - 4539*m^2*x*e^(m*ln(3*x + 2)) - 5436*m*x^2*e^(
m*ln(3*x + 2)) + 7560*x^3*e^(m*ln(3*x + 2)) - 1314*m^2*e^(m*ln(3*x + 2)) - 9870*
m*x*e^(m*ln(3*x + 2)) - 3888*x^2*e^(m*ln(3*x + 2)) - 2644*m*e^(m*ln(3*x + 2)) -
5832*x*e^(m*ln(3*x + 2)) - 1520*e^(m*ln(3*x + 2)))/(m^4 + 10*m^3 + 35*m^2 + 50*m
+ 24)