3.3161 \(\int (1-2 x) (2+3 x)^m (3+5 x) \, dx\)

Optimal. Leaf size=55 \[ -\frac{7 (3 x+2)^{m+1}}{27 (m+1)}+\frac{37 (3 x+2)^{m+2}}{27 (m+2)}-\frac{10 (3 x+2)^{m+3}}{27 (m+3)} \]

[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))

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Rubi [A]  time = 0.0467358, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ -\frac{7 (3 x+2)^{m+1}}{27 (m+1)}+\frac{37 (3 x+2)^{m+2}}{27 (m+2)}-\frac{10 (3 x+2)^{m+3}}{27 (m+3)} \]

Antiderivative was successfully verified.

[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))

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Rubi in Sympy [A]  time = 8.08648, size = 44, normalized size = 0.8 \[ - \frac{10 \left (3 x + 2\right )^{m + 3}}{27 \left (m + 3\right )} + \frac{37 \left (3 x + 2\right )^{m + 2}}{27 \left (m + 2\right )} - \frac{7 \left (3 x + 2\right )^{m + 1}}{27 \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0365385, size = 63, normalized size = 1.15 \[ \frac{(3 x+2)^{m+1} \left (-9 m^2 \left (10 x^2+x-3\right )+m \left (-270 x^2+84 x+141\right )-180 x^2+93 x+100\right )}{27 (m+1) (m+2) (m+3)} \]

Antiderivative was successfully verified.

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

[Out]

((2 + 3*x)^(1 + m)*(100 + 93*x - 180*x^2 + m*(141 + 84*x - 270*x^2) - 9*m^2*(-3
+ x + 10*x^2)))/(27*(1 + m)*(2 + m)*(3 + m))

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Maple [A]  time = 0.005, size = 69, normalized size = 1.3 \[ -{\frac{ \left ( 2+3\,x \right ) ^{1+m} \left ( 90\,{m}^{2}{x}^{2}+9\,{m}^{2}x+270\,m{x}^{2}-27\,{m}^{2}-84\,mx+180\,{x}^{2}-141\,m-93\,x-100 \right ) }{27\,{m}^{3}+162\,{m}^{2}+297\,m+162}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/27*(2+3*x)^(1+m)*(90*m^2*x^2+9*m^2*x+270*m*x^2-27*m^2-84*m*x+180*x^2-141*m-93
*x-100)/(m^3+6*m^2+11*m+6)

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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*x - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.239321, size = 101, normalized size = 1.84 \[ -\frac{{\left (270 \,{\left (m^{2} + 3 \, m + 2\right )} x^{3} + 9 \,{\left (23 \, m^{2} + 32 \, m + 9\right )} x^{2} - 54 \, m^{2} - 3 \,{\left (21 \, m^{2} + 197 \, m + 162\right )} x - 282 \, m - 200\right )}{\left (3 \, x + 2\right )}^{m}}{27 \,{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Sympy [A]  time = 1.30094, size = 488, normalized size = 8.87 \[ \begin{cases} - \frac{360 x^{2} \log{\left (x + \frac{2}{3} \right )}}{972 x^{2} + 1296 x + 432} + \frac{333 x^{2}}{972 x^{2} + 1296 x + 432} - \frac{480 x \log{\left (x + \frac{2}{3} \right )}}{972 x^{2} + 1296 x + 432} - \frac{160 \log{\left (x + \frac{2}{3} \right )}}{972 x^{2} + 1296 x + 432} - \frac{134}{972 x^{2} + 1296 x + 432} & \text{for}\: m = -3 \\- \frac{90 x^{2}}{81 x + 54} + \frac{111 x \log{\left (x + \frac{2}{3} \right )}}{81 x + 54} + \frac{74 \log{\left (x + \frac{2}{3} \right )}}{81 x + 54} + \frac{47}{81 x + 54} & \text{for}\: m = -2 \\- \frac{5 x^{2}}{3} + \frac{17 x}{9} - \frac{7 \log{\left (x + \frac{2}{3} \right )}}{27} & \text{for}\: m = -1 \\- \frac{270 m^{2} x^{3} \left (3 x + 2\right )^{m}}{27 m^{3} + 162 m^{2} + 297 m + 162} - \frac{207 m^{2} x^{2} \left (3 x + 2\right )^{m}}{27 m^{3} + 162 m^{2} + 297 m + 162} + \frac{63 m^{2} x \left (3 x + 2\right )^{m}}{27 m^{3} + 162 m^{2} + 297 m + 162} + \frac{54 m^{2} \left (3 x + 2\right )^{m}}{27 m^{3} + 162 m^{2} + 297 m + 162} - \frac{810 m x^{3} \left (3 x + 2\right )^{m}}{27 m^{3} + 162 m^{2} + 297 m + 162} - \frac{288 m x^{2} \left (3 x + 2\right )^{m}}{27 m^{3} + 162 m^{2} + 297 m + 162} + \frac{591 m x \left (3 x + 2\right )^{m}}{27 m^{3} + 162 m^{2} + 297 m + 162} + \frac{282 m \left (3 x + 2\right )^{m}}{27 m^{3} + 162 m^{2} + 297 m + 162} - \frac{540 x^{3} \left (3 x + 2\right )^{m}}{27 m^{3} + 162 m^{2} + 297 m + 162} - \frac{81 x^{2} \left (3 x + 2\right )^{m}}{27 m^{3} + 162 m^{2} + 297 m + 162} + \frac{486 x \left (3 x + 2\right )^{m}}{27 m^{3} + 162 m^{2} + 297 m + 162} + \frac{200 \left (3 x + 2\right )^{m}}{27 m^{3} + 162 m^{2} + 297 m + 162} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-360*x**2*log(x + 2/3)/(972*x**2 + 1296*x + 432) + 333*x**2/(972*x**2
+ 1296*x + 432) - 480*x*log(x + 2/3)/(972*x**2 + 1296*x + 432) - 160*log(x + 2/
3)/(972*x**2 + 1296*x + 432) - 134/(972*x**2 + 1296*x + 432), Eq(m, -3)), (-90*x
**2/(81*x + 54) + 111*x*log(x + 2/3)/(81*x + 54) + 74*log(x + 2/3)/(81*x + 54) +
47/(81*x + 54), Eq(m, -2)), (-5*x**2/3 + 17*x/9 - 7*log(x + 2/3)/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 + 1
62) - 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))

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GIAC/XCAS [A]  time = 0.235533, size = 252, normalized size = 4.58 \[ -\frac{270 \, m^{2} x^{3} e^{\left (m{\rm ln}\left (3 \, x + 2\right )\right )} + 207 \, m^{2} x^{2} e^{\left (m{\rm ln}\left (3 \, x + 2\right )\right )} + 810 \, m x^{3} e^{\left (m{\rm ln}\left (3 \, x + 2\right )\right )} - 63 \, m^{2} x e^{\left (m{\rm ln}\left (3 \, x + 2\right )\right )} + 288 \, m x^{2} e^{\left (m{\rm ln}\left (3 \, x + 2\right )\right )} + 540 \, x^{3} e^{\left (m{\rm ln}\left (3 \, x + 2\right )\right )} - 54 \, m^{2} e^{\left (m{\rm ln}\left (3 \, x + 2\right )\right )} - 591 \, m x e^{\left (m{\rm ln}\left (3 \, x + 2\right )\right )} + 81 \, x^{2} e^{\left (m{\rm ln}\left (3 \, x + 2\right )\right )} - 282 \, m e^{\left (m{\rm ln}\left (3 \, x + 2\right )\right )} - 486 \, x e^{\left (m{\rm ln}\left (3 \, x + 2\right )\right )} - 200 \, e^{\left (m{\rm ln}\left (3 \, x + 2\right )\right )}}{27 \,{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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