3.3189 \(\int (1-2 x) (2+3 x)^m (3+5 x) \, dx\)

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 = 0.0121969, 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, Rules used = {77} \[ -\frac{7 (3 x+2)^{m+1}}{27 (m+1)}+\frac{37 (3 x+2)^{m+2}}{27 (m+2)}-\frac{10 (3 x+2)^{m+3}}{27 (m+3)} \]

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)
)

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) \, dx &=\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]  time = 0.0286581, size = 47, normalized size = 0.85 \[ \frac{1}{27} (3 x+2)^{m+1} \left (-\frac{10 (3 x+2)^2}{m+3}+\frac{37 (3 x+2)}{m+2}-\frac{7}{m+1}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 69, normalized size = 1.3 \begin{align*} -{\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}} \end{align*}

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(-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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.29577, size = 203, normalized size = 3.69 \begin{align*} -\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 0.799982, size = 490, normalized size = 8.91 \begin{align*} \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{180 x^{2}}{162 x + 108} + \frac{222 x \log{\left (x + \frac{2}{3} \right )}}{162 x + 108} - \frac{141 x}{162 x + 108} + \frac{148 \log{\left (x + \frac{2}{3} \right )}}{162 x + 108} & \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} \end{align*}

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)), (-180*x**2/(162*x + 108) + 222*x*log(x + 2/3)/(162*x + 108) - 141*x/(162*x + 108) + 148*log(x +
2/3)/(162*x + 108), 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 + 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))

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Giac [B]  time = 1.91869, size = 220, normalized size = 4. \begin{align*} -\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)