3.58 \(\int (e x)^m (2-2 a x) (1+a x)^2 \, dx\)

Optimal. Leaf size=76 \[ -\frac{2 a^2 (e x)^{m+3}}{e^3 (m+3)}-\frac{2 a^3 (e x)^{m+4}}{e^4 (m+4)}+\frac{2 a (e x)^{m+2}}{e^2 (m+2)}+\frac{2 (e x)^{m+1}}{e (m+1)} \]

[Out]

(2*(e*x)^(1 + m))/(e*(1 + m)) + (2*a*(e*x)^(2 + m))/(e^2*(2 + m)) - (2*a^2*(e*x)^(3 + m))/(e^3*(3 + m)) - (2*a
^3*(e*x)^(4 + m))/(e^4*(4 + m))

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Rubi [A]  time = 0.0259322, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {75} \[ -\frac{2 a^2 (e x)^{m+3}}{e^3 (m+3)}-\frac{2 a^3 (e x)^{m+4}}{e^4 (m+4)}+\frac{2 a (e x)^{m+2}}{e^2 (m+2)}+\frac{2 (e x)^{m+1}}{e (m+1)} \]

Antiderivative was successfully verified.

[In]

Int[(e*x)^m*(2 - 2*a*x)*(1 + a*x)^2,x]

[Out]

(2*(e*x)^(1 + m))/(e*(1 + m)) + (2*a*(e*x)^(2 + m))/(e^2*(2 + m)) - (2*a^2*(e*x)^(3 + m))/(e^3*(3 + m)) - (2*a
^3*(e*x)^(4 + m))/(e^4*(4 + m))

Rule 75

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] &&  !(ILtQ[n
+ p + 2, 0] && GtQ[n + 2*p, 0])

Rubi steps

\begin{align*} \int (e x)^m (2-2 a x) (1+a x)^2 \, dx &=\int \left (2 (e x)^m+\frac{2 a (e x)^{1+m}}{e}-\frac{2 a^2 (e x)^{2+m}}{e^2}-\frac{2 a^3 (e x)^{3+m}}{e^3}\right ) \, dx\\ &=\frac{2 (e x)^{1+m}}{e (1+m)}+\frac{2 a (e x)^{2+m}}{e^2 (2+m)}-\frac{2 a^2 (e x)^{3+m}}{e^3 (3+m)}-\frac{2 a^3 (e x)^{4+m}}{e^4 (4+m)}\\ \end{align*}

Mathematica [A]  time = 0.0576997, size = 83, normalized size = 1.09 \[ \frac{\left (\frac{2 (2 m+5) x \left (m \left (3 a^2 x^2+8 a x+5\right )+2 a^2 x^2+(a m x+m)^2+6 a x+6\right )}{(m+1) (m+2) (m+3)}-2 x (a x+1)^3\right ) (e x)^m}{m+4} \]

Antiderivative was successfully verified.

[In]

Integrate[(e*x)^m*(2 - 2*a*x)*(1 + a*x)^2,x]

[Out]

((e*x)^m*(-2*x*(1 + a*x)^3 + (2*(5 + 2*m)*x*(6 + 6*a*x + 2*a^2*x^2 + (m + a*m*x)^2 + m*(5 + 8*a*x + 3*a^2*x^2)
))/((1 + m)*(2 + m)*(3 + m))))/(4 + m)

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Maple [A]  time = 0.004, size = 143, normalized size = 1.9 \begin{align*} -2\,{\frac{ \left ( ex \right ) ^{m} \left ({a}^{3}{m}^{3}{x}^{3}+6\,{a}^{3}{m}^{2}{x}^{3}+11\,{a}^{3}m{x}^{3}+{a}^{2}{m}^{3}{x}^{2}+6\,{a}^{3}{x}^{3}+7\,{a}^{2}{m}^{2}{x}^{2}+14\,{a}^{2}m{x}^{2}-a{m}^{3}x+8\,{a}^{2}{x}^{2}-8\,a{m}^{2}x-19\,amx-{m}^{3}-12\,ax-9\,{m}^{2}-26\,m-24 \right ) x}{ \left ( 4+m \right ) \left ( 3+m \right ) \left ( 2+m \right ) \left ( 1+m \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m*(-2*a*x+2)*(a*x+1)^2,x)

[Out]

-2*(e*x)^m*(a^3*m^3*x^3+6*a^3*m^2*x^3+11*a^3*m*x^3+a^2*m^3*x^2+6*a^3*x^3+7*a^2*m^2*x^2+14*a^2*m*x^2-a*m^3*x+8*
a^2*x^2-8*a*m^2*x-19*a*m*x-m^3-12*a*x-9*m^2-26*m-24)*x/(4+m)/(3+m)/(2+m)/(1+m)

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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((e*x)^m*(-2*a*x+2)*(a*x+1)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.1812, size = 286, normalized size = 3.76 \begin{align*} -\frac{2 \,{\left ({\left (a^{3} m^{3} + 6 \, a^{3} m^{2} + 11 \, a^{3} m + 6 \, a^{3}\right )} x^{4} +{\left (a^{2} m^{3} + 7 \, a^{2} m^{2} + 14 \, a^{2} m + 8 \, a^{2}\right )} x^{3} -{\left (a m^{3} + 8 \, a m^{2} + 19 \, a m + 12 \, a\right )} x^{2} -{\left (m^{3} + 9 \, m^{2} + 26 \, m + 24\right )} x\right )} \left (e x\right )^{m}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*((a^3*m^3 + 6*a^3*m^2 + 11*a^3*m + 6*a^3)*x^4 + (a^2*m^3 + 7*a^2*m^2 + 14*a^2*m + 8*a^2)*x^3 - (a*m^3 + 8*a
*m^2 + 19*a*m + 12*a)*x^2 - (m^3 + 9*m^2 + 26*m + 24)*x)*(e*x)^m/(m^4 + 10*m^3 + 35*m^2 + 50*m + 24)

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Sympy [A]  time = 0.895672, size = 668, normalized size = 8.79 \begin{align*} \begin{cases} \frac{- 2 a^{3} \log{\left (x \right )} + \frac{2 a^{2}}{x} - \frac{a}{x^{2}} - \frac{2}{3 x^{3}}}{e^{4}} & \text{for}\: m = -4 \\\frac{- 2 a^{3} x - 2 a^{2} \log{\left (x \right )} - \frac{2 a}{x} - \frac{1}{x^{2}}}{e^{3}} & \text{for}\: m = -3 \\\frac{- a^{3} x^{2} - 2 a^{2} x + 2 a \log{\left (x \right )} - \frac{2}{x}}{e^{2}} & \text{for}\: m = -2 \\\frac{- \frac{2 a^{3} x^{3}}{3} - a^{2} x^{2} + 2 a x + 2 \log{\left (x \right )}}{e} & \text{for}\: m = -1 \\- \frac{2 a^{3} e^{m} m^{3} x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac{12 a^{3} e^{m} m^{2} x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac{22 a^{3} e^{m} m x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac{12 a^{3} e^{m} x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac{2 a^{2} e^{m} m^{3} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac{14 a^{2} e^{m} m^{2} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac{28 a^{2} e^{m} m x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac{16 a^{2} e^{m} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{2 a e^{m} m^{3} x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{16 a e^{m} m^{2} x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{38 a e^{m} m x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{24 a e^{m} x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{2 e^{m} m^{3} x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{18 e^{m} m^{2} x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{52 e^{m} m x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{48 e^{m} x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**m*(-2*a*x+2)*(a*x+1)**2,x)

[Out]

Piecewise(((-2*a**3*log(x) + 2*a**2/x - a/x**2 - 2/(3*x**3))/e**4, Eq(m, -4)), ((-2*a**3*x - 2*a**2*log(x) - 2
*a/x - 1/x**2)/e**3, Eq(m, -3)), ((-a**3*x**2 - 2*a**2*x + 2*a*log(x) - 2/x)/e**2, Eq(m, -2)), ((-2*a**3*x**3/
3 - a**2*x**2 + 2*a*x + 2*log(x))/e, Eq(m, -1)), (-2*a**3*e**m*m**3*x**4*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m
+ 24) - 12*a**3*e**m*m**2*x**4*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) - 22*a**3*e**m*m*x**4*x**m/(m**4 +
10*m**3 + 35*m**2 + 50*m + 24) - 12*a**3*e**m*x**4*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) - 2*a**2*e**m*
m**3*x**3*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) - 14*a**2*e**m*m**2*x**3*x**m/(m**4 + 10*m**3 + 35*m**2
+ 50*m + 24) - 28*a**2*e**m*m*x**3*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) - 16*a**2*e**m*x**3*x**m/(m**4
+ 10*m**3 + 35*m**2 + 50*m + 24) + 2*a*e**m*m**3*x**2*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 16*a*e**m*
m**2*x**2*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 38*a*e**m*m*x**2*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m
+ 24) + 24*a*e**m*x**2*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 2*e**m*m**3*x*x**m/(m**4 + 10*m**3 + 35*
m**2 + 50*m + 24) + 18*e**m*m**2*x*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 52*e**m*m*x*x**m/(m**4 + 10*m
**3 + 35*m**2 + 50*m + 24) + 48*e**m*x*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24), True))

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Giac [B]  time = 1.19875, size = 309, normalized size = 4.07 \begin{align*} -\frac{2 \,{\left (a^{3} m^{3} x^{4} x^{m} e^{m} + 6 \, a^{3} m^{2} x^{4} x^{m} e^{m} + a^{2} m^{3} x^{3} x^{m} e^{m} + 11 \, a^{3} m x^{4} x^{m} e^{m} + 7 \, a^{2} m^{2} x^{3} x^{m} e^{m} + 6 \, a^{3} x^{4} x^{m} e^{m} - a m^{3} x^{2} x^{m} e^{m} + 14 \, a^{2} m x^{3} x^{m} e^{m} - 8 \, a m^{2} x^{2} x^{m} e^{m} + 8 \, a^{2} x^{3} x^{m} e^{m} - m^{3} x x^{m} e^{m} - 19 \, a m x^{2} x^{m} e^{m} - 9 \, m^{2} x x^{m} e^{m} - 12 \, a x^{2} x^{m} e^{m} - 26 \, m x x^{m} e^{m} - 24 \, x x^{m} e^{m}\right )}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(a^3*m^3*x^4*x^m*e^m + 6*a^3*m^2*x^4*x^m*e^m + a^2*m^3*x^3*x^m*e^m + 11*a^3*m*x^4*x^m*e^m + 7*a^2*m^2*x^3*x
^m*e^m + 6*a^3*x^4*x^m*e^m - a*m^3*x^2*x^m*e^m + 14*a^2*m*x^3*x^m*e^m - 8*a*m^2*x^2*x^m*e^m + 8*a^2*x^3*x^m*e^
m - m^3*x*x^m*e^m - 19*a*m*x^2*x^m*e^m - 9*m^2*x*x^m*e^m - 12*a*x^2*x^m*e^m - 26*m*x*x^m*e^m - 24*x*x^m*e^m)/(
m^4 + 10*m^3 + 35*m^2 + 50*m + 24)