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

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

[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 = 0.0824718, 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 \[ -\frac{2 a^3 (e x)^{m+4}}{e^4 (m+4)}-\frac{2 a^2 (e x)^{m+3}}{e^3 (m+3)}+\frac{2 a (e x)^{m+2}}{e^2 (m+2)}+\frac{2 (e x)^{m+1}}{e (m+1)} \]

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))

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Rubi in Sympy [A]  time = 17.7893, size = 66, normalized size = 0.87 \[ - \frac{2 a^{3} \left (e x\right )^{m + 4}}{e^{4} \left (m + 4\right )} - \frac{2 a^{2} \left (e x\right )^{m + 3}}{e^{3} \left (m + 3\right )} + \frac{2 a \left (e x\right )^{m + 2}}{e^{2} \left (m + 2\right )} + \frac{2 \left (e x\right )^{m + 1}}{e \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**m*(-2*a*x+2)*(a*x+1)**2,x)

[Out]

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

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Mathematica [A]  time = 0.0562629, size = 116, normalized size = 1.53 \[ -\frac{2 x \left (6 a^3 x^3+8 a^2 x^2+m^2 \left (6 a^3 x^3+7 a^2 x^2-8 a x-9\right )+m \left (11 a^3 x^3+14 a^2 x^2-19 a x-26\right )+m^3 (a x-1) (a x+1)^2-12 a x-24\right ) (e x)^m}{(m+1) (m+2) (m+3) (m+4)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 143, normalized size = 1.9 \[ -2\,{\frac{ \left ( ex \right ) ^{m} \left ({a}^{3}{m}^{3}{x}^{3}+6\,{a}^{3}{m}^{2}{x}^{3}+11\,{a}^{3}m{x}^{3}+{a}^{2}{m}^{3}{x}^{2}+6\,{a}^{3}{x}^{3}+7\,{a}^{2}{m}^{2}{x}^{2}+14\,{a}^{2}m{x}^{2}-a{m}^{3}x+8\,{a}^{2}{x}^{2}-8\,a{m}^{2}x-19\,amx-{m}^{3}-12\,ax-9\,{m}^{2}-26\,m-24 \right ) x}{ \left ( 4+m \right ) \left ( 3+m \right ) \left ( 2+m \right ) \left ( 1+m \right ) }} \]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.218068, size = 176, normalized size = 2.32 \[ -\frac{2 \,{\left ({\left (a^{3} m^{3} + 6 \, a^{3} m^{2} + 11 \, a^{3} m + 6 \, a^{3}\right )} x^{4} +{\left (a^{2} m^{3} + 7 \, a^{2} m^{2} + 14 \, a^{2} m + 8 \, a^{2}\right )} x^{3} -{\left (a m^{3} + 8 \, a m^{2} + 19 \, a m + 12 \, a\right )} x^{2} -{\left (m^{3} + 9 \, m^{2} + 26 \, m + 24\right )} x\right )} \left (e x\right )^{m}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-2*(a*x + 1)^2*(a*x - 1)*(e*x)^m,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 = 2.49559, size = 668, normalized size = 8.79 \[ \begin{cases} \frac{- 2 a^{3} \log{\left (x \right )} + \frac{2 a^{2}}{x} - \frac{a}{x^{2}} - \frac{2}{3 x^{3}}}{e^{4}} & \text{for}\: m = -4 \\\frac{- 2 a^{3} x - 2 a^{2} \log{\left (x \right )} - \frac{2 a}{x} - \frac{1}{x^{2}}}{e^{3}} & \text{for}\: m = -3 \\\frac{- a^{3} x^{2} - 2 a^{2} x + 2 a \log{\left (x \right )} - \frac{2}{x}}{e^{2}} & \text{for}\: m = -2 \\\frac{- \frac{2 a^{3} x^{3}}{3} - a^{2} x^{2} + 2 a x + 2 \log{\left (x \right )}}{e} & \text{for}\: m = -1 \\- \frac{2 a^{3} e^{m} m^{3} x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac{12 a^{3} e^{m} m^{2} x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac{22 a^{3} e^{m} m x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac{12 a^{3} e^{m} x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac{2 a^{2} e^{m} m^{3} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac{14 a^{2} e^{m} m^{2} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac{28 a^{2} e^{m} m x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac{16 a^{2} e^{m} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{2 a e^{m} m^{3} x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{16 a e^{m} m^{2} x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{38 a e^{m} m x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{24 a e^{m} x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{2 e^{m} m^{3} x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{18 e^{m} m^{2} x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{52 e^{m} m x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{48 e^{m} x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} & \text{otherwise} \end{cases} \]

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 + 2
4) + 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/XCAS [A]  time = 0.216983, size = 352, normalized size = 4.63 \[ -\frac{2 \,{\left (a^{3} m^{3} x^{4} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 6 \, a^{3} m^{2} x^{4} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + a^{2} m^{3} x^{3} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 11 \, a^{3} m x^{4} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 7 \, a^{2} m^{2} x^{3} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 6 \, a^{3} x^{4} e^{\left (m{\rm ln}\left (x\right ) + m\right )} - a m^{3} x^{2} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 14 \, a^{2} m x^{3} e^{\left (m{\rm ln}\left (x\right ) + m\right )} - 8 \, a m^{2} x^{2} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 8 \, a^{2} x^{3} e^{\left (m{\rm ln}\left (x\right ) + m\right )} - m^{3} x e^{\left (m{\rm ln}\left (x\right ) + m\right )} - 19 \, a m x^{2} e^{\left (m{\rm ln}\left (x\right ) + m\right )} - 9 \, m^{2} x e^{\left (m{\rm ln}\left (x\right ) + m\right )} - 12 \, a x^{2} e^{\left (m{\rm ln}\left (x\right ) + m\right )} - 26 \, m x e^{\left (m{\rm ln}\left (x\right ) + m\right )} - 24 \, x e^{\left (m{\rm ln}\left (x\right ) + m\right )}\right )}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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