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

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

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Rubi [A]  time = 0.03, antiderivative size = 76, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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)} \end {gather*}

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*}

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Mathematica [A]  time = 0.09, size = 83, normalized size = 1.09 \begin {gather*} \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} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.08, size = 0, normalized size = 0.00 \begin {gather*} \int (e x)^m (2-2 a x) (1+a x)^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.11, size = 130, normalized size = 1.71 \begin {gather*} -\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 {gather*}

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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giac [B]  time = 1.23, size = 229, normalized size = 3.01 \begin {gather*} -\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 {gather*}

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)

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maple [A]  time = 0.00, size = 143, normalized size = 1.88 \begin {gather*} -\frac {2 \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 a m x -m^{3}-12 a x -9 m^{2}-26 m -24\right ) x \left (e x \right )^{m}}{\left (m +4\right ) \left (m +3\right ) \left (m +2\right ) \left (m +1\right )} \end {gather*}

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/(m+4)/(m+3)/(m+2)/(m+1)

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maxima [A]  time = 1.20, size = 73, normalized size = 0.96 \begin {gather*} -\frac {2 \, a^{3} e^{m} x^{4} x^{m}}{m + 4} - \frac {2 \, a^{2} e^{m} x^{3} x^{m}}{m + 3} + \frac {2 \, a e^{m} x^{2} x^{m}}{m + 2} + \frac {2 \, \left (e x\right )^{m + 1}}{e {\left (m + 1\right )}} \end {gather*}

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]

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

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mupad [B]  time = 0.42, size = 165, normalized size = 2.17 \begin {gather*} {\left (e\,x\right )}^m\,\left (\frac {x\,\left (2\,m^3+18\,m^2+52\,m+48\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}+\frac {2\,a\,x^2\,\left (m^3+8\,m^2+19\,m+12\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}-\frac {2\,a^3\,x^4\,\left (m^3+6\,m^2+11\,m+6\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}-\frac {2\,a^2\,x^3\,\left (m^3+7\,m^2+14\,m+8\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(e*x)^m*((x*(52*m + 18*m^2 + 2*m^3 + 48))/(50*m + 35*m^2 + 10*m^3 + m^4 + 24) + (2*a*x^2*(19*m + 8*m^2 + m^3 +
12))/(50*m + 35*m^2 + 10*m^3 + m^4 + 24) - (2*a^3*x^4*(11*m + 6*m^2 + m^3 + 6))/(50*m + 35*m^2 + 10*m^3 + m^4
+ 24) - (2*a^2*x^3*(14*m + 7*m^2 + m^3 + 8))/(50*m + 35*m^2 + 10*m^3 + m^4 + 24))

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sympy [A]  time = 1.11, size = 668, normalized size = 8.79 \begin {gather*} \begin {cases} \frac {- 2 a^{3} \log {\relax (x )} + \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 {\relax (x )} - \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 {\relax (x )} - \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 {\relax (x )}}{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 {gather*}

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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