3.356 \(\int (e x)^m (2-2 a x)^2 (1+a x)^3 \, dx\)

Optimal. Leaf size=116 \[ \frac{4 a^5 (e x)^{m+6}}{e^6 (m+6)}+\frac{4 a^4 (e x)^{m+5}}{e^5 (m+5)}-\frac{8 a^3 (e x)^{m+4}}{e^4 (m+4)}-\frac{8 a^2 (e x)^{m+3}}{e^3 (m+3)}+\frac{4 a (e x)^{m+2}}{e^2 (m+2)}+\frac{4 (e x)^{m+1}}{e (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.138221, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ \frac{4 a^5 (e x)^{m+6}}{e^6 (m+6)}+\frac{4 a^4 (e x)^{m+5}}{e^5 (m+5)}-\frac{8 a^3 (e x)^{m+4}}{e^4 (m+4)}-\frac{8 a^2 (e x)^{m+3}}{e^3 (m+3)}+\frac{4 a (e x)^{m+2}}{e^2 (m+2)}+\frac{4 (e x)^{m+1}}{e (m+1)} \]

Antiderivative was successfully verified.

[In]  Int[(e*x)^m*(2 - 2*a*x)^2*(1 + a*x)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 28.0979, size = 104, normalized size = 0.9 \[ \frac{4 a^{5} \left (e x\right )^{m + 6}}{e^{6} \left (m + 6\right )} + \frac{4 a^{4} \left (e x\right )^{m + 5}}{e^{5} \left (m + 5\right )} - \frac{8 a^{3} \left (e x\right )^{m + 4}}{e^{4} \left (m + 4\right )} - \frac{8 a^{2} \left (e x\right )^{m + 3}}{e^{3} \left (m + 3\right )} + \frac{4 a \left (e x\right )^{m + 2}}{e^{2} \left (m + 2\right )} + \frac{4 \left (e x\right )^{m + 1}}{e \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0592756, size = 72, normalized size = 0.62 \[ 4 x \left (\frac{a^5 x^5}{m+6}+\frac{a^4 x^4}{m+5}-\frac{2 a^3 x^3}{m+4}-\frac{2 a^2 x^2}{m+3}+\frac{a x}{m+2}+\frac{1}{m+1}\right ) (e x)^m \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.01, size = 340, normalized size = 2.9 \[ 4\,{\frac{ \left ( ex \right ) ^{m} \left ({a}^{5}{m}^{5}{x}^{5}+15\,{a}^{5}{m}^{4}{x}^{5}+85\,{a}^{5}{m}^{3}{x}^{5}+{a}^{4}{m}^{5}{x}^{4}+225\,{a}^{5}{m}^{2}{x}^{5}+16\,{a}^{4}{m}^{4}{x}^{4}+274\,{a}^{5}m{x}^{5}+95\,{a}^{4}{m}^{3}{x}^{4}-2\,{a}^{3}{m}^{5}{x}^{3}+120\,{a}^{5}{x}^{5}+260\,{a}^{4}{m}^{2}{x}^{4}-34\,{a}^{3}{m}^{4}{x}^{3}+324\,{a}^{4}m{x}^{4}-214\,{a}^{3}{m}^{3}{x}^{3}-2\,{a}^{2}{m}^{5}{x}^{2}+144\,{a}^{4}{x}^{4}-614\,{a}^{3}{m}^{2}{x}^{3}-36\,{a}^{2}{m}^{4}{x}^{2}-792\,{a}^{3}m{x}^{3}-242\,{a}^{2}{m}^{3}{x}^{2}+a{m}^{5}x-360\,{a}^{3}{x}^{3}-744\,{a}^{2}{m}^{2}{x}^{2}+19\,a{m}^{4}x-1016\,{a}^{2}m{x}^{2}+137\,a{m}^{3}x+{m}^{5}-480\,{a}^{2}{x}^{2}+461\,a{m}^{2}x+20\,{m}^{4}+702\,amx+155\,{m}^{3}+360\,ax+580\,{m}^{2}+1044\,m+720 \right ) x}{ \left ( 6+m \right ) \left ( 5+m \right ) \left ( 4+m \right ) \left ( 3+m \right ) \left ( 2+m \right ) \left ( 1+m \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*(e*x)^m*(a^5*m^5*x^5+15*a^5*m^4*x^5+85*a^5*m^3*x^5+a^4*m^5*x^4+225*a^5*m^2*x^5
+16*a^4*m^4*x^4+274*a^5*m*x^5+95*a^4*m^3*x^4-2*a^3*m^5*x^3+120*a^5*x^5+260*a^4*m
^2*x^4-34*a^3*m^4*x^3+324*a^4*m*x^4-214*a^3*m^3*x^3-2*a^2*m^5*x^2+144*a^4*x^4-61
4*a^3*m^2*x^3-36*a^2*m^4*x^2-792*a^3*m*x^3-242*a^2*m^3*x^2+a*m^5*x-360*a^3*x^3-7
44*a^2*m^2*x^2+19*a*m^4*x-1016*a^2*m*x^2+137*a*m^3*x+m^5-480*a^2*x^2+461*a*m^2*x
+20*m^4+702*a*m*x+155*m^3+360*a*x+580*m^2+1044*m+720)*x/(6+m)/(5+m)/(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(4*(a*x + 1)^3*(a*x - 1)^2*(e*x)^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.220208, size = 389, normalized size = 3.35 \[ \frac{4 \,{\left ({\left (a^{5} m^{5} + 15 \, a^{5} m^{4} + 85 \, a^{5} m^{3} + 225 \, a^{5} m^{2} + 274 \, a^{5} m + 120 \, a^{5}\right )} x^{6} +{\left (a^{4} m^{5} + 16 \, a^{4} m^{4} + 95 \, a^{4} m^{3} + 260 \, a^{4} m^{2} + 324 \, a^{4} m + 144 \, a^{4}\right )} x^{5} - 2 \,{\left (a^{3} m^{5} + 17 \, a^{3} m^{4} + 107 \, a^{3} m^{3} + 307 \, a^{3} m^{2} + 396 \, a^{3} m + 180 \, a^{3}\right )} x^{4} - 2 \,{\left (a^{2} m^{5} + 18 \, a^{2} m^{4} + 121 \, a^{2} m^{3} + 372 \, a^{2} m^{2} + 508 \, a^{2} m + 240 \, a^{2}\right )} x^{3} +{\left (a m^{5} + 19 \, a m^{4} + 137 \, a m^{3} + 461 \, a m^{2} + 702 \, a m + 360 \, a\right )} x^{2} +{\left (m^{5} + 20 \, m^{4} + 155 \, m^{3} + 580 \, m^{2} + 1044 \, m + 720\right )} x\right )} \left (e x\right )^{m}}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(4*(a*x + 1)^3*(a*x - 1)^2*(e*x)^m,x, algorithm="fricas")

[Out]

4*((a^5*m^5 + 15*a^5*m^4 + 85*a^5*m^3 + 225*a^5*m^2 + 274*a^5*m + 120*a^5)*x^6 +
(a^4*m^5 + 16*a^4*m^4 + 95*a^4*m^3 + 260*a^4*m^2 + 324*a^4*m + 144*a^4)*x^5 - 2
*(a^3*m^5 + 17*a^3*m^4 + 107*a^3*m^3 + 307*a^3*m^2 + 396*a^3*m + 180*a^3)*x^4 -
2*(a^2*m^5 + 18*a^2*m^4 + 121*a^2*m^3 + 372*a^2*m^2 + 508*a^2*m + 240*a^2)*x^3 +
(a*m^5 + 19*a*m^4 + 137*a*m^3 + 461*a*m^2 + 702*a*m + 360*a)*x^2 + (m^5 + 20*m^
4 + 155*m^3 + 580*m^2 + 1044*m + 720)*x)*(e*x)^m/(m^6 + 21*m^5 + 175*m^4 + 735*m
^3 + 1624*m^2 + 1764*m + 720)

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Sympy [A]  time = 5.12744, size = 1928, normalized size = 16.62 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise(((4*a**5*log(x) - 4*a**4/x + 4*a**3/x**2 + 8*a**2/(3*x**3) - a/x**4 -
4/(5*x**5))/e**6, Eq(m, -6)), ((4*a**5*x + 4*a**4*log(x) + 8*a**3/x + 4*a**2/x**
2 - 4*a/(3*x**3) - 1/x**4)/e**5, Eq(m, -5)), ((2*a**5*x**2 + 4*a**4*x - 8*a**3*l
og(x) + 8*a**2/x - 2*a/x**2 - 4/(3*x**3))/e**4, Eq(m, -4)), ((4*a**5*x**3/3 + 2*
a**4*x**2 - 8*a**3*x - 8*a**2*log(x) - 4*a/x - 2/x**2)/e**3, Eq(m, -3)), ((a**5*
x**4 + 4*a**4*x**3/3 - 4*a**3*x**2 - 8*a**2*x + 4*a*log(x) - 4/x)/e**2, Eq(m, -2
)), ((4*a**5*x**5/5 + a**4*x**4 - 8*a**3*x**3/3 - 4*a**2*x**2 + 4*a*x + 4*log(x)
)/e, Eq(m, -1)), (4*a**5*e**m*m**5*x**6*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m*
*3 + 1624*m**2 + 1764*m + 720) + 60*a**5*e**m*m**4*x**6*x**m/(m**6 + 21*m**5 + 1
75*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 340*a**5*e**m*m**3*x**6*x**m/(m
**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 900*a**5*e**m*
m**2*x**6*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720)
+ 1096*a**5*e**m*m*x**6*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2
+ 1764*m + 720) + 480*a**5*e**m*x**6*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3
+ 1624*m**2 + 1764*m + 720) + 4*a**4*e**m*m**5*x**5*x**m/(m**6 + 21*m**5 + 175*m
**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 64*a**4*e**m*m**4*x**5*x**m/(m**6 +
21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 380*a**4*e**m*m**3*
x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 10
40*a**4*e**m*m**2*x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 +
1764*m + 720) + 1296*a**4*e**m*m*x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3
+ 1624*m**2 + 1764*m + 720) + 576*a**4*e**m*x**5*x**m/(m**6 + 21*m**5 + 175*m**
4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 8*a**3*e**m*m**5*x**4*x**m/(m**6 + 21
*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 136*a**3*e**m*m**4*x**
4*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 856*a
**3*e**m*m**3*x**4*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764
*m + 720) - 2456*a**3*e**m*m**2*x**4*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3
+ 1624*m**2 + 1764*m + 720) - 3168*a**3*e**m*m*x**4*x**m/(m**6 + 21*m**5 + 175*m
**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 1440*a**3*e**m*x**4*x**m/(m**6 + 21
*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 8*a**2*e**m*m**5*x**3*
x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 144*a**
2*e**m*m**4*x**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m
+ 720) - 968*a**2*e**m*m**3*x**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1
624*m**2 + 1764*m + 720) - 2976*a**2*e**m*m**2*x**3*x**m/(m**6 + 21*m**5 + 175*m
**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 4064*a**2*e**m*m*x**3*x**m/(m**6 +
21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 1920*a**2*e**m*x**3*
x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 4*a*e**
m*m**5*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 72
0) + 76*a*e**m*m**4*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2
+ 1764*m + 720) + 548*a*e**m*m**3*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**
3 + 1624*m**2 + 1764*m + 720) + 1844*a*e**m*m**2*x**2*x**m/(m**6 + 21*m**5 + 175
*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 2808*a*e**m*m*x**2*x**m/(m**6 + 2
1*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 1440*a*e**m*x**2*x**m
/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 4*e**m*m**5
*x*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 80*e
**m*m**4*x*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720
) + 620*e**m*m**3*x*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 176
4*m + 720) + 2320*e**m*m**2*x*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*
m**2 + 1764*m + 720) + 4176*e**m*m*x*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3
+ 1624*m**2 + 1764*m + 720) + 2880*e**m*x*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*
m**3 + 1624*m**2 + 1764*m + 720), True))

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GIAC/XCAS [A]  time = 0.212899, size = 817, normalized size = 7.04 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(4*(a*x + 1)^3*(a*x - 1)^2*(e*x)^m,x, algorithm="giac")

[Out]

4*(a^5*m^5*x^6*e^(m*ln(x) + m) + 15*a^5*m^4*x^6*e^(m*ln(x) + m) + a^4*m^5*x^5*e^
(m*ln(x) + m) + 85*a^5*m^3*x^6*e^(m*ln(x) + m) + 16*a^4*m^4*x^5*e^(m*ln(x) + m)
+ 225*a^5*m^2*x^6*e^(m*ln(x) + m) - 2*a^3*m^5*x^4*e^(m*ln(x) + m) + 95*a^4*m^3*x
^5*e^(m*ln(x) + m) + 274*a^5*m*x^6*e^(m*ln(x) + m) - 34*a^3*m^4*x^4*e^(m*ln(x) +
m) + 260*a^4*m^2*x^5*e^(m*ln(x) + m) + 120*a^5*x^6*e^(m*ln(x) + m) - 2*a^2*m^5*
x^3*e^(m*ln(x) + m) - 214*a^3*m^3*x^4*e^(m*ln(x) + m) + 324*a^4*m*x^5*e^(m*ln(x)
+ m) - 36*a^2*m^4*x^3*e^(m*ln(x) + m) - 614*a^3*m^2*x^4*e^(m*ln(x) + m) + 144*a
^4*x^5*e^(m*ln(x) + m) + a*m^5*x^2*e^(m*ln(x) + m) - 242*a^2*m^3*x^3*e^(m*ln(x)
+ m) - 792*a^3*m*x^4*e^(m*ln(x) + m) + 19*a*m^4*x^2*e^(m*ln(x) + m) - 744*a^2*m^
2*x^3*e^(m*ln(x) + m) - 360*a^3*x^4*e^(m*ln(x) + m) + m^5*x*e^(m*ln(x) + m) + 13
7*a*m^3*x^2*e^(m*ln(x) + m) - 1016*a^2*m*x^3*e^(m*ln(x) + m) + 20*m^4*x*e^(m*ln(
x) + m) + 461*a*m^2*x^2*e^(m*ln(x) + m) - 480*a^2*x^3*e^(m*ln(x) + m) + 155*m^3*
x*e^(m*ln(x) + m) + 702*a*m*x^2*e^(m*ln(x) + m) + 580*m^2*x*e^(m*ln(x) + m) + 36
0*a*x^2*e^(m*ln(x) + m) + 1044*m*x*e^(m*ln(x) + m) + 720*x*e^(m*ln(x) + m))/(m^6
+ 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)