3.57 \(\int (e x)^m (2-2 a x)^2 (1+a x)^3 \, dx\)
Optimal. Leaf size=116 \[ -\frac{8 a^2 (e x)^{m+3}}{e^3 (m+3)}-\frac{8 a^3 (e x)^{m+4}}{e^4 (m+4)}+\frac{4 a^4 (e x)^{m+5}}{e^5 (m+5)}+\frac{4 a^5 (e x)^{m+6}}{e^6 (m+6)}+\frac{4 a (e x)^{m+2}}{e^2 (m+2)}+\frac{4 (e x)^{m+1}}{e (m+1)} \]
[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))
________________________________________________________________________________________
Rubi [A] time = 0.0421191, 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, Rules used =
{88} \[ -\frac{8 a^2 (e x)^{m+3}}{e^3 (m+3)}-\frac{8 a^3 (e x)^{m+4}}{e^4 (m+4)}+\frac{4 a^4 (e x)^{m+5}}{e^5 (m+5)}+\frac{4 a^5 (e x)^{m+6}}{e^6 (m+6)}+\frac{4 a (e x)^{m+2}}{e^2 (m+2)}+\frac{4 (e x)^{m+1}}{e (m+1)} \]
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))
Rule 88
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))
Rubi steps
\begin{align*} \int (e x)^m (2-2 a x)^2 (1+a x)^3 \, dx &=\int \left (4 (e x)^m+\frac{4 a (e x)^{1+m}}{e}-\frac{8 a^2 (e x)^{2+m}}{e^2}-\frac{8 a^3 (e x)^{3+m}}{e^3}+\frac{4 a^4 (e x)^{4+m}}{e^4}+\frac{4 a^5 (e x)^{5+m}}{e^5}\right ) \, dx\\ &=\frac{4 (e x)^{1+m}}{e (1+m)}+\frac{4 a (e x)^{2+m}}{e^2 (2+m)}-\frac{8 a^2 (e x)^{3+m}}{e^3 (3+m)}-\frac{8 a^3 (e x)^{4+m}}{e^4 (4+m)}+\frac{4 a^4 (e x)^{5+m}}{e^5 (5+m)}+\frac{4 a^5 (e x)^{6+m}}{e^6 (6+m)}\\ \end{align*}
Mathematica [A] time = 0.0334767, 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 = 0.006, size = 340, normalized size = 2.9 \begin{align*} 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 ) }} \end{align*}
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^4-614*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*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)
________________________________________________________________________________________
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)^2*(a*x+1)^3,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 2.16223, size = 690, normalized size = 5.95 \begin{align*} \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} \end{align*}
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, 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)
________________________________________________________________________________________
Sympy [A] time = 1.77798, size = 1928, normalized size = 16.62 \begin{align*} \text{result too large to display} \end{align*}
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*log(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 + 176
4*m + 720) + 60*a**5*e**m*m**4*x**6*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 3
40*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) + 1040*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 + 72
0) - 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 + 1624*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 + 176
4*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 + 720) + 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 + 21*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 + 1764*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))
________________________________________________________________________________________
Giac [B] time = 1.17151, size = 720, normalized size = 6.21 \begin{align*} \frac{4 \,{\left (a^{5} m^{5} x^{6} x^{m} e^{m} + 15 \, a^{5} m^{4} x^{6} x^{m} e^{m} + a^{4} m^{5} x^{5} x^{m} e^{m} + 85 \, a^{5} m^{3} x^{6} x^{m} e^{m} + 16 \, a^{4} m^{4} x^{5} x^{m} e^{m} + 225 \, a^{5} m^{2} x^{6} x^{m} e^{m} - 2 \, a^{3} m^{5} x^{4} x^{m} e^{m} + 95 \, a^{4} m^{3} x^{5} x^{m} e^{m} + 274 \, a^{5} m x^{6} x^{m} e^{m} - 34 \, a^{3} m^{4} x^{4} x^{m} e^{m} + 260 \, a^{4} m^{2} x^{5} x^{m} e^{m} + 120 \, a^{5} x^{6} x^{m} e^{m} - 2 \, a^{2} m^{5} x^{3} x^{m} e^{m} - 214 \, a^{3} m^{3} x^{4} x^{m} e^{m} + 324 \, a^{4} m x^{5} x^{m} e^{m} - 36 \, a^{2} m^{4} x^{3} x^{m} e^{m} - 614 \, a^{3} m^{2} x^{4} x^{m} e^{m} + 144 \, a^{4} x^{5} x^{m} e^{m} + a m^{5} x^{2} x^{m} e^{m} - 242 \, a^{2} m^{3} x^{3} x^{m} e^{m} - 792 \, a^{3} m x^{4} x^{m} e^{m} + 19 \, a m^{4} x^{2} x^{m} e^{m} - 744 \, a^{2} m^{2} x^{3} x^{m} e^{m} - 360 \, a^{3} x^{4} x^{m} e^{m} + m^{5} x x^{m} e^{m} + 137 \, a m^{3} x^{2} x^{m} e^{m} - 1016 \, a^{2} m x^{3} x^{m} e^{m} + 20 \, m^{4} x x^{m} e^{m} + 461 \, a m^{2} x^{2} x^{m} e^{m} - 480 \, a^{2} x^{3} x^{m} e^{m} + 155 \, m^{3} x x^{m} e^{m} + 702 \, a m x^{2} x^{m} e^{m} + 580 \, m^{2} x x^{m} e^{m} + 360 \, a x^{2} x^{m} e^{m} + 1044 \, m x x^{m} e^{m} + 720 \, x x^{m} e^{m}\right )}}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720} \end{align*}
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, algorithm="giac")
[Out]
4*(a^5*m^5*x^6*x^m*e^m + 15*a^5*m^4*x^6*x^m*e^m + a^4*m^5*x^5*x^m*e^m + 85*a^5*m^3*x^6*x^m*e^m + 16*a^4*m^4*x^
5*x^m*e^m + 225*a^5*m^2*x^6*x^m*e^m - 2*a^3*m^5*x^4*x^m*e^m + 95*a^4*m^3*x^5*x^m*e^m + 274*a^5*m*x^6*x^m*e^m -
34*a^3*m^4*x^4*x^m*e^m + 260*a^4*m^2*x^5*x^m*e^m + 120*a^5*x^6*x^m*e^m - 2*a^2*m^5*x^3*x^m*e^m - 214*a^3*m^3*
x^4*x^m*e^m + 324*a^4*m*x^5*x^m*e^m - 36*a^2*m^4*x^3*x^m*e^m - 614*a^3*m^2*x^4*x^m*e^m + 144*a^4*x^5*x^m*e^m +
a*m^5*x^2*x^m*e^m - 242*a^2*m^3*x^3*x^m*e^m - 792*a^3*m*x^4*x^m*e^m + 19*a*m^4*x^2*x^m*e^m - 744*a^2*m^2*x^3*
x^m*e^m - 360*a^3*x^4*x^m*e^m + m^5*x*x^m*e^m + 137*a*m^3*x^2*x^m*e^m - 1016*a^2*m*x^3*x^m*e^m + 20*m^4*x*x^m*
e^m + 461*a*m^2*x^2*x^m*e^m - 480*a^2*x^3*x^m*e^m + 155*m^3*x*x^m*e^m + 702*a*m*x^2*x^m*e^m + 580*m^2*x*x^m*e^
m + 360*a*x^2*x^m*e^m + 1044*m*x*x^m*e^m + 720*x*x^m*e^m)/(m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*
m + 720)