3.369 \(\int (e x)^m (a+b x) (a c-b c x)^2 \, dx\)

Optimal. Leaf size=93 \[ \frac{a^3 c^2 (e x)^{m+1}}{e (m+1)}-\frac{a^2 b c^2 (e x)^{m+2}}{e^2 (m+2)}-\frac{a b^2 c^2 (e x)^{m+3}}{e^3 (m+3)}+\frac{b^3 c^2 (e x)^{m+4}}{e^4 (m+4)} \]

[Out]

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

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Rubi [A]  time = 0.134304, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{a^3 c^2 (e x)^{m+1}}{e (m+1)}-\frac{a^2 b c^2 (e x)^{m+2}}{e^2 (m+2)}-\frac{a b^2 c^2 (e x)^{m+3}}{e^3 (m+3)}+\frac{b^3 c^2 (e x)^{m+4}}{e^4 (m+4)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 29.9057, size = 82, normalized size = 0.88 \[ \frac{a^{3} c^{2} \left (e x\right )^{m + 1}}{e \left (m + 1\right )} - \frac{a^{2} b c^{2} \left (e x\right )^{m + 2}}{e^{2} \left (m + 2\right )} - \frac{a b^{2} c^{2} \left (e x\right )^{m + 3}}{e^{3} \left (m + 3\right )} + \frac{b^{3} c^{2} \left (e x\right )^{m + 4}}{e^{4} \left (m + 4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0781005, size = 110, normalized size = 1.18 \[ \frac{c^2 x (e x)^m \left (a^3 \left (m^3+9 m^2+26 m+24\right )-a^2 b \left (m^3+8 m^2+19 m+12\right ) x-a b^2 \left (m^3+7 m^2+14 m+8\right ) x^2+b^3 \left (m^3+6 m^2+11 m+6\right ) x^3\right )}{(m+1) (m+2) (m+3) (m+4)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 174, normalized size = 1.9 \[{\frac{{c}^{2} \left ( ex \right ) ^{m} \left ({b}^{3}{m}^{3}{x}^{3}-a{b}^{2}{m}^{3}{x}^{2}+6\,{b}^{3}{m}^{2}{x}^{3}-{a}^{2}b{m}^{3}x-7\,a{b}^{2}{m}^{2}{x}^{2}+11\,{b}^{3}m{x}^{3}+{a}^{3}{m}^{3}-8\,{a}^{2}b{m}^{2}x-14\,a{b}^{2}m{x}^{2}+6\,{b}^{3}{x}^{3}+9\,{a}^{3}{m}^{2}-19\,{a}^{2}bmx-8\,a{b}^{2}{x}^{2}+26\,{a}^{3}m-12\,{a}^{2}bx+24\,{a}^{3} \right ) x}{ \left ( 4+m \right ) \left ( 3+m \right ) \left ( 2+m \right ) \left ( 1+m \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c^2*(e*x)^m*(b^3*m^3*x^3-a*b^2*m^3*x^2+6*b^3*m^2*x^3-a^2*b*m^3*x-7*a*b^2*m^2*x^2
+11*b^3*m*x^3+a^3*m^3-8*a^2*b*m^2*x-14*a*b^2*m*x^2+6*b^3*x^3+9*a^3*m^2-19*a^2*b*
m*x-8*a*b^2*x^2+26*a^3*m-12*a^2*b*x+24*a^3)*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((b*c*x - a*c)^2*(b*x + a)*(e*x)^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.22265, size = 279, normalized size = 3. \[ \frac{{\left ({\left (b^{3} c^{2} m^{3} + 6 \, b^{3} c^{2} m^{2} + 11 \, b^{3} c^{2} m + 6 \, b^{3} c^{2}\right )} x^{4} -{\left (a b^{2} c^{2} m^{3} + 7 \, a b^{2} c^{2} m^{2} + 14 \, a b^{2} c^{2} m + 8 \, a b^{2} c^{2}\right )} x^{3} -{\left (a^{2} b c^{2} m^{3} + 8 \, a^{2} b c^{2} m^{2} + 19 \, a^{2} b c^{2} m + 12 \, a^{2} b c^{2}\right )} x^{2} +{\left (a^{3} c^{2} m^{3} + 9 \, a^{3} c^{2} m^{2} + 26 \, a^{3} c^{2} m + 24 \, a^{3} c^{2}\right )} x\right )} \left (e x\right )^{m}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.75496, size = 821, normalized size = 8.83 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)**m*(b*x+a)*(-b*c*x+a*c)**2,x)

[Out]

Piecewise(((-a**3*c**2/(3*x**3) + a**2*b*c**2/(2*x**2) + a*b**2*c**2/x + b**3*c*
*2*log(x))/e**4, Eq(m, -4)), ((-a**3*c**2/(2*x**2) + a**2*b*c**2/x - a*b**2*c**2
*log(x) + b**3*c**2*x)/e**3, Eq(m, -3)), ((-a**3*c**2/x - a**2*b*c**2*log(x) - a
*b**2*c**2*x + b**3*c**2*x**2/2)/e**2, Eq(m, -2)), ((a**3*c**2*log(x) - a**2*b*c
**2*x - a*b**2*c**2*x**2/2 + b**3*c**2*x**3/3)/e, Eq(m, -1)), (a**3*c**2*e**m*m*
*3*x*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 9*a**3*c**2*e**m*m**2*x*x**m/
(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 26*a**3*c**2*e**m*m*x*x**m/(m**4 + 10*m
**3 + 35*m**2 + 50*m + 24) + 24*a**3*c**2*e**m*x*x**m/(m**4 + 10*m**3 + 35*m**2
+ 50*m + 24) - a**2*b*c**2*e**m*m**3*x**2*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m
+ 24) - 8*a**2*b*c**2*e**m*m**2*x**2*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24)
- 19*a**2*b*c**2*e**m*m*x**2*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) - 12*a
**2*b*c**2*e**m*x**2*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) - a*b**2*c**2*e
**m*m**3*x**3*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) - 7*a*b**2*c**2*e**m*m
**2*x**3*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) - 14*a*b**2*c**2*e**m*m*x**
3*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) - 8*a*b**2*c**2*e**m*x**3*x**m/(m*
*4 + 10*m**3 + 35*m**2 + 50*m + 24) + b**3*c**2*e**m*m**3*x**4*x**m/(m**4 + 10*m
**3 + 35*m**2 + 50*m + 24) + 6*b**3*c**2*e**m*m**2*x**4*x**m/(m**4 + 10*m**3 + 3
5*m**2 + 50*m + 24) + 11*b**3*c**2*e**m*m*x**4*x**m/(m**4 + 10*m**3 + 35*m**2 +
50*m + 24) + 6*b**3*c**2*e**m*x**4*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24),
True))

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GIAC/XCAS [A]  time = 0.214632, size = 454, normalized size = 4.88 \[ \frac{b^{3} c^{2} m^{3} x^{4} e^{\left (m{\rm ln}\left (x\right ) + m\right )} - a b^{2} c^{2} m^{3} x^{3} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 6 \, b^{3} c^{2} m^{2} x^{4} e^{\left (m{\rm ln}\left (x\right ) + m\right )} - a^{2} b c^{2} m^{3} x^{2} e^{\left (m{\rm ln}\left (x\right ) + m\right )} - 7 \, a b^{2} c^{2} m^{2} x^{3} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 11 \, b^{3} c^{2} m x^{4} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + a^{3} c^{2} m^{3} x e^{\left (m{\rm ln}\left (x\right ) + m\right )} - 8 \, a^{2} b c^{2} m^{2} x^{2} e^{\left (m{\rm ln}\left (x\right ) + m\right )} - 14 \, a b^{2} c^{2} m x^{3} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 6 \, b^{3} c^{2} x^{4} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 9 \, a^{3} c^{2} m^{2} x e^{\left (m{\rm ln}\left (x\right ) + m\right )} - 19 \, a^{2} b c^{2} m x^{2} e^{\left (m{\rm ln}\left (x\right ) + m\right )} - 8 \, a b^{2} c^{2} x^{3} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 26 \, a^{3} c^{2} m x e^{\left (m{\rm ln}\left (x\right ) + m\right )} - 12 \, a^{2} b c^{2} x^{2} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + 24 \, a^{3} c^{2} x e^{\left (m{\rm ln}\left (x\right ) + m\right )}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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