3.69 \(\int (e x)^m (a+b x) (a c-b c x)^3 \, dx\)

Optimal. Leaf size=94 \[ -\frac{2 a^3 b c^3 (e x)^{m+2}}{e^2 (m+2)}+\frac{a^4 c^3 (e x)^{m+1}}{e (m+1)}+\frac{2 a b^3 c^3 (e x)^{m+4}}{e^4 (m+4)}-\frac{b^4 c^3 (e x)^{m+5}}{e^5 (m+5)} \]

[Out]

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

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Rubi [A]  time = 0.0520564, antiderivative size = 94, 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, Rules used = {75} \[ -\frac{2 a^3 b c^3 (e x)^{m+2}}{e^2 (m+2)}+\frac{a^4 c^3 (e x)^{m+1}}{e (m+1)}+\frac{2 a b^3 c^3 (e x)^{m+4}}{e^4 (m+4)}-\frac{b^4 c^3 (e x)^{m+5}}{e^5 (m+5)} \]

Antiderivative was successfully verified.

[In]

Int[(e*x)^m*(a + b*x)*(a*c - b*c*x)^3,x]

[Out]

(a^4*c^3*(e*x)^(1 + m))/(e*(1 + m)) - (2*a^3*b*c^3*(e*x)^(2 + m))/(e^2*(2 + m)) + (2*a*b^3*c^3*(e*x)^(4 + m))/
(e^4*(4 + m)) - (b^4*c^3*(e*x)^(5 + m))/(e^5*(5 + 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 (a+b x) (a c-b c x)^3 \, dx &=\int \left (a^4 c^3 (e x)^m-\frac{2 a^3 b c^3 (e x)^{1+m}}{e}+\frac{2 a b^3 c^3 (e x)^{3+m}}{e^3}-\frac{b^4 c^3 (e x)^{4+m}}{e^4}\right ) \, dx\\ &=\frac{a^4 c^3 (e x)^{1+m}}{e (1+m)}-\frac{2 a^3 b c^3 (e x)^{2+m}}{e^2 (2+m)}+\frac{2 a b^3 c^3 (e x)^{4+m}}{e^4 (4+m)}-\frac{b^4 c^3 (e x)^{5+m}}{e^5 (5+m)}\\ \end{align*}

Mathematica [A]  time = 0.0469745, size = 112, normalized size = 1.19 \[ -\frac{c^3 x (e x)^m \left (2 a^3 b \left (m^3+10 m^2+29 m+20\right ) x+a^4 \left (-\left (m^3+11 m^2+38 m+40\right )\right )-2 a b^3 \left (m^3+8 m^2+17 m+10\right ) x^3+b^4 \left (m^3+7 m^2+14 m+8\right ) x^4\right )}{(m+1) (m+2) (m+4) (m+5)} \]

Antiderivative was successfully verified.

[In]

Integrate[(e*x)^m*(a + b*x)*(a*c - b*c*x)^3,x]

[Out]

-((c^3*x*(e*x)^m*(-(a^4*(40 + 38*m + 11*m^2 + m^3)) + 2*a^3*b*(20 + 29*m + 10*m^2 + m^3)*x - 2*a*b^3*(10 + 17*
m + 8*m^2 + m^3)*x^3 + b^4*(8 + 14*m + 7*m^2 + m^3)*x^4))/((1 + m)*(2 + m)*(4 + m)*(5 + m)))

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Maple [A]  time = 0.005, size = 175, normalized size = 1.9 \begin{align*}{\frac{{c}^{3} \left ( ex \right ) ^{m} \left ( -{b}^{4}{m}^{3}{x}^{4}+2\,a{b}^{3}{m}^{3}{x}^{3}-7\,{b}^{4}{m}^{2}{x}^{4}+16\,a{b}^{3}{m}^{2}{x}^{3}-14\,{b}^{4}m{x}^{4}-2\,{a}^{3}b{m}^{3}x+34\,a{b}^{3}m{x}^{3}-8\,{b}^{4}{x}^{4}+{a}^{4}{m}^{3}-20\,{a}^{3}b{m}^{2}x+20\,{x}^{3}a{b}^{3}+11\,{a}^{4}{m}^{2}-58\,{a}^{3}bmx+38\,{a}^{4}m-40\,bx{a}^{3}+40\,{a}^{4} \right ) x}{ \left ( 5+m \right ) \left ( 4+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*(b*x+a)*(-b*c*x+a*c)^3,x)

[Out]

c^3*(e*x)^m*(-b^4*m^3*x^4+2*a*b^3*m^3*x^3-7*b^4*m^2*x^4+16*a*b^3*m^2*x^3-14*b^4*m*x^4-2*a^3*b*m^3*x+34*a*b^3*m
*x^3-8*b^4*x^4+a^4*m^3-20*a^3*b*m^2*x+20*a*b^3*x^3+11*a^4*m^2-58*a^3*b*m*x+38*a^4*m-40*a^3*b*x+40*a^4)*x/(5+m)
/(4+m)/(2+m)/(1+m)

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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*(b*x+a)*(-b*c*x+a*c)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.84773, size = 433, normalized size = 4.61 \begin{align*} -\frac{{\left ({\left (b^{4} c^{3} m^{3} + 7 \, b^{4} c^{3} m^{2} + 14 \, b^{4} c^{3} m + 8 \, b^{4} c^{3}\right )} x^{5} - 2 \,{\left (a b^{3} c^{3} m^{3} + 8 \, a b^{3} c^{3} m^{2} + 17 \, a b^{3} c^{3} m + 10 \, a b^{3} c^{3}\right )} x^{4} + 2 \,{\left (a^{3} b c^{3} m^{3} + 10 \, a^{3} b c^{3} m^{2} + 29 \, a^{3} b c^{3} m + 20 \, a^{3} b c^{3}\right )} x^{2} -{\left (a^{4} c^{3} m^{3} + 11 \, a^{4} c^{3} m^{2} + 38 \, a^{4} c^{3} m + 40 \, a^{4} c^{3}\right )} x\right )} \left (e x\right )^{m}}{m^{4} + 12 \, m^{3} + 49 \, m^{2} + 78 \, m + 40} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*(b*x+a)*(-b*c*x+a*c)^3,x, algorithm="fricas")

[Out]

-((b^4*c^3*m^3 + 7*b^4*c^3*m^2 + 14*b^4*c^3*m + 8*b^4*c^3)*x^5 - 2*(a*b^3*c^3*m^3 + 8*a*b^3*c^3*m^2 + 17*a*b^3
*c^3*m + 10*a*b^3*c^3)*x^4 + 2*(a^3*b*c^3*m^3 + 10*a^3*b*c^3*m^2 + 29*a^3*b*c^3*m + 20*a^3*b*c^3)*x^2 - (a^4*c
^3*m^3 + 11*a^4*c^3*m^2 + 38*a^4*c^3*m + 40*a^4*c^3)*x)*(e*x)^m/(m^4 + 12*m^3 + 49*m^2 + 78*m + 40)

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Sympy [A]  time = 1.31816, size = 838, normalized size = 8.91 \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*(b*x+a)*(-b*c*x+a*c)**3,x)

[Out]

Piecewise(((-a**4*c**3/(4*x**4) + 2*a**3*b*c**3/(3*x**3) - 2*a*b**3*c**3/x - b**4*c**3*log(x))/e**5, Eq(m, -5)
), ((-a**4*c**3/(3*x**3) + a**3*b*c**3/x**2 + 2*a*b**3*c**3*log(x) - b**4*c**3*x)/e**4, Eq(m, -4)), ((-a**4*c*
*3/x - 2*a**3*b*c**3*log(x) + a*b**3*c**3*x**2 - b**4*c**3*x**3/3)/e**2, Eq(m, -2)), ((a**4*c**3*log(x) - 2*a*
*3*b*c**3*x + 2*a*b**3*c**3*x**3/3 - b**4*c**3*x**4/4)/e, Eq(m, -1)), (a**4*c**3*e**m*m**3*x*x**m/(m**4 + 12*m
**3 + 49*m**2 + 78*m + 40) + 11*a**4*c**3*e**m*m**2*x*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) + 38*a**4*c*
*3*e**m*m*x*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) + 40*a**4*c**3*e**m*x*x**m/(m**4 + 12*m**3 + 49*m**2 +
78*m + 40) - 2*a**3*b*c**3*e**m*m**3*x**2*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) - 20*a**3*b*c**3*e**m*m
**2*x**2*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) - 58*a**3*b*c**3*e**m*m*x**2*x**m/(m**4 + 12*m**3 + 49*m*
*2 + 78*m + 40) - 40*a**3*b*c**3*e**m*x**2*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) + 2*a*b**3*c**3*e**m*m*
*3*x**4*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) + 16*a*b**3*c**3*e**m*m**2*x**4*x**m/(m**4 + 12*m**3 + 49*
m**2 + 78*m + 40) + 34*a*b**3*c**3*e**m*m*x**4*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) + 20*a*b**3*c**3*e*
*m*x**4*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) - b**4*c**3*e**m*m**3*x**5*x**m/(m**4 + 12*m**3 + 49*m**2
+ 78*m + 40) - 7*b**4*c**3*e**m*m**2*x**5*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) - 14*b**4*c**3*e**m*m*x*
*5*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) - 8*b**4*c**3*e**m*x**5*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m +
40), True))

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Giac [B]  time = 1.22333, size = 413, normalized size = 4.39 \begin{align*} -\frac{b^{4} c^{3} m^{3} x^{5} x^{m} e^{m} - 2 \, a b^{3} c^{3} m^{3} x^{4} x^{m} e^{m} + 7 \, b^{4} c^{3} m^{2} x^{5} x^{m} e^{m} - 16 \, a b^{3} c^{3} m^{2} x^{4} x^{m} e^{m} + 14 \, b^{4} c^{3} m x^{5} x^{m} e^{m} + 2 \, a^{3} b c^{3} m^{3} x^{2} x^{m} e^{m} - 34 \, a b^{3} c^{3} m x^{4} x^{m} e^{m} + 8 \, b^{4} c^{3} x^{5} x^{m} e^{m} - a^{4} c^{3} m^{3} x x^{m} e^{m} + 20 \, a^{3} b c^{3} m^{2} x^{2} x^{m} e^{m} - 20 \, a b^{3} c^{3} x^{4} x^{m} e^{m} - 11 \, a^{4} c^{3} m^{2} x x^{m} e^{m} + 58 \, a^{3} b c^{3} m x^{2} x^{m} e^{m} - 38 \, a^{4} c^{3} m x x^{m} e^{m} + 40 \, a^{3} b c^{3} x^{2} x^{m} e^{m} - 40 \, a^{4} c^{3} x x^{m} e^{m}}{m^{4} + 12 \, m^{3} + 49 \, m^{2} + 78 \, m + 40} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*(b*x+a)*(-b*c*x+a*c)^3,x, algorithm="giac")

[Out]

-(b^4*c^3*m^3*x^5*x^m*e^m - 2*a*b^3*c^3*m^3*x^4*x^m*e^m + 7*b^4*c^3*m^2*x^5*x^m*e^m - 16*a*b^3*c^3*m^2*x^4*x^m
*e^m + 14*b^4*c^3*m*x^5*x^m*e^m + 2*a^3*b*c^3*m^3*x^2*x^m*e^m - 34*a*b^3*c^3*m*x^4*x^m*e^m + 8*b^4*c^3*x^5*x^m
*e^m - a^4*c^3*m^3*x*x^m*e^m + 20*a^3*b*c^3*m^2*x^2*x^m*e^m - 20*a*b^3*c^3*x^4*x^m*e^m - 11*a^4*c^3*m^2*x*x^m*
e^m + 58*a^3*b*c^3*m*x^2*x^m*e^m - 38*a^4*c^3*m*x*x^m*e^m + 40*a^3*b*c^3*x^2*x^m*e^m - 40*a^4*c^3*x*x^m*e^m)/(
m^4 + 12*m^3 + 49*m^2 + 78*m + 40)