3.64 \(\int (e x)^m (a+b x)^2 (a d-b d x) \, dx\)

Optimal. Leaf size=85 \[ \frac{a^2 b d (e x)^{m+2}}{e^2 (m+2)}+\frac{a^3 d (e x)^{m+1}}{e (m+1)}-\frac{a b^2 d (e x)^{m+3}}{e^3 (m+3)}-\frac{b^3 d (e x)^{m+4}}{e^4 (m+4)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0410381, antiderivative size = 85, 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} \[ \frac{a^2 b d (e x)^{m+2}}{e^2 (m+2)}+\frac{a^3 d (e x)^{m+1}}{e (m+1)}-\frac{a b^2 d (e x)^{m+3}}{e^3 (m+3)}-\frac{b^3 d (e x)^{m+4}}{e^4 (m+4)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.071291, size = 87, normalized size = 1.02 \[ \frac{d (e x)^m \left (\frac{a (2 m+5) x \left (a^2 \left (m^2+5 m+6\right )+2 a b \left (m^2+4 m+3\right ) x+b^2 \left (m^2+3 m+2\right ) x^2\right )}{(m+1) (m+2) (m+3)}-x (a+b x)^3\right )}{m+4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.005, size = 172, normalized size = 2. \begin{align*}{\frac{d \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 ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d*(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)

________________________________________________________________________________________

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)^2*(-b*d*x+a*d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 1.76826, size = 381, normalized size = 4.48 \begin{align*} -\frac{{\left ({\left (b^{3} d m^{3} + 6 \, b^{3} d m^{2} + 11 \, b^{3} d m + 6 \, b^{3} d\right )} x^{4} +{\left (a b^{2} d m^{3} + 7 \, a b^{2} d m^{2} + 14 \, a b^{2} d m + 8 \, a b^{2} d\right )} x^{3} -{\left (a^{2} b d m^{3} + 8 \, a^{2} b d m^{2} + 19 \, a^{2} b d m + 12 \, a^{2} b d\right )} x^{2} -{\left (a^{3} d m^{3} + 9 \, a^{3} d m^{2} + 26 \, a^{3} d m + 24 \, a^{3} d\right )} x\right )} \left (e x\right )^{m}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*(b*x+a)^2*(-b*d*x+a*d),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 1.05697, size = 768, normalized size = 9.04 \begin{align*} \begin{cases} \frac{- \frac{a^{3} d}{3 x^{3}} - \frac{a^{2} b d}{2 x^{2}} + \frac{a b^{2} d}{x} - b^{3} d \log{\left (x \right )}}{e^{4}} & \text{for}\: m = -4 \\\frac{- \frac{a^{3} d}{2 x^{2}} - \frac{a^{2} b d}{x} - a b^{2} d \log{\left (x \right )} - b^{3} d x}{e^{3}} & \text{for}\: m = -3 \\\frac{- \frac{a^{3} d}{x} + a^{2} b d \log{\left (x \right )} - a b^{2} d x - \frac{b^{3} d x^{2}}{2}}{e^{2}} & \text{for}\: m = -2 \\\frac{a^{3} d \log{\left (x \right )} + a^{2} b d x - \frac{a b^{2} d x^{2}}{2} - \frac{b^{3} d x^{3}}{3}}{e} & \text{for}\: m = -1 \\\frac{a^{3} d e^{m} m^{3} x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{9 a^{3} d e^{m} m^{2} x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{26 a^{3} d e^{m} m x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{24 a^{3} d e^{m} x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{a^{2} b d e^{m} m^{3} x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{8 a^{2} b d e^{m} m^{2} x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{19 a^{2} b d e^{m} m x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{12 a^{2} b d e^{m} x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac{a b^{2} d e^{m} m^{3} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac{7 a b^{2} d e^{m} m^{2} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac{14 a b^{2} d e^{m} m x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac{8 a b^{2} d e^{m} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac{b^{3} d e^{m} m^{3} x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac{6 b^{3} d e^{m} m^{2} x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac{11 b^{3} d e^{m} m x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac{6 b^{3} d e^{m} x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**m*(b*x+a)**2*(-b*d*x+a*d),x)

[Out]

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

________________________________________________________________________________________

Giac [B]  time = 1.18665, size = 369, normalized size = 4.34 \begin{align*} -\frac{b^{3} d m^{3} x^{4} x^{m} e^{m} + a b^{2} d m^{3} x^{3} x^{m} e^{m} + 6 \, b^{3} d m^{2} x^{4} x^{m} e^{m} - a^{2} b d m^{3} x^{2} x^{m} e^{m} + 7 \, a b^{2} d m^{2} x^{3} x^{m} e^{m} + 11 \, b^{3} d m x^{4} x^{m} e^{m} - a^{3} d m^{3} x x^{m} e^{m} - 8 \, a^{2} b d m^{2} x^{2} x^{m} e^{m} + 14 \, a b^{2} d m x^{3} x^{m} e^{m} + 6 \, b^{3} d x^{4} x^{m} e^{m} - 9 \, a^{3} d m^{2} x x^{m} e^{m} - 19 \, a^{2} b d m x^{2} x^{m} e^{m} + 8 \, a b^{2} d x^{3} x^{m} e^{m} - 26 \, a^{3} d m x x^{m} e^{m} - 12 \, a^{2} b d x^{2} x^{m} e^{m} - 24 \, a^{3} d x x^{m} e^{m}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*(b*x+a)^2*(-b*d*x+a*d),x, algorithm="giac")

[Out]

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