\(\int (e x)^m (a+b x)^2 (a d-b d x) \, dx\) [64]

Optimal result

Integrand size = 22, antiderivative size = 85 \[ \int (e x)^m (a+b x)^2 (a d-b d x) \, dx=\frac {a^3 d (e x)^{1+m}}{e (1+m)}+\frac {a^2 b d (e x)^{2+m}}{e^2 (2+m)}-\frac {a b^2 d (e x)^{3+m}}{e^3 (3+m)}-\frac {b^3 d (e x)^{4+m}}{e^4 (4+m)} \]

[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] (verified)

Time = 0.03 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {76} \[ \int (e x)^m (a+b x)^2 (a d-b d x) \, dx=\frac {a^3 d (e x)^{m+1}}{e (m+1)}+\frac {a^2 b d (e x)^{m+2}}{e^2 (m+2)}-\frac {a b^2 d (e x)^{m+3}}{e^3 (m+3)}-\frac {b^3 d (e x)^{m+4}}{e^4 (m+4)} \]

[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 76

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*} \text {integral}& = \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] (verified)

Time = 0.08 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.02 \[ \int (e x)^m (a+b x)^2 (a d-b d x) \, dx=\frac {d (e x)^m \left (-x (a+b x)^3+\frac {a (5+2 m) x \left (a^2 \left (6+5 m+m^2\right )+2 a b \left (3+4 m+m^2\right ) x+b^2 \left (2+3 m+m^2\right ) x^2\right )}{(1+m) (2+m) (3+m)}\right )}{4+m} \]

[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 [A] (verified)

Time = 0.40 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.99

method	result	size
norman	\(\frac {a^{3} d x \,{\mathrm e}^{m \ln \left (e x \right )}}{1+m}+\frac {a^{2} b d \,x^{2} {\mathrm e}^{m \ln \left (e x \right )}}{2+m}-\frac {b^{3} d \,x^{4} {\mathrm e}^{m \ln \left (e x \right )}}{4+m}-\frac {a \,b^{2} d \,x^{3} {\mathrm e}^{m \ln \left (e x \right )}}{3+m}\)	\(84\)
gosper	\(\frac {d \left (e x \right )^{m} \left (-b^{3} m^{3} x^{3}-a \,b^{2} m^{3} x^{2}-6 b^{3} m^{2} x^{3}+a^{2} b \,m^{3} x -7 a \,b^{2} m^{2} x^{2}-11 m \,x^{3} b^{3}+a^{3} m^{3}+8 a^{2} b \,m^{2} x -14 a \,b^{2} m \,x^{2}-6 b^{3} x^{3}+9 a^{3} m^{2}+19 a^{2} b m x -8 a \,b^{2} x^{2}+26 a^{3} m +12 a^{2} b x +24 a^{3}\right ) x}{\left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\)	\(172\)
risch	\(\frac {d \left (e x \right )^{m} \left (-b^{3} m^{3} x^{3}-a \,b^{2} m^{3} x^{2}-6 b^{3} m^{2} x^{3}+a^{2} b \,m^{3} x -7 a \,b^{2} m^{2} x^{2}-11 m \,x^{3} b^{3}+a^{3} m^{3}+8 a^{2} b \,m^{2} x -14 a \,b^{2} m \,x^{2}-6 b^{3} x^{3}+9 a^{3} m^{2}+19 a^{2} b m x -8 a \,b^{2} x^{2}+26 a^{3} m +12 a^{2} b x +24 a^{3}\right ) x}{\left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\)	\(172\)
parallelrisch	\(-\frac {x^{4} \left (e x \right )^{m} b^{3} d \,m^{3}+6 x^{4} \left (e x \right )^{m} b^{3} d \,m^{2}+x^{3} \left (e x \right )^{m} a \,b^{2} d \,m^{3}+11 x^{4} \left (e x \right )^{m} b^{3} d m +7 x^{3} \left (e x \right )^{m} a \,b^{2} d \,m^{2}-x^{2} \left (e x \right )^{m} a^{2} b d \,m^{3}+6 x^{4} \left (e x \right )^{m} b^{3} d +14 x^{3} \left (e x \right )^{m} a \,b^{2} d m -8 x^{2} \left (e x \right )^{m} a^{2} b d \,m^{2}-x \left (e x \right )^{m} a^{3} d \,m^{3}+8 x^{3} \left (e x \right )^{m} a \,b^{2} d -19 x^{2} \left (e x \right )^{m} a^{2} b d m -9 x \left (e x \right )^{m} a^{3} d \,m^{2}-12 x^{2} \left (e x \right )^{m} a^{2} b d -26 x \left (e x \right )^{m} a^{3} d m -24 x \left (e x \right )^{m} a^{3} d}{\left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\)	\(274\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 176 vs. \(2 (85) = 170\).

Time = 0.23 (sec) , antiderivative size = 176, normalized size of antiderivative = 2.07 \[ \int (e x)^m (a+b x)^2 (a d-b d x) \, dx=-\frac {{\left ({\left (b^{3} d m^{3} + 6 \, b^{3} d m^{2} + 11 \, b^{3} d m + 6 \, b^{3} d\right )} x^{4} + {\left (a b^{2} d m^{3} + 7 \, a b^{2} d m^{2} + 14 \, a b^{2} d m + 8 \, a b^{2} d\right )} x^{3} - {\left (a^{2} b d m^{3} + 8 \, a^{2} b d m^{2} + 19 \, a^{2} b d m + 12 \, a^{2} b d\right )} x^{2} - {\left (a^{3} d m^{3} + 9 \, a^{3} d m^{2} + 26 \, a^{3} d m + 24 \, a^{3} d\right )} x\right )} \left (e x\right )^{m}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \]

[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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 741 vs. \(2 (75) = 150\).

Time = 0.32 (sec) , antiderivative size = 741, normalized size of antiderivative = 8.72 \[ \int (e x)^m (a+b x)^2 (a d-b d x) \, dx=\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 m^{3} x \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {9 a^{3} d m^{2} x \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {26 a^{3} d m x \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {24 a^{3} d x \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {a^{2} b d m^{3} x^{2} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {8 a^{2} b d m^{2} x^{2} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {19 a^{2} b d m x^{2} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {12 a^{2} b d x^{2} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac {a b^{2} d m^{3} x^{3} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac {7 a b^{2} d m^{2} x^{3} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac {14 a b^{2} d m x^{3} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac {8 a b^{2} d x^{3} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac {b^{3} d m^{3} x^{4} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac {6 b^{3} d m^{2} x^{4} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac {11 b^{3} d m x^{4} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac {6 b^{3} d x^{4} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} & \text {otherwise} \end {cases} \]

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

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.96 \[ \int (e x)^m (a+b x)^2 (a d-b d x) \, dx=-\frac {b^{3} d e^{m} x^{4} x^{m}}{m + 4} - \frac {a b^{2} d e^{m} x^{3} x^{m}}{m + 3} + \frac {a^{2} b d e^{m} x^{2} x^{m}}{m + 2} + \frac {\left (e x\right )^{m + 1} a^{3} d}{e {\left (m + 1\right )}} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 273 vs. \(2 (85) = 170\).

Time = 0.29 (sec) , antiderivative size = 273, normalized size of antiderivative = 3.21 \[ \int (e x)^m (a+b x)^2 (a d-b d x) \, dx=-\frac {\left (e x\right )^{m} b^{3} d m^{3} x^{4} + \left (e x\right )^{m} a b^{2} d m^{3} x^{3} + 6 \, \left (e x\right )^{m} b^{3} d m^{2} x^{4} - \left (e x\right )^{m} a^{2} b d m^{3} x^{2} + 7 \, \left (e x\right )^{m} a b^{2} d m^{2} x^{3} + 11 \, \left (e x\right )^{m} b^{3} d m x^{4} - \left (e x\right )^{m} a^{3} d m^{3} x - 8 \, \left (e x\right )^{m} a^{2} b d m^{2} x^{2} + 14 \, \left (e x\right )^{m} a b^{2} d m x^{3} + 6 \, \left (e x\right )^{m} b^{3} d x^{4} - 9 \, \left (e x\right )^{m} a^{3} d m^{2} x - 19 \, \left (e x\right )^{m} a^{2} b d m x^{2} + 8 \, \left (e x\right )^{m} a b^{2} d x^{3} - 26 \, \left (e x\right )^{m} a^{3} d m x - 12 \, \left (e x\right )^{m} a^{2} b d x^{2} - 24 \, \left (e x\right )^{m} a^{3} d x}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 174, normalized size of antiderivative = 2.05 \[ \int (e x)^m (a+b x)^2 (a d-b d x) \, dx=-{\left (e\,x\right )}^m\,\left (\frac {b^3\,d\,x^4\,\left (m^3+6\,m^2+11\,m+6\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}-\frac {a^3\,d\,x\,\left (m^3+9\,m^2+26\,m+24\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}+\frac {a\,b^2\,d\,x^3\,\left (m^3+7\,m^2+14\,m+8\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}-\frac {a^2\,b\,d\,x^2\,\left (m^3+8\,m^2+19\,m+12\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}\right ) \]

[In]

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

[Out]

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